GIFT  OF 
V 


MECHANICS' AND  ENGINEERS' 
POCKET-BOOK 

OF 

TABLES,  RULES,  AND  FORMULAS 

PERTAINING  TO 

MECHANICS,  MATHEMATICS,  AND  PHYSICS: 

INCLUDING 
AREAS,  SQUARES,  CUBES,  AND  ROOTS,  ETC.; 

LOGARITHMS,  HYDRAULICS,  HYDRODYNAMICS,  STEAM   AND 

THE  STEAM-ENGINE,  NAVAL  ARCHITECTURE, 

MASONRY,  STEAM  VESSELS, 

MILLS,  ETC. ; 

COMPRESSED  AIR,  GAS,  AND  OIL  ENGINES; 
LIMES,  MORTARS,  CEMENTS,  ETC.; 

ORTHOGRAPHY   OF  TECHNICAL   WORDS   AND  TERMS,  ETC,  ETC 


Seventy-second.    Ifidition. — 146th.    Thousand.. 

BY    CHAS.    H.     HASWELL, 

CIVIL,  MARINE,  AND    MECHANICAL    LNOINEHS.    HONORARY  MBMBBR    UP    TCB  AM.  SOC.  OF   NATAL    ARCHI 


AL    ARCHITECTS    OF 


MBMBBR    OF  THK  AM.   INSTITUTE    OF  ARCHITECTS,  AND 
ASSOCIATE    MEMBER    OF  THE    NEW    YORK     MI- 
CROSCOPICAL    SOCIETY,    ETC.,    ETC. 

An  examination  of  facts  is  the  foundation  of  science. 

A  resultant  effect,  physical  or  mechanical,  cannot  be  renewed  without  an 

expenditure  of  power  in  addition  to  that  which  originated  it. 


NEW    YORK: 
HARPER    &    BROTHERS,  PUBLISHERS, 

FRANKLIN     SQUAB  K. 

1906. 


By  the  Author  of  "Mechanics'  and  Engineers'  Pocket-Book." 


MENSURATION. 

For  Tuition  and  Reference,  containing  Tables  of  Weights  and  Measures;  Mensura- 
tion of  Surfaces,  Lines,  and  Solids,  and  Conic  Sections,  Centres  of  Gravity,  etc. 
To  which  is  added  Tables  ot  the  Areas  of  Circular  Segments,  Sines  of  a  Circle, 
Circular  and  Semi  elliptical  Arcs,  etc.  By  CHAS.  H.  HASWKLL,  Civil  and  Marine 
Engineer.  Sixth  Edition.  12mo,  90  cents.—  American  Book  Concern,  N.  Y. 


REMINISCENCES  OF  AN  OCTOGENARIAN. 

1816  to  1860. 
Second  Edition $3  00 


Copyright,  1884,  1887,  1890,  1892,  1893,  1894,  1895,  1896,  1897,  1898,  1899,  1901, 
1903,  by  ILvKi'KR  &  BROTHERS. 


INSCRIBED 
TO 

CAPTAIN  JOHN  ERICSSON,  LL.D., 

AS  A  SLIGHT  TRIBUTE  TO  HIS  GENIUS  AND  ATTAINMENTS, 

AND  IN  TESTIMONY  OF  THE  SINCERE  REGARD 

AND  ESTEEM  OF  HIS  FRIEND, 

THE  AUTHOR 


380362 


PREFACE 

To    the    ITorty-fiftlx    Edition. 


THE  First  Edition  of  this  work,  consisting  of  284  pages, 
was  submitted  to  the  Mechanics  and  Engineers  of  the  United 
States  by  one  of  their  number  in  1843,  who  designed  it  for 
a  convenient  reference  to  Rules,  Results,  and  Tables  con- 
nected with  the  discharge  of  their  various  duties. 

The  Twenty-first  Edition  was  published  in  1867,  consisted 
of  664  pages,  and,  in  addition  to  the  original  design  of  the 
work,  it  was  essayed  to  embrace  some  general  information 
upon  Mechanical  and  Physical  subjects. 

The  Tables  of  Areas  and  Circumferences  of  Circles  have 
been  extended,  and  together  with  those  of  Weights  of  Metals, 
Balls,  Tubes,  Pipes,  etc.,  of  this  and  some  preceding  editions 
were  computed  and  verified  by  the  author. 

This  edition  is  a  revision  and  an  entire  reconstruction  of 
all  preceding,  embracing  amended  and  much  new  matter,  as 
Masonry,  Strength  of  Girders,  Floor  Beams,  Logarithms,  etc., 
etc. 

To  the  young  Mechanic  and  Engineer  it  is  recommended 
to  cultivate  a  knowledge  of  Physical  Laws  and  to  note  re- 
sults of  observations  and  of  practice,  without  which  eminence 
in  his  profession  can  never  be  attained ;  and  if  this  work 
shall  assist  him  in  the  attainment  of  these  objects,  one  great 
purpose  of  the  author  will  be  well  accomplished. 

NOTE  i. — Mechanical  and  Physical  subjects,  commencing  at  p.  427  and  ending  at 
p.  870,  are  given  in  alphabetical  order. 
2. — Tons  are  given  and  computed  at  2240  Ibs. 
3. — Degrees  of  temperature  are  given  by  the  Scale  of  Fahrenheit. 


INDEX. 


A.  Pag 

ABUTMENTS  AND  ARCHES  (See  Arches 

and  Abutments) 604-605 

ACCELERATED    BODY,  Distances,  Ve- 
locities, and  Acceleration  of.  .921-92 

Acids 18! 

Acreage,  To  Compute 337 

Adulteration  in  Metals,  Proportion 
of  Two  Ingredients  in  a  Compound.  21 

A  erodynamics 61 

AEROMETRY 673-676 

(See  also  Pneumatics.) 

"  Course  of  Wind,  etc 675 

"  Cyclones,  Direction  of. 675 

"  Degree  of  Rarefaction 67; 

"  Discharge  Pipes,  Diameter  of.  676 
"  Distance  of  Audible  Sounds. ..  674 
"  ForceofWindonaSurface.bn-tijs 
"  Height  of  a  Column  of  Mer- 
cury to  Induce  an  Efflux  of 

Air,  To  Compute 675 

"  Resistance  of  A  Plane  Surface 

to  Air,  To  Compute 675 

"  Resistance  to  a  Steam  Vessel  in 

Air  or  Water 911 

"  Weight  or  Pressure   of  Air 

under  a  Given  Height  of 

Barometer  and  Temperature 

Discharged  in  One  Second. .  675 

"  Wind,  Velocity  and  Pressure 

of. 674, 911,  924 

"  Volume    of   Air    discharged 
Through  an  Opening,  etc., 

into  a  Vacuum 674 

AEROSTATICS 427-431, 614 

"  Clouds,  Classification  of. 430 

"  Distances,  by  Velocity  of  Sound  428 
' '  Elevations,  by  a  Barometer.  428-429 
"  Thermometer..  429 

"  Lightning,  Classification  of. ..  430 
"  Velocity  of  Air  flowing  into 

a  Vacuum,  To  Compute.  42 

"  of  Sound 428 

"  Weather  Glasses  and  Barom- 
eter Indications 430-431 

Age  of  Horses,  To  Ascertain 186 

Ages  of  Animals IQ2)  Ig6 

AIR,  ATMOSPHERIC 431-432,  912 

"    and  Steam,  Mixture  of. 737 

"    Carbonic  Acid  Exhaled  by  Man.  432 

"    Compressors 940 

"    Consumption  of,  by  Candle 432 

"    Decrease  of  Temperature  by  Al- 
titudes   522 

"    Discharge  of,  Coefficients  of  Ef- 

,       flux.   674 

"    Expansion  of. 520 

"    Flow  of,  in  Pipes,  Loss  of.  .745,  909 
"    Head  of,  to  Resist  Friction  in 

Long  and  Rectilineal  Pipes . .  925 
Pressure  of,  in  Rear  ofaProjectile  6481 


Page 

AIR,  ATMOSPHERIC,  Pressure,Velocity, 
and  Resistance  of  a  Plane  Sur- 
face, To  Compute 648 

"  Pressure  of  a  Weight  of,  or  other 
Gas  at  62°  and  14. 7  Lbs.  Press- 
ure, with  Constant  Volume  for 

a  Given  Temperature 522 

"  Proportion  of  Oxygen  and  Car- 
bonic Acid  at  Various  Loca- 
tions   432 

"    Rarefaction  of. 430 

"  Resistance  of  different  Figures 
in,  at  different  Velocities,  and 

to  Falling  Bodies 646-647,  949 

"    Specific    Heat    of,   and    Other 

Gases 505-506 

"  Temperature,  for  a  Given  Lati- 
tude and  Elevation 676 

"     Volume  of  a  Weight  of,  or  Per- 
manent   Gas  for    any 
Pressure,  To  Compute. .   520 
' '    and  Weight  of  Vapor  in .  68-69 
"        "    of  a  Weight  of,  or  Perma- 
nent Gasforany  Pressure 
and  Temperature. .  .521-522 
"        "    of  and  Gas  in  a  Furnace..  760 
"        "    of  Enclosed,  at  o°  that  may 
be  Heated  by  One  Sq. 
Foot  of  Iron  Surface. . .  925 
"        "    of,  or    Gas     Discharged 
through  an  Opening  and 
under  a  Pressure  above 
that  of  External  Air. . .  676 
"        "    Pressure,  and  Density  of 
at    Various     Tempera- 
tures  521-522 

"        "    Required   per   Hour,  for 
each    Occupant    in    an 

Enclosed  Space 525 

Ajutage,  Cylindrical 549 

ALCOHOL 194 

"        Elastic  Force  and  Tempera- 
ture, of  Vapor  of. 707 

' '        Proportion  of,  in  Liquors.  191, 204 

Ale  and  Beer  Measures 45 

Water  in   201 

ALGEBRAIC  SYMBOLS  AND  FORMULAS,  22-25 

Alimentary  Principles 200 

Alligation 106 

Alloy,  Expands  in  Cooling 952 

ALLOYS  AND  COMPOSITIONS 634-637 

"  Bronze 637 

"   Compounds, Fusible  and  Solders  634 

"   Of  Steel 643 

"  Soldering  Fluid  and  Fluxes. . .  636 

"  Solders 636 

< '    Welding  Cast  Steel 634 

ALMANAC,  Epacts  and  Dominical  Let- 
ters, 1800  to  1901 73 


Vlll 


INDEX. 


Page 

ALMANAC,  Altitude  of  Sun  at  New 

York 932 

Altitudes,  Decrease  of  Temperature  by  522 

Aluminum J55}  938,  976 

Amalgam 054 

Ammeters  and  Voltmeters 961-962 

ANALYSIS  OF  ORGANIC  SUBSTANCES..  190 

"  of  Foods  and  Fruits 201 

"  of  'Meat,  Fish,  and  Vegetables, etc.  200 

ANCHORS  AND  KEDGES   and  Units, 
To  Determine  Weight  and 
Number  of,  U.S.N. . .  174-175 
"       "Number   and    Weight  of, 

U.S.N. 174 

"      Cables,   Chains,   etc.,  for    a 
Given  Tonnage,  Am.  Ship 

M.  Ass'n I73-I74 

*     Experiments  on  and  Compar- 
ative Resistance  to  Dragging  175 

"     Proof  Strain  of. 175 

Ancient  and  Scripture  Lineal  Meas- 
ures and  Weights 53 

ANGLES  (See  also  Trigonometry)..  385-389 
"      and  Distances  Corresponding 
to  Opening  of  a  Rule  of 

Two  Feet 160 

Chord  of,  to  Plot  and  Compute.  359 

Critical  and  Visual 669 

Sines  of,  To  Compute 402 

To  Describe,  etc 222 

To  Plot  without  a  Protractor,  359 

ANIMAL  and  Human  Sustenance 203 

FOOD 200-207 

Power  and  Work 432-440 

Birds  and  Insects. .  438,  440 
Camel  and  Crocodile. .  438 
Coursing  and  Chasing .  440 

Day's  Work 434 

Dog 438 

Horse.  435-437, 439-440,918 
Llama  and  Ox 438 

Men 433-435,  438-439 

Mule  and  Ass 437,  918 

on  Street  Rails  or  Tram- 
ways  435 

"    Stage  Coaching 440 

ANIMALS,  Proportion  of  Food  for. . .  -  205 

"        Ages  of,  etc 192,196 

Annealing 786 

ANNUITIES no-m 

Amount  of,  To  Compute. . .  no 
"          at    Compound 

Interest in 

Present  Worth  of. . . . .  no-m 
"         Yearly  Amount,  that  will 

Liquidate  a  Debt... . .  no-m 

Anti-attrition  Metal,  Babbitt's 636 

ANTIDOTKS  for  Poisons 185,  03 s 

Anvils,  Weight  of. 0x8 

Apartments,  Buildings,  Ventilation  of  524 

Apothecaries'  or  Fluid  Measure 46 

Weight 32 

Appold's  Pump  and  Wheel 579-580 

APPENDIX 1.  .919-955  1024-1029 

Aqueducts,  Roads,  and  Railroads,i78, 939 


Page 

Arc,  To  Describe 225,  227-228 

Arch,  Radius  of,  To  Compute 604 

"•      Depth  of,  To  Compute 605 

ARCHES  AND  ABUTMENTS 604-605 

(See  also  Masonry,  604-605, ) 
"  "  Chords,  etc. ,  Safe  Weight  of  7  76 
"  "  Minimum  Thickness  of ,  for 

Bridges 605 

"    AND  WALLS 602-603 

Area   and   Population  of  Divisions 

and  Countries 188 

Arenes , 589 

AREAS  of  CIRCLES  by  Sths 231-236 

"  "  and  Circumferences  of  Cir- 
cles by  ioths 243-252 

"      "  and  Circumferences  Greater 

than  in  Table,  To  Compute  252 

'  byi2ths 253-257 

'  by  Birmingham  Wire  Gauge  236 

'  by  Logarithms 236,  252 

"        '  Greater  than  in  Table.  .235,  252 

"        '  In  Feet  and  Inches 235-236 

"  '  When  Diameter  is  composed 
of  an  Integer  and  a  Frac- 
tion, To  Compute 236 

1     OF  SEGMENTS  OF  A  CIRCLE.  .  267-269 

'     OF  ZONES  OF  A  CIRCLE 269-271 

"  "         To  Compute 271 

ARITHMETICAL  PROGRESSION 101-103 

Artesian  Well 179,  198 

ASBESTOS 913 

"        Felting,  Cement,  etc 1032 

Ash,  Proportion  of,  in  Woods 482 

ASPHALT 481,  515,  689-690 

and  Pavement 944-945 

"       Composition 593 

"       Mortars  and  Concrete 913 

Astronomical  Day 70 

Atlantic  and  Pacific  Oceans 937 

"  "   Tides  of. 191 

Atmosphere 912 

AVOIRDUPOIS  WEIGHT.  .    32 

Axle,  Compound,  or  Chinese  Windlass,  627 
(See  also  Wheel  and  Axle,  626-627.) 

B. 

Babbitt's  A  nti-attrition  Metal 636 

Bacteria  in  Earth  soil 942 

Baking  of  Meats,  Loss  by 206 

Balances,  Fraudulent 65 

Balks  and  Battens,  Dimensions  of. . .     62 
Balloons,  Capacity  and  Diameter  of..  218 

BALLS,  Cast  Iron  and  Lead 153 

Balls,  Lead,  Weight  and  Dimensions  of  501 

BARBED  WIRE  Fencing 947 

Barker's  Mill 577 

Barley,  Value  of,  Compared  to   100 

Lbs.  of  Hay 203 

BAROMETER  (See  also  Aerostatics).  427-431 
Elevations  by  Readings .  429 
Height  of,  To  Compute. .  429 

Indications 429-430 

Weather  Glasses 430 

Weather  Indications..,  ,  431 
Barrel,  Dimensions  of. ,      30 


INDEX.  IX 


Page 
BARS,  Wro't  Iron,  Square  and. Rolled, 

125-128 
"     Steel,  Flat  and  Rolled 134-135 

BEAMS,  BARS,  OR  GIRDKRS,  Transverse 

Strength  of. . . ._.  802-8 1 1 ,  813-820 
(See also  Girders,  Beams,  etc.) 
"    Box,  Plate  and  Bars. .  806,  817,  827 
"    Comparative  Strength  and  De- 
flection of  Cast  Iron 809 

44    Comparative    Value    of  Bars, 

Girders,  or  Tubes 824 

44    Common  Centre  of  Gravity,  etc.  819 
44     Cylinder  and  Cylindrical,  Di- 
ameter and  Weight,  To  Com- 
pute   805 

44    Cylindrical,  or  Tubes,  Trans- 
verse Strength  of. 810 

4 '    Deflection  of. 770-782,  840-8  \  i 

"     Depths  and Weight,  etc.,  of  'Steel  134 
' '    Dimensions  and  Weigh  t  of  Roll- 
ed Steel,  and  To  Compute.  134,  644 
44     Rolled  Steel  Beams  and  Bulb 

Angles 807,  808 

"    Elliptical  Sided,  To  Determine 

Side  or  Curve  of. 826 

41    Factors  of  Safety 821,841 

44    Flanged,  Dimensions  and  Pro- 
portions of  Wro't  Iron 809 

41    Floor  Beams,  Headers,  Trim- 
mers, etc.,  To  Compute. .  .835-838 
•    Formulas  for  TransverseStress, 

To  Compute 801,  816 

1    General  Deductions 824-825 

4    Inclined,  Formula  for 811 

'    Lintels,  etc 822-823 

'    Moments  and  Stress  of... .  .621-622 

'          "        of  Resistance 818 

'    Rectangular,  Girders  or  Tubes.  809 

4    Scarfs,  Resistance  of 841 

4    Shearing  Stress 623 

1    Solid  Cylinder,  Diameter   of, 

To  Compute 804-805 

4    Symmetrical  Forms  and  Sec- 
tions, Conditions  of. 825-826 

4    Trussed,  Notes  on 823-824 

4     Unequally  Loaded 810 

'     Various  Figures  and  Sections, 

Dimensions  of. .  147,  805,  813,  814 

1     Wrought  Iron  Rolled 809 

BEAMS,  Inertia  and  Resistance,  Mo- 
ments of  and  Neutral  Axis, 

To  Compute 818-820 

4     Circular  or  Elliptic 815 

'    Hollow  or  Annular 815 

'    Miscellaneous  Illustrations. . .  826 
4    of  Unsymmetrical  Section,  and 

Ultimate  Strength  0/817,  820-821 

44    OR  TUBES,  Elliptical 810,  815 

Bearings,  for  Propeller  Shaft 473 

Beef,  Lean,  Water  in 201 

Beer  and  Ale  Measure 45 

44       Water  in 201 

Beet  Root,  Ratio  of  Flesh  -formers 

and  Sugar,  Analysis  of. 207 

Beeves  and  Beef,  Weights  of 35 

Bell  Ringing 433 


Page 

Bells,  Weight  of. 180,  936 

Belt,  Equivalent,  and  Wire  Rope 167 

Belting,  and  Destructive  Stress  of,  907,  935 

"  or  Hose,  To  Preserve 877 

BELTS  AND  BELTING, 441-443,877.960,1018 
"    Adhesive  Compound  for  Rubber.  877 

Dressing  for 878 

Dynamo  and  Link  Leather 960 

General  Notes  and  Memoranda, 

442-443,  872,  877,  989 

India  Rubber 442 

Width  of,  To  Compute 44i~442 

Bench  Marks 85, 1035 

Beton  or  Concrete 593 

Bibliotheque  National 936 

Birds,  Flying 440 

Incubation  of. 192 

and  Insects 196,  438 

Bissextile  or  Leap  Year 70 

Blacking 877 

Blast  Furnace 529 

44  Pipe  of a  Locomotive..  907 

BLASTING 443-445,  9'3,  9l6 

Boring  Holes  in  Granite . .  444 
44        Charge  of  Gunpowder  for, 

and  Effects  of. 444 

44        Weight  of  Explosive  Mate- 
rials in  Holes 444 

44        Gelatine,  Composition  of...  916 

Paper,  Composition  of. 912 

44        Tunnels  and  Shafts,  Cost  of '.  444 

44        Drilling  and  Mining 445 

Blasts   and   Draughts,  Comparative 

746 


Blower  and  Exhausting  Fan 89 

BLOWERS,  Fan 447-448,  898, 1015 

14    Elements  of,  To  Compute 448 

44    Memoranda 448-449 

44    Power  of  a  Centrifugal  Fan.  448 
44    Pressure  of  Blast,  To  Compute  447 
BLOWING  ENGINE,  Friction  of  Air  in 

Long  Pipes,  etc 925 

ENGINES 445-449,  898, 1018 

44        Dimensions  of  a  Driving 

Engine,  To  Compute 446 

44        Elements  of,  To  Compute. . .  447 

44        Memoranda 448 

44        Power  of,  and  Power  Re- 
quired to  Drive 446 

44        Root's  Rotary 449 

44        Volume  of  Air  transmitted.  447 

Blowing  Off  of  Steam 726-727 

Board  and  Timber  Measure 61 

BOARDS,  Volume  that  can  be  sawed 

from  a  Round  Log 947 

BOILER,  Steam 526,  739-745 

"     and  Ship  Plates 828 

' 4     Abut  Straps  and  Stays. . .  753-754 
4  4     A  reas  and  Ratio  of  Grate  and 
Heating  Surface,  Volume  of 
Water andWeightqf 'Fuel.  741-742 
44      Coal,  Utilization  of,  in  a  Ma- 
rine  726 

4  4      Comparative  Result  of  Experi- 
ments with  a  Steam  Jet. . . .  746 


INDEX. 


Page 

BOILER,  Consumption  of  Fuel  in  a 

Furnace,  To  Compute. . .  725-726 

"      Draught 739,  744-74^ 

*«            "       and  Blasts,  Compar- 
ative Effect  of 746 

"  "        Velocity  of 746 

u       Efficiency,  Nominal  and  IIP 

and  Economy  of. 758,  976 

44     Evaporative  Capacity  of  Tubes  742 
"      Evaporative    Effects    of,  for 
Different  Rates  of  Combus- 
tion and  Surface  Ratios. . .  743 
"     Evaporation,  Power  of....  757-758 

"     Eyes,  Stays,  Rods,  etc 754 

44     Fuel  that  may  be  Consumed 

per  Sq.  Foot  of  Grate 742 

44      Girders  for  Furnaces 754 

"      Grate,  Heating  Surface,Water, 

Fuel,  etc.,  To  Compute.  .741,  927 
44     Heating  Surfaces  and  Rela- 
tive Value  of. 740 

"     Mean    Strength    of    Riveted 

Joints  Compared  to  Plate.j  51-752 
44     Minimum   Volumes   of  Fuel 
Consumed  per  Sq.  Foot  of 

Grate 740-741 

"     Plates  and  Bolts 749-750 

"          u   Thickness  of  for  a  Given 

Pressure  and  Pitch,  etc.  753 

44     Power  and  IPof. 526,  760 

"     Proportion    and     Capacities 

of,  and  Firing 739-74° 

44     Rate  of  Combustion 760 

44     Relation   of   Grate,   Heating 

Surface,  and  Fuel 741 

'*  Results  of  Operation  of  and 
under  Varying  Proportions 
of  Grate,  Surface,  Draught, 

Combustion,  etc 743,  924 

"  Results  of  Operation  of  Va- 
rious Designs  of  Boiler  and 
Varying  Proportions  of  Area 

of  Grate  Surface,  etc 744 

44     Return  Tubular,  Elements  of 

a  Test 726 

"     Riveting 755-757, 9°7 

11          "      General  Formulas  and 

Illustrations 757 

"     Safety  Valves 746-747 

"     Saline  Saturation  in 726 

44  Scale,  Removal  of  Incrusta- 
tion of. 726 

• '     Stay  Bolts,  Diameter  and  Pitch 

of,  To  Compute 754 

"  Tensile  Strength  of.  753-754 

44     Steam  Heating 526,  957-958 

41      Steam  Room 748 

"      Tubes,  Evaporating  Capacity 

of  Various  Lengths. . . .  742 
44         "  Lap-welded  Charcoal,  Di- 
mensions of. 139 

'•      Volume  of  Water  per  Lb.  of 

Coal,  To  Compute 725 

"      Weights  of , and  with  Water.  7 59. 929 

BOILBRS,  Area  of  Grate  per  Lb.  of 

Coal...,, 748 


BOILERS,  Blowing   off,  To    Compute 

Loss  of  by  .................  727 

"    Bottoms  of,  To  Preserve  .......   878 

Corrugated  Flues  ............  941 


Cylindrical  Shells  ----  751-752,91 
Elements  of.  ................. 

Flued,  Arched,  or    Circular 


13 
58 


Furnaces,  U.  S.  Law  ____  754=755 
44    Shell    Plates,  Pressure    ana 

Thickness  of,  U.  S.  Law.  7  50-751 
44    Horse-Power  of.  ..........  914,  936 

44    Incrustation  or  Scale,  To  Re- 

move ......................  726 

44    Magnesia  Covering  of.  ----  918,  921 

44    Plates,  Straps,  and  Stays  ......  753 

44    Proportion  of  Grate  and  Heat- 

ing Surfaces,  Result  of  Ex- 

periments ..................  926 

44    Saline,  Saturation  in  .........  726 

*4         "      Matter,    Proportionate 

Volumes  of.  .........  727 

44    Smoke     Pipes     and      Chim- 

neys.. .................  748-749 

14    Steam  in  Foreign  Countries  .  .  935 
44    Steam  Water  Tube,  Efficiency 

and  Results  of.  ............  947 

4  4     Volume  of  Furnace  Gas  per  Lb. 

of  Coal  ..............  76o 

44          4'  of  Water  Slowed  off  to 

that  Evaporated  ......  727 

44     Water  Tube  and  Efficiency  of, 

926,  947 

BOILING  -  POINTS    of   Various    Sub- 

stances ...............  517 

44      "    at  Different  Degrees   of 

Saturation  ............  815 

BOLTS,  Adhesion  of  Drifted  .........  949 

4'      and  Plates,  U.  S.  Test  ____  749-750 

4  4      Rods  of  Copper,  Weight  of....   148 

44      Round  and  Square,  Relative 

Driving  Resistance  of  Steel.  970 
44      Tenacity  and   Resistance    of, 

Round,  Square,  and  Screiv.  .  970 
<t      Wro't  Iron,  Experiments  on.  .  783 
44      AND   NUTS,   Dimensions   and 
Weights  of,  and  ofTobin 
Bronze  .........  1  56-1  59,  929 

44        4<  English  and  French  Stand- 

ard ....................  158 

44        "  In  Wood,  Tenacity  of.  .....  198 

44       u  of  Wro't  Iron  as  effected  by 

the  Thread  .............  916 

44        4'  Square  Heads  ............   159 

Boring  and  Turning  Metal.  ........  197 

44       Instruments,  Tempering  of.  .  197 
"       Well  .......................  197- 

Boyden  Turbine  ...................  574 

Brain,  Relative  Weights  of.  .........  192 

BRAKES,  Looomotive  ...............  923 

BRASS  .............................  636 


of  a  Given  Section,  Weight  of.  136, 149 

Ornaments,  To  Clean 877 

Plates,  Weight  of. 118-119,  146 

44       Thickness  of. 121 

Seamless  Pipes 143 

Sheet,  Weight  of. 14* 


INDEX. 


XI 


Page 

BRASS  Castings,  Weight  of,  To  Compute  1 5  5 

"    Tubes,  Weight  of. 142 

"     Weight  of. 136 

"    Wire,  Weight  of. 120-121 

Braziers'  Sheets  and  Sheathing 155 

"  "          Copper 131 

Bread,  Wheat,  Water  Lost  by  Drying.  207 

Breakwaters 181 

Breast- wheel 568-570 

"        Proportions  and  Effect. .  569 

Brick  Walls,  Thickness  of. 603 

Brickwork 595,  597-598,  600,  801 

"  Masonry 595 

Brick  or  Compressed  Fuel 

BRICKS , 

"      Stones,  etc '. .  800 

u      Crushing  Resistance  of. 908 

"       Volume  of  and  Number  in  a 

Cube  Foot  of  Masonry. .  599-600 

BRIDGE,  Britannia  Tubular 178 

"         Highest 907 

"  Iron  over  Kentucky  River. .  178 
"  N.Y.,  Erie,  andW.  Railroad  178 
"  New  York  and  Brooklyn ...  178 

"         over  Oxus 930 

"  Resistance  of  from  a  Model.  645 
"  Suspension, Elements  of. ...  842 

"         Plates  and  Rivets 830 

BRIDGES 178,  930,  936 

u  Lengths  and  Spans  of. . .  181-182 
"  Suspension  and  Length  of 

Spans  oj. 178, 199,  842 

Bridles  or  Stirrups,  for  Beams 838 

British  and  Metric  Measures,  Com- 
mercial Equivalents  of. 906 

Broccoli,  Value  of. 207 

BRONZE 637 

Malleable 907 

u      Manganese 832 

"     TO*TS,  Yacht  Shafting.Platcs, 

and  Pump  Piston  Rods. . .    929 

Browning  or  Bronzing  Liquid 874 

Builders'  Measure 46 

Building  Department,  Requirements 


of. 


907 


BUILDING    STONES,  Expansion    and 
Contraction  of. 184 

BUILDINGS,  Walls  of  English 189 

"     Protection  of  from  Lightning  907 
Burns  and  Stings,  Application  for. . .  196 

Bushel,  Pounds  in 34 

Buttermilk,  Sugar  and  Water  in 201 

Buttress  and  Counterfort 696 

C. 

Cabbage,  Value  oj 207 

"  Water  in 201 

CABLES,  ROPES,  HAWSERS,  ANCHORS, 

AND  CHAINS 163-175 

Chain 169,  930 

"       Diameter    of,  for    a    Given 

Weight  of  Anchor 175 

"       Galvanized     Steel,    Strength 

and  Weight  of. 163 

"       Hawsers,  Hemp    and    Wire 

Rope,  Comparison  of. 169 


Page 

CABLES,  Ropes  and  Hawsers,  Circum- 
ference of,  To  Compute. .   171 
"          "  Weights  of,  To  Compute. .   172 

Ropes-  and  Hawsers 170 

"           "    Strain  Borne  with  Safe- 
ty, To  Compute 171 

Calculus 24 

Calendar,  Ecclesiastical 70 

"         Gregorian  or  N.  S 70-71 

Caloric 504,  614 

u      Engine,  Ericsson's 903 

Darnel,  Load  of  and  Travel 438,  918 

Canal,  Suez,  and  Via 177,  183,  912 

' '  or  Conduit,  To  Discharge  a  Given 
Volume  of  Water,  and  To  Compute 

Fall  of. 920 

CANALS,  Dimensions,  etc 181 

"     Flow  of  Water  in 550 

"     Locks  and  Capacities  of,  To 

Compute 183,  553-555 

"     Power  of  a  Horse  on 848 

''     Traction  on 848 

"      Transportation  of. 193 

Candles,  Lamps,  Fluids,  &udG&s<  Light 

of,  etc .583-584 

Cane  Sugar  or  Saccharose 207 

Cannon  Ball,  Flight  of. 495 

Capillary  Tube 358 

Cargoes,  To  Ascertain  Weight  of.  .176-177 
Units  for  Measurement  of  ..  176 

Carrot,  Ratio  of  Flesh-formers 207 

'4  Value  of,  Compared  to  100  Lbs. 

of  Hay 203 

Cascades  and  Waterfalls 184 

Case  Hardening 644,  786 

CASK  GAUGING 377 

Casks,  Buoyancy  of. ; . .  192 

u       Ullage,  To  Compute  Volume  0/378 
CAST  AND  WROUGHT  IRON  of  a  Given 

Sectional  Area,  Weight  of 136 

"    Weight  of,  To  Compute 155 

CAST  IRON. 1 31,  637-638,  765,  783-786, 798 
"  and  Lead  Balls,  Weight  of. ...  153 
"  Balls,  Weight  and  Diameter  of.  153 

•'   Bar  or  Rod,  Weight  of. 131 

"   Bars,  Experiments  on 780 

"    Characteristics  of 637 

' '   Column  s,  We  igh  t  Borne  Safely 

by 768 

"  Malleable  Castings 639 

"  of  a  Given  Sectional  Area, 

Weight  of.  136,  149 

"   Pipe,  To  Resist  Oxidation 927 

u  Pipes  and  Tubes,  Dimensions 

and  Weight  of. .. . 147 

"         "  Weight  of. 132-133 

"   Plates,  Weight  of  a  Sq.  Foot. ..   146 

"    Ultimate  Strength  of 781 

Castings,  Shrinkage  of. 218 

"  Holes  in 872 

"    Weight  of,  by  Pattern 217 

Catenary,  To  Delineate 230 

Cathedral,  St.  Peter's 179 

Cattle  and  Horses,  Transportation  of.  192 
"     Dressed   Weight   of,  To  Com- 

PVte 35 


Xll 


INDEX. 


Page 

Cauliflower,  Value  of.  ..............  207 

Cave,  Mammoth,  of  Kentucky  .......  936 

Cement  Mortar  ....................  595 

"    Cloth,  etc.,  for  Covering  Steam 

Pipes  ......................  956 

CEMKNTS.  515,  589-590,871-873,907,956,958 
(See  Limes,  Cements,  Mortars,  and 

Concretes,  588-597.) 
CENTRAL  FORCES  ................  449-454 

44  "    Formulas  for  Various 

Elements  ..........  450 

CENTRE  OP  GRAVITY  .............  605-608 

44        4l    and  Vertical  Distance  be- 

tween Centres  of  Crush- 

ing and  TensiteStrength 

of  a  Girder  or  Beam.  .  .  819 

11        "    Centre  of  any  Plane  Fig- 

ure, To  Ascertain.  .605-606 
"        "of   Displacement    of    a 

Vessel  .............  653,  658 

CENTRE  OF  GYRATION  ...........  609-61  1 

44      "  Elements  and  Centre  0/610,611 
CENTRES  OF  OSCILLATION  AND  PER- 

CUSSION, and  To  Compute.  612-614 
44    Centre  of,  in  Bodies  of 

Various  Figures.  .  .  613 
CENTRIFUGAL  FAS,  Elements  and  Pow- 

er of  a  Fan  Blower,  etc.  448 
(See  Fan  Blowers,  447-448.  ) 
44        FORCE  of  any  Body  ........  450 

44        Forces  ...................  499 

'  *        Formulas,  to  Determine  Va- 

rious Elements  .........  450 

44    PUMPS  ..............  579,  9JI>  9*7 

CHAIN,  To  Set  out  a  Right  Angle  with  .     69 
14     Cable,   Diameter   and   Length 
foraGiven  Weight  of  Anchor, 


4  4    CABLES,  Breaking  Strain,  Proof 

and  Strength  of.  .  169,  930 
44      "  Anchors,  etc.  ,  for  a  Given 

Tonnage,  Am.  S.Ass'n.  173-174 
"      "  Length  of,  for  Anchors  ____   175 

44      u  Stud  -  Link,    Weight    and 

Strength  of.  .........  168,  930 

41      "  Stowage  of  ...............  913 

Chaining  over  an  Elevation  .....  69-1034 

CHAINS  AND  RopES/or  Cranes,  Weight 

of  and  Proof.  .........  457 

14         "  of  Equal  Strength  ........   165 

*  '    Safe  Working  Load  of.  .......  168 

CHARACTERS  AND  SYMBOLS  .....  21-22,  973 

CHARCOAL  ...........  33,  194.  480-481,  485 

44         Produce  of  from  Woods.  .  .  481 
Cheese,    Composition    of    Different 
Countries  ........................  205 

Chemical  Composition  of  some  Com- 

pound Combustibles  .....  461 

"        Formulas,  To  Convert  .....  190 

Chimney  Draught  and  Chimney  s.goj,  918 
44         Velocities  of  Current  of  Air 

in  One  of  100  Feet  ........  749 

CHIMNEYS  ......  179-180,  904,  916,  918,  925 

44        and  Smoke  Pipes  ......  748-749 

44        Height  of  and  Commercial 
fffor  a  Given  Diameter  of  Flue.  .  925 


Chinese  Wall 179,  936 

44       Windlass,  To  Compute 627 

44       or  Indian  Ink 907 

CHRONOLOGICAL  ERAS  and  Cycles 26 

CHRONOLOGY 70-74 

44    To  Ascertain  Years  of  Coinci- 
dence of  a  Given  Day  of  the  Week, 

etc 74 

Churches,  Opera-Houses,  and  Thea- 
tres, Capacity  of. 180 

CIRCULAR  Av.cs,from  i°  to  180° 26* 

44    Lengths  of  up  to  a 

Semi-circle 260-261 

44    Length  of,  To  Ascer- 
tain   261 

44      Motion 618 

44      Measure  of  an  Angle 113-114 

Circulating  Pumps 749 

CIRCLE,  to  Ascertain  Square  that  has 

the  Same  Area  of. 259 

44    Side*  of  a  Square  Equal  in 

Area  to 258-259 

CIRCLES,  Areas  of,  by  8ths,  ioths,  and 

i2ths 231-257 

44  by  Logarithms 236 

44          44  by  Wire  Gauge 236 

CIRCUMFERENCE    of  a   Circk  when 
Greater  than  Contained  in 
Tables,  To  Compute.  24 1-242,  252 
44    of  Birmingham  Wire  Gauge.  242 
4  4    Wh  en  Diameter  consists  of  an 

Integer  and  a  Fraction. . . .  242 
CIRCUMFERENCES  OF  CIRCLES  by  Sths, 

237-242 

44  and  Area  of  by  10^5.243-252 
44        44     44       "   by  Inches  and 

izths 252-257 

44  by  Logarithms 242,  252 

14        44  In  Feet  and  Inches.  .241-242 
Cisterns  and  Wells,  Excavation,  Lin- 
ing, and  Capacity  of. 63 

Civil  Day 37,70 

44    Year 70 

Cloth  Measure 27 

Clouds,  Classification  of. 430 

Clover,  Value   of,  Compared  to   100 

Lbs.  of  Hay 203 

COAL,  Anthracite 33, 480,  483,  485-486 

44     Composition  of  Average., 485 

44     Fields  of  U.  S. ,  Areas  of. 191 

COALS,  Average  Composition  of  and 
Fuels,  Heat  of  Combustion 
and  Evaporative  Power  of. . .  486 

44    Bituminous 33,  479,  483,  485-486 

44       44  and  Natural  Gas,  Relative 

Water  Evaporating  Powers  913 
44      "  Caking,  Splint    or    Hard, 

Cherry  or  Soft,  and  Cannel  479 
44      44  Classification,     Chemical 
Composition  and  Varieties 

of. 479 

;  4     Consumed  per  Hour,  to  Heat  100 

Feet  of  Pipe 527-528 

44    Effective  Value  of 908 

COALS,  Elements  of  Various  American  480 
"    Fields,  Areas  of  U '.  S. 191 


INDEX. 


Xlll 


COALS,  Gas 484 

44    Japan 909 

"    Lignite 479>  481 

"    Measures 33,  46 

"    Mine 936 

"    Miscellaneous  Experiments 487 

44    Production   and    Consumption 

of  the  World 955 

Coast  and  Bay  Service  and  Scour. . .  908 
Cocks,    Composition,    and     Copper 

Pipes,  Dimensions  of. 150 

Coffee  and  Tea,  Water  in 201 

COHESION 614 

"  Modulus  of 763-764 

Coins,  British  Standards 38 

"      To   Convert   U.  S.  to   British 

Currency,  and  Contrariwise    31 

"      Tolerance  of. 3! 

"     U.  S.,  Weight  and  Fineness  of.     38 

"     Value  of,  To  Compute 39 

"     Foreign  Silver  and  Gold,  and 
Weight,  Fineness,  and  Mint 

Value  of 39, 43 

COKE,  Evaporative  Power  of,  etc 480 

Cold,  Extremes  of,  in  Various  Coun- 
tries and  Snow  Line 191-192 

"     Greatest,  Artificial. 908 

College,  Oxford 179 

COLLISION  OR  IMPACT 580-582 

"    Velocities  of  ..  58 1-582 

Color  Blindness 195 

Colors  for  Drawings 913 

*      Proportion  of,  for  Paints. ....    66 
Columns,  Towers,  Domes,  Spires,  etc. , 

1 80,  936 

41  Crushing  and  Safe  Load  of. . .  766 
"  Long  Solid,  Comparative  Value 

of. 769 

"  of  Cast  Iron 768-769 

"        "  Weight  Borne  with  Safety, 

768-769 
COMBINATION 112-113 

COMBUSTION 458-466 

4  4    Chemical  Composition  of  some 

Compound  Combustibles. . .  461 
~  x  **    Composition  and  Equivalents 
of  Gases  Combined  in  Com- 
bustion of  Fuel 460 

M    Consumption  of  Fuel  to  Ileat 

Air,  To  Compute 466 

**    Evaporative  Power  of  i  Lb. 

of  a  Given  Combustible. . .  462 

44    Heat  of. 463 

44    Heating  Powers  of  Combus- 

tibles,and  To  Compute. 461-462 

"    Of  Fuel,  Ratio,  etc 463-465 

*'    Products  of  Decomposition  in 

the  Furnace 458-459 

11    Rate  of,  in  a  Furnace 760 

**    Relative  Evaporation  of  Sev- 
eral Combustibles 465 

'«         "  Volumes    of  Gases    or 
Products  of,  per  Lb. 

of  Fuel 465 

"    Temperature    of,  To  Ascer- 
tain.  462-463 


COMBUSTION,  Volume  of  Air  Chemical- 
ly Consumed  in  Complete 
Combustion  of  i  Lb.of  Coal  459 
1 '     Volumes  of  Air  Required  for 

Combustion 464-465 

44    Weight  and  Specific  Heat  of 
Products  and  Temperature 

of  Combustion,  etc. 462-463 

4  4     Weight  and  Volume,  of  Gaseous 

Products  of  i  Lb.  Fuel 460 

Compass,  Degrees,  <£  Graduation,  54, 1023 
COMPOSITION  Cocks,  Dimensions  of. . .  150 

COMPOSITION  for  Welding  Steel. 634 

COMPOSITIONS  AND  ALLOTS 634-637 

(See  Alloys  and  Compositions.) 

Composition  Sheathing  Fails 135 

Compound  Axle  or  Chinese  Windlass  627 

*•        INTEREST 108-109 

44        PROPORTION 95-96 

.'*..       Weights  of  Ingredients...  218 

CONCRETE 588-597 

(See  Limes,  Cements,  Mortars, 
and  Concretes. ) 

44  CoigneVs 914 

"   Compositions  of. 593 

44  or  Beton. 593 

Concretes,  Cements,  etc 800 

CONES. 353-354 

CONDENSATION,  Surface 967 

44  4t  Experiments  on.  911 

CONDENSER,  Results  of  an  Operation 


of.. 


967 

CONIC  SECTIONS 379-380 

4  4     Conoid  and  Ellipse,  Elements  of  380 
44       44  To  Describt.and  Area,  Or - 
dinate,  Abscissa,  Diam- 
eters, Circumference,  Seg- 
ment,and  Length  of  Curve, 

To  Compute. 380-382 

44     General  Definitions 379 

44  Hyperbola,  To  Describe,  and 
Abscissa*,  Diameters,  Length 
of  an  Arc,  and  Area,  To  Com- 
pute  379-38o 

44     Parabola 379-380, 382-383 

44  To   Describe    Ordinate, 
Abscissa,  Curve,  Area, 
and  Segment  of.. .  .382-383 
Constructions  and  Natural  Forma- 
tions, Largest 936 

Contractility  and  Elasticity 614 

Cooking  of  Meats 206 

COPPER,  Tensile  Strength  of. 750,  788 

and  Iron  Riveted  Pipes,  Wevght 

of. 148 

Braziers1  and  Sheathing 131 

"    Rods  or  Bolts,  Weight  of .   148 
Given  Sectional  Area,  Weight  of  1 36 

Plates,  Thickness  of. 121 

44        Weight  of. 118-119 

44      per  Sq.  Feet. .  146 
Seamless  Drawn  Tubes.  Weight 

of. 140-142,  144-M5 

Sheathing  and  nraziers*. .  .131-155 
Sheet,  Weight  of  a  Sq.  Foot ...  135 


XIV 


INDEX. 


Page 

COPPER,  Weight  of,  and  To  Compute, 
i36> 

"     Wire,  Cord 123 

"        "      Weight  of 120-121 

Copying,  Words  in  a  Folio 

Cord,  Copper  Wire 123 

CORDAGE,  Friction  andRigidity  0/472-47; 

Corn  Measure 19! 

11  Value  of,  Compared  to  100  Lbs. 

of  Hay 203 

Corrosive  Effects  of  Salt-water   on 

Steel  and  Iron 916,  971 

CO-SECANTS  AND  SECANTS 403-414 

"  "        To  Compute,  etc.  414 

COSINES  AND  SINES 390-402 

it  ic        TO  compute,  etc. 401-402 

CO-TANGENTS  AND  TANGENTS 415-426 

"  "To  Compute,  etc.  426 

COTTON  FACTORIES 8« 

Couple,  Constitution  of. 6 

Coupling  or  Sleeves  of  Shafts 71 

Coursing  and  Chasing 440 

CRANE,  Railroad 962 

"      Steam  Dredgers,  Elements  of 

and  Dredging 899-900 

CRANEB 179,  433,  455-457,  962 

'     Chains  and  Ropes  for 457 

"  Dimensionsof  Post,  To  Compute  456 
"  Machinery  and  Proportion  of.  457 
"  Post,  Stress  and  Conditions  of.  455 
"  Stress  on  Jib,  Stay,  or  Strut, 

455-457 

Crank,  Turning 433 

Cream,  Percentage  of,  in  Milk 205 

Creosoting,  Effects  of. 869 

Crocodile,  Power  of. 438 

CROPS,  Mineral  Constituents  Absorbed 

or  Removed  from  an  Acre  of  Soil. .   189 
Cross-ties,  Railroad,  Duration  of. . .  970 

Croton  Aqueduct 178, 939 

Crusher,  Ore  and  Stone  Breaker 957 

CRUSHING  STRENGTH 764-769,  1021 

(See  Strength  of  Materials.) 
Cube  Measures 30-31 

CUBE  ROOT,  To  Extract. 97 

"  AND  SQUARE  ROOT  of  a  dum- 
ber consisting  of  Integers 
and  Decimals,  To  Ascer- 
tain   301-302 

"  "of  Decimals  alone,  To  As- 
certain   302 

"        "of  any  Number  over  1600, 

To  Ascertain 301 

*'  "or  Square  Root  of  Roots, 
Whole  Numbers  and  of 
Integers  and  Decimals,  To 

Ascertain 97-98 

"         "  of  a  Higher  Number  than  is 

Contained  in  Table 301 

CUBES,  SQUARES,  AND  ROOTS 272-302 

(See  Squares,  Cubes,  and 

Roots.) 

"  To  Compute  and  to  As- 
certain, etc 300-302 

Cucumber,  Water  in 207 

Currency,  To  Convert  U.  S-  to  British    39 


Page 

Current  Wheel 570 

"       of  Rivers 193 

Curvature  and  Refraction  of  Earth. .     55 

Curves,  Caustic,  or  Lines 669 

Cut  Nails,  Tacks,  Spikes,  etc 154 

Cutters,  Yachts,  Pilot  Boats,  Launches  895 

Cycle,  Dominical  or  Sunday  Letter. .     70 

"     Lunar  or  Golden  Number. ...     71 

"     of  the  Sun,  To  Compute 70-71 

CYCLES  and  Chronological  Eras 26 

Cycloid,  To  Describe 228 

CYCLONES,  Direction  of. 675 

CYLINDERS,  FLUES,  AND  TUBES,  Hollow  827 
u  Solid  and  Hollow,  of  Various 
Metals 801 

D. 
DAMS,  EMBANKMENTS,  AND  WALLS  (See 

Embankments,  etc.) 700-703 

DAY,  Astronomical,  Marine  or  Sea.  .37,  70 

"    Sidereal,  Solar,  and  Civil 37,  70 

Day^s  Work 434 

Dead  Sea  and  Valley  of  the  Jordan. .  934 

Deals  and  Local  Standards  of 62 

DECIMALS 92-94 

Deer  Park,  Copenhagen 1 79 

DEFLECTION  (See  Strength  of  Mate- 
rials. ) 770-781 

Delta  Metal 384,  913 

Departures,  Table  of. 54 

Depths,  Sea 184 

Derrick  Guys 163 

Desert  of  Sahara 936 

Desiccation 513 

DETRUSIVE  OR  SHEARING  STRENGTH 

(SeeStrength  of  Materials,  782-783. ) 
"  and  Transverse,  Comparison  of.  782 

"  Strength  of  Woods 782 

"  Wood,  Surface  of  Resistance  of.  782 

Dew  Point,  and  To  Ascertain 68 

Diamond  Weight 32 

Diamonds,  Weight  of. 193 

Diet,  Daily,  of  a  Man 202,  207 

u        "      of  an  Esquimau 914 

Differentiation,  Integration,  and  Cal- 
culus  24-25 

DIGESTION  OF  FOOD,  Time  Required 

for 206-207 

DISCOUNT  or  Rebate 109 

DISPLACEMENT  of  a  Vessel 653 

DISTANCES,  STEAMING 86 

and  Angles,  Corresponding  to 
Opening  of  a  Rule  of  2  Feet .  1 60 

between  Cities  of  U.  S. 184 

"           "  East  and  West. .  187 
"  Principal  Ports  of  World    87 
"  "     of  U.S..     88 

"  Various  Ports  of  Eng- 
land, Canada,  and  U. S.,and 

N.  Y.  and  London 86 

Velocities  and  Acceleration  of 

a  Body,  To  Compute 921-922 

Geographic,  and  Measures 54 

Distemper  ( Coloring) 593 

DISTILLATION 514 

of  Fresh  Water 955 


INDEX. 


XV 


Page 

Distillers  and  Evaporators,  Capaci- 
ties of. 950 

Dog,  Power  of,  Coursing  and  Chasing, 

438,  440 
DOMES  and  Towers,  Diameter  and 

Heights  of. 179-180,  932 

Domestic  Remedials 938 

Dominical  Letters  and  Epacts 73 

u          or  Sunday  Letter 70 

DOVETAILS,  Tenacity  of. 948 

DRAINAGE  OF  LANDS  by  Pipes 691 

Drains,  Diameter  and  Grade  of,  to 

Discharge  Rainfall 906 

4 '  and  Sewers,  Velocity  and  Grade 

of. 692 

DRAUGHT,  A  rtificial 745-746 

Natural 739-74Q,  744 

Steam  Jet  and  Blast,  Com- 
parative Effects  of,  and  Result 

of  Experiments  with 746 

Drawing  and  Tracing  Paper 29,  964 

u        or  Pushing 433 

Drawings,  Colors  for 196,  913 

' 4  Dimensions  of,  for  U.  S.  Patents  198 
Dredger,  Steam   Hopper,  and   Ma- 
chines  899-900 

DREDGING,  and  Cost  of. 197 

44         Machines  and  Crane. ... ..  899 

DRILLING  in  Rock 445,  940 

"        in  Metals 477 

Drills,  Mountings,  etc 940 

DROWNING  PERSONS,  Treatment  of. . .  187 

DRY  MEASURES 30,  31 

DUALIN 503 

DUODECIMALS 94 

DYNAMITE  and  Cellulose 443-444 

DYNAMICS 614, 616-620 

"      Circular  Motion 618 

44      Decomposition  of  Force 620 

44      Motion  on  an  Inclined  Plane  619 

44       Uniform  Motion 617-618 

"      Work  A  ccumulated  in  Moving 

Bodies,  an  d  To  Compute  619 
44         "  By  Percussive  Force. . . .  620 

DYNAMO  Leather  Belts 960 

E. 

EARTH,  Diameters  and  Density. . .  188, 198 
"     and    Rock    Excavation    and 

Embankment 192 

44      Area  and  Population  of. 188 

"     Boring  and  Heat  of  Mines 955 

"      Conductivity  of  Temperature  in  914 
44      Curvature  and  Refraction  of.     55 

44      Elements  of  Figure  of. 6* 

"     Influence  of  the  Rotation  of,  on 

Moving  Bodies 942 

44      Motion  of. 70 

"      Weight  of,  per  Cube  Yard 468 

"      Weights  of. 33 

EARTHWORK 467-468 

44   Bulk  of  Rock,  etc.,  Original 

Excavation  Assumed  at  i. .  468 
44  Number  of  Barrow  and  Horse- 
cart   Loads   and   Shovelling,  and 
Volume  of,  Transported  per  Day..  908 


Page 
Easter  Day , 71 

Ecclesiastical  Year 70 

Egg,  Fowls1,  Composition  of. 207 

Egyptian  and  Hebrew  Measures 53 

ELASTIC  FLUIDS,  Specific  Gravity  of.  215 
ELASTICITY  and  Strength.  195, 614, 761-763 

44  Coefficient  of. 761 

"  Modulus  of,  and  To  Compute  762-763 
4 4  Relative,  of  Materials 780 

ELECTRIC  AND  GAS  LIGHT 198 

44    Dynamo  Engine 954 

44    Elevator s,  Power  Required. ..  959 

44    Launch 900 

"    Light,  Candle- Power  of. 908 

44    FANS,  MOTORS,  Power,  Pumps.  959 
"    WIRES    AND     CABLES,    Tele- 
graph, Telephone,  and  Light  Wires 

and  Cables 960 

Electrical  Engineering,  Units  in,  Re- 
sistance and  Expressions. 987-988,  1033 
ELEMENTARY  BODIES,  with  their  Sym- 
bols and  Equivalents 190 

Elephant,  Power  and  Weight  of. 918 

Elevations  by  a  Barometer 428-429 

44          and  Heights  of  Various 

Places  above  the  Sea 183,1035 

ELLIPSE,  To  Describe  and  Construct, 

etc 226-227,  380 

(See  Conic  Sections,  379-380.) 
ELLIPTIC  ARCS,  Lengths  up  to  a  Semi- 

44          "          ellipse  of. 263-266 

44          "      To  Ascertain  Length  of  266 

EMBANKMENTS, WALLS,  AND  DAMS,£7«- 

ments  of. 700-703 

(See  also  Revetment  Walls,  694- 

699,  and  Stability,  693-703. ) 
u    "  Equilibrium,   Stability   and 

Moment,  To  Compute 701 

44    "  Form  of  a  Pier,  To  Determine  700 

14     "  High  Masonry  Dams 703 

44     "  Materials,  Weight  of  a  Cube 

Foot  of. 694 

14    "  Surcharged  Revetments 699 

44    "  Various  Elements,  To   Com- 
pute and  Determine. . . . 702-703 
Endless  Ropes 167 

ENGINES   AND    MACHINES,  Elements 

and  Cost  of. 898- 

44       and  Sugar -mills,  Weights  of '. 

Engravings,  To  Clean  Soiled 875 

Ensigns,  Pennants,  and  Flags,  U.S  .  199 
EPACTS,  AND  DOJIINICAL  LETTERS.  ...  73 
EQUATION  OF  PAYMENTS 109 

EQUILIBRIUM,  Angles  of,  at  which  Va- 
rious Substances  will  Repose  694 

41  Of  Forces 616 

Ericsson's  Caloric 903 

Esquimau,  Daily  Food  of. 914 

Establishment  of  the  Port  for  Several 

Locations  in  Europe 85 

Ether,  Elastic  Force  of  Vapor 707 

Evaporation 747-748,  1024 

44         of  Water  per  Sq.  Foot 514 

44  u    per  Month  of  Year  916 


XVI 


INDEX. 


Evaporative  Power  of  Tubes  per  De- 
gree of  Heat,  etc 51 

Evaporators  and  Distillers,  Capaci- 
ties of. 95 

EVOLUTION 

"    To  Extract  Square  and  Cube 

Roots 9- 

"         any  Root 97-9; 

EXCAVATION  AND  EMBANKMENT,  Ele- 
ments of,  etc 466-468 

"    "  Bulk  of  Rock,  Earthwork, 
etc.,  Original  Excavation 

Assumed  as  i 468 

'    "  Cost  of,  per  Cube  Yard. ...  467 
««    "  Earth,  Rocks,  etc.,  Weight  of.  468 
'*    "  Earthwork  by   Carts    and 
Barrow  Loads  Removed 
by  a  Laborer  per  Day.  467-468 
"    "  Labor  and  Work  upon  and 

Estimate  of  Cost  07.466-467 
"     "     *'  in  Blasting  and  Hauling 

Stone  or  Earthwork,  etc.  468 
"    "  Loads  or  Tripsin  Cube  Yards 

per  Cart  per  Day 466-467 

"     "  of  Earth  and  Rock 192 

Expansion 614 

**  and  Contraction  of  Building 

Stones,  etc 184 

Expenditures  in  England  for  Various 
Purposes  and  of  Articles,  compared 

with  Spirituous  Liquors 938 

EXPLOSIVES,   Relative    Strength    of, 

Fired  under  Water.  946 
"  High,  Firing  Point  and  Rela- 
tive Strength 953)9^6 

F. 

FALLING  BODIES,  Resistance  of  Air  to,  941 
Family  of  Mechanics,  Costof,inFrance,gol 

FAN  BLOWERS , ...... . .447-448 

"    Elements  of,  and  Power  of. 448 

"    Exhausting  and  Blower 898 

"    Memoranda 448 

Farms,  Sustaining  Production  of...  207 

Fascines 690 

FELLOWSHIP 99 

FELTING,  Covering,  Lagging,  etc.  . . .  1032 
FENCE  WIRE,  Strength  and  Weight  of 
Single  Thread  and  Cable  Laid. ...  164 

FENCING,  Barbed  Steel  Wire 947 

ffig,  Value  of. 207 

files,  Repair  of. 878 

Filter  Beds 851 

Filtering  Stone 909 

FILTERS  for  Waterworks 184 

Fire  Bricks 515,  600 

"     Clay 597 

FIRE-ENGINE,  Steam.. 904,  909 

F\sh,Meat,and  Vegetables,Analysis  of  200 
Flags,  Ensigns,  and  Pennants,  U.  S..  199 

FLAX  MILL 476 

Floating  Bodies,  Velocities  of 909 

Flexible  Paint  for  Canvas 915 

Flood  Wave  of  Ohio  River 912 

FLOOR  BEAMS,  of  Wrought  Iron,  and 
Distances  from  Centres, . . . , 931 


Page 

FLOORS  AND  LOADS,  Factor  of  Safety, 

and  Weights  of  and  on 841 

(See  Strength  of  Materials,  795-841.) 
FLOUR,  Consumption  and  Tests  of.  206-207 

"       Mills 900 

FLUES,  TUBES,  and  Cylinders 747,  827 

"     Arched  or  Circular  Furnaces .  754 
"     and  Furnace,  Corrugated,  and 

Formulas  for 909,  941 

Fluid  and  Liquid  Measures 30,  46 

Fluids,  Candles,  Lamps,  and  Gas 584 

"      Percussion  of. 579 

FLUTTER  WHEEL 571 

Fluxes  for  Soldering  or  Welding 636 

FLY  WHEEL,  Weight  and  Dimensions 

of  Rim,  To  Compute 451 

FLYING  of  Birds 440 

Fontaine  Turbine 574 

FOOD,  Animal 200-207 

"     Comparative  Values,  for  Sheep.  938 

"    Daily,  of  an  Esquimau 206,  914 

"  Digestion  of,  and  Time  Re- 
quired for 200 

' '    Milk,  Relative  Richness  of. 207 

"    Nutrient  Value  and  Ratio  of 

100  Parts 202 

"      "  Equivalents  of  from  Amount 

of  Nitrogen  in,  when  Dried  205 
<l    Proportion  of,  Appropriated  and 

Expended  by  Animals 205 

"      "  of  Starch  in  Sundry  Veg- 
etables   205 

"    Ratio  of  Flesh -formers  from 

Tubers 207 

"  Thermometric  Power  and  Me- 
chanical Energy  of.  when  Ox 
idized  in  the  Animal  Body. .  205 

FOODS  and  Fruits,  Analysis  of. 201 

"          "in  Reference  only  to  Heat 

and  Strength 203 

"    Elements  of  Various 207 

*  *    M ilk,  Nutritive  Values  and  Con- 

stituents of. 202 

*  *    Nutritious  Properties  of  Differ- 

ent Vegetables  and  Oil- cake 

Compared 204 

11    Relative  Values  of. 202,  204 

**        '*    "  to    make    an    Equal 
Quantity  of  Flesh  in 

Cattle  or  Sheep 202 

"        "    "  Comparedwith  iooLbs. 

of  Hay 203 

"        "    "  as  Productive  of  Force 
when  Oxidized  in  the 

Body 204 

"    Required  by  a  Man 207 

"     Volume  of  Oxygen  to  Oxidize 

as  Consumed  in  the  Body. . .  204 
"     Weight  of,  to  Develop  Power  in 

Human  System 204 

"          ' '   to  Furnish  1 220  Grains  of 

Nitrogenous  Matter . . .  202 

'OOT- POUND 947 

'ORCE,  Decomposition  of. 620 

'ORCES,  Composition  and  Resolution 

of. 6x5 


INDEX. 


XV11 


Page 

FORCES,  Division  of. 614-616 

"        Equilibrium  of 616 

"        Inertia  of  a  Revolving  Body.  616 

u        Percussive,  Work  by 620 

FOREIGN  MEASURES  AND  WEIGHTS.  .48-52 
Formations,  Natural,  and  Construc- 
tions, Largest. 936 

Formulas  and  Algebraic  Symbols.  .22-23 

Fortress  Monroe 179 

FOUNDATION  PILES 198,  909 

Foundations,  Pressure,  S.  Load.  781-1040 

FRACTIONS 89- 

Fraudulent  Balances 65 

Freeboard  of  Vessels 666,  913 

Freezing,  Effects  of,  to  the  Resistance 

of  Stones,  etc : 

FRESH  WATER,  Distillation  of. 955 

FRICTION,  Elements  of,  etc... 469-478,  571 

"    and  Delivery  in  Hose 922 

u  and  Rigidity  of  Cordage .  472-473 

"    Application  of  Results 474 

"  Bearings  for  Propeller  Shaft.  473 

"    Coefficients  of  Axle 471 

*«  "  of  Masonry  on  Masonry, 

Clay,  and  Earths 696 

"  "  of  Motion  and  Repose ....  470 

"        "  To  Determine 471 

"    Elements  of. 469-470 

"    Grain  Conveyers 478 

"    in  Launching  of  Vessels 478 

u  Mechanical  Effect  of. . ...  471-472 

*  *  of  Air  in  Long  and  Rectilineal 

Pipes,  Head  of  in  Lbs 925 

"    of  Bottoms  of  Vessels 909 

4 '    of  Engines  and  Propellers .  662-663 
"    of  Journals  or  Gudgeons  of  a 

Water-wheel 571 

"    of  Journals  of  a  Water- Wheel.  571 
"    of  Machinery,  Results  of  Ex- 
periments upon. .  .475-478,  ioio 
1 '    of  a  Non-condensing  Engine.^jB,g  1 8 
"    of  Pivots  and  Relative  Value 

of  Angles  of. 472 

"    of  Planed  Brass  Surfaces ooo 

"    ofRoads 847 

[  4    of  Steam- Engines ....  47  5,  478,  918 

"    of  Screw  Steamer 478 

"    of  Tools 476 

11    of  Water  in  Pipes 925 

*  *    of  Winding  Engines,  Shearing, 

Flax  Mill,  Tools,  Planing, 
Molding,  Slotting,  Turnings, 

Grindstones,  etc 476-478 

"    of  Wall  and  Earth 698 

"    Relative  Value  of  Unguents. ..  471 
'•     Results  of  Experiments  on  Sev- 
eral Instruments 474 

ft     Rolling 473 

"     Wood-sawing 477 

Frictional  Resistance  of  a  Railway 

Train 916 

Frigorific  Mixtures 193,  516 

FRUITS,  Analysis  of. 201 

"     Proportion  of  Acid  and  Sugar  203 

FUEL,  Elements  of. 479-487 

"    Anthracite  Coal 480-486 


Pag* 
FvRL,Area  of  Grate  and  Consumption 

of,  To  Compute 513 

"    Ash,  Peat,  and  Tan 482 

"    Asphalt 481 

"    Average  Composition  of,  Heat, 

and  Evaporative  Power  0/485-486 

u    Bituminous  Coal 479-487 

"    Brick  or  Compressed 907 

"     Charcoal  and  Coke 480,  483 

"     Classification  of  Coal 479 

"     Comparative     Value    of,    and 

Weights 484 

"    Elements  of. 486 

"    Lignite  and  Wood. 481-482 

"    Liquid,  Petroleum,   Coal   Gas, 

and  OiU 484 

"    Miscellaneous 483.  485-487 

"    Produce  of  Charcoal 481 

"    Relative  Values  of. 483 

"     Units  of  Heat  in 927 

"     Values,  Weights,  and  Evapora- 
tive Power  of  per  Bulk 483 

FURNACES 528-529,  754-755 

FUSIBLE  COMPOUNDS. 634 

G. 
GALVANIZED  SHEET  IRON,  Thickness 

and  Weight  of. 124,  129 

' '  Charcoal  Iron 163 

"  Iron  Wire  Rigging  and  Guys 

and  Steel  Cables 162-163 

"  Iron  Wire  and  Diameter  of. ..  123 
Galvanizing 786 

GAS,  Elements  of,  etc 585-587,  676 

"  and  Electric  Light,  etc 198. 969 

' '  A  tmospheric  Engine  and  Engines, 

587,  912,  990 

"  Candles,  Lamps,  Fluids,  etc 584 

u  Coal,  Composition  of. 484 

"  Diameter  and  Length  of  Pipe  to 

Transmit  Given  Volume  of...  586 

"   Flow  of,  in  Pipes 586 

' '  Light,  Intensity  of,  equal  Volumes 

from  Different  Burners 585 

"  Mains,  Dimensions  of,  etc 587 

"  Natural-  and  Bituminous  Coal, 

Relative  Evaporating  Powers.  913 

"  Pipe  Threads 160 

"     Elements  of  and  Weight 123 

"  Steam  and  Hot-air  Engines 909 

"   Tubing,  and  Number  of  Burners, 

Regulation  and  Length  of. 586 

"  Volume  of  per  Lb.  of  Coal  and  Air  760 
"  "  of  a  Weigh  t  of  A  ir,  or  Perma- 
nent Gas  for  any  Pressure.  520 
"  "  of  Discharged  through  a  Pipe  587 
"  "  of  per  Hour  under  Pressure.  587 
"  "  fromaTonofCoal,Resin,etc.  586 

"   Weight  of  a  Cube  Foot  of '. 215 

JASES,  Expansion  of  and  Volume  ofi 
Lb.  at  32°  under  One  Atmosphere  520 

"    Permanent,  Volumes  of '. 193 

"    Temperature  of. 587 

"    of  Solidification 516 

Gauge,  Mercurial 910 

Gauges,  Wire. ., 118-123 


XV111 


INDEX. 


Page 
GAUGING  Cask  and  Varieties  of.....  377 

44         of  Weirs 922 

Gear,  Spur 911 

Gelatine,  Blasting 916 

GEOGRAPHIC    MEASURES    and    Dis- 
tances and  Levellings 54-56 

Geographical  and  Nautical  Measures    26 
GEOMETRICAL  PROGRESSION 103-105 

GEOMETRY,  Elements  of,  etc 219-230 

44  Angles * 222 

41        "     and  Distances 160 

44  Arcs 225-227 

44  Catenary 230 

44   Circles 224-225 

' 4  Cycloid  and  Epicycloid —  22  8-2  2  9 

44  Definitions 219-220 

44  Ellipse 226-227 

44   Gnomon 219 

44  Hexagon 223 

44  Hyperbola  and  Spiral 230 

44  Involute 229 

* '  Length  of  Elements  and  Lines, 

221-222 

44   Octagon  and  Polygon 223 

14  Parabola 229-230 

44  Rectilineal  Figures 222-224 

44  Scales 221 

Geostatics  and  Geodynamics 614 

GESTATION,  Periods  of  and  Number 

of  Young  of  Animals 192 

GIRDER,  Dimensions  of  and  Greatest 

Load,  To  Compute 839-840 

44       Bowstring 812 

GIRDERS,  Beams,  etc.,  General  Deduc- 
tions from  Experiments. .  824 
(See  Beams,Bars,or  Girders,  802-820. ) 
44      4'  and      Beams,     Destructive 
Weight     of,  of    Various 
Figures  and  Sections. 805-806 
44      44   Centre  of,  and  Vertical  Dis- 
tance of  Centres  of  Crush- 
ing and  Tensile  Stress. . .  819 
44      t4   Comparative  Value  of  Bars, 

Tubes,  etc 824 

44      "  Deflection  of. 840-841 

44      "  Factors  of  Safety 821,  841 

44      44  Lintels.  Beams,  etc 822-826 

44      4  4  Moments  of  Stress 621-622 

44      "  Plate 811-812 

44      "   Shearing  Stress  Deduced  by 

Graphic  Delineation  of. .  623 

«      < «  Trussed 823 

44      "   Tubular 775 

"      "     "  Deflection  and  Weight  0/775 
GLASS  GLOBES  and  Cylinders,  Resist- 
ance to  Internal  Pressure  and  Col- 
lapse   831 

Glass,  Window,  Thickness,  etc 124 

GLAZING 197 

GLUES,  Mucilage,  etc 874 

Gold  Sheet,  Thickness  of. 119 

"     Value  of,  from  1501  to  1889 934 

Golden  Number  or  Lunar  Cycle,  To 

Compute 7 

Gooseberry,  Value  of. 207 


Page 

GOVERNORS 452 

"    Revolutions  and  Elements  of ',  452 

GRADE,  Reduction  of,  to  Degrees 359 

GRAIN,  AND  ROOTS,  Weights  of. 34 

'      Conveyers,  Operation  of,  etc..  478 
'      Standard  Weights  per  Bushel.    32 

Granite  Masonry 595 

Graphic  Operation,  Solution  by 905 

Graphite  and  Pencils 978 

GRAVEL 690 

14      or  Earth  Roads 688 

GRATES  of  Boilers,  To  Compute  Areas,  927 

GRAVITATION 487-496 

44    Accelerated  and  Retarded  Mo- 
tion   494 

44    Action  of,  by  a  Body  Pro- 
jected Upward  or  Down- 
ward, To  Compute. .  .490-491 
44    Average  Velocity  of  a  Moving 
Body,  Uniformly  Acceler- 
ated or  Retarded 495 

44    Formula  for  Flight  of  a  Can- 
non Ball'. 495-496 

44    Formulas  to  Determine   the 

Various  Elements  of  490 
44          44  to  Determine  Various 
Elements  of  on  an 
Inclined  Plane.  .493-494 
44  44   of  Retarded  Motion ...  49? 

44     General   Formulas  for  Ac- 
celerating  and  Retarding 

Forces 495 

44     Gravity  and  Motion   at  an 

Inclination 492-493 

44    Miscellaneous  Illustrations. .  496 
44    Relation  of  Time,  Space,  and 

Velocities 488-492 

44    Retarded  Motion 490 

44    Space  Fallen  Through.  .489,  491 
44     Velocity  of  a  Falling  Stream 

of  Water,  To  Compute  496 
44          4'  due  to  a  Given  Height 
of  Fall  and  Height 
due  to  Given  Velocity  488 
GRAVITY,  Action  of,  To  Compute. 488-489 
44    at  Various  Locations  at  Level 

of  Sea 487 

44     Centre  of. 605-608 

44     OF  BODIES 208 

4  4    Promiscuous  Examples  of.  489-490 

44     Various  Formulas  for 488-489 

Grease.  Anti-friction 877 

4'   from  Stone  or  Marble,  To  Re- 
move   878 

Grecian  Measures  and  Weights 53 

GregorianYear  and  Calendar 70-71 

Grindstones  and  Friction  of. 478 

Grouting 593-594>  598 

GUDGEONS,    Diameter  of,  To    Com- 
pute   795 

Gun  Barrels,  Length  of. 198 

"    Browning 875 

44    Cotton 443 

44    Steel,  Kruppi s ....« 913 

Gun-metal,  Weight  of,  and  of  a  Given 
Sectional  Area 136,  149 


INDEX. 


XIX 


Page  ' 
,  Elements  of,  etc 457-503 

* '     Charge,  Range,  Elevation,  and 

Velocity, and  To  Compute^gj^gg 

"     Comparison   of  Force    of  a 

Charge  in  Various  Arms. .   502 

"  Experiments  with  Ordnance 
and  Penetration  of  Shot 
and  Shells 498,  5°° 

"    Initial  Velocity  and  Ranges 

of  Shot  and  Shells 498-499 

"  Lead  Balls,  Weight  and  Di- 
mensions of. 501 

"    Number  of  Percussion  Caps 

corresponding  to  B  Gauge  502 

"      "  of  Pellets  in  an  Oz.  of  Lead 

Shot  of  all  Sizes 501 

«    Penetration  of  Lead  BaUs  in 

Small  Arms 5°° 

4 '    Ranges  for  Small  A  rms 502 

•  *    Report  of  Board  of  Engineers, 

U.S.  A.,  Fortifications,  etc.  499 

"  Summary  of  Practice  in  Eu- 
rope with  Heavy  Guns 500 

•«     Time  of  Flight,  Rute  for 497 

"    Velocity  of  a  Shot  or  Shell ...  497 

"    Windage  and  Waddings,  Loss 

and  Effect  of. 501 

GUNPOWDER 443)  502 

u  Charges  of,  and  To  Compute.  444 
11  Heat   and  Explosive   Power 

of. 503 

"  Manufacture  of. 503 

"  Proof  of. 502 

"  Properties  and  Results  of,  De- 
termined by  Experiments ..  503 

"   Proportion  of,  to  Shot 502 

"  Relative  Strength  of  Different, 

for  Use  under  Water 

GUTTER'S  CHAIN 

Guys,  Derrick,  Strength^  etc 162-163 

Gwynne's  Pump,  Centrifugal 579,  917 

GYRATION,  and  Centres  of. 609-611 

"   Centre  of,  of  a  Water-wheel. . .  6n 

"   General  Formulas 61 1 

' '  Moment  of  Inertia  of  a  Revolv- 
ing Body,  To  Compute 609 

"  Radius,  To  Compute 609 

H. 


HAMMERS,  Steam 179 

Hancock  Inspirator 901 

Hand- cars  and  Portable  Railroad. .  908 
HAWSERS,  WIRE,  AND   HEMP  ROPES 

AND  CABLES,  Comparison  of.  169 
(See  Cables,  Ropes,  etc. ,  163-178. ) 

"    and  Warps,  Length  of 173 

u    Cables  and  Ropes 170 

"    Circumference  of.  To  Compute  171 
u    Units  for    Computing    Safe 
Strain  Borne  by  and  for 
New  Ropes   and  Hawsers, 

170-171 

'    Weight  of,  To  Compute 172 

HAY,  Relative  Value  of  Foods  com- 
pared with  TOO  Lbs.  of. 912 

"     and  Straw,  Weights,  etc 198 


Page 
HEAT,  Elements  of,  etc  ...........  504-529 

'•    Absolute  Temperature  .........  504 

"    Absorption  of  ,  in  Generation  of 

i  Lb.  Steam  at  212°  .........  705 

u    Altitudes,  Decrease  of  by  .......  522 

1  '    A  vailable  Expended  per  IIP  —  909 
"    Boiling-  Points  of  Pure  Water, 
etc.,   Corresponding   to  Alti- 
tudes of  Barometer  ..........  518 

"    Capacity  for  ...............  505,  507 

"    Communication  and  Transmis- 
sion of,  and  Relative  Power 
of,  of  Various  Substances  ----  510 

u    CONDENSATION  and  of  Steam  in 

Cast-iron  Pipes  ......  515-516 

"      "  ofSteamper  Sq.  Foot  and  per 

Degree  per  Hour  .........  516 

4  '    Conducting  Powers,  Relative,  of 
Various  Substances  and  De- 
ductions from  Results  ----  514-515 

"    CONDUCTION  or  Convection  of.  .  .  514 
"    CONGELATION  and  LIQUEFACTION  516 
"    Degrees  of  Different  Scales,  To 

Reduce,  and  Contrariwise  ----  523 

"    Densities  of  Some  Vapors  ......  521 

"    Density  of  Water,  To  Compute.  .  520 
««    DESICCATION  ...............  513-5*4 

"    DISTILLATION  .................  514 

"    Effect  upon  Various  Bodies  by.  518 
"    EVAPORATION  or  Vaporization, 

Elements  of,  etc.  .512-513 
"  "    A  rea  of  Grate  and  Fuel 

for,  To  Compute  .....  513 

'  '    of  Water  per  Sq.  Foot  of 

Surface  per  Hour.  .  .  514 
"  "    To  Evaporate  i  Lb.  of 

Water  ..............  5" 

*  '   Evaporative  Power  of  Tubes  per 

Degree  of  Heat,  etc  ..........  513 

"    Expansion  of  Water,  Liquids, 

Gases,  and  Air  ...........  519-520 

"    Extremes  of  and  Cold  in  Va- 

rious Countries  ...........  191-192 

"    Fluids,  Expansion   of,  in  Vol- 

ume, To  Compute  ........  523-524 

'  '    Frigorific  Mixtures  ........  193,  516 

"    Heating  and  Evaporating  Water 

by  Steam  in  Pipes  and  Boilers  511 
"    Height  Corresponding  to  Boil- 

ing-Point of  Water  ..........  519 

"    LATENT,  and  To  Compute.  .  .  508-509 
"        "  of  Fusion  of  Solids,  and  of  a 

Non-  Metallic  Substance.  509 
"        "  of  Steam,  To  Compute  .....  707 

"    Length  of  /(-Inch  Pipe  to  Heat 

looo  Cube  Feet  of  Air  .......   526 

'  '    Linear  Expansion  or  Dilatation 

of  a  Bar,  Prism,  or  Substance.  519 
"    Liquids,  Volume  of  Several  at 

their  Boiling-  Point  ..........  518 

"    Mean  Temperatures  of  Various 

Localities  ...................  192 

"    Mechanical  Equivalent  (Joules).  504 
"        "      "  of  ,  Contained  in  Steam.  705 
"    Melting  and  Boiling  Points  of 

Various  Substances  .........   517 

<«« 


of  Mines 


XX 


INDEX. 


Page 

HEAT,  of  the  Sun 193 

<• «   Proper  Temperatures  of  Enclosed 

Spaces 526 

"    RADIATION  of. 509-510,  1027 

4 '    Radiating  or  A  bsorben  I  and  Re- 
flecting Powers  of  Substances, 

and  in  Units  of. 509-510 

"    Reduction  of,  by  Surfaces 525 

"    REFLECTION  of. 510-512 

"    Refrigerator,  Surface  of. 512 

"    Relative  Capacities  of  Various 

Bodies  for 507 

"   Required  to  Evaporate  i  Lb.  Wa- 
ter Below  212°  from  Air  at  32°.  512 
"    Saturated  Vapors,  Pressure  of, 

under  Various  Temperatures  518 

"      "  Steam,  Total  of. 705, 707 

"   SENSIBLE 504,507 

"    Sensible   and  Latent,  Sum  of, 

and  Latent  of  Vaporization. .  508 
"   Snow  Line,  or  Perpetual  Con- 
gelation   192 

"   SPECIFIC,  To  Ascertain,  etc. .  505-507 
"         "  for  Equal  Volumes  of  Gas 

and  Air,  To  Compute —  507 

"    Temperature  by  Agitation 524 

"      u  of  a  Mixture  of  Like  and 

Unlike  Substances 506 

"      "of  Solidification  of  Several 

Gases 516 

•'  "to  which  a  Substance  of  a 
Given  Length  must  be  Sub- 
mitted or  Reduced  to  Give  it 
a  Greater  or  Less  Length  or 
Volume  by  Expansion  or 

Contraction 522-523 

*'    Transmission  of,  through  Glass 

of  Different  Colors 511 

•*  "  Quantities  Transmitted  from 
Water  to  Water  through 
Metals  and  other  Solid 
Bodies  i  Inch  Thick,  per 

Sq.Foot. 511 

"  Units  of,  To  Compute. . . .  511-512 

"    Underground 519 

"    Unit 504 

"    Units  of,  in  Fuels 927 

*'    Vegetation,  Limit  of 192 

"    Volume  of  Water  Evaporated  in 

a  Given  Time,  To  Compute. .  .513 

HEATING,^  ir,  Length  of  Pipe  Required 
to  Heat  Air  in  an  Enclosed 

Space  by  Water 525 

'  by  Steam,  Illustration  of. 527 

1  by  Hot-air  Furnaces,  Stoves,  or 

or  Open  Fires, 528-529 

'    by  Hot  Water 524,  1028 

*    by  Steam 527,  913,  1027 

*  Coal  Consumed  per  Hour  to 

Heat  ioo  Feet  of  Pipe 527-528 

"  Length  of  Pipe  Required  to  Heat 
Air  by  Steam  at  5  L bs.  per 

Sq.  Inch 527 

"  Temperature  of  Enclosed  Spaces  526 
"  VENTILATION    of    Buildings, 
Apartments,  etc 524-525 


HBATING,  Volume  of  Air  by  i  Sq.  Foot 

of  Iron  Surface 925 

"         "    of  Air  Heated   by  Ra- 
diators, Fuel,    Grate, 
and  Heating  Surfaces  528 
"  Warming  Buildings,  etc. . .  524-529 
Hebrew  and  Egyptian  Measures,  etc.    53 
Height   Corresponding    to    Boiling- 

Points  of  Water 519 

HEIGHTS  and  Elevations  of  Various 

Places  Above  the  Sea 183 

"       Measurement  of 60 

HEMP  AND  WIRE  ROPE,  Circumference 

?/>  for  Rigging,  U.S.N....  172 
"     "  Circumference  and  Breaking 

Weight  of,  U.S.N. 168 

"  General  Notes 167 

u     "  Hawsers  and  Cables,  Com- 
parison of. 169 

*'     "  Weight  and  Strength  of. 172 

'     "  Weight  of 166 

"    ROPE,  IRON  AND   STEEL,  Safe 

Load  and  Strength  of.  164-165 
"     "  Iron  and  Steel,  Relative  Di- 
mensions of. 172 

"     "  Safe  Strain  Borne  by,  Units 

for  Computing. 170-171 

"    Shrouds  and  Wire 173 

*  *     Tarred,  Destructive  and  Break- 
ing Strength  of. 171 

' '    ROPES  (See  Ropes,  Hawsers,  and 

Cables) 166-173 

Hewing  and  Sawing  Timber,  Loss  in.  62 
High  Water,  Time  of,  To  Compute.  .74-75 
Hills  or  Plants  in  Area  of  an  Acre. .  193 
Historical  Events  and  Notable  Facts.  939 

Hitches,  Knots,  etc 972 

Hoggin 690 

Hoisting  Engines,  Details  of,  etc 901 

Honey,  Analysis  of. 207 

Hoop  Iron,  Weight  of. 129,131 

Hopper  Dredgers,  Steam 899-900 

Horizon,  Dip  of. 60 

Horizontal  Wheels 572 

HORSE 435-437 

'     Cart,  Volume  of  Earth  Trans- 
ported   908 

Transportation 918 

POWER 441,  733-734,  1028 

"    British  Admiralty  Rule.  734 

Cost  of,  by  Steam 950-951 

Notes  on 758 

of  Boilers 914 

On  a  Canal 848 

Transmission  of. 188 

TEAM,  Tractive  Power  of. 436 

HORSES,  Age  of. 186 

"       Labor  of,  etc 435-437 

' '       Performance  of. 439-440 

u       and    Cattle,  Transportation 

of 192 

"        Weight  of. 35 

Horseshoe  Nails,  Length  of. 1 53 

Horseshoes  and  Spikes 152 

HOSE,  Delivery  and  Friction  in 922 

877 


INDEX. 


XXI 


HOT-AIR  Furnaces  or  Stove*  .........  528 

44  Gas,  and  Steam  Engines,  Rel- 
ative Cost  of.  ..............  909 

Human  and  Animal  Sustenance  .....  203 

HYDRAULIC  RADIUS  or  Mean  Depth.  .  552 
"       Cement  ..................  958 

"  "     or  Turkish  Plaster  .  591 

"       Paint  ....................  872 

4  4       RAM,  Elements  and  Efficien- 

cy of  .  .......  561,  917,923 

"          "  per  Cent,  of  Volume  of 
Water  Expended,  To 
Compute  ............  917 

HYDRAULICS,  Elements  of,  etc  ----  529-557 

"     Canal  Locks,  and  Times  of 

Filling  and  Discharg- 

ing, To  Compute.  .553-555 

"         "  M  iscell.  Illustrations.szfi-ssj 

44     Circular  Bent   or  Angular, 

Circular    or    Cylindrical 

Curved  Pipes,  Valve  Gates 

or   Slide  Valves,   Throttle, 

Clack  or  Trap  Valve  Cock, 

or   Imperfect   Contraction, 


Circular  Openings  or  Sluices, 
Coefficients  of.  ............  536 

Circular  Sluices,  etc  ........  537 

Circular,  Triangular,  Trape- 
zoidal, Prismatic    Wedges, 
Sluices,  Slits,  etc  .........  538 

Curvatures,  Radii  of.  .......  544 

Curves  and  Bends  ..........  545 

Cylindrical  Ajutage  .........  549 

Deductions  from  Experiments 
on  Discharge,  from  Reser- 
voirs, Conduits,  or  Pipes.  529-531 
Depth  of  Flow  over  a  Sill  that 
will  Discharge  a  Given  Vol- 
ume, To  Compute  .........  534 

Discharge  from  a  Notch  in 

Side  of  a  Vessel  ......  541 

"from  Conduits  or  Pipes, 

and  Friction  of.  .  .530-531 
"  from  Irregular  -  shaped 
Vessels,  as    a    Pond, 
Lake,  etc.,  Time,  Flow, 
Fall,  Velocity,  and  Vol- 
ume Discharged  ......  542 

"from  Vessels  not  Receiv- 

ing any  Supply.  .  .538-539 
"  from  Vessels  of  Commu- 
nication .............  541 

"  of  inPipesf  or  any  Length 
and    Head,  etc.,  and 
Elements  of,  To  Com- 
pute ..............  547-548 

"  of  under  Variable  Press- 
ures, and  Time,  Rise, 
Fall,  and  Volume  ____  540 

"  of  when  Form  and  Di- 
mensions of  Vessel  of 
Efflux  are  not  Known  539 
*'  or  Effiux  for  Various 
Openings  and  Aper- 
tures, and  Relative  Ve- 
locity under  like  Heads  532 


HYDRAULICS,  Distance  of  a  Jet  of 
Water,  Projected  from  an 
Opening  in  Side  of  a  Vessel, 

14    Fall  of  a  Canal  or  Conduit 
to  Conduct  and  Discharge 
a  Given  Quantity  of  Water 
per  Second,  To  Compute. . .  920 
<4    Flow  and  Velocity  in  Rivers, 
Canals,   and    Streams, 
and    To  Compute  Ele- 
ments of. 550-553 

44       "  and  Velocities  at  which 

Materials  will  Move 916 

44       "  in  Lined  Channels..  .551-552 

u       "  of  Water  in  .Beds,  Fall 

and  Velocity  of,  as  in 

Rivers,  Canals,  Streams, 

etc.,  and  Coefficients  of 

Friction  of. 542-543 

11    Flowing  Water,  Head  of,  To 

Compute 552 

44    Forms  of  Transverse  Sections 

of  Canals,  etc 543 

4  4  Friction  in  Pipes  and  Sewers, 
and  Head  Necessary  to  Over- 
come, To  Compute 543-544 

u      "  of  Liquids  through  Pipes .  531 
"      "of  Water  in  Beds,  as  Rivers, 

etc. ,  Coefficients  of. 543 

"    Head  and  Discharge  in  Pipes 

of  Great  Length 920 

44       4 '  from  Surface  of  Supply  to 

Centre  of  Discharge 544 

44    Height  of  a  Jet  in  a  Conduit 

Pipe,  To  Compute 919 

44    Inspirator,  Hancock 901 

"    Jets    d'Eau,  and   Formulas 

for 550 

44  Journals  or  Gudgeons,  Fric- 
tion of. 571 

44    Miner's  and  Water  Inch 557 

44    Obstruction  in  Rivers 551 

4  4  Pipe,  Inclination  of,  and  Ele- 
ments of  Long,  To  Com- 
pute   548 

44    Prismatic  Vessels 539-54° 

44          "  Fall  of  in  a  Given  Time, 

To  Compute 540 

44    IP  under  Different  Heads. . .  557 
44    Rectangular  Notches  or  Ver- 
tical Apertures  or  Slits.  534 
44       "  Openings   or   Sluices    or 
Horizontal    Slits,   and 
Discharge,  To  Compute.  535 
44       "  Weir,  Volume  of  Dis^Jiarge, 

To  Compute 532-534 

44  Relative  Velocity  of  Effiux, 
through  Different  Apertures 

and  under  Like  Heads 532 

44  Reservoirs  or  Cisterns,  Time 
of  Filling  and  Emptying, 

To  Compute 541 

4  4  Short  Tubes,  Mouthpieces,  and 
Cylindrical  Prolongations 
or  Aiutagts,and  Coefficients 
for  Discharge  of. 536-537 


XX11 


INDEX. 


Page 
HTDRAUJJCS,  Sluice  Weirs  or  Sluices .  535 

44  u  Impeded  or  Unim- 

peded Discharge  of. . .  .535-536 

44  Submerged  or  Drowned  Ori- 
fices and  Weirs 553 

44    Variable  Motion  of  Water  in 

Beds  of  Rivers  or  Streams.  543 

"  Velocity  in  Profile  of  a 
Navigable  River,  To  Com- 
pute   551 

4  4    Velocity  of  Water  or  of  Fluids, 

Coefficients  oj 'Discharge.  531-532 

44    Vena  Contracta 529 

44  Vertical  Height  of  a  Stream 
Projected  from  Pipe  of  a 
Fire-Engine,  To  Compute. .  549 

44    Volume  of  Water  Flowing  in 

a  River,  To  Compute. .  543 

44    Weirs,  Gauging  of. 922 

44        "       or  Notches 539 

HYDRODYNAMICS  AND  HYDROSTATICS, 

Elements  of,  610.558-580,  614 

44        44  AppoWs  Wheel 580 

44        "  Barker's  Mill 577 

44        u  Boy  den  Turbine 574 

44        44  Breast  Wheel,  Proportion, 

Effect,  and  Power  of.  569-570 
44        44  Centrifugal  Pumps —  579-580 

44        "   Current  Wheel 570 

44        "  Flutter  or  Saw-mill  Wheel  571 

44        4t  Fontaine  Turbine 574 

44        "  Horizontal  Wheels 572-577 

44        44  Hydraulic  Ram,  Elements 

of  and  Operation 561-562 

44        44  Hydrostatic  Press 561,  901 

tt        it      u  Thickness  of  Metal. ..  561 
44         x      4'  Motors,  Effective  Pow- 
er of  Water 563 

44        "  Impact  and  Reaction  Wheel.  576 
44        "  Impulse  and  Resistance  of 

Fluids 577-578 

44        4k  Inward -Flow    Turbines, 

Description  of. 575 

44        "  Jonval  Turbine,  Elements 

and  Results 575 

44        "  Low-Pressure  Turbines. . .  575 
44        44  Memoranda   on    Water- 
Wheels 571-572 

44        44   Overshot  Wheel,  Elements, 

Power  of,  etc. .  563-566 
44        4<        "  Power  and  Effect  of, 

To  Compute. . .  565-566 

44         44   Percussion  of  Fluids 579 

44         44  Pipes.  Elements  and  Weight 

of,  etc.,  To  Compute. 560-561 
44         "   PonceleVs  Wheel,  Propor- 
tion and  Power  of.  567-568 
44        '4       "  Turbine, Elements  of. .   574 
44        44  Power  of  a  Fall  of  Water, 

To  Compute 562 

44  44  Pressure  and  Centre  of '.  558-560 
44  44  "  of  a  Fluid  upon  Bottom 
of  Vessel,  Vertical,  In- 
clined, Curved,  or  any 
Surface,  and  also  on 
a  Sluice 559-560 


ftp 

HYDRODYNAMICS  AND  HYDROSTATICS, 
u        u   pressure  of  a  Column  of 

a  Fluid  per  Sq.  Inch . . .  560 

44        "  Rankine  Wheel 580 

41        "  Ratio  of  Effect  to  Power 

of  Several  Turbines 577 

44        u   Reaction  Wheel 576 

44        u   Swain  Turbine 575-576 

4'         44    Tangential  Wheel 576 

u        u    Tremont  Turbine 576 

44        44   Turbine  and  Water  Wheels, 

Comparison  Between. . .  579 
44        "   Turbines, Elements,  Power, 

and  Results..  .572-577 
44        "         "  High  Pressure  and 

Downward  Flow.  574 
44         4'    Undershot  Wheel,  Power  of  566 

44        4C    Victor  Turbine 576 

44        44    Water  Power 562 

44        "       "  Fall  of. 563 

44        "       44  Motors,  Ratio  of  Ef- 
fective Power 563 

«        u       «  pressure  Engine 579 

44        "       4'   Wheels,   Dimensions 

of  Arms,  etc 571 

44        "       44   Wheels,  Divisions  of, 

etc 563 

44        "    Whitelaw's  Wheel 576-577 

Hydrometers 67 

4  Strength  of  or  Volume  of  a  Spirit, 

To  Compute,  etc 67 

HYDROSTATIC  RAIL  OR  SLIP  RAILWAY, 
Power  Required  to  Draw  a  Vessel.  910 

HYDROSTATIC  PRESS 561,  901 

Hygrometer 68 

"  Dew -point,  and  to  Ascertain 
Volume  of  Vapor  in  Atmos- 
phere   68 

%4   Existing  Dryness,  To  Ascertain    68 

44   Vapor,  Weight  of,  in  Air 69 

44   Volume  of  Vapor  in  Air 68 

Hyperbola,  To  Describe 230 

HYPERBOLIC  LOGARITHMS — 331-334,  712 


ICE,  Strength  of,  etc. ...  195,  912,  939,  943 
44  and  Snow,  Weight  and  Volume. . .  849 

44  Boats  and  Speed  of. 896,  909 

4  4  Making  and  Refrigeration.  943 , 965  -968 

44  Manufacture  of. 943,  967 

IMPACT  and  Reaction  Wheel 576 

44      OR  COLLISION 580-582 

44       Velocities  of  Inelastic  Bodies 

after,  To  Compute 581-582 

Impenetrability 195 

INCLINED  PLANE,  Motion  on 619 

44    Elements  of,  To  Compute. 625-630 

Incubation  of  Birds,  Periods  of. 192 

India-  Rubber,  To  Cut 877 

Indicator,  To  Compute  Pressure  by. .  724 

INERTIA,  Moment  of  a  Revolving  Body.  609 

"  Moment   of,  Approximately  to 

Ascertain 659 

"        44    of  a  Solid  Beam 819 

44  of  a  Revolving  Body,  To  Com- 
pute   616 


INDEX. 


XX111 


Page 

Injector,  Steam 736 

"   Size  of,  To  Compute 736 

"    Volume  of  Water  required  per 

IIP  per  Hour,  To  Compute  736 
"      "  of  Feed  Water  required  per 

IIP  per  Hour 736 

Ink,  Chinese  or  India 907 

"    Stains,  To  Remove 935 

Inks,  Indelible,  etc 875 

Insects  and  Birds 196 

Inspirator,  Hancock's 901 

Integration 24-25 

INTEREST,  Simple  and  Compound.  107-109 
INVENTIONS,  Origin  and  Period  of 

Great 937 

Involute,  To  Describe 229 

INVOLUTION 96 

IRON,  Elements  of,  etc 637-640 

(See  Wr -ought-iron,  130-136,  639-640, 
765,768,773,780,785-786.) 

"  and  Steel,  Corrosion  of. 908 

"  Bolts  in  Wood,  Tenacity  of. 198 

* '  Bridges,  and  Iron  Pipe  Bridge ..  178 

"  Mold,  To  Remove 871 

OR  STEEL,  Corrosive  Effects  of 

Salt  Water  on 916 

"  Pig,  Ton  of,  Requirement  of  Air.  445 

"  Preservation  of. 955 

' '  Rust,  To  Remove 935 

Iron  Steamer,  First  Built 915 

Irregular  Body,  Volume  of,  To  Compute  870 
IRRIGATION,  Cost  of,  per  Acre 952 

J. 

Jarrah  Wood 913 

Jets  d'Eau 550 

Jewish  Measures 53 

Jonval  Turbine 575 

Jordan,  Valley  of,  and  Dead  Sea. . . .  934 

Joules'  Equivalent 504 

Julian  Calendar 70 

Jumping,  Leaping,  etc.,  by  Men 439 

K. 

KEDGES  AND  ANCHORS,  Weight  and 
Number  of,  Units  to  Determine. ...  174 

Kerosene  Lamps,  etc 872 

Khorassar,  or  Turkish  Mortar 592 

KNOT 27 

Knots,  Hitches,  etc 9go 


LABOR,  Man  and  Horse. 433-434,  436,  468 

Lacquers 875 

Laitance 592 

Lake,  Highest  Elevation  of. 96? 

Lakes,  Areas  of,  in  Europe,  Asia,  and 
Africa,  and  Depths  and  Heights  of 

Great  Northern  of  U.  S. 181-182 

Lamps,  Candles,  Fluids,  and  Gas. . .  584 

Laud  Measure 29 

Larrying 598 

Laths,  Dimensions,  etc 60 

LATITUDE,  Length  of,  etc 19! 

"  and   Longitude    of  Principal 
Locations  and  Observations.j6-&; 


Page 

LATITUDE  N.  reached  by  Explorers. . .  939 
Launching  Vessels,  Friction  of 478 

EAD,  Sheet,  Cast  or  Milled 640 

"    Balls,  Weight  and  Dimensions 

of. 501 

"    Encased  Tin  Pipe,  Weight  of...  151 
"     Given  Sectional  Area,  Weight 

of- 136 

"    Measure 32 

"  Pipe,  Resistance,  Thickness, 
Weight,  and  Bursting  Press- 
ure of,  To  Compute 831 

"    Pipes  and  Tin  lined,  Weight  of , 

per  Foot  and  Thickness.  .137,  150 
"    Plates,  Weight  of  per  Sq.  Foot .  146 

"        "  Thickness  of. 121 

"    Sheet,  Weight  of. 151 

"    Shot,  Number  of  Pellets  in  an 

Ounce 501 

"    AND  CAST  IRON  BALLS,  Weight 

and  Volume  of 153 

"     Weight  of,  To  Compute 155 

cap  or  Bissextile  Year 70 

raping.  Jumping,  etc 439-440 

.eaves,  Value  of,  etc 207,  481 

,ee-way  or  Drift  of  a  Vessel 910 

Legal  Tenders 38 

lENSESAND  MlRRORS,  Element* 0/670-67 1 

iCvel,  Apparent,  of  Objects  at  or  upon 

Surface  of  Land  or  Sea 56 

Bevelling,  Geographic 55-57.  1035 

"  by  Boiling  -  Point    of  Water, 

Table  of,  etc 55~57 

' '  Height  of  Above  or  Below  Level 
of  Sea,  To  Compute 55 

..EVER, Elements  of, To  Compute.. 624-626 
lifting  by  Men 439 

LIGHT,  Elements  of,  etc 583-587 

"  Sun's  Rays 195 

"  Candles,  Lamps,  Fluids,  and  Gas  584 
"  Consumption  and   Comparative 

Intensity  of,  of  Candles 583 

' '  Decomposition  of. 583 

"  Electric,  Candle  Power  of. 908 

"  Gas  and  Electric 198 

"     "  Consumption,    Volume,   and 

Flow  of. 585-587 

"  Intensity  of,  with  Equal  Volumes 

of  Gas  from  Different  Burners  585 
"  Loss  of,  by  Use  of  Glass  Globes.  584 

"  Penetration  of,  in  Water 915 

' '  Refraction,  Mean  Indices  of. . . .  584 
"  Relative  Intensity,  Consumption, 
Illumination,  and  Cost  of  Va- 
rious Modes  of  Illumination . .   584 

"  Services  for  Lamps 587 

"  Standard  of. 910 

"  Volume  of  Gas  from  a  Ton  of 

Coal,  Resin,  etc 586 

Lighting  Power  in  Streets,  To  Deter- 
mine Coefficients  of. 969 

Lightning.'CTossi/icatfon  of. 430 

' '       Protection  of  Buildings .  907-956 

Lignite 479,  481 

Lime,  Hydraulic  ofTeil 589 


XXIV 


INDEX. 


Page 

LIMES,  CEMENTS,  MORTARS,  AND  CON- 
CRETES, Elements  of 588-597 

and  Cements 594-595 

Asphalt  Composition 593 

Cements  and  Mortars,  Experi- 
ments of  Gen.  Gillmore  596 
"  Transverse  Strength  of. . .  596 
Conclusions  from  Experiments. .  590 

Concrete  or  Beton 593 

General  Deductions  and  Notes, 
by  Gens.Totten  and  Gilmore.  596-59  7 
Limestones,  Indication  of,  etc. .  588 

Mortars 590-592,  595 

Mural  Efflorescence 593 

Pozzuolana 589 

Slaking  Lime,  etc 594 

Stucco,  Exterior  Plaster. .  .591-592 

Trass  or  Terras 589 

Turkish  Plaster  or  Hydraulic 

Cement 591 

LINES,  To  Draw,  Bisect,  etc 221 

Linseed  Cake,  Value  of,  Compared  to 

100  Lbs.  of  Hay 203 

LIQUID  MEASURE SQ-S1*  46 

LIQUIDS,  Expansion  of. 520 

u     Volume  of,  at  Boiling- Point. .  518 
LIQUORS,  Proportion  of  Alcohol  in. . .  204 

"        Proof  of  Spirituous 218 

Lithro-fracteur 443 

Llama,  Load  of. 438 

LOCOMOTIVE,  Elements  of 902 

u      Axles,  Friction  of. 910 

' '      Brakes,  Operation  of. 923 

' '      Comparative  Operations  of  a 

Simple  and  Compound. . .  953 

LOCOMOTIVES,  Operation  of.  .681-685,  912 

"    Adhesion 681-685 

"    in  Foreign  Countries 935 

"     Tractive  Power 681 

"    Freight  Train  Resistance. ..  682 

Log  Lines 27 

Logarithm  of  a  Number 23-24 

LOGARITHMS 3°5-3IO>  1030-1031 

"      Hyperbolic 33i~334>  712 

"      of  Numbers 311-330 

LONGITUDE,  To  Reduce  Time  into 54 

(See  Latitude,  76-80. ) 

' '     Lengths  of  a  Degree 60 

Lucifer  Match,  First  in  Use 915 

Luminous  Point 195 

Lunar  Cycle  or  Golden  Number. ...     71 
Lunar  Month 70 

M. 

MACADAMIZED  ROADS 687-690 

Machinery,  Friction  of. 475 

MACHINES  AND  ENGINES,  Mills,  etc., 

Elements  of. 898-904 

Magnesia  Covering  of  Steam -pipes 

and  Boilers 918,  921 

"       Boilers,  Pipes,  etc 921 

Magnetic  Variation 57-59,  i°35 

"        Bearings  of  N.  Y. 184 

Magnetism 614 

Magnifying  Power 669,  968 

Malleable  Castings 639 


Page 

Malleable  Cast  Iron 785 

Maltha,  or  Greek  Mastic 873 

Manganese  Bronze 832 

Manures,  Fertilizing  Properties    of, 
and  Relative  Value  of  Covered  and 

Uncovered 188 

Marble,  To  Clean,  etc 873,  878 

Marine  Day 37 

MARINE  STEAMERS  AND  ENGINES.  .  886-893 

"   Composite  Yachts 891 

"  Electric  Launch 900 

"  Ferry,  Passenger,  Team,  and 

Tow-Boats 890 

"  Fire-Boat 891 

"   Glue 874 

"  Launches,  Wood  and  Steel 895 

' '  Naval  Cruisers,  Iron-dads,  and 

Protected 886-887 

"  Oil- Engine  Launch 893 

"  Passenger  and  Freight 888-890 

"  Petroleum,  Refrigerator,  Fruit, 

and  Fishing 888 

"  River  and  Inland 888-889 

"  Side  Wheels,  Wood 892 

' '  Stern  Wheels,  Iron  and  ^00^.892-893 
«  Torpedoes  and  Dynamite. . .  886-887 
"  SAILING  VESSELS, 'Yachts,  Wood.  895 

"  "    Cutter,  Wood 895 

"  "    Pilot-Boat 895 

"  "    Steel,  Iron,  and  Wood.  894 

MASONRY 197,  597-605,  913 

"  Arch,  Depth  of.  To  Compute. . .  605 
"  Arches  and  Abutments,  Depth, 
Radius,  and  Thickness  of, 

To  Compute 602,  604-605 

"  "  Walls,  and  Abutments. 602, 604-605 
"  Brick,  Stone,  and  Granite. .  595-600 

"  Brickwork 597-600 

u  Designation  of. 602-603 

"  Estimate  of  Materials  and  La- 
bor for  ioo  Sq.  Yds.  Lath  and 

Plaster 604 

"   Grouting 594 

"  Plastering 604 

"  Pointing 598 

"  Rubble 600-601 

"  Stone 600-603 

"  Technical  Terms.  .597-599,  602-603 
"  Volume  of  Bricks  and  Number 

of  in  a  Cube  Foot  of. 599 

"    Walls,  Thickness  of  Brick  for 

Warehouses 600 

"    Working  Load 781 

Mason  and  Dixon's  Line 188 

Mastic 593 

MATERIALS,  Strength  of. 761-841 

"  Non-  Conducting 911,  914 

' '  Relative  Non-  Conductibility  0/911 

MATTER  and  Minuteness  of. 194 

Mean  Proportion 94 

MEASURES,  Length,  etc 26-54 

"  and  WEIGHTS 27~35 

"  and  Mint  Values 38-43 

"  and  Weights,  Foreign  ofValue.^g-^ 
"      "        "       Memoranda 43 


INDEX. 


XXV 


Page 

MKASURBS,  a  Barrel. 30 

"  Ale  and  Beer. 45 

"  Apothecaries' 32,  46-47 

"  Avoirdupois 32,  47 

"  Board  and  Timber 61 

"  British  and  Metric,  and  Com- 
mercial Equivalents  of. 906 

"  Builder's 46 

"  by  Act  of  Congress  and  Metric 

Computation 36 

11  Cables  and  Ropes 26 

"  Circular. 113-114,  34- 

"  Cloth I..?    27 

"  Coal 33,  46 

"  Copying. 29 

"  Corn 198 

"  Cube 30,934 

"  Day,  Civil,  Solar,  and  Sidereal.    37 

"  Diamond 32 

"  Dry 3o 

"  English  and  French 44-45 

"  Electrical,  British  Ass'n 34 

"  Equivalents  of  Old  and  New 
U.  S.  and  of  Metric  Denom- 
inations, To  Compute 36 

"  Fluid 30 

"  Foreign 48-53 

"  French  Old  System 47 

u  Geographic,  and  Distances. . .     54 
' '  and  Nautical. ...     26 

Grain 32,  45 

Grecian 53 

Gunter's  Chain 26 

Hebrew  and  Egyptian 53 

Land 29 

Lead 32 

Lineal  and  U.  S.  Standard.  26,  934 

Liquid 30,  46,  934 

Log  Lines 27 

Metric 27-33,  36,  44,  47 

' '  Equivalent  Value,  U.  S. ,  and 

Old  and  New  U.S. 28,  33 

'  Power  and  Work 36 

'  Temperatures 37 

'  Velocities 37 

'  Volumes 36,  46 

'  Weights  and  Pressures 36 

Mint,  and  Weight  of  Value.  .40-43 

Miscellaneous 27,  29,  31,  44,  46 

Nautical 26,  30,  44 

of  Earth,  Clay,  Sand,  etc 33 

of  Length 26,44 

of  Surface 29,  44,  934 

of  Timber,  Local   Standards, 

and  To  Compute 61-62 

of  Value 38-39 

of  Vessels 45 

of  Volume 30-31,  45,  934 

of  Weight 32,  47,  934 

Old  and  New  U.  S. ;  Approx- 
imate Equivalents. . .     33 

u    "  Equivalents  of. 36 

Paper 29 

Pendulum 27 

Roman. 53 

Ropes  and  Cables 26 

Scriptw  e  and  Ancient 53 


Page 

MEASURES,  Shoemaker's 27 

"  Timber,  English 62 

"Time 37 

"  Troy 32,47 

"  Vernier  Scale 27 

"  Wine  and  Spirit 45 

"   Wood 33,47 

Meat,  Analysis  of,  and  of  Fish  and 

Vegetables 200-202 

Meat,  To  Preserve ...  196 

Meats,  Loss  of,  by  Boiling,  Roasting, 
or  Baking 206 

MECHANICAL  CENTRES 605-614 

"  and  Physical  Elements,  Con- 
structions and  Results. .  .907-918 

"  Centres  of  Gravity 605-608 

' '  Centre  of  Gravity  of  any  Plane 

Figure,  To  Ascertain. . .  605 

"  of  Gyration 609-611 

"  of  a  Water-wheel 6n 

"      "      "  General  Formulas  for  6n 

"  Elements  of  Gyration 610 

u  Moment  of  Inertia  of  a  Revolv- 
ing Body  and  Radius  of  Gy- 
ration of,  To  Compute 609 

11  POWERS 624-634 

"  Compound  Axle,   or    Chinese 

Windlass 627 

"      Screw 631 

( '  Inclined  Plane 628-630 

"  Lever. 624-626 

"  Pulley 632-634 

"  Rack  and  Pinion 628 

"  Screw  and  Wedge 630-631 

"      "    Differential 632 

"  Wheel  and  Axle 626-627 

"       "      and  Pinion,  Combina- 
tion, Chinese  Windlass,  etc 627-628 

MECHANICS 614-634 

"  Accumulated  Work  in  Moving 

Bodies,  and  To  Compute 619 

"  Couple 614 

u  Decomposition  of  Forces 620 

"  Dynamics 614,  616-620 

"  Moment 614 

"  Moments  of  Stress  on  Girders, 

621-623 

u  Motion  on  an  Inclined  Plane.  619 
"  Solid,  Fluid,  and    Aeriform 

Bodies 614 

"  Uniform  and  Variable  Motion  617 

"  Work  by  Percussive  Force 620 

Mechanic's,  Cost  of  Family  in  France  908 
Melting- Points 517 

MEMORANDA  .  Physical  and  Mechanical 

Elements  and  Results.  .907-918 
' '    Cast  and  Wro '  t  Iron  and  Steel  832 

>f en  and  Women,  Weights  of 35 

"    Performances  of 438-439 

"    Power  of. 433-435 

Meniscus  and  Concave- Convex 669 

MENSURATION  OP  AREAS,  LINES,  SUR- 
FACES, AND  VOLUMES 335-378 

"  Acreage,  To  Compute 337 


XXVI 


INDEX. 


Page 

MENSURATION  OP  AREAB,  etc.,  Any 

Figure  of  Revolution 358,  376 

"  Arc  and  Chord,  etc. ,  of  a  Circle, 

343-345 

"  Area  Bounded  by  a  Curve 342 

"      "     of  any  Plane  Figure. . .  359 

"  Capillary  Tube 358 

"  Cask  Gauging  and  Ullaging, 

S77-378 
"  Chord  of  an  Angle,  To  Compute  359 

"  Circle 342 

"      "  Sector  and  Segment  of.  346-347 

"  Circular  Zone 349 

"  Cones 353-354,  363,  365 

"  Cubes  and  Parallelopipedon. .  360 

"  Cycloid 352 

"  Cylinder 350,  363 

"         "      Sections 357 

"  Ellipsoid,  Paraboloid,  or  Hy- 
perboloid  of  Revolution, 

357>  375-376 

"  Gnomon 335 

"  Helix  (Screw) 354-355 

"  Irregular  Bodies 377 

u        "         Figures 341,  358 

"  Link 353,370 

"  Lune 352 

"  Parallelograms 335 

"  Plot  Angles  without   a  Pro- 
tractor   359 

"  Polyaons 338-341 

"  Polyhedrons 362 

"  Prismoids 351,  361 

"  Prisms 350,  360 

14  Pyramids 354,  365-366 

"  Reduction  of  Ascending  or  De- 
scending Line  to  Horizontal 

Measurement 359 

"  Regular  Bodies. .  .340-341,  362-364 
"  Rings,  Circular    and   Cylin- 
drical  353,  368 

"  Side  of  Greatest  Square  in  a 

Circle 343 

"  Sphere 347-348,  367-368 

"       "     Segment  of. 347 

"  Spherical  Sector 370 

"         "         Triangle 387 

"         "         Zone 368 

"  Spheroids  or  Ellipsoids, 

348-349,  368-369 

"  Spindles 355,  370-374 

u  Spirals 355 

"  To  Plot  Angles  without  a  Pro- 
traction   359 

"  Trapezium 337-338 

"  Trapezoid 338 

"  Triangles 335~337 

(see  Trigonometry,  385-389. ) 

"  Ungulas 351-352, 366-367 

"  Useful  Factors 343 

"  Volume  of  an  Irregular  Body .  870 

u  Wedge 350,  361-362 

"  Zone,  Spherical  and  Circular, 

348-349 

Mercurial  Gauge. 910 

Meta-Centre  of  Hull  of  a  Vessel.  .659,  919 
Metal  Products  of  U.S. 910 


Pugg 

METALS,  Alloys  and  Compositions. 634-637 
"  Adulteration  in,  To  Discover. . .   216 

"  and  Elements  of. 637-644 

"  Comparative  Quality  of  Various  821 

"  Lustre,  Degrees  of. 194 

"  Values  of  some  Precious 938 

"  Various  Weight  of. 155 

"  Weight  of,  To  Compute 131 

u        u       by  Pattern,  To  Compute  217 
"  of  a    Given    Sectional   Area, 

Weight  of. 149 

METER,cmcZ  Value  of. 27,  934 

Meters,  Water 942 

METRIC  Measures 27-33,  IOI3 

"        Factors 923 

MILK,  Nutritive  Values  and  Constit- 
uents of. 202 

Percentage  of  Cream 205 

and  Relative  Richness  of,  of  Sev- 
eral Animals 207 

To  Detect  Starch  in 196 

Mineral  Constituents  Absorbed  from 

an  Acre  of  Soil 189 

"       Waters,  Analysis  of,  etc.  .850-851 

Minerals,  Relative  Hardness  of. 193 

Miner's  Inch 557 

Mines,  Temperature  of 918,  955 

Mining,  for  Blasting 445 

''       Engines  and  Boilers 901 

"       Flat  Ropes „ . .  165 

Mirage 195,  669 

Mirrors  and  Lenses 670 

Miscellaneous  Elements 188-198 

'    Mixtures,  Cements,  Glue,  Inks, 
Lacquers,  Soldering,  Varnish, 

Staining,  etc 871-879 

'    Operations   and   Illustrations, 

879-885,  935 

Mississippi  River,  Silt  in 910 

Models,  Strength  of. 644-645 

"  Bridge,  Resistance  of,  from 645 

"  Dimensions  of  a  Beam,  etc.,ivhich 

a  Structure  can  Bear 644-645 

Molasses,  Analysis  of. 207 

u  Sugar  and  Water  in., 201 

Molding  and  Planing 476 

Molecules,  Velocity,  Weight,  and  Vol- 
ume of. 194 

Moment,  Quantity  of,  etc 614 

Momentum 195 

Monoliths 179 

Month,  Mean  Lunar 70 

Months,  Numbers  of. 74 

MOON'S  AGE,  To  Compute 74-75 

Mortar 590-592,  951,  971 

"    Sugar  in 951 

MORTARS,  Limes,  Cements,  and  Con- 
cretes,   588-597 

MOTION,  Accelerated,  Retarded,  and 

Uniform  Variable. .  .494-495,  617 

"  of  Bodies  in  Fluids 645-648 

"  Pressure,  Velocity,  Time,  etc.. . .  648 
"  Resistances  of  Areas  and  Dif 
ferent  Figures  in  Water  or  Air. . .  646  ; 

Motive  Power 910  j 

"          of  the  World 935 


INDEX. 


XXV11 


Page 
Motors.  Experiments   on,  for   Street 

Railways 915 

Mountains,  Volcanoes,  and  Passes, 

Heights  of. 182-183 

Mowing 433 

u      Machine 910 

Mule,  Load  and  Work  of. 437,  918 

Mural  Efflorescence 593 

N. 
NAILS,  Length  and  Number  of. . .  153-154 

"  and  Spikes,  Retentiveness  of 159 

"  Composition  Sheathing 135 

"  Tacks,  Spikes,  etc 154 

National  Road, 178 

Natural  Formations  and  Construc- 
tions, Largest 936 

"      Powers 198 

NAUTICAL  MEASURE 30 

NAVAL  ARCHITECTURE 649-667 

u  Angles  of  Course  and  Sails. . .  665 
"  Bottom,  Side,  and   Immersed 

Surface  of  Hull 653 

*'  Centre  of  Gravity  of  Bottom 

Plating  of  a  Vessel  658 
*'      "      "  Common  of  Hull,  Ar- 
mament,    Engines, 
etc.,  To  Compute. .  656 
u      u      tl  Depth  of  or  Buoyancy 
below  Meta-  Centre, 
and  Approximately.  656-657 
"      "  of  Effort,  and  Lateral  Re- 
sistance, Relative  Posi- 
tions of. 659 

"  Centres  of  Lateral  Resistance 

and  Effort,  To  Compute. 6 58-6 59 

"Dead  Flat 22 

"  Displacement,  and  its  Centre 
of  Gravity,  To  Compute, 

653-655 

"  Approximately,  and  Coeffi- 
cients of,  To  Compute. . .  655 

"     "  Coefficients  of. 655,  657 

"     "  Curve  of,  To  Delineate,  and 

Coefficients  of. 657 

"  Elements  of  a  Vessel,  To  Com- 
pute  653-660 

"  "  of  Capacity  and  Speed  of 
Several  Types  of  Steam- 
ers of  R.N.  660 

"      "  of  a  Steam  Frigate,  Weight, 

Moments  of,  etc 656 

"  Experiments  upon  Forms  of 

Vessels,  Results  of. 649 

"  Freeboard 666,  913 

"  Heel  and  Steady  Heel,  An- 
gles of,  To  Compute.  . .  .664-665 
"  Lee- way,  Angle,  Ardency  and 

Slackness 666 

"  Length  of  Vessel 909 

"  Masts  and  Spars 667 

"      "  Location  of. 664 

"  Memoranda   of  Weights  and 

Elements 667 

"  Meta- Centre  of  Hull  of  a  Ves- 
sel, To  Compute 659,  919 


Page 

NAVAL     ARCHITECTURE,    Metalling, 
Loss  of  Weight  per  Sq.  Foot 

of  on  a  Vessel's  Bottom 667 

"  Moment   of  Inertia   Approx- 
imately, To  Ascertain. .  .659-660 
"  Pitch  of  Screw  Propeller  and 

Slip  of  Side  Wheels 662 

"  Plating  Iron  Hulls 667 

'-  Proportion  of  Power  Utilized 
in  a  Steam  Vessel  and  Fric- 

tion  of  Engines 663 

"  Resistance  of  Bottoms  of  Hulls.  662 

"          "  of  Air  to  a  Vessel 666 

"          "  to  Wet  Surface  of  Hull.  653 

"  Rudder  Head 667 

"  Sailing  Power  and  Careening 

Power  of  a  Vessel 665 

"      "  Ratio  of  Effective  Area 
of  Sails,  etc.,  and  of 
Vessel1  s  Speed  to  Wind.  663 
"  Sails,  Propulsion  and  Area  of  663 
"     "  Area  and  Trimming  of.  664-665 
"  Screw  Propeller,  Experiments 
upon  Resistance  of  at  High 

Velocities,  etc 666 

"  Slip  of  Propeller   and  Side- 
wheels 666 

"  Speeds,  Relative  of  Forms  of 

Vessels 649 

"  Stability,  Elements  of,  010.649-653 
"       "  and  Speed  of  Models  of 

different  Sections,  etc. .  650 
*'       "  Elements  of  Power  Re- 
quired to  Careen  a  Body 
or  Vessel,  To  Compute, 

652-653 

"       "  Power    Required    in    a 
Steam -vessel,  Capacity 
of  another  being  given.  66 1 
"       u  Results   of  Experiments 

upon  Bodies 649-650 

"  "  Statical,  Statical  Surface 
and  Dynamical  Surface 
Stability,  To  Compute, 

651-652 
"       "  Measure  of,  of  Hull  of  a 

Vessel,  ToDetermine.65&-652 
"  Steam    Vessels,   Approximate 
Rule  for  Speed  and  IP  of, 

To  Compute 662-663 

"  Trim,  Change  of. 655-656 

"   Weight,  Curve  of. 657 

"  Wind,  Effective  Impulse  of. . .  665 
"       "      Course    and  Apparent 

Course  of. 666 

Needle,  Magnetic,  Variation  of. 1035 

"    Decennial  Variation  of. 58 

"     Variation  of  it  in  U.  S.  and 

Canada 59 

Needles,  First  Introduced 72 

Neutral  Axis  of  a  Beam,  To  Compute  820 

New  and  Old  Style 37,  70 

Niagara,  Falls,  Height  of,  etc.  198,  930,  952 
u    Volume  of  Water  and  Power ..   952 

Nitro-Glycerine 443 

Non-conductibility  of  Materials.  .911,  914 
"  -condensing  Engine,  Friction  of '.  918 


XXV111 


INDEX. 


Page 

Non-conductors  of  Temperature,  and 
Comparative  Efficiency  of. 933 

NOTATION 25 

Number  of  Direction 71 

Numbers,  Properties  and  Powers  of.     98 

"  ^th  and  $th  Powers  of. 303,  304 

u  ^th,  $th,  and  6th  Power,  and  ^th 

and  ^th  Root  of,  To  Compute.  304 
Nutritive  Equivalents,  Computedfrom 
amount  of  Nitrogen  in  Human  Milk 
at  i 205 

0. 

Oats  and  Oat  Straw,  Value  of  com- 
pared to  ioo  Lbs.  of  Hay  , 203 

Obelisks,  Egypt  and  New  York 179 

Objective    Glasses,  Diameter  of  the 

Principal 942 

Observatories,  Latitude  and  Longi- 
tude      80 

Ocean,  Depth  of. 912 

Oceans  and  Seas,  Depths  and  Areas.  182 

"    Atlantic  and  Pacific 937 

Offal,  Weight  of,  in  a  Beef  and  Sheep.    35 

Oil,  Yield  of,  from  Seeds 189,  939 

"   Cake  and  Vegetables,  Nutritious 

Properties  of  Compared 204 

"  -Engine  Launch 893 

"  Proportions  of,  in  Air-dry  Seeds.  203 

"   To  Remove  from  Leather 878 

"    Watchmakers' 878 

Oils,  Petroleum,  Schist,  and   Pine- 
wood 484 

Old  and  New  Style 37,  70 

Omnibus,  Weight,  etc 844 

Onion,  Proportion  of  Gluten,  and  Ra- 
tio of  Flesh-formers 207 

Opera-Glasses,  Telescopes,  etc 671 

"    -Houses 180,  936,  954 

Operations,  Miscellaneous 879-885 

OPTICS,  Elements,  etc 668-671 

u   Critical  and  Visual  Angles,  Mi- 
rage and  Caustic  Curves  or 

Lines 669 

1 '  Elemen  ts  of  Mirrors  and  Lenses, 

To  Compute 670 

"  Focus  and  Focal  Distance,  etc..  668 
"  Refraction,  Index  of  and  Indices 

of,  To  Compute 668-669 

44  Dimensions  or   Volume  of  an 

Image,  To  Compute 668 

Ordnance,  Energy  of.. 910 

Ore  and  Stone  Breaker .903,  951 

ORGANIC  SUBSTANCES,  Analysis  of,  by 

Weight 190 

ORTHOGRAPHY  OF  TECHNICAL  WORDS 
AND  TERMS 1042-1052 

OSCILLATION  AND  PERCUSSION,  Centres 

of. 612-614 

"      "  Centre  of,  in  Bodies  of  Va- 
rious Figures 613 

"      "  Centres  of,  To  Compute.  612-61 4 
44       "        "      of ,  Experimentally, 

To  Ascertain 613 

OVBRSHOT  WHEEL 563 


Oxford  College 179 

Oxidation  of  Cast-iron  Pipe,  To  Resist  927 

P. 

Pacific  and  Atlantic  Oceans 937 

Paint,  Flexible  for  Canvas 915 

"    for  Window-Glass 879 

"     Hydraulic 872 

44     To  Clean  and  Remove 878 

Painting,  and  Proportion  of  Colors  for    66 

"       Iron  Rods 956 

Paper,  Blasting 912 

44      Drawing,    Tracing,    Profile, 

Photo-printing,  Cloths,  etc 29,  964 

PARABOLA,  To  Describe 229 

Park,  Deer 179 

Parsnips,  Ratio  of  Flesh-formers 207 

Paste,  Durable 878 

il     Preservation  of. 878 

Passages  of  Steamboats 896 

' l    Ice-boats 896 

44  Steamer  and  Sailing  Vessels .  897 
Passes,  Mountains,  and  Volcanoes. . .  182 
Pastils  for  Fumigating 879 

PAVEMENT,  Asphalt 690,  944-945 

"    Block  Stone 689-690,  944-945 

"    Comparative  Merits  of. 945 

u    Granite 690 

44    Macadam  and  Brick 944 

"    Miscellaneous  Notes 690 

44    Neufchatel 945 

"    Rubble  Stone 689 

"    Telford 688 

44    Voids  in  a  Cube  Yara  of  Stone  690 

"    Wood 689,  690,  944 

PAVEMENTS,  Roads,  and  Streets. . .  686-690 

Payments,  Equation  of. 109 

PEAT 482 

Pendulum  Measure 27 

PENDULUMS,  Elements  of,  etc 452-454 

44  Centre  of  Gravity  of. 453-454 

"  Conical,  Number   of  Revolu- 
tions of,  To  Compute 454 

44  Lengths  and  Number  of  Vibra- 
tions of,  To  Compute 453,  454 

"  Vibrations,  Number  and  Time 

of,  To  Compute 454 

Pennants,  Ensigns,  and  Flags,  U.  S..  199 
PERCUSSION  AND  OSCILLATION,  Centres 
of  (see  Oscillation  and  Per- 
cussion)   612-614 

1 '    Caps,  Number  of,  Correspond- 
ing to  Birmingham  Gauge.  502 
PERFORMANCES  of  Men,  Horses,  010.438-440 

Perimeter  of  a  Figure 912 

PERMUTATIONS ioo 

PERPETUITIES 112 

Petroleum,  Elastic  Force  of  Vapor. .  707 

"    Evaporative  Effects  of. 910 

PHYSICAL  AND  MECHANICAL  ELEMENTS, 
Construction  and  Results 907 

PILE  DRIVING 433,  671-673,  902,  972 

44    Coefficient  of Resistance  of  Earth  972 

44    Pneumatic 673,  972 

"    Resistance  of  Formations 673 


INDEX. 


XXIX 


Page 

PILE  DRIVING,  Ringing  Engine 972 

"  Safe  Load,  To  Compute 672 

"  Sheet  Piling 672 

"  Sinking. 673 

"  Weight  of  Ram 972 

PILES,  Foundation 198,  781,  909 

"    Extreme  Load  a  Pile  will  Bear  912 

"    Retaining  Walls  of  Iron 196 

PILING  OF  SHOT  AND  SHKLLS 65 

Pillar  at  Delphi 179 

Pillars  or  Columns 936 

Pins,  First  in  Use 915 

PIPES,  Dimensions,  etc 747 

and  Tubes,  Weight  of. 147-148 

Copper,  Dimensions  of. 150 

Gas,Thickness  of 123 

"    Threads 160 

Lead,  and  Tin-lined,  Weight  of. .  137 

"     Encased 151 

"      «     Weight  of. 150 

"  Metal,  and  Weight  of. 147 

u  of  Cast-iron  to  Resist  Oxidation.  927 

u  or  Cylinders  of  Cast-iron 132-133 

"  Riveted  Iron  and  Copper,  Weight 

of  One  Foot  in  Length 148 

"  Steam,  Gas,  and  Water 138 

"  Tin,  Weight  of. 151 

"  Thickness  of,  To  Compute 560 

u  Water,  Standard  of  Cast-iron. . .  147 

Pise 593 

Pivots,  Friction  of. 472 

Planing,  Cast-iron  and  Molding.  .476-477 

PLANK  ROADS 688 

Plants  or  Hills,  in  an  Acre 193 

u     Weather  Foretelling 185 

Plaster,  Turkish 591 

PLASTERING,  Measuring  of. 197 

"    Volumes  Required,  Materials 
and  Labor  for  100  Sq.  Yards  of. . .  604 

PLATE  BENDING,  Iron 476 

PLATES  and  Bolts,  U.S.  Test  Q/j etc. 749-753 

"  of  Metals,  by  Gauge 121 

"  Thickness  of,  To  Compute 751 

"  Wrought-iron  Shell 750 

PLATING  IRON  HULLS 667 

Ploughing 433 

PNEUMATICS.— AEROMETRY 673-676 

(See  also  Aerometry.) 

POINTING  in  Masonry. ., 598 

POISONS,  Antidotes  and  Treatment.  185, 935 

POLES  AND  SPARS 62 

PONCELET'S  WHEEL 567-568 

"     Turbine 574 

POPULATION,  and  Area  of  Divisions 

and  Countries 188 

"  Comparative  Density  of,  and 
Number  of  Persons  in  a 
House  in  Different  Cities.  .910 

'  *    of  Principal  Cities 187 

POSITION 98-99 

Pozzuolana,  Elements  of. 589 

Potato,  Anti  -  Scorbutic  Power   and 

Ratio  of  Flesh-formers 206-207 

POWDER,  Smokeless 952 

"    Forcite 966 

"    Gun,  Proportions  of  to  Shot..  502 


POWER  AND  WORK,  Metric 36 

Motive 910 

"     of  the  World 935 

Movers  and  Transmitters  of... .  797 

of  a  Quantity,  Value  of. 359 

Ordinary  Distribution  of,  in  a 

Propeller  Steamer 911 

Required  to  Draw  a  Vessel  up  an 

Inclined  Plane 910 

Thermometric  and  Mechanical 
Energy  of  10  Grains  of  Va- 
rious Substances  when  Oxi- 
dized in  Human  Body 205 

To  Sustain  a  Vehicle  on  an  In- 
clined Road,  To  Compute.. 84 5-846 

Transmission,  Elements  of 176 

POWERS,  Natural 198 

'    ^th,  $th,  and  6th  of  a  Number.  304 

'    of  first  9  Numbers 98 

'    of  ^th  and  $th  Numbers. . . .  303-304 

"    of  6th  Number 304 

PRECIOUS  METALS,  Values  of  some . . .  938 

Pressures  and  Weights,  Metric 36,  923 

PROBABILITY  and  Illustrations. .  .114-117 
"    Odds    between   Results    and 

Chances,  etc 117 

PROGRESSION 101-105 

Proof  of  Spirituous  Liquors , .  218 

Propeller  Steamers,  Ordinary  Distri- 
bution of  Power  in 911 

PROPELLERS •. . .  .730-731,  886-891 

Properties  of  Numbers 98 

PROPORTION 94-96 

PULLEY 433 

"  Compound,  etc 633-634 

"  Power,  Weight  it  will  Raise,  and 
No.of  Cords  to  Sustain  Lower  Block  632 

PUMP,  Working  of  a 433 

Appold's  and  Gwynne's 579 

Steam, Elements  and  Capacities  of  738 

Water,  First  in  Use 932 

Pumping 433 

"      ENGINES.  . . 738, 902-903, 954,  963 

PUMPS,  Direct  Acting 738 

'    Centrifugal. .  579-580,  911,  917,  1031 

'     Circulating 749 

'     Water  and  Vacuum 932,  963 

'     Worthington 738 

Pushing  or  Drawing 433 

Pyramids,  Statues,  etc 178, 936 

Q. 
Quartermasters,  Service  Train  of. . .  198 

R. 
Race-Courses,  English,  Length  of....  930 

Rack  and  Pinion,  Power  of. 628 

Radius  Vector 449 

RAIL,  Weight  of,  To  Compute 679 

RAILROAD  Crane 962 

"  Horse,  First  in  Use 915 

"  Portable,  and  Hand  Cars 908 

"  Signals  and  Significations —  954 

"   Speed 969 

"       "  in  England 930,  951 

u  Ties,  Duration  of. 968 


XXX 


INDEX. 


Page 

RAILROADS,  Street,  and  Cost  of  Main- 
tenance of. 915,  918 

"  Result  of  Experiments  on  Mo- 
tors for 915 

<r"  Street  or  Tramway 848,915 

"          u  "    Engine 902 

flails,  Iron  and  Steel,  Strength  of,  To 

Compute 812 

*'    Tangential  Angles  for   Chords 

and  Curves 677-678 

"    Elevation  of  Outer  Hail,  To  De- 
termine  679 

RAILWAY  TRAINS,  Frictional  Resist- 
ance of. 916 

4 '  Power  and  Resistance  of. 911 

RAILWAYS,  Elements  of,  etc 677-685 

'  *  Adhesion  on  a  Given  Grade  and 

Curve,  To  Compute 685 

"  Curve,  To  Define,  etc 677,  685 

"  Curves  by  Offsets 678 

"  Curving,  Versed  Sines  and  Or- 

dinates  of,  To  Compute 678 

"  Driving-wheels,  Coefficients  of 

A  dhesion  on,  per  Ton 68 1 

"  Gradient 685 

44  Load  a  Locomotive  will  Draw 

up  an  Inclination 680 

"  Locomotives,  Adhesion  of. 68 1 

44  Maximum  Load  that  can  be 
Drawn   up   the   Maximum 

Grade  it  can  Attain 680 

"  Memoranda  on  English 685 

44  Operation  of  Locomotives,  Ele- 
ments of, 681,  685,  1031 

u  Points  and  Crossings 678 

"  Radii  of  Curves 679 

"  Radius  of  Curve  and  Wheel 

Base 679 

41  Regulations  for  Crossing  Roads, 

Parliamentary,  English. ...  685 
**  Relation  of  Base  of  Driving  or 

Rigid  Wheels  to  Curve 678 

"  Resistance  of  Gravity  upon  an 

Inclination,  To  Compute —  679 
*4  Rise  per  Mile  and  Resistance 

to  Gravity  per  Ton 679 

44  Sidings 677-678 

44  Tangential  Angle  for  Curves. .  678 
41  Traction,  Retraction,  and  Ad- 
hesive Power  of  a  Locomotive 

or  Train,  To  Compute 680 

44  Tractive  Power 681 

44  Train  Resistances 682-684 

44  Tangential  Angles  for  Chords 

of  One  Chain 677 

44  Turnout  of  Unequal  Radii. . .  677 
44  Velocity  of  Trains.  680,930,951, 1031 
Rainfall,  Annual  and  Volume  of.....  850 
' 4    Diameter  and  Grade  of  Drains, 

To  Discharge , 906 

Rand  Rock  Drilling  Co 940 

Rankine  Wheel. 580 

Rations  of  an  Esquimau „  206 

REACTION  AND  IMPACT  WHEEL 576 

Reaping 433 

REBATE  OR  DISCOUNT 109 


Page 

Reciprocals 304 

Rectilineal  Figures,  To  Describe.  .222-223 

REFRACTION  and  Indices  of 584,  668 

44    and  Curvature  of  the  Earth.     55 

44    of  Light 584 

REFRIGERATING 943,  965-967 

1 4    and  Ice-making 967-968 

44    and  Machine 943,  1028 

44    Elements  of  a  Test  of. .  .943,  966 

Refrigerator,  Surface  of. „  512 

Remedials,  Domestic 938 

Remedy  of  the  Mint 38 

Rendering,  Laying,  etc 598 

Resilience  of  Woods 763 

RETAINING  WALLS,  Memoranda 695 

4  4    of  Iron  Piles 196 

REVETMENT  WALLS,  Elements  of  ^94-700 
41  Elements  of,  To  Compute.  696-698 

44  Friction,  Effects  of. 698 

44  Memoranda .695-696 

44  Moment  of  Pressure,  Point  of.  698 

44  Pressure  of  Earth 695,  698 

"  Surcharged 699 

(See  Embankments, Walls,  and  Dams, 

700-703,  and  Stability,  693-703.) 
Rigging,  Circumference  of  Hemp  or 

Wire  Rope  for  Fore  and  Main.  172 

44  Vessels  and  Guys 163 

Ringing  Engine 672 

River  Steamboat,  Side-wheels 919 

44     Steamboats  and  Engines.  .892,  919 

RIVERS,  Current  of. 193 

44    Descent  of  Western 188 

44    Flow  of  Water  in 550,  551,  933 

44    Lengths  of... 183 

44    Obstruction  in 551 

RIVETED  JOINTS,  Comparative  Strength 

of,  etc 751,  752,  755,  829 

44  Experiments  on 783 

RIVETING  of  Plates 755-757 

RIVETS,  Pitch,  Lap,  etc. ,  Dimensions 

and  Proportions  of... . 829-830 

44  in  Steel  Plates 830 

44  Memoranda 830 

44  Pitches,  Diameter  of,  etc 756,  829 

44       4  4  Diameter  and  Length  of,  for 

Hulls  of  Vessels 830 

44  Proportion  of  Single  for  Wro't- 

iron  Joints 829 

44  Rivet  Heads 830 

44  Result  of  Experiments  on  Double 
Riveted  and  Strapped  Plate 
Joints 829 

ROADS,  STREETS,  AND  PA YEMBXTS.  686-690 
'  Aqueducts,  and  Railroads. ....  178 
Asphalt  and  Wood  Pavement.  689-690 

Bituminous 690 

Blinding  and  Darning 690 

Block  Stone 689-690 

Classification  of. 686 

Concrete 689 

Construction,  and  Estimate  of 

Labor 686 

Corduroy 688 


INDEX. 


XXXI 


ROADS,  etc.,  Ditches 686 

44  English 689 

44  Getters,  Fillers,  and  Wheelers, 
Proportion  of,  in  Different 

Soils 688 

44   Grade  of. 686 

"  Granite  and  Hoggin 690 

"   Gravel  or  Earth 688 

44  Macadamized 687-690 

44  Metalling  and  Metalled 690 

44  Miscellaneous  Notes 690 

«  Plank 688 

14  Resistance  to  Traction  of  a  Stage 

Coach 848 

"  Rubble  Stone 689 

44  Ruts  and  Stone-breaking. 687 

"  Sweeping,  Sprinkling,  Watering, 
Rolling,  Washing,  and  Fas- 
cines   690 

"  Telford 688 

"    Voids  in  a  Cube  Yard  of  Stone.  690 
ROADWAY,  Central  Width  of,  in  Cut.  917 

44    Construction  of. 687 

ROADWAYS,   Relative    Resistance    to 

Traction 945 

ROCK,  Weight  and  Volume  of. 467 

44  AND  EARTH,  Excavation  and  Em- 
bankment of 192 

"  and  Bulk  of,  and  Earthwork. ...  468 

44  Drilling 940 

44  Weight  of,  per  Cube  Yard 468 

Roman  Calendar 71 

44    Indiction 71 

44    Long  Measures 53 

Roof  Plates,  Corrugated,  Weight  of. . .  131 

ROOFS  of  Buildings 179,  952 

44  Stress  on,  To  Compute 952 

44  Weights  and  Pressure  on. .  .952,  1020 

14  Wooden 189 

ROOT,  of  an  Even  Power  Greater  than 

Contained  in  Table 98 

44  To  Extract  any  whatever 97 

ROOTS  and  Grains,  Weights  of. 34 

44    SQUARES,  AND  CUBES 272-302 

44    ^th,  $th, and  6th,  To  Compute.  302-304 

ROPES,  Working  Load 782 

44  AND  CABLES,  Measure  of. 26 

44  and  Chains  of  Equal  Strength. .  165 
44  at  an  Inclination,  Stress  of....  166 

44  CABLES,  CHAINS,  etc 161-175 

44  Circumference  of  Wire,  etc.,  To 

Compute 169 

44  44  of,  and  of  Hawser  or  Cable 

for  a  Given  Strain  of...  171 

44  Endless 167 

44  Hawsers  and  Cables 170-172 

44  4t  Circumference  of,  To  Compute  171 

44  44  Weight  of,  To  Compute 172 

44  Hemp  and  Wire,  General  Notes.  167 
44  "and  Wire,WeightandStrengthofij2 

44  "  Iron,  and  Steel 164 

•4  Iron,  Wire,  and  U.  S.  Hemp  ....  168 

44  Mining,  Flat 165, 1029 

"  of  Corresponding  Strength  to 

kHemp,  and  of  Hemp  to  Circum- 
Jvrencc  of  Wire  Rope .: 169 


Page 
ROPES,  Stress,  Tension,  and  Deflection 

of,  To  Compute 166 

44  Tarred  Hemp  and  Wire,Circum- 

ference  of. 169 

44       44        44  and  Destructive  and 

Breaking  Strength  of. 171 

44  Transmission  of  Power 167 

44  Units  for  Computing  Safe  Strain 
for  New,  Hawsers,  and  Cables, 

U.  S.  N. 170-171 

"  WIRE,  Elements  of,  etc 161-172 

44          4t  and  Equivalent  Belt 167 

44          44  Experiments  on,  U.  S.  N. . .  166 
44          "  and  Hemp,  Circumference 

of  for  Standing  Rigging  1 72 
*4  4  4  Iron,  Steel,  and  Hemp,  Rel- 

ative   Dimensions    of, 

U.S.N. 172 

4'  White,  Durability  of. 170 

Rotation  of  the  Earth,  Influence  of. .  942 

Rowing 433 

Rubble-Stone  Pavement. . . 689 

RULE  OP  THREE 95 

Running,  Men  and  Horses 438-440 

Rye  Straw,  Relative  Value  of  com- 
pared with  loo  Lbs.  of  Hay 203 

S. 

Saccharose,  or  Cane  Sugar 207 

SAFETY  VALVE,  Adjustable  Pop 986 

44       VALVES 746,  921,  933 

Sago,  Value  of 207 

SAILING,  Area  of  Sails  and  VesseVs 

Speed 663 

44     Vessels,  Iron 894-895 

SAILS,  Propulsion  and  Area  of 663 

44      To  Preserve 870 

44      Trimming  of. 665 

Saline  Saturation 726 

' 4    Proportional  Volumes  of  Matter 

in  Sea-water. 727 

SALT  WATER,  Corrosive  Effects  of,  on 

Steel  or  Iron 916 

Sand,  Composition,  etc 599 

Sandstones 193 

SAW- MILL,  Elements  of. 904,  913 

Sawing  and  Hewing  Timber,  Loss  in,    62 

4  4     Stone  and  Wood 196,  904 

SAWS,  Circular 197,  477,  911-912 

44      Vertical  and  Band 477 

Scale  or  Sediment,  Removal  of  Incrus- 
tation in  Boilers 726 

SCALES,  To  Divide  a  Line,  etc 221 

u        Weighing  without 66 

Scarfs  Resistance  of 841 

SCREW,  Length,  Power,  Weight,  Pitch, 

etc. ,  To  Compute 630-631 

44  Bolts,  Power  of. 968 

44   Cutting 477 

u    Compound 631 

44  Differential 632 

44  Propeller,  Pitch  and  Speed  of. .  662 

44          44     Friction  of  Engines ....  663 

44         "     Elements  of  To  Compute  917 

Scripture  and  Ancient  Measures. ...     53 

SeaDepths -    184 


XXX11 


INDEX. 


Page 
Seas  and  Oceans,  Depths  and  Area 

of. .182 

SECANTS  AND  COSECANTS 403-414 

u       u  Degrees,  Minutes,  etc.,  of, 

To  Compute 414 

SEEDS,  Number  of,  in  a  Bushel,  and 

per  Sq.  Foot  per  Acre 193,  938 

"  Proportion  of  Oil  in  Air-dry .  203,  939 

"  Yield  of  Oil  in  Several 189 

SEGMENTS  OF  A  CIRCLE,  Area  of.  ,267,  269 
"        "  Area  of,  To  Compute. 268-269 

Sewage,  Volume  of,  etc 692,  1029 

SEWERS,  Classification  of. 691, 692 

"  Drains,  Diameter  and  Grade  of, 

to  Discharge  Rainfall 906 

"  Drainage  of  Lands  by  Pipes. . .  691 
4 '  Material  per  Lineal  Foot  of  Egg- 
shaped,  Dimensions  of. 692 

"  Minimum  Velocity  and   Grade 

of. 691 

"  Sewer  Pipes 692 

"  Surface,  which  will  Discharge 
a  Volume  of  an  Inch  and  also 
of  Two  Inches  per  Hour,  etc. .  692 

Shaft,  Bearings  for  Propeller 473 

SHAFTING  for  Lathes  and  Mills.  .948, 1010 

SHAFTS 778,  793,  794~797>  9J4 

(See  also  Torsion,  790-797.) 

"  and  Gudgeons 790 

"  Deflection  of  Shafts 778-779 

"  Diameter  and  Journal  of,  Stress 
Uniformly  along  its  Length, 

To  Compute 571 

"  Journals  or  Bearings  of,  etc 796 

"  Loaded  Transversely  and  Jour- 
nals of. 914 

SHEARING  or  Detrusive  Strength. 782-783 
u  Experiments  in  Cast-iron,Steel, 

Treenails,  and  Wood 783 

"  Power  to  Punch  Iron,  Brass,  or 

Copper,  To  Compute ...  782 

"  Results  of  Experiments  on,  with 

a  Punch 782 

"      "     "  with  Parallel  Cutters, 
Wro't  -  iron  Bolts,  Riveted  Joints, 

and  Various  Materials 783 

Sheathing  and  Braziers'1  Sheets 155 

"  Copper....  131 

"    Nails,  Weight  of. 135 

Sheet-  Iron,  Blackand  Galvanized.  124, 129 

"     Weight  of. 129 

SHEET  PILING 672 

SHELLS  and  Shot,  Piling  of. 65 

Shingles 63 

Shoemaker's  Measure 27 

SHOT  and  Shells,  Piling  of. 65 

"  Chilled  and  Drop 906 

"  No.,  Diameter,  and  Numbers  of.  906 

"  Number  of  Pellets  in  an  Oz 501 

Shrinkage  of  Castings 218 

Shrouds,  Hemp  and  Wire 173 

SIDE  LIGHTS,  Visibility  of  a  Vessel's,  918 
*'    WHEELS,  A  rea  of  Blades  and  Slip  662 

"    Friction  of  Engines 662 

Sidereal  Day  and  Year. 37 


Page 

Sides  of  Squares,  Equal  in  Area  to  a 

Circle 258-259 

Signals,  Night,  U.S.N. 199 

"  Railroad,  and  Significations..  954 

Silt,  in  Mississippi  River 910 

Silver  Sheet,  Thickness  of. 1 19 

SIMPLE  INTEREST 107 

Simpson's  Rule, Area,  To  Compute..  342 
''      "  Volume  of  an  Irregular  Body  870 

SINES  AND  COSINES 390-402 

"    Number  of  Degrees,  Min- 
utes, etc. ,  of,  To  Compute 402 

SIPHON,  Steam 1010 

Sixth  Power  of  a  Number 304 

Skating  Performances 439 

Slackwater,  Canal,  etc.,  Traction  on.  848 

Slaking  of  Lime 594 

Slate,  Surface  of,  and  Number  of 

Squares,  To  Compute 64 

Slating,  Weight  of  One  Sq.  Foot 64 

SLATES  and  Slating 64 

"  Dimensions  of. 64 

' '  English 64 

"  Weight  per  1000  and  Number 

Required  to  Cover  a  Square 64 

SLIDE  VALVES,  Elements,  etc 731-733 

Smelting  of  Iron  Ore 445 

Slotting 477 

SMOKE  PIPES  AND  CHIMNEYS 748-749 

SNOW,  Pressure  of,  on  Roofs 952 

"  and  Ice 849 

"  Flakes 195 

"  Line  or  of  Perpetual  Congela- 
tion   192 

"  Melted,  Volume  of. 195 

Solar  Day  and  Year 37,  70 

Solders 634-636 

Soldering 875 

SOUND,  Velocity  of. 195 

"  Distances  by  Velocity  of,  To  Com- 
pute   428 

"  Velocity  of,  in  Several  Solids. . .  428 
Soundings,  to  Reduce  to  Low  Water.  60 
Spars  and  Poles 62 

SPECIFIC  GRAVITY  AND  WEIGHT.  .  .208-215 
"    Given  Weight  of  a  Body,  To 

Compute 215 

"   of  a  Body  Soluble  in  Water.  209 
' '    of  a  Body  Heavier  or  Lighter 

than  Water. 209 

"    of  Elastic  Fluids 215 

"    of  a  Fluid 209 

"    of  Liquids 214-215 

"    of  Miscellaneous  Substances  214 

"    of  Solids 210-214 

"    or  Density  of  Steam 706 

"    Proportions  of  Two  Ingre- 
dients in  a  Compound,  or 
to  Discover  A  duller ution .  216 
"    Weight  of  Ingredients,  that 
of  Compound  being  Given, 

To  Compute 218 

' '  Weights  and  Volumes  of  Va- 
rious Substances  in  Ordi- 
nary Use .216-215 


INDEX. 


XXX111 


P*.] 

Speed  of  Vessels 971, 1010 

SPIKES,  SHIP,  BOAT,  and  Railroad.  152, 154 

and  Horseshoes 152 

and  Nails, Retentiveness  of....  159 

General  Remarks 159-160 

Ship  and  Railway 970 

WroH-iron  Nails  and  Tacks. ..  154 

Spiral,  To  Describe 230 

Spires,  Towers,  Columns,  etc 180,  932 

Spirits,  Strength  of,  To  Compute 67 

Spirituous  Liquors,  Dilution  per  Cent.  191 

"  Dilution,  To  Reduce 191 

"  Proof  of. 218 

44  Proportion  of  Alcohol. ....  191 

Springs,  Deflection  of 779 

Spur  Gear 911 

SQUARE  AND  CUBE  ROOT,  Square  or 
Cube,  and  when  Number  is 

an  Odd  Number 300-302 

**  of  Decimals  alone,To  A  scertain,302 
"  of  a  Higher  Number  than  con- 
tained in  Table,  To  Compute  301 
44  of  a  Number  consisting  of  In- 
tegers and  Decimals,  To  As- 
certain  301-302 

"  ROOT,  To  Extract. 97 

"  *'  or  CUBE  ROOTS  of  Roots, 
Whole  Numbers,  and  of 
Integers  and  Decimals, 

To  Ascertain 97-98 

"  To  Ascertain  One   that  has 
Same  Area  as  a  given  Circle 259 

SQUARES,  CUBES,  AND  SQUARE  AND 

CUBE  ROOTS 272-302 

"  Sides  of,  Equal  in  Area  to  a 
Circle 258-259 

STABILITY,  Elements,  etc 693-703 

44  Angles  of  Equilibrium  of.....  694 

44  Dynamical  and  Statical 651 

44  Earthwork,  Centre  of  Pressure 

of. 696 

44  Equilibrium  and  Stability,  To 

Compute 701 

44  Memoranda 695-696 

"  Moment  of, and  To  Compute.  693, 701 
44  of  a  Body  on   a  Horizontal 
Plane  or  on  an  Inclination, 

694-695 

44  of  a  Fixed  or  Floating  Body. .  693 
44  of  Hull  of  a  Vessel  or  Floating 

Body,  To  Determine 650 

"  of  Varying  Models 649 

"  Statical   and   Dynamical,  To 

Compute 651 

"  Weight  of  a  Body,  To  Sustain  a 

Given  Thrust 693-694 

Staging,  Coach 440 

Staining,  Wood  and  Ivory 876 

Stains,  To  Remove 878 

Starch,  Proportion  of,  in  Vegetables.  205 
Stars,  Velocity  of. 198 

STATICS 615-616 

"  Composition  and  Resolution  of 
Forces 615 


Page 

STATICS,  Equilibrium  of  Force 616 

"  Inertia  of  a  Revolving  Body. . .  616 

"  Spherical  Triangles 387 

"  Specific  Heat 505-507 

"         of  Air  and  Gases..  505 

Statues,  Pyramids,  etc 178 

STAY  BOLTS,  Diameter,  Pitch,  etc. ...  754 

STEAM,  Elements  of,  etc.. 640-643,  704-727 

44  and  Air,  Mixture  of. 737 

"  Blowing  off,  Saturated  Water,  Loss 

of  Heat  by,  To  Compute. .  726-727 
"     "  Volume   Btoivn   off  to    that 

Evaporated,  To  Compute. . .  727 

"  Clearance,  Effect  of.....   715 

44  Coal,  Utilization  of,  in  a  Boiler.  726 
44  Combined  Ratio  of  Expansion 
and  Final  Pressure  in  zd  Cyl- 
inder, To  Attain 723 

"  Condensation  of,  in  Cast-iron 

Pipes 515-516 

"  Condensed  per  Sq.  Foot  and  per 

Degree  per  Hour 516 

"        "  of  Expanded,  per  H»  of  Ef- 
fect per  Hour 716 

"  Consumption  of  Fuel  in  a  Fur- 
nace, To  Compute 725-726 

"  Cutting  Off,  Point  of,  for  a  Given 

Ratio  of  Expansion 711 

"        ««  Point  of,  to  Attain  Limit 

of  Expansion 710 

44  Cylinder,  Net  Volume  of  for 
Given  Weight  of  Steam,  etc., 

To  Compute 715 

44  Density  or  Specific  Gravity  of. .  706 
"  Effect  for  One  Stroke  and  a 
Given  Combined  Ratio  of 

Expansion 723-724 

"      *•    Relative,of  Equal  Volumes.  714 
1    Total  ofiLb.  of  Expanded, 

714-715 

"  Effective  Work  in  One  Stroke  as 
Given  by  an  Indicator  Dia- 
gram, To  Compute 714 

44  Efficiency,  Actual,  Conclusions  on  724 
u  Elastic  Force  and  Temperature 
of  Vapors  of  Alcohol,  Ether, 
Sulphuret  of  Carbon,  Petro- 
leum, and  Turpentine 707 

44  Expanded,  Consumption  of  per 

IP  of  Effect  per  Hour 716 

44  Expansion,  Points  of 712 

"        "  Effects  of. 713 

"  "  Point  of  Cutting  off,  Actual 
Ratio  of,  Pressure  at  any  Point 
of.  Mean  or  Average,  and  Final 
Effective  or  Initial,  To  Com- 
pute  710-711 

44  Expansive  Force  of. 704 

44  Feed  Water,  Gain  in  at  High 

Temperature,  To  Compute 719 

u  *'  Gain  in,  and  Initial  Pressure, 
when  Acting  Expansively, 
compared  with  Non-eoepan- 

sion  or  Full  Stroke 725 

41  Gaseous,  Total  Heat,  and  Veloc- 
ity of,  To  Compute 710 


XXXIV 


INDEX. 


Page 

STKAM  Heating  Co.  of  N.  T. 904 

44        "  and  Bolters.  913,  957,  1025, 1027 
44  Incrustation  of  Scale  or  Sedi- 
ment, To  Remove 726 

44  Indicator,  Mean  Pressure  by,  To 

Compute 724 

44  -Injector 736 

44  Mean  Pressure  by  Hyperbolic 
Logarithms,  To  Compute.  .712-713 

"  Mechanical  Equivalent  of. 705 

"  Notes  of. 936,  954 

14  Pipes  and  Casing 515 

44  Pipes,  Gas,  etc.,  Dimensions  and 

Weights  of. 138 

44  Plant,  Cost  of  Coal  and  Labor  in 

Operation  of  1000  IP 951 

44  Pressure  in  Ins.  of  Mercury 706 

"  "  Weight  of  a  Cube  Foot, 
Pressure  and  Tempera- 
ture, To  Compute 705 

44        *'  in  a  Cylinder,  at  any  Point 

of  Expansion,  or  at  End 

of  Stroke,  To  Compute..  711 

"  Pressures,  Mean,  Final.  Effective, 

Initial,  or  Total  Average,  To 

Compute 711 

14  Saline  Matter,  Proportion  of  in 

Sea-water 727 

44      "   Saturation  in  Boilers 726 

44  Specific  Gravity  of. 704,  706 

"         "  of,  compared  with  Air,  To 

Compute 706 

44  Surface  Condensation,  Experi- 
ments on 911 

44  Velocity  of., 704,  913,  936 

44  of,  into  a  Vacuum 704 

44  Volume  of  Cylinder  for  a  Given 

Effect,  etc 715 

•«  4<  of  Water  at  any  Given 
Temperature  Mixed  with 
it,  to  Raise  or  Reduce 
Mixture  to  any  Required 
Temperature,  To  Compute  707 
"  "  of  Water  Evaporated  per 
Lb.  of  Coal,  To  Com- 
pute   725 

••  "  of  Water  in  a  Given  Volume 
of  and  of  a  Cube  Foot  of, 

To  Compute 706 

•*  "  of  to  Raise  a  Given  Volume 
of  Water  to  any  Given 

Temperature. 706-707 

«  Weight  and  Effect  of,  for  other 
Pressures  than  looLbs.j  Multi- 
pliers for 719 

"  Wire  drawing  of. 718 

"  Yachts,  Relative  Velocities  of, 
from  Elements  of  their  Con- 
struction, To  Compute 928 

44  COMPOUND  EXPANSION,  Elements 

of,  etc 712,  720-724 

"  "  Combined  Ratio  of  and  Final 
Pressure  in  2d  Cylinder, 
To  Attain 723 

*  *'  Comparative  Effect  in  Receiv- 
er and  Woolf  Engine 724 


Pag 

STEAM,  COMPOUND  EXPANSION,  Effect 
for  One  Stroke  and  a  Given 
Ratio  of  in  ist  Cylinder, 

To  Compute 721-722 

44      "  Effect  for  One  Stroke  and  a 
Given    Combined    Actual 
Ratio  of,  To  Compute.  .713-724 
41     "  Expansion  in  a  Compound 

Engine,  To  Compute. .  721 

"      "      "  From  Receiver 720-724 

44     4 '  Final  Pressure 720-72 1 

44     "  Woolf  Engine,  Ratio  of  Ex- 
pansion, etc 722 

44  SATURATED,  Total  Heat  and  Ab- 
sorption of. 705 

44        "  Energy  and  Efficiency  of, 

To  Compute 716-717 

44        *4  Latent  and  Total  Heat  of, 

To  Compute 707 

44        "  Pressure, Temperature,Vol- 

ume.  and  Density. . . .  708-709 
u        it  Properties  of,  of  Maximum 

Density 717 

44        "  Vapors,  Pressure  of. 518 

44  SUPERHEATED,  Energy  and  Effi- 
ciency of,  To  Compute.  7 1 7-7 1 8 
44        "  Expansion,   Effects    with 
Equal  Volumes  and  One  Lb.  of 
ioo  Lbs.  Pressure 718-719 

STEAMBOAT,  Iron,  First  built 915 

STEAMBOATS,  RIVER,  AND  ENGINES.  892-893 

44  Passages  of. 896 

44  Wood  and  River  Side-wheels.  892,919 
44     "  Ferry,  Passenger,  Team,  and 

Tow-Boats 890 

44     4'  Passenger  and  Deck  Freight.  893 
44     "  Stern-wheels 892-893 

STEAM-ENGINE,  Elements  of,  etc.  .  727-760 
*•  and  Sugar- Mill,  Weights  of. . .  908 
44  -Boilers  in  Foreign  Countries.  935 
44  and  Boilers,  Cost  of  Operating 

per  Day  of  10  Hours ....  904 

44  Circulating  Pumps,  Volume  of, 

etc 749 

44  Condenser  or  Reservoir,  Tem- 
perature of  Water  in,  To 

Compute 707 

4  4  Dimensions  of  Cylinder,  Grate, 
and  Heating  Surfaces,  To 

Compute 927,  1024 

44  Distance  of  Piston  from  End 
of  Stroke,  when  Lead  pro- 
duces its  Effect,  and  when 
Steam  is  Admitted  for  Re- 
turn Stroke 732-735 

44  Feed  Pump,  Area  «>/,  To  Com- 
pute   736 

44  Fire,  Elements,  etc 904 

44  General  Rules  for 728-730 

"  H*  of,  To  Compute 733-734 

44      4t  Admiralty  and  French ..  734 
44  Injection   Pipe,  Area  of,   To 

Compute 73S-736 

44  Notes  of 936,  954 


INDEX. 


XXXV 


Page 

STEAM  -  ENGINE,  Portable,  Standard 
Operation  of,  and  Elements 

of 737 

"  Propeller  Slip  and  Thrust,  To 

Compute 730-731 

44  Proportion  of  Parts,  Condens- 
ing and  Non-condensing.  727-729 

44  Receiver 721 

"  Results    of   Experiments    on 

Operation  of. 933 

"  Screw,  Friction  of 478 

44  Steam- Injector  and  Volume  of 

Water  Discharged  per  Hour  736 
44  "  -Pumps,  Elements  and  Ca- 
pacities of. 738 

44  Vertical  Beam,  Jet  Condens- 
ing, Weight  of,  To  Compute. .  759 
44  Volume  of  Water  Required  to 

be  Evaporated  in..  .734-735 
"  Volume  of  Circulating  Water 

Required  in 735 

"  Volume  of  Feed  Water  and  In- 
jection Water  Required 

per  IP  per  Hour 736 

"       "  of  Flow  through  an  Injec- 
tion Pipe 735 

"  Water-wheels,  Radial  and 
Feathering,  and  Elements  of.  730 

w  SLIDE  VALVES,  To  Compute  and 
Ascertain  Lap  and  Breadth 
of  Ports 731 

"  Distance  of  Piston  from  End 
of  Stroke  given,  To  Compute 
Lead,  etc 732-733 

"  Lap  and  Lead  of  Locomotive 
Valves 733 

44  Part  of  Stroke  any  Given  Lap 
will  Cut  off,  To  Compute. ...  731 

44  Stroke  at  which  Exhausting 

Port  is  Closed,  etc 732 

44      "  of,  To  Compute 732 

STEAM  ENGINES,  Results  of  Operation 

°f- 737.  924»  933.  954 

"  and  Boilers,  Weights  of  with 

Water 929 

44        "  Results  of  Performances 

of. 924,  927 

"  Duty  of  and  Relative  Cost  of 

for  Equal  Effects 757 

44  Practical  Efficiency  of. 737 

"  Side- wheels,  Propeller  and  Ma- 
rine, Weights  of. 758-759 

"  Weights  of 758-759,  9" 

STEAM  VESSEL,  Power  Utilized  in...  662 
4 '    Resistance  to,  in  Air  and  Water  9 1 1 
"    -PROPELLER,  Ordinary  Distri- 
bution of  Power  in 91 1 

"     Velocity  of. 1010 

Steamer  "Great  Eastern " 173 

STEAMERS 478 

u    Iron,  First  Built 915 

"    Relative  Velocities  of  Yachts, 
from    Elements    of    Construction, 

and  Large 928 

Steaming  Distances 86 


TEEL 640-643,  750,  783,  787-788,  827 

44  and  Iron,  Corrosion  of. 908 

'4  and  Iron,  Corrosive  Effects  of 

Salt  Water 916 

44  Guns 913 

4  4  Hemp,  Iron  and  Steel  Wire  Ropes, 

Relative  Dimensions  of...  172 
44  l4  and  Iron  Rope,  Round  and 

Flat  and  Safe  Load. . .  164-166 
1  4<  Iron  and  Steel  Wire  Rope. ..  164 
"  Hexagonal,  Octagonal,  and  Oval  135 

'4  Locomotive  Tubes 1 38 

44  Manufacture  of,  Remarks  on. ...  642 
44  of  a  Given  Section,  Weight  of.  136, 149 

"  Plates 750,  830 

44       "  Thickness  of 121 

44       44  Weight  of ...118-119,146 

"  and    Iron,    Rolled    Bars,    and 
Weight  of. . .  125, 126, 128,  134,  135 

44  Wire,  Weight  of. 120-121 

Sterling,  Pound,  etc 38 

Stings  and  Burns,  Application  for. . .  196 

Stirling's  Mixed  Iron 785 

Stirrups  or  Bridles,  for  Beams 838 

STONE  and  Ore  Breakers  and  Crusher, 

903,  957 

4  4  Dressed,  Modes  of. 603 

44  Hauling 468 

44  Load  per  Sq.  Foot 915 

44  Masonry,  Elements  of. 595-600 

44        "  A  shlar  and  Rubble 600-601 

44  Resistance  of,  to  Freezing 184 

44  Sawing 196,  904 

•4       44   and  Dressing,  Cost  of.....  949 

44  Voids  in  a  Cube  Yard  of. 690 

STONES,  CEMENTS,  etc.,  Crushing  of. .  766 

(See  Crushing  Strength,  764-769.) 
' '  Building,  Expansion  and  Con- 
traction of. 184 

Straw  and  Hay,  Weight  of. 198 

Streams,  Rivers,  and  Canals,  Flow  of 
Water  in 550 

STREET  RAILS  OR  TRAMWAYS.  435,  915,  918 
44  RAILROADS,  Experiments  on  Mo- 
tors, Result  of. 915 

44         44  Cost  of  Maintenance  ....  918 
41  ROADS  and  Pavements 686-690 

STRENGTH  OF  MATERIALS,  Elements 

of. 761-841 

"  ELASTICITY  AND  STRENGTH,  Co- 
hesion and  Resilience.  761-763 

44     44  Coefficient  of. 761-762 

44  44  Modulus,  Height  of,  Weight 

of,  Various  Materials.  762-763 

u     <(  oj-  rp0  Compute 762 

44  4t  of  Elasticity,  Height  of,  To 

Compute 763-764 

44  "  Resilience,  Comparative,  of 

Woods 763 

41  4  4  Weight  a  Material  will  Bear 
without  Permanent  Alter- 
ation of  its  Length 763 

"  COHESION,    Modulus    of    and 

Weight.  To  Compute. .  =763-764 


XXXVI 


INDEX. 


Page 

STRENGTH  OP    MATERIALS,  CRUSHING, 

ly\  \ments  of,  etc 764-769 

"    "  Bricks 908 

"     "  Ca*    tnd  Wro't  Iron,  Woods 

a    \  Various  Metals 765 

"  "  Coll.:  ins  of  Iron  and  Steel, 
Sn.'e  Load  of  and  Coeffi- 
cients of,  To  Compute. . .  769 

"  "  Cohans,  Arches,  Chords, 
etc. ,  of  Cast  Iron,  Safe 
Load  of. 766-767 

M     u     "  Weight  of,  To  Compute.   769 

14    "  Cylinders  and  Rectangular 

' Tubes  of  Wro't  Iron 767 

11    "  Elastic  Limit  compared  to  764 

"     "  Granite,  Limestone,  Marble, 

and  Sandstone 767,  1029 

"    "  Hollow  Columns  or  Tubes, 

Safe  Load  of. 768-769 

"    "  Ice 912 

•*  "  Long  Solid  Columns,  Com- 
parative Value  of. 976 

"  "  Notes  and  Effects,  Season- 
ing, etc 764 

"    "  of  Cements  and  Mortars. ..  596 

••  "  Relative  Value  of  Woods, 
Strength  and  Stiffness 
Combined 976 

**    "  Sandstones, Stones,  Cements, 

Masonry,  etc 765-766 

"    "  Various  Materials 765-769 

"  DEFLECTION,  Elements  of,  etc. , 

770-781 
"     "  and  Weight  Borne  by  a  Bar 

or  Beam  of  Wro't  Iron. .  773 
"    "  Bars,  Beams,  Girders,  etc., 

770-771 

"     "  Beams  of  Rectangular  Sec- 
tion, Formulas  for.  771-773 
"     "  "and  Comparative  Strength 

of  Flanged 778 

"     "    "  and  Girders 840-841 

"    "    "  Elastic  Strength  of,  of 

Unsymmetrical  Section  778 
"     "    "  Flanged,  and  Weight  of 
that  may  be  Borne  by 
One  of  Cast  Iron. .  777-778 
ii     ii    (t  q^  @ast  anft  Wro^t-iron, 

and  Woods 772-774 

it     u    «  or  Girders,  Continuous.  772 
"    ««  Bearings,  Admissible  Dis- 
tances between 778 

"  "  Cast  Iron  Flanged  Beams 
and  Comparative  Strength 

of. 809 

*•    *'  General  Deductions —  779-780 
«•    •«  Girders,  Tubular,  ofWro't 

Iron 775,  809 

"    "  Rails,  Flanged,  Iron  and 

Steel 775-776 

"  "  Rectangular  Beam  of  Iron 
and  Woods,  Load  that 

may  be  Borne  by 773 

M  "  "  Bars  and  Beams  of  Cast 
Iron  and  Various  Sec- 
«on*,etc 777 


STRENGTH  OF  MATERIALS,  Results  of 
Experiments  on  Subjec- 
tion of  Cast-iron  Bars  to 
Continued  Strains 780 

"  "  Riveted  Beams  of  Wro't 

Iron  and  Weight  of. .  774-775 

"  "  Rolled  Beams  of  WroH 

Iron 774 

"     "  Shaft  from  its  Weight  alone  778 

"  "  Shafts  and  Distributed 

Weight  for  Limit 778 

«  <t  i(  Mm  an(i  factory,  and 

To  Compute 779 

"  "  Springs,  Carriage  and  In- 
dia-rubber  779 

"  "  Working  Strength  or  Fac- 
tors of  Safety 781-782 

11  DETRUSIVB  OR  SHEARING,  Ele- 
ments of,  etc 782-783 

"  "  Comparison  between  Trans- 
verse  782 

1  *     "  Length  of  Surface  of  Wood 

to  Horizontal  Thrust. . . .  783 
1  "  of  Metals  with  a  Punch. . .  782 
"  * '  of  Riveted  Joints,  Cast  Iron, 

Steel,  Treenails,  and  Woods  783 
"  of  Rivets  and  Memoranda.  830 
"  of  Various  Metals  by  Par- 
allel Cutters 783 

"  of  Woods 782 

"  of  Wro't-iron 783 

"         "     Bolts,  Experiments 

on  and  by  U.  S.  N. 783 

u     u  Power  to  Punch  Iron,  Brass, 

or  Copper,  To  Compute. .  782 

"  TENSILE,  Elements  of,  etc. .  784-790 

"  "  Average  Elasticity  of  Steel 

Bars  and  Plates 788 

"  "  Cast  Iron,  Stirling's  Mal- 
leable, and  Wrought. .  784-785 

"  "  Ductility  and  Malleability 

of  Metals,  Ratio  of. 787 

"  "  Elements  connected  with 

Various  Substances. ....  786 

u     "  Manganese  Bronze 832 

"  "  Memoranda  on  Reheating, 
Temperature,  Annealing, 
Cold  Rolling,  Hammer- 
ing, Welding,  Cutting 
Threads,  Case  Harden- 
ing, and  Galvanizing. . .  786 

u  "of  Various  Metals,  Woods, 
and  Miscellaneous  Sub- 
stances  788-790 

"  "of  Wood  in  Various  Posi- 
tions   870 

"  "  Result  of  Experiments  on 

by  Soc.  ofC.E. 787-788 

"  "  Riveted  and  Welded  Joints 

of  Wro't-iron  Plates 828 

"  "  Steel,  Crucible,  Bessemer, 
Fagersta's  and  Siemens 
Experiments  on 787 

u  "  Wro't  Iron  Tie-rods, Exper- 
iments on 781 


INDEX. 


XXXV11 


Pag 

STRENGTH  OF  MATERIALS,  Relative 
Resistance  of  Wro't-iron 
and  Copper  to,  and  Com- 
pression  787 

"  TORSIONAL,  Elements  of,  etc.  790-797 
"  "  Shafts  and  Gudgeons.. 790-792 
"  "  Cast  Steel  and  Coefficients 

of. 795 

4<    "  Gudgeons,  Diameter  of,  To 

Compute 795 

"    * c  Hollow,  Round,  and  Square 
Shafts  and  Cylinders,  To 

Compute 792-794 

"    "  Shafting,  Coefficients   for 

Ultimate  Resistance  of. .  797 
"    "  Shafts  of  Oak  or  Pine,  To 

Compute 793-794 

1     "  Couplings   or    Sleeves, 

and  Supports  for. ...  796 
"    "    "  Diameter  of,  to  Resist 
Lateral  Stress,  To 

Compute 701-792 

*'    "    "   "and  Journal  of  for 

a  Water --wheel. ...  571 
u    u    ((   «  Oj-a  wroit  Iron  Cen- 
tre Shaft 794 

«     n     u   «  Of  t0  Resist  Torsion 

and  Weight 792 

"    "    "  Hollow,  Diameter  of  to 

Sustain  its  Load,  etc.  792 

"    "     "  Journals  of,  etc 796 

"  "  "  Mill  and  Factory,  For- 
mulas for 797 

"    "     "  of  Writ  Iron  for  Ma- 
rine   Engines,   For- 
mulas for  Diameter.  796 
"    "    "  Round  and  Square,  Ul- 
timate Strength  of,  To 

Compute 795 

"  "  "  toResistLateralStressofjqj 
"  "of  Variant  Metals,  Woods, 

and Tobin  Bronze,  793-794, 929 

"  TRANSVERSE,  Elements  of,  etc. , 

798-841 
"    "  Coefficient    or  Factor    of 

Safety  of  Materials 802 

"     "  Bar  or  Beam,Fixed  or  Sup- 
ported, bears  Weights 
at  Unequal  Distances.  803 
"     "  Bars  of  Steel,  To  Compute, 

817,  827 

u  Hollow  Girdersor  Tubes, 
Comparative  Value  of. .  824-825 
"  Beam  or  Girder  of  any  Sec- 
tion or  Material,  For- 
mulas for 817-818 

1  "  "or  Shaft,  Rectangular, 
Diagonal,  etc.,  For- 
mulas for 817 

5     u  Beams  of  Various  Woods, 
Safe  Statical  Load  of  and 

Coefficients  for 834-835 

'     "  Bottom  Plate  Area  and  De- 
structive Weight  of. 810 

'     "  Bowstring  Girder,  Diameter 

of  Tie-rod,  To  Compute. .  8x2 


Page 

STRENGTH  OP  MATERIALS,  Cast  Iron 
Beams  or  Girders  of  Va- 
rious Figures,  To  Com- 
pute ................  814-821 

"  "of  Various  Figures  of, 

798,  800,  813-817 
"  Centres  of  Gravity  and  of 
Crushing    and     Tensile 
Strength  of  a  Girder  or 
Beam,  To  Compute  ......  819 

u    "  Concretes,  Cements,  etc  ____  800 

"     "  Cylinders,  Flues,  and  Tubes, 
Elements  of  within  Elas- 
ticity, To  Compute.  .  .827-828 
u     "  Cylindrical  and  Elliptical 
Beams  or  Tubes  of  Wro't 
Iron  ...................  810 

"     "  Diameter  of  a  Cylinder  to 
Support  a  Given  Weight, 

804-805 

"  Depth  of  Beam  to  Support 
a  Uniform  Load  ........  808 

"     "  Elastic  of  Wro't  Iron  Bars.  808 
"     "  Elliptical-sided  Beams,  Side 

or  Curve,  To  Determine.  .  826 
"    "  Equilateral  Triangle  or  T 

Beam,  To  Compute  ......  804 

"    "  Flanged  Beams  of  Cast  Iron, 
Comparative  Strength 
and  Deflection  of  .  —  809 
44    "     4t  Dimensions   and  Pro- 

portions of.  .........  809 

"     "     u  Hollow    or    Annular 
Beams  of  Symmetri- 
cal Section  ..........  815 

"     "  Floor  Beams,  Girders,  etc., 
of  Wood,  Capacity  and  El- 
ements of,  To  Compute.Sjs-^ 
"     "  Formulas  and  Rules  for  Rec- 
tangular Bars,  Beams, 
or  Cylinders  ......  801-803 

"     "     "/or  Destructive  Weight 
of  Solid  Beams  of  Un- 
symmetrical  Section, 

816-819 
"     "  General   Deductions  from 

Experiments  .........  824-825 

"     "  GirderorCylindricalShaft, 
Lower  Flange  of,  Sec- 
tion to  Sustain  a  Safe 
Load  in  its  Middle,  To 
Compute  ............  817 

"    "     "  Beam,  etc.,  Factors  of 

Safety  ..............  821 

"     "     "  Dimensions  of  and  Load 

on,  To  Compute.  .  .839-840 
"     "     "  Graphic  Delineation  of 

Stress  ..............  840 

"  "  Girders  and  Beams  of  Va- 
rious Figures  and 
Sections,  Symmetrical 
and  Unsymmetrical, 


. 
"  Beams,    Lintels,    etc., 

Elements  of.  .....  822-823 

u  Qff    £eams   Of  Unsym- 

metrical Section.  .810-811 


XXXV111 


INDEX. 


Page 

STRENGTH  OP  MATERIALS,  Girders  of 

Wro't-iron,  Plate 811-812 

a    "  Glass   Globes   and    Cylin- 
ders, Resistance  to  Inter- 
nal Pressure  and  Collapse  831 
"     "  Headers  and  Trimmers  or 
Carriage  Beams,  Elements 

of,  To  Compute 836-838 

"  "  Homogeneous  Beams  of  Un- 
symmetrical  Section,  Ul- 
timate Strength  of,  To 

Compute 820-821 

"    "  Inertia  of  Beams,  Moment 

of. 818-819 

u     "Inclined    Beams,   etc.,   of 

Wrd't  Iron 811 

"  "  Lead  Pipes,  Resistance  to, 
Thickness  of,  Weight,  and 
Bursting  Pressure,  To 

Compute 831 

"    "  Manganese  Bronze 832 

"  Memoranda  on  Joints,  etc. .  830 

u     "  on  Metals. 832 

"  Metals 798 

"  Miscellaneous  Illustrations  826 
"  Neutral  Axis  of  a  Beam  of 
Unsymmetrical    Section, 

To  Compute 820 

"  of  a  Beam  in  an  Inclined 
or  Oblique  Position.  .799,  804 

"  of  Brick-work 801 

"  of  Woods 798-799*  800 

"  Plate  Joints,  Double  Rivet- 
ed and  Strapped,  Experi- 
ments on 829 

"  Pressure  on  Ends  or  Sup- 
ports of  a  Bar,  Beam,  etc.  803 
"  Rails,  Iron  and  Steel,  To 

Compute 812 

"  Rectangular  and  Cylindri- 
cal Beams,  Formulas 
and  Coefficients  for. .  805 

"     "  Girders  or  Tubes 809 

"Resistance,  Moment    and 

Work  of,  To  Compute. ..  818 
"  Riveted  Joints,  Compara- 
tive Strength  of. 828 

"  Rivets,  Pitch,  Lap,  and  Di- 
ameter of 829 

"  Rolled  Beams  and  Channel 
Bars,  Safe  Load  and 

Weight  of. 807-808 

"  "  and  Girders  or  Riveted 
Tubes  of  Wro't  Iron, 
of  Various  Figures 
and  Sections, Destruc- 
tive Weight,  To  Com- 
pute  80 

"  "  "  and  Channel  Bars  of 
Wro't  Iron  of  Various 

Sections 806-80? 

;     "     "  Steel  Beams .147,  80 

;    "  Scarfs,  Relative  Resistance 

of. 84 

1    *'  Seasoning,    Increase      in 

Strength  of  Woods 8oc 

k    "  Solid  Cylinder^  To  Compute  80 


Page 

STRENGTH  OP  MATERIALS,  Solid  Cylin- 
der and  Hollow  Cyl- 
inders of  Various  Ma- 
terials   801 

u     "    u  Diameter  of  to  Support 

a  Given  Weight. . .  804-805 
"     "  Statical,  Dead  or  Live  Load, 

Factors  of  Safety 841 

"     "  Steel  and  Bridge  Plates  and 

Rivets 830 

"     "     "  Bars,  To  Compute 827 

"     "  Stiffness  of  Materials 798 

u     ' '  Stirrups  or  Bridles,  Dimen- 
sions of.  To  Compute. . . .  838 
"     "  Symmetrical     Girder     or 
Beam,    Conditions    of 

Forms,  etc 825-826 

"     "  Tobin  Bronze 929 

"  Trussed  Beams  or  Girders,  823 
"     "  Unequally  Loaded  Beams, 

etc 810 

"     "  Various  Materials  and  of 

Various  F fox.  .7Q8-8oi,  1029 
< <    < i    "Metals,  Comparative 

Qualities  of. 821 

"     "  Woods,  Large  Timber,  To 

Compute 833 

u     u  wroit-iron  Joints,  Propor- 
tion of  Single  Rivets .  829 

STRENGTH  OF  MODELS 644-645 

u  Dimensions  of  a  Beam  which  a 
Structure  will  Bear,  To  Com- 
pute   644 

"  Resistance  of  a  Bridge  from  a 

Model,  To  Compute 645 

Stress,  Moment  of,  and  on  Rods. 62 1-1041 
Street  Railroads  or  Tramways,  and 
Result   of  Experiments   on 

Motors  for 915 

Cost  of  Maintenance 918 

Stucco 591-592 

Suez  Canal,  Via 912 

Sugar  Cane,  and  Beet  Root 207 

and  Acid  in  Fruits 203 

and  Water  in  Various  Products  201 

in  Mortar 951 

-Mill  Rollers 911 

-Mills,  and  Engines,  Weights  of, 

759»  9°3>  9°8 
Sulphuret  of  Carbon,  Elastic  Force 
and  Temperature  of  Vapor  of. .. .  707 

Sun,  Heat,  Diameter,  etc 188 

"  Heat  of. 193 

Sunday  Cycle  or  Cycle  of  the  Sun. . .     70 

u  or  Dominical  Letter 70 

Sun  dial,  To  Set 69 

Sunstroke  Remedy 938 

Surveying,  Useful  Number  in 69 

SUSPENSION  BRIDGES 178,  199,  842 

u  "  Elements,  Stress,  and 

Pressure,  To  Compute  842 
"            "  Horizontal  Stress  and 
Vertical  Pressure  on 
Piers,  To  Compute  . .  842 
"  "  Steel  Cables  for 163 


INDEX. 


XXXIX 


Page 

SUSPENSION  BRIDGES,  Ratio  of  Stress 
on  Chains  or  Cables  at  Point  of  Sus- 
pension Bears  to  ivhole  Weight  of 
Structure  and  Load,  To  Compute. .  842 

Sustenance,  Human  and  Animal 203 

4'  Requirements  of  a  Workman  207 

Sweet  Potato 207 

Swimming 439 

SYMBOLS,  Algebraic,  and  Formulas. 22-23 

41  and  Characters 21-22 

44  and  Equivalents 190 

44  for  Elements  and  Formulas. . .  981 
Symbolic  Hatching  and   Designa- 
tions      932 

T. 
Tacks,  Nails,  Spikes,  etc.,  Wro't  Iron  154 

Tan,  Elements  of. 482 

Tangential  Wheel 576 

TANGENTS  AND  CO-TANGENTS 415-426 

44          44  To  Compute 426 

«          44        44  Degrees,  Minutes, 

etc. ,  of  Given 426 

Tannin,  Quantity  of,  in  Substances. .  190 
Tape  Line  or  Chain,  to  Set  out  a  Right 

Angle  with 69 

TECHNICAL  TERMS,  Orthography  of.  1042-51 

44  44      in  Masonry 597-599 

Tee  and  Angle  Iron,  Weight  of. 130 

Teeth  of  Wheels 859-862 

44  Dimensions  of  a  Tooth,  etc.,  To 

Compute 860-861 

"  IP  and  Stress  of,  To  Compute. . .  86 1 

1 '  Involute 859 

Telegraph  Wire,  Span  of. 179,  936 

44  Telephone,  Wires  and  Cables. .  960 
Telescopes,  Opera- Glasses,  etc — 671,  942 

Telford  Roads 688,  690 

TEMPERATURE  and  Extremes  of.  .914,  952 

44  Absolute 504 

44  Artificial  and  of  Earth 195 

44  by  Agitation 524 

44  Decrease  of,  by  Altitude 522 

44  Extremes  of. 952 

44  Non-conductors  of. 933 

44  of  Enclosed  Spaces. ........  526 

44  of  Mines 918 

"  of  Saturated  Steam,  Latent 
and  Total  Heat  of, 

To  Compute 707 

"  of  Steam,  To  Compute 705 

44  of  Various  Localities 192 

*4  To  Reduce  Degrees  of  Differ- 
ent Scales 523 

44  Transmission  or  Conductiv- 
ity of 914 

44  Underground 519 

Temperatures,  Metric 37 

Tempering  Boring  Instruments 197 

Tenacity  of  Iron  Bolts  in  Woods  —    198 

TENSILE  STRENGTH  (See  Strengtfi  of 

Materials) 784-790 

Terne  Plates 124 

Terra  Cotta 602 

Tests,  Simple,  of  Water 928 


Theatres  and  Opera-Houses 180 

Thermometers,  Reduction  of. 523 

Throwing  Weights  by  Men 439 

Thrust,  Weight  of  a  Body,  to  Sustain  a 

Given.  To  Compute 693 

Tidal  Phenomena  and  Current.  .75-1010 

Tide  Table  for  Cocat  of  U.  S 84-85 

TIDES 84,  198 

of  Atlantic  and  Pacific 191, 198 

of  Pacific  Coast 85 

Rise  and  Fall  of,  Gulf  of  Mexico.    85 

Time  of  High  Water 74-75 

Tie-rods,  Experiments  on 787 

TIMBER  AND  BOAKD  MEASURE 61 

;i  AND   WOODS,   Elements,    Notes, 

Treatment,  etc 865-870 

4  Comparative   Weight   of  Green 

and  Seasoned 217 

(See  Wood  and  Timber,  865-870.) 
4  Measure,  and  Volume  of,  To  Com- 
pute   61-62 

1  Strengthof. 870 

1  Volume  of  Squared,  To  Compute    62 
'  Waste  in  Hewing  or  Sawing. ...     62 
TIME,  after  Apparent   Noon,  before 

Moon  next  passes  Meridian. ..     75 

Civil  and  Marine 37 

Difference  in 81-83 

44  of,  between  New  York  and 
Greenwich,  and  any  Location, 

To  Compute* 83 

Measures  of,  and  New  Style 37 

Sidereal  and  Solar 37 

To  Reduce  to  Longitude 54 

TIN,  Plate  and  Block 644 

"  Lined,  and  Lead  Pipes,  Weights  of 

per  Foot. i37>  X5i 

44  Pipes,  Lead  Encased,  Weight  of..  151 

4  k  Plates,  Marks  and  Weights 137 

TOBIN  BRONZE,  YachtShaflingtetc..  929 

Tolerance,  of  Coins 38 

Tonite,  or  Cotton  Powder 443 

TONNAGE,  of  Vessels,  To  Compute.  175-177 

u  Approximate  Rule 176 

41  Builder's  Measurement 176 

44  Corinthian,  New  Thames,  and 
Royal  Thames  Yacht  Clubs. .   177 

"  English  Registered 1 75-176 

44  Freight  or  Measurement 177 

41  of  Suez  Canal 177 

44  To  Compute 173 

44  Units  for    Measurement,  and 

Dead- weight  Cargoes 176-177 

"  Weight  of  Cargo,  To  Ascertain.  177 

Tools,  Friction  of. 476 

Tornado,  Pressure  of. 911 

TORPEDOES,  Submarine 946 

TORSIONAL  STRENGTH  (See  Strength 

of  Materials) 79°~797 

Towers,  Spires,  and  Domes 1 80,  932 

Towing,  on  Erie  Canal  and  Hudson 

River 193 

TRACTION,  Elements  of,  etc 843-849 

44  and  Statical  Resistance  of  Ele- 
vations  846 

4  4  Coefficients  of,  for  Roads 845 


xl 


INDEX. 


Page 

TRACTION,  Friction  of  Roads  and 
Coefficients  of  in  Propor- 
tion to  Load 847 

44  Maximum  Power  of  a  Horse  on 

a  Canal 848 

44  of  Omnibus  and  Speed 844 

"  on  Canal,  Slack-water,  River, 

and  on  Street  Railroads 848 

u  on  Various  Roads  and  of  Va- 
rious Vehicles 845,  847-848 

u  Power  Necessary  to  Sustain  a 
Vehicle  on  an  Inclined  Road, 

To  Compute 845-846 

44  Power  Necessary  to  Move  and 
Sustain  a  Vehicle  Ascending 
or  Descending  an  Elevation, 

To  Compute 846 

"  Resistance  of  a  Car 849 

44          u  of  Gravity  and  Grade..  847 

44          "  on  an  Inclined  Road 846 

44         u  on   Paved,  Rough,  and 

Common  Roads 843-844 

44          "  to  on  Common  Roads.  843-845 
44  Results  of  Experiments  of,  on 

Roads  and  Pavements 843 

Tramways  or  Street  Railroads, 

435,848,915 

Transportation,  Canal 193 

44  of  Horses  and  Cattle 192 

TRANSVERSE  STRENGTH  (See  Strength 

of  Materials) 798-841 

Trass  or  Terras. 589 

Treadmill 433 

Treenails,  Strength  of,  etc 783 

Trees,  Large,  in  California 184 

"          u      in  Australia 971 

Trigonometrical  Equivalents 387 

Trigonometry,  Plane,  Angles,  Sides, 

etc.,  To  Compute 385-389 

41  Distances  of  Inaccessible  Ob- 
jects, To  Ascertain 388-389 

44  Height  of  an  Elevated  Point, 

To  Compute 389 

Tripolith,  Composition  of. 198 

Troops,  Marine  Transportation  of...  914 

Trotting. 439-44° 

Troy  Measure 32 

Truss,  Iron  and  Stress  on 178-1041 

Tubers,  Ratio  of  Flesh-formers 207 

Tubes,  and  Flues 747,  827 

14  and  Pipes,  Weight  of,  To  Compute, 

147-148 
* 4  Brass  and  Seamless  Brass,  Weight 

of. 142 

44  Copper,  Seamless  Drawn,  Weight 

of. 140-142,  144 

"  English  Wro'tlron,  Weight  0/143,  *45 
44  Evaporative  Capacity  :>f  Varying 

Length 742 

44  Lap-welded  Iron  Boiler 139 

41  or  Girders,  Dimensions  and  Pro- 
portions of. 809 

44  Steel  and  Semi- Steel  Locomotive  138 

"  Thickness  of  B  W  G 748 

Tubular  Bridge,  Britannia 178 

Tunnels,  Lengths  of. 179,  936 


Page 

TURBINES  (See  Hydrodynamics).  .572-577 
"  and  Water -Wheels  Compared.  579 

Turkish  Plaster  and  Mortar 591-592 

Turning,  Friction  of. 477 

44  and  Boring  Metal 197 

Turnips,  Ratio  of  Flesh-formers 207 

Turpentine,  Elastic  Force  and  Tem- 
perature of  Vapor  of. 707 

U. 

Underground  Temperature 519 

UNDERSHOT  -  WHEEL  (See  Hydrody- 
namics)   566-571 

Unguents,  Relative  Value  of. 471 

Units  for  Computing  Safe  Strain  of 
New  Ropes,  Hawsers,  etc 170 

V. 

Value  of  Coins,  To  Compute 39 

•    and  Weight  of  Foreign  Coins .  40-45 
of  the  Metre  in  terms  of  the  Brit- 
ish Imp.  Yard 934 

VAPOR  IN  ATMOSPHERE,  Volume  and 

Weight  of,  To  Compute 68-69 

"  Elastic  Force  of,  of  Alcohol, 
Ether,  Sulphuret  of  Carbon,  Petro- 
leum, and  Turpentine 707 

VAPORS,  Relative  Density  of  Some. . .  521 
Variation  of  Magnetic  Needle. . .  57-i°39 

1 '  Decennial,  of  Needle 58 

rt  of,  in  U.  S.  and  Canada 59 

Varnishes 876 

Vegetable  Marrow,  Composition  of. .  207 
Vegetables,  Analysis   of  Meat   and 

Fish  and  Foods 200-201 

4 4  and  Oil-cake,  Nutritious  Prop- 
erties of  Compared 204 

44  Elements  of  Various 207 

44  Proportion  of  Starch  in 205 

"   Tubers,  Ratio  of  Flesh-formers  207 

Vegetation,  Limits  of. 192 

Velocities,  Metric 37,923 

44  Acceleration  and  Distance  of  a 

Body,  To  Compute 921-922 

41  of  Different  Figures  in  Air, 

'  Resistance  of. 646-648 

Velocity  Lost  by  a  Projectile,  To  Com- 
pute    648 

"  and  Time,  To  Compute 648 

44  and  Volume  of  Molecules 194 

44  of  Current  of  a  Bay  or  River. .  971 
VENTILATION,  Buildings,  Apartments, 

etc 524 

"  Length  of  Iron  Pipe  required 

to  Heat  Air,  To  Compute.  52  5-526 

4'  of  Mines 449 

44  Proper  Temperatures  of  En- 
closed Spaces 526 

44    Volume    of  Air   Discharged 

through  a  Ventilator.  524 
44        4i  of  Air  per   Hour  for 

each  Occupant  of  a  House.  525 

Vernier  Scale 27 

VESSEL,  Elements  of,  To  Compute. . .  653 
44  Hulls    of  Iron,    Thickness    of 
Plates  and  Rivets 830 


INDEX. 


xli 


Vcuel'i  Side  Lights,  Visibility  of.... 

Vessels,  Mean  Speed  of. 971 

Veterinary,  Treatment  in 186 

Victor  Turbine. 576 

Volcano,  Power  of. 910 

Volcanoes,  and  Heights  of. 182,  936 

Voltmeters  and  Ammeters 961-962 

VOLUME  AND  WEIGHT  of  Various  Sub- 
stances in  Ordinary  Use 216 

"    of  Molecules 194 

VOLUMES,  Mensuration  of. 360-378 

W. 

Walking 433,438 

Wall,  Chinese 179 

WALLS  and  Arches,  Elements  of.  .602-603 
(See  Walls,  Dams  and  Embank- 
ments, 700-703. ) 

4 '  and  Earth,  Friction  of,  To  Ascer- 
tain and  Compute 698 

"       •'  Moment  of,  To  Compute 701 

44       u        "  of  Pressure,  Point  of, 

To  Ascertain 698 

"  of  Buildings,  Thickness  of. i&g,  1020 
44  or  Dams,  Centre  of  Gravity  of. .  702 

44  Retaining,  of  Iron  Piles 196 

14          "or  Dam  Stability  of,  To 

Determine 702 

44  Revetment,  Elements  of. 694 

Warehouses,  Brick  Walls  for 603 

WARMING  BUILDINGS 527-528 

44  by  Hot-air  Furnaces  or  Stoves  528 

44  by  Hot  Water 524 

4  by  Steam 527 

4   Coal  Consumed  per  Hour. . . .  527 
4  Furnaces  and  Open  Fires. . . .  528 

4  Illustrations  of  Heating 527 

4  Volume  of  Air  Heated  by  Ra- 
diators, Consumption  of  Coal,  Areas 
of  Grate,  and  Heating  Surface  of 

Boiler,  etc. ,  per  100  Sq.  Feet 528 

Warps  and  Hawsers 173 

Washington  Aqueduct 178 

Watches,  First  Constructed 915 

WATER,  Elements  of 849-852,  916 

*4  Approximate  Bottom  Velocities 
of  Flow  of  in  Channels,  at 
which  Materials  are  Moved. . .  916 
44  Boiling  -  Points  of,  at  Different 

Degrees  of  Saturation 851 

41   Column,  Height  of. 849 

;     4'   Density  of  To  Compute 520 

44  Deposits  of.  at  Different  Degrees 
of  Saturation  and  Tempera- 
ture   852 

44  Evaporation  of. 916 

4  Expansion  of. 519 

4  Fresh  and  Sea 849-851 

14  Distillation  of. 955 

4  Friction  of  in  Pipes 925 

4  Head  and  Discharge  of  in  Pipes  920 

'  Inch,  Miner's 557 

4  Motors,  Ratio  of  Effective  Power  563 
4  Pipe,  Dimensions  and  Weight  of, 
from  .375  to  5  Inches 137-138 


Page 
WATER  and  Metal  Pipes,  To  Compute 

Weightof. 147 

44  Power  and  of  a  Fall  of 562 

44       44  Cost  of  on  Driving  Shaft. ..  950 

44  Pressure  Engine 579 

44  Rainfall  and  Volume  of 850 

44  Resistance  of,  to  an  Area  of  One 
Sq.  Foot  Moving  through,  or 

Contrariwise 646 

44  Saline  Contents  of 852 

4  4  Salt,  Corrosive  Effects  of,  on  Steel 

or  Iron 916 

44  Sea,  Composition  of. 851 

44  Tests,  Simple 928,  974 

44    Velocity  of  a  Falling  Stream 

of. 496 

Volumes  of  Pure,  and  at  32°. . .  849 
44    Weights  of,  and  To  Compute.  852,  923 

44  -METERS,  IVorthington's 942 

44  -RAISING,  Cost  of. 949 

44  -TUBE  BOILER,  Elements,  Tests, 

and  Average  Results  of. .  926 

44      44  Efficiency  of 947 

"  -WHEEL,  Centre  of  Gyration...  611 
44         4  4  Diameter  and  Journal  of  a 

Shaft,  etc 571 

44  Dimensions  of  Arms 571 

44  -WHEELS,  of  Steamboats. 730 

44         44  Compared  with  Turbines. .  579 
(See  Hydrodynamics  and 
Hydrostatics,  563-579.) 

Waterfalls  and  Cascades 184 

Watermelon,  Water  in 207 

Waterproof,  To  Render,  Wood,  Iron, 

Walls,  Paper,  etc 875 

Waters,  Mineral,  Analysis,  etc. .  .850-852 

Waterworks,  Filters  for 184 

Wave,  Flood 912 

Waves  of  the  Sea 852-853 

44  Height  of,  in  Reservoirs,  etc.,  To 

Compute 853 

44   Tidal,  and  Length  of. 853 

44  Velocity  of  To  Compute 853 

Weather-Foretelling  Plants 185 

"  Glasses 430 

44  Indications 431 

WEDGE,  and  To  Compute  Power 630 

Weighing  without  Scales 66 

WEIGHT,  Diameter  and  Volume  of,  oj 

Cast-iron  and  Lead  Balls....  153 

44  Aluminum 155 

44  Anchors,  Cables,  Chains,  etc...  173 
44  and  Dimensions  of  Lead  Balls .  501 
44  44  of  Gas  and  Water  Pipe..  138 

44        44  of  Water  Pipe 137 

"  and  Fineness  of  U.  S.  Coins. . .     38 

44  and  Mint  Value  of  Coins 40-43 

44  and   Mint   Value   of  Foreign 

Coin,  1888 40-43 

44  and  Specific  Gravity 208 

44  and  Strength  of  Wire,  Iron.  etc.  114 
44  44  of  Stud-link  Chain  Cable 

per  Fathom 168, 930 

44  4l  of  Hemp  and  Wire  Ropes  172 
44  Angle  and  T  Iron  and  Steel, 

125,  126,  130 


xlii 


INDEX. 


Page 

WEIGHT,  Apothecaries' 47 

"         Brass  and   Various  Metals 
per  Cube  Inch  and  Foot, 

and  Wire i 

"      "  and  Gun- Metal  of  a  Given 

Sectional  Area 136, 149 

u      1 1    1 1  yjrro  it  and  Cast  Iron  and 

Steel,  Lead,  Copper,  and 

Zinc  Plates  per  Sq.  Foot. .  1 46 

••      "  Cast  and  Wro't  Iron,  Steel 

and  Gun-Metal,  of  a  Given 

Sectional  Area. . , 149 

«'      "  Castings 155 

"      "  Copper,   Wro't    and    Cast 
Iron,  and  Steel,  of  a  Given 

Sectional  Area 136 

"      "       * '    Iron,  etc. ,  Wire. . .  1 20-1 2 1 

"      u  of  Sheets 142 

"      *'  Pipes  corresponding  to  Iron 

and  Iron  Pipe  Fittings ..  142 
"      "  Wrought  Iron,  Steel,  and 

Copper  Plates 1 18-1 1 

"   Cables,    Galvanized   Steel,  for 

Bridges 163 

"  Cattle,    To    Compute    Dressed 

Weightof. 35 

"  Centrifugal  Pump 917 

"  Crane  Chains  and  Ropes 457 

"  Diamond,  and  of  Diamonds.  32, 193 

"  Earth 33 

"  Electrical  Resistance 34 

"  Fire-Engine 904 

"  Gun-Metal 155 

;  Hay  and  Straw i 

Hemp  and  Wire  Rope 162,  i 

Lead, and  of,  To  Compute.  32,151,155 

"  Pipes 137,  150,  831 

"  Plates,  Weight  of  Sq.  Foot.  146 

"  Sheet. 151 

Length  and  Gauge  of  Iron  Wire  172 

of  Anvils 918 

of  Articles  of  Food  Consumed 
in  Human  System  to  Develop 
Power  of  Raising  140  Lbs.  to 

a  Height  of  10,000  Feet 204 

"  of  a  Body  or  Substance  when 
Specific  Gravity  is  given,  To 

Compute 215 

"  of  Beeves  and  Beef,  Comparative    35 

"  of  Bolts  and  Nuts .-156-157, 159 

"  of  Cast  and  Wrought  Iron  Bar 

or  Rod,  To  Compute —  131 
"         u  Pipes  or  Cylinders. . .  132-133 
41  of  Cast  Metal  by  Weight  of  Pat- 
tern   217 

"  of  Composition  Sheathing  Nails  135 
"  of  Copper,  Cast  and  Wro'tlron, 

and  Lead,  To  Compute..  155 
"  of  Copper,  Braziers' and  Sheath- 
ing   131 

* '  Pipes  and  Composition  Cocks  1 50 

"      "  Rods  or  Bolts. 148 

**      "  Seamless  Tubes 140-142,  144 

"      "  Sheet  per  Sq.  Foot 135 

"  of  Corrugated  Roof  Plates. ....   131 
"   of  a   Cube  Foot   of  Oak  and 

Yellow  Pine 870 , 


Page 

WEIGHT  of  a  Cube  Foot  of  Steam,  To 

Compute 705 

u  of  a  Solid  or  Liquid  Substance 
or  an  Elastic  Fluid,  To  Ascer- 
tain   217 

"  of  Cube  Foot  of  Gases  at  32°.. .  215 
"  of  Embankments,  Walls,  and 

Dams,  per  Cube  Foot 694 

"  of  Fence  Wire 164 

"  of  Flat  Mining  Ropes 165 

"  of  Flat  Rolled  Iron  and  Steel, 

and  Steel  Angles 126,  127,  128 

"  of  Food,  Articles  of. 204 

"  of  Foods,  to  Furnish  Nitroge- 
nous Matter 202 

"  of  Galvanized  Iron  Wire.  ..162,  163 
Sheet  Iron. . .  124, 129 
"  of  Gaseous  Products  of  Combus- 
tion, To  Compute 460 

"  of  Gun- Metal  of  a  Given  Sec- 
tional Area,  and  per  Cube 

Inch  or  Foot 149,  155 

"  of  Hemp,  Iron,  and  Steel  Rope.   164 

**  of  Hoop  and  Sheet  Iron 129, 131 

**  of  Horses 35 

* '  of  Ingredients,  that  of  Compound 

being  given,  To  Compute 218 

"  of  Iron  and  Steel,  Round  Rolled  126 

"        Square  Rolled  125 

"        "        Wire  and  Strength  of '..   124 

"  of  Lead  and  Tin-lined  Pipe.. .  137 

"  of  Men  and  Women 35 

"   of  Metals  of  a  Given  Sectional 

Area, per  Lineal  Foot.. .  136, 149 
"  of  Molecules,  Volume  and  Ve- 
locity   194 

"  of  Oak  and  Yellow  Pine 870 

"  °f Offal  in  a  Beef  and  Sheep..  35 
u  of  Products  of  Combustion. ...  462 
"  of  Riveted  Iron  and  Copper 

Pipes. 148 

"  of  Ropes,  Hawsers,  and  Cables, 

To  Compute 172 

"  of  Silver  and  Tin 155 

"  of  Steam- Engine,  Vertical  Beam, 

Condensing,  To  Compute 759 

"  of  Steel,   Round,   Hexagonal, 

Octagonal,  and  Oval 135 

"  of  Stud-link  Chain  Cables 168 

"  of  Timber,  Green  and  Seasoned  217 

"  of  Tin  Pipes 151 

"      "       Plates  and  Marks 137 

"  of  Tubes  of  Copper,  Brass,  and 

Iron 140-147 

"       4*  of  Brass,  To  Compute 142 

"  of  Various  Materials 763 

"  of  Various  Substances  per  Cube 

Foot  in  Bulk 217 

' '  of  Volume  of  A  ir  Consumed  per 

Lb.  of  Combustible 461 

"   of  Water-Pipes,  To  Compute. . .   561 

"   of  Wire  and  Wire  Rope 162 

"       "  Length  and  Gauge 163 

"       u   Iron,  and  Steel 164,  172 

"  of  Wro't  and  Cast  Iron  per  Sq. 

Foot 146 

"  of  Zinc.  Rolled 146, 155 


INDEX. 


xliii 


Page 
WEIGHT  of  Wrought  and  Cast  Iron 

Tubes I43-M5 

"  "  "  Steel,  Copper,  and 
Brass  Plates,  per 
Sq.  Ft.  per  Gauge, 

118-119 
"  on  Floors  and  of  Structures —  841 

"  Rocks,  Earth,  etc 467-468 

"  Silver 155 

44  Special,  Locomotive 138 

44  Steel,  Copper,  Iron,  and  Brass.  136 
41      "  Copper,  Brass,   and  Wro't 

Iron  Plates 118-119, 155 

"      "  Copper,  Brass,  and    Iron 

Wire 120-121 

44   Terne  Plates,  and  Thickness. ..  124 

"   Tin  Cast 155 

"   U.S.  and  Standard  Measures. .  934 

"    Water  Pipe 137 

"    Wrought  and  Cast  Iron,  Steel, 
Copper,  Lead,  Brass,  and 
Zinc  Plates,  per  Sq.  Foot.  146 
"      "Steel,   Copper,  and    Brass 

Wire 120-121 

44  Wire,  and  Strength 124 

44  Zinc  Sheets 123,  151-152 

"     "  Plates  and  Dimensions  of.  1 46, 151 
44     u  Rolled 146,  155 

WEIGHTS,  Various  Materials 118-175 

44  AND  MEASURES,  U.  S.  Stand- 
ard  26-36,  934 

44  A-ncient  and  Scripture 53 

44  and  Pressures,  Metric 923 

44  and  Volumes  of  Various  Sub- 
stances in  Ordinary  Use —  216 

Apothecaries1 32 

Avoirdupois 32 

Bushel,  Pounds  in 34 

Coal,  Earth,  and  Wood 33 

Engine  and  Sugar  Mill. 908 

English  and  French 44,  47 

Foreign  Countries 48-53 

Grain,  Lead,  and  Troy 32,  47 

Grecian 53 

Hebrew,  Jewish ,  and  Egyptian .  5 3 

Measures  of. 32-47 

44  and  Pressures,  Metric. .  36 
Metric. . .  .27-33,  36-37,  46-4?,  923 
Miscellaneous  Articles  and 

Substances. 33,  46,  214-217 

of  Bells. 180 

of  Boilers 759 

of  Brain,  Relative 192 

of  Chains  and  Anchors 174 

of  Fuels,  Coals  and  Woods, 

483-484,  486 

of  Guns  (Ordnance) 498 

of  Lead  Balls,  and  Shot. . .  500-502 

"      Pipe,  To  Compute 831 

of    Rope,  Hemp,  Iron    and 

Steel 164-166, 172 

of  Slating  per  Sq.  Foot 64 

of  Steam-Engines, 

758-759,  9",  929i  954 
of  Steam- Engines  and  Boilers  929 
of  Steamers'  Engines 911,  954 


WEIGHTS  of  Steamers,  Steamboats, 
Engines,  Boilers,  Launches, 
Yachts,  Cutter,  Pilot -Boat, 
Sailing  Vessels,and  Dredgers, 

888-895,900 

"  of  Substances  in  Bulk. 217 

"  of  Sugar-Mills 903,  908 

"  of  U.S.  Coins. 38 

44  of  Wire,  Iron 162-164 

44  on  Roofs. 952 

"  on  Structures  per  Sq.  Foot. . .  841 
44  Pipes,  Steam,  Gas,and  Water, 

Standard  Dimensions.  138 

Cast-iron 132-133 

Iron  and  Copper,  Riveted  148 
Lap-welded,  Steam,  Gas, 

and  Water 138 

Lead  and  Tin.. .  137,  150-151 
Metal,  To  Compute. . .  147-148 
Seamless  Brass,  to  Corre- 
spond with  Iron 142 

Roman 53 

Tubes,  Lap-  welded,  Steel,  Semi- 
Steel,  Special  Locomotive, 

and  Boiler 138-139 

44  Charcoal  Iron,  Boiler 139 

If.  S.  Old  <&  New,  Approximate, 
Equivalents,  <t  To  Compute.  33, 36 

Water. 852 

Flat,  Square,  and  Round  Roll- 
ed Iron  and  Steel 125-128 

Weirs,  Guaging  of 922 

Welding. 786 

Cast  Steel,  Composition  for . . .  634 

or  Soldering,  Fluxes  for 636 

Well,  Artesian 179,  198 

44     Boring. 197 

Wells  or  Cisterns,  Excavation  of,  etc.    63 
Whales,  Length  and  Weight  of 918 

WHEAT,  Yearly  Consumption  of. 206 

Straw  and  Bran,  Relative  Value 
of,  Compared  with  100  Lbs.  Hay  203 

WHEEL  AND  AXLE .626-627 

44    and   Pinion,  Combinations  or 

Complex  Wheel-work 627-628 

44  GEARING,  Elements  of. 854-862 

44  General  Illustrations  of. 858 

44  IP  of  a  Tooth,  To  Compute 861 

44  Pitch,  Diameter,  Number  of  Teeth, 
Velocity,  Circumference  of,  Rev- 
olutions, etc. ,  To  Compute.  .855-858 
44  Spur  Gear 911 

WHEELS,  Toothed,  Proportions  of...  862 
44  Depth,  Pitch,  Breadth,  Propor- 
tions,and  Resistance  to  Stress, 

To  Compute 859-862 

44  Teeth,  Elements  of, To  Compute, 

859-862 

Whitelaw's  Wheel 576 

Whitewash  or  Grouting 594 

WIND,  Course  of. 675 

"    Effective  Impulse  of '. 665 

Force  of,  To  Compute. 674 

Pressure  of. 91 1, 1035 

Velocity  and  Pressure  of.. .  .674,  924 


xliv 


INDEX. 


Page 

WINDING  ENGINES 476,  862-863 

"  Diameter  of  a  Drum  and  Roll, 
and  Number  of  Revolutions, 

To  Compute 862-863 

Windlass 433 

"  Chinese,  Elements  of,  To  Com- 
pute   627 

"  Power  of. 433 

WINDMILLS,  Elements  of. 863-865,  924 

"  Deductions  and  Results  of, 
from  Experiments  on  Effect 
and  of  Operation. .  .864-865, 924 

"  Elements  of.  To  Compute 864 

Window-Glass,  Thickness  and  Weight  124 

Wine  and  Spirit  Measures 45 

Wire  Gauge,  French 123 

"        "  Area  of  Circles  by 236 

"        "  Circumference  of  Circles  by  242 

..        '•••for  Galvanized  Iron 123 

41       "  Standard  of  Great  Britain  122 

"    Gauges 122-123 

k  •    Iron,  Gauge,  Weight  and  Length 

of. 163 

WIRE,  Lengths  of  100  Pounds 124 

"    ROPE,  HEMP,  IRON,  AND  STEEL, 
Dimensions,  Safe  Load,  and 

Strength  of. 164 

"   ROPE 161-162,  948 

44       44  and  Equivalent  Belt 167 

44       44  Elements  of  Standing  and 

Running 162 

"       "  Results  of  an  Experiment 

with  Galvanized 161 

44       4<       "  of  Experiments  on,  at 

U.  S.  Navy-  Yard 169 

44       "  Rolling  Friction  of. 473 

4  4    and  Hemp  Rope,  Iron  and  S  teel, 

Relative  Dimensions  of. 172 

44    and  Hemp  Ropes,  Weight  and 

Strength  of. 172 

44   and  Tarred  Hemp  Rope,  Haw- 
sers, and  Cables,  Comparison 

of. 169 

44   and  Hemp  Ropes,  General  Notes  167 

44         »  "    Weight  of. 166 

"    and   U.  S.  Navy  Hemp  Rope, 

Breaking  Weight  of. 168 

44    Brass,  Weight  of 155 

"    Cables,  Galvanized  Steel 163 

"    Diameter  and  Weight  of....  120-121 
41    Fence,  Weight  and  Strength  of.  164 

44    Galvanized  Iron 162-163 

44  Iron  Gauge,  Weight  and  Length  163 
"  Rope,  Circumference  of  with 
Hemp  Core  of  Correspond- 
ing Strength  to  Hemp, 
and  of  Hemp  to  Circum- 
ference of  Wire,  To  Com- 
pute  169 

"       "  for  Standing  Rigging,  Cir- 
cumference of,  To  Compute. . .  17: 
"    Ropes  Endless,  Transmission  of 

Power  of. 167 

"    Shrouds 173 

44    Steel,  Weight  of 155 

44    Weight  and  Strength  of 124 


WIRES  AND  CABLES,  Telegraphic,  Tele- 
phone, Electric,  etc 960 

VOOD,  Elements,  etc 48 

Bituminous  or  Lignite 479 

Evaporative  Power  of. 482 

Floor  Beams 835 

Measure 47 

Pavement 689~69<» 

Tensile  Strength  of. 870 

Working   Strength,  or  Factors 
of  Safety 781 

WOOD  AND  TIMBER,  Elements  of. .  865-870 
(See  also   Timber  and   Wood, 
865-870.) 

and  Stone  Sawing 196 

Creosoting,  Effect  of. 869 

Decrease  by  Seasoning 869 

Defects  of. 866 

Impregnation  of 868-869 

Jarrah 913 

Seasoning   and   Preserving, 

866-868 

Selection  of  Standing  Trees..  865 
Transverse  Strength  of  Large, 

To  Compute 833 

Weight  of  Oak  and   Yellow 
Pine  per  Cube  Foot 870 

WOODS 765,  769*  78°.  782-783,  986 

Absolution  of  Preserving  Solu- 
tion by. 869 

Coefficients  for  Safety 781,  835 

Composition  of. 485 

Detrusive  Strength  of 782 

Durability  of  Various 934,  969 

Extension  of,  by  Water 952 

Proportion  of  Water  in 869 

Relative  Value  of  their  Crush- 
ing  Strength   and   Stiffness 

combined 769 

Safe  Statical  Loads  for  Beams.  834 

Strength  of. 833,  986 

Weights  of. 33 

41    and  Comparative  Values 

of  Various 484 

Woolf  Engine 722 

WORK,  Animal  Power 432,  440 

Accumulated  in  Moving  Bodies.  619 

and  Power,  Metric 36,  923 

Works  of  Magnitude,  Miscellaneous, 

178-179,  936 

WROUGHT- IRON,  Elements  of.  — 130-136, 
639-640,  765,  768.  773,  780,  785-786 

' '  and  Cast,  Weight  of. 155] 

11  Angle  and  T,  Weight  of.  125,  130 
"  BOLTS,  as  Affected  by  the 

Thread 916 

44       "    and  Nuts,  Dimensions 

and  Weights 156-159 

Corrosion  of. 955 

Crushing  Weight  of  Columns  768 
Deflection  of  Bars,  Beams, 

Girders,  etc 773~775 

"   of  Rails 77S-776 

Flat  Rolled 126-128 


INDEX. 


xlv 


Page 

WROUGHT  -  IRON  of  a  Given  Sec- 
tional Area,  Weight  of, 

136, 149 

"  Plates,  Weight  of  a  Sq. 

Foot 146 

41  Plates  and  Bolts,  Strength 
and  Test  of. 749 

44       "   Thickness  of '. 121 

"       u    Weight  of. 118-119,146 

"  Rope,  Hemp,  and  Steel, 
Strength  and  Safe  Load 
of. 164-165 

"  Round  Rolled 126 

"  Sheet  and  Hoop,  Thickness 
and  Weight  of. 129,  131 

44  Shell  Plates,  Pressure  and 

Thickness  (U.  S.  Law] 750 

44  Square  Rolled,  and  To  Com- 
pute Weight  of. 125 

'•    Weight  of,  To  Compute 155  . 

"    Wire,  Weight  of. 120-121  I 


Yachts,  Steam,  Relative  Velocities, 
from  Elements  of  their  Construc- 
tion, To  Compute 928 

Yam,  Ratio  of  Flesh- formers. . .- 207 

Tear,  Bissextile  or  Leap,  Civil,  Ec- 
clesiastical       70 

*'      Solar 37,  70 

Years  of  Coincidence,  To  Ascertain.  •     74 


ZENITH  AND  MERIDIAN  Distances  and 
Altitude  of  Sun  at  New  York. ....  932 

ZINC,  Elements  of. 644 

44   Malleability  of. .. 644 

44  Plates,  Weight  of  per  Sq.  Foot. ..  146 
"  Sheets,  Thickness  and  Weight  of, 

123,  151 
44  Foil  in  Steam  Boilers 912 

ZONES  OF  A  CIRCLE,  Areas  of. 269-271 

44        "         44   To  Compute  Area  of .  271 


ADDENDUM. 


Absorptive  Power  of  Charcoal 986 

Acetylene  Formula 1036 

Air,Dryness  of,  Dew -point,  and  Hu- 
midity, To  Ascertain 1011-1013 

AIR  TUMPS.     Of  Condensing  Steam 

Engines 1041 

ALMANAC,  Perpetual 1012 

Aluminum 976 

Area  of  a  Circle  or  Volume  of  a  Cube, 
Increase  of,  To  Compute 988 

BEAM?,  Headers  and  Trimmers 1021 

Bearings  without  Lubricants 974 

BELT  DRIVING 1038 

Bi-sulphide  of  Carbon  Engine,  Rela- 
tive Efficiency  of,  and  a  Steam-En- 
gine   982 

BJ.ASTAND  EXHAUST  FAN  BLOWERS.  1019 

44    Dimensions  of  Fan 1019 

"    Loss  of  Pressure  of  Flow 
of  Air  for  Varying  Di- 
ameter of  Pipe,  etc.. . .  1020 
Blast  Draught  in  a  Marine  Boiler..     974 

Boiler  Setting 976 

Holts  in  Stone,  Anchorage  of. 974 

BRASS,  COPPKK,  ETC.,  To  Color 972 

CAST  IRON,  Strength  of. 101 1 

Cement,  Portland  and  Cement  Mor- 
tar   983 

CEMENT  AND  MORTAR,  Relative  Hard- 
ening in  fresh  and  Salt  Water. .  1035 

Centrifugal  Pump.  Required  Veloc- 
ity of  Edge  of  Blades,  To  Compute  953 

CHIMNEY,  Height  and  Draught  of, 
To  Ascertain 1014 

CORK  AND  COAL,  Evaporative  Power 
of 1035 

COLUMNS,  Safe  Crushing  Strength  of  ion 
44        Computations  of. 1022 

Compass,  to  Graduate,  a  Transit 
Theodilite 1023 


COMPRESSION  OP  AIR.  Flow,  Opera- 
tion, Effect.  Power  of, 
Pressure,  Temperature, 

and  Compression 994 

44  Volume,  Mean  Pressure  and 

Temperature  of. 995-966 

44  To  Compute  IIP 997 

44   Friction  of,  in  Long  Pipes    997 
44  Efficiency   of    Engine    of 

Operation 997 

44   Loss  of  Head,  To  Compute    998 
44  Loss  of  Pressure  per  Mile 

of  Pipe 998 

44  Heating  Compressed  Air..    999 

**  Loss  dj  Efficiency 999 

44  Efficiency  of  Cooling 999 

44   Air  Receivers 1000 

44  Efficiency  of  Engines 1000 

44  Adiabatic  Expansion 1000 

44  Compression,  1007-1008 
44  Hardie  Motor.      Operation 

of,  and  Mean  Results 1000 

44  Power    required  to   Com- 
press Air looo 

44  Flow  of  Compressed  Air 

through  Pipes 1001 

**  Volume    of  Free  Air  Re- 
quired   in  a  Motor  per 

IIP  per  Minute 1002 

«4  Loss  of  Pressure  by  Fi-ic- 

ton  in  Pipes 1002,  1005 

44  Dimensions  and  Elements 

of  Air  Compression 1003 

44  IP   Required  to  Compress 
one  Cube  Foot  of  A  ir  per 

Minute,  etc 1003 

44  Mean  and  Terminal  Press- 
ures      of      Compressed 

Air 1003 

44  Heat    procured    by    Com- 
pression of  Dry  Air. ...  1004 
44  Efficiency  of  an  Engine,..  1004 


xlvi 


INDEX. 


Page 
COMPRESSION  OP  AIR.      Work  Lost 

,    by  Heat  of  Compression . .  1004 

"  Steam  Pressure  and  Point 
vf  Cutting  off,  To  Com- 
pute   1006 

* '  Volume  of  One  Pound  of  Dry 
Air  in  Cube  Feet,  Weight 
of,  etc. ,  To  Compute 1006 

"  Single  and  Compound 
Compression,  Compari- 
son of 1006 

44  Mean  "Effective  Pressures 
in  Compressing  and  De- 
livering of  Air,  etc 1005 

"  Work  per  Pound  of  Air  in 
Compressing  it,  To  Com- 
pute   1007 

"  Stean.  Pressure  required  in 
the  Steam  Cylinder,  To 
Compute 1007-1008 

••  Weight  of  Air  per  Minute 
for  a  Given  Amount  of 
Work,  To  Compute.  1008-1009 

11  Isothermal  Compression  .   1007 

"  Dimensions  of  Valves, 
Pipes,  and  Clearance  of 
Air  in  Cylinders 1009 

11  Compound  Air  Cylinder..  1007 

*'  Work  which  may  be  At- 
tained in  a  Motor  by  One 
Cube  Foot  of  Stored  Air, 

To  Compute 1009 

CRUSHING  AND  TRANSVERSE  Strength 
and  Coefficients  of  Safety 1022 

Depths  and  Heights,  Greatest 983 

Draught,  Blast 1036 


Earth,  Repose  of 982 

Eddy  Valve 975 

ELECTRICAL,    Units    in    Electrical 

Engineering 987 

"  British  Association.  988 

Electrical  IP,  Cost  of 983 

Elevation  of  Localities  in    Upper 

Mississippi,  etc 982 

ENERGY  AND  MOTION 989 

FLOOR  BEAMS,  Computatious  of.....  1021 

Flume,  To  Compute  Elements  of. ...  984 

Food  Substances,  Composition  of...  34 

FOOTINGS  of  Buildings 1020 

FOUNDATIONS,  Computation  of. 1022 

"            Safe  Static  Load  of. .  1040 

Freestone,  Brownstone,  Tests  of....  982 

Friction  of  Engines  and  Gearing. . .  976 

Gas  and  Steam  Engine,  Comparison 

"/•• 953 

GAS  ENGINES 990-992 

* '      Results  of  Trials ...  99 1  -992 
"      Pressures  produced  by 
Explosion  of  Gaseous 

Mixtures 992 

Gate  Valves 975 

Geological  Strata,  Absorption  of .c, 79-101 5 


Girders  and  Floor  Beams,  Capacity 

of. 984 

Computations  of. 1021 

Glue 976 

GROUND,  Consolidation  of  Loose  or 
Made 1037 

HAND  BRAKES,  Efficiency  of. 1036 

Heat,  Coefficients  of  Radiation  of..  1014 
' '  Radiated  per  Sq.  Ft.  per  Hour  1014 
"  Relative  Efficiency  of  Non- 

Conductor 1023 

Heights  and  Depths,  Greatest 983 

Hydrants,  Eddy 975 

Hydraulics  of  a  Fire- Engine 1035 

Hydro-Geology 983 

IRON  AND  STEEL,  Effects  of  a  Low 
Temperature 1035 

KILN  DRYING,  Effects  of,  on  Pine  and 
Hemlock 1037 

KINETICS,  Force  and  Mass 989-990 

Lightest  Known  Substance 1036 

Lightning  Conductors 907,  956 

Liquid  Fuel 1040 

Long-leaf  Pine,  Effect  of  Tapping. .     985 

Magnalium 1037 

Magnesium 976 

MARINE  BOILER,  To  Compute  Weight 

of. 985 

Forced  Draught  in .  1015 
MASONRY,  Execution  of,  during  Se- 
vere Frost 1038 

"         Mortar  for,  below  Freez- 
ing-point      982 

Maximite 1037 

MEMORANDA  (additional). 

974,  976,  979,982-985, 1024,  1025 
Metal  Bearings,  Lubrication  of,..,  1038 

Metric  Measures,  Reduction  of. 1013 

Mill  Race,   Ratio  between  Surface 

and  Mean  Velocity 1036 

MINES,  Temperature  in 1038 

Monolith 982 

Mortar,  Portland  and  Cement 983 

NAILS  AND  DRIFT  BOLTS,  Tenacity 

of. 1038 

"     Cut  and  Wire,  Tenacity  of. . .     983 

OIL  ENGINES 1019 

OIL  OR  TALLOW  IN  A  STEAM  BOILER, 
Effects  of. 1041 

Paint,  Estimate  of  Quantity  required    984 

PIER,  Computation  of. 1022 

PIERS,  Stress  of,  etc 1020 

Piles,  Dimensions  of 1020 

PILING,  Computation  of. 1022 

PUMPS,  Relative  Efficiency  of,  and  a 
Siphon. 1038 

Railway,  Highest,  in  Europe 983 


INDEX. 


xlvii 


Page 

Rifle,  Muzzle  Velocity  of. 1036 

RIVETS,  Resistance  of,  to  a  Lap. . . .  1036 
Ropes,  To  Compute  IPof. 979 

Sand  and  Clay,  Supporting  Power 

of 979 

SAW,  Effect  of  a  Diamond-edged. . . .  1036 

Shaft,  To  Compute  IP  of 983 

Shafting,  Nickel  Steel 986 

Slates,  Roofing 979 

South  Point,  To  Determine,  by  the. 

Hand  of  a  Watch 983 

Spirally  Riveted  Iron  Pipes 977 

Staff. 976 

Steam  Boilers  and  Pipes,  Insulation 

of. 1040 

STEAM-ENGINES,  Cylinder  Ratios  for 

Compound  and  Triple 1037 

Steam-Launch,  First 974 

Steam-Power,  Relative  Cost  of  per 

IP 1023 

STEAM  VESSELS.  Resistance  of..  ...  1014 

Steel  Springs,  Elements  of. 973 

STONES,  Testing  of 1036 

Stream  of  Water,  To  Compute  IP 

of. 984 


Page 

Tire  Cement 1013 

To  Determine  the  East  and  West 

Meridian ;.*.     983 

TRAIN  RESISTANCE,  a  New  Formula  1037 
Trees,  Age  of. 983 

Vegetation,  Absorption  by. 34 

WALLS,  Width  of. 1020 

Water,  Friction  and  Flow  of,  in 
Smooth  Metal  Pipes  and 
Loss  of  Head,  To  Com- 
pute    1016 

"      Friction  and   Flow  of,  in 

Cast-Iron  Pipes 1016 

*  *       Flow  of,  from  a  GivenHead, 

To  Compute 1016-1017 

*  *       Velocity  of  Flow,  Discharge, 

and  Loss  of  Head  due  to 
Friction  of  Flow,  etc.,  To 

Compute 1017 

•       Tests  for 983 

Water  Pipes  of  Cast  Iron,  Thickness 

of,  To  Compute 1015 

WIRE  ROPE,  Test  of. 1037 

Wrought  Iron,  Effect  of  Repeated 
Stress  on io4c 


NOTE. — Tons  are  given  and  computed  at  2240  Ibs. 

Degrees  of  temperature  are  given  by  the  scale  of  Fahrenheit 


EXPLANATIONS   OP   CHARACTERS   AND   SYMBOLS 

Used  in  Formulas,  Computations,  etc.,  etc. 

=  Equal  to,  signifies  equality ;  as  12  inches  =  i  foot,  or  8  x  8  =  16  X  4, 

-f  Plus,  or  More,  signifies  addition  ;  as  4  -f  6  +  5  =  15. 

—  Minus,  or  Less,  signifies  subtraction ;  as  15  —  5  =  10. 

X  Multiplied  by,  or  Into,  signifies  multiplication;  as  8  x  9=  72.  a  X  di 
a.d,or  ad,  also  signify  that  a  is  to  be  multiplied  by  d. 

-f-  Divided  by,  signifies  division ;  as  72  -f-  9  =  8. 

:  Is  to,  ::  So  is, :  To,  signifies  Proportion,  as  a  :  4  : :  8 :  16 ;  that  is,  as  2  is 
to  4,  so  is  8  to  16. 

.-.  signifies  Therefore  or  Hence,  and  v  Because. 

Vinculum,  or  Bar,  signifies  that  numbers,  etc.,  over  which  it  is 

placed,  are  to  be  taken  together ;  as  8  —  2  -f  6  =  12,  or  3  X  5  +  3  =  24- 

.  Decimal  point,  signifies,  when  prefixed  to  a  number,  that  that  number 
has  some  power  of  10  for  its  denominator ;  as  .1  is  ^,  .15  is  ^J,  etc. 

co  Difference,  signifies,  when  pkced  between  two  quantities,  that  their 
difference  is  to  be  taken,  it  being  unknown  which  is  greater. 

V  Radical  sign,  which,  prefixed  to  any  number  or  symbol,  signifies  that 
square  root  01  that  number,  etc.,  is  required ;  as  \/9,  or  Va +6.  The  degree 
of  the  root  is  indicated  by  number  placed  over  the  sign,  which  is  termed 
index  of  the  root  or  radical;  as  J',  v^etc. 

>  1 1  <  L  signify  Inequality,  or  greater,  or  less  than,  and  are  put  between 
two  quantities ;  as  a  1 6  reads  a  greater  than  b,  and  a  L  6  reads  a  less  than  6. 

( )  [  ]  Parentheses  and  Brackets  signify  that  all  figures,  etc.,  within  them 
are  to  be  operated  upon  as  if  they  were  only  one ;  thus,  (3  -h  2)  x  5  =  25 ; 
[8-2]  X5  =  3o. 

±  ip  signify  that  the  formula  is  to  be  adapted  to  two  distinct  cases,  as 
c  =p  v  =  a,  either  diminished  or  increased  by  v.  Here  there  are  expressed 
two  values :  first,  the  difference  between  c  and  v ;  second,  the  sum  of  c  and  v. 

In  this  and  like  expressions,  the  upper  symbol  takes  preference  of  the  lower. 

p  or  TT  is  used  to  express  ratio  of  circumference  of  a  circle  to  its  diameter 
=  3.1416;  ^p=.7854,and-g-/>  =  .5236. 

'"  signify  Degrees,  Minutes,  Seconds,  and  Thirds. 

'  "  set  superior  to  a  figure  or  figures,  signify,  hi  denoting  dimensions,  Feet 
and  Inches. 

a'  a"  a'"  signify  a  prime,  a  second,  a  third,  etc. 

1,  2,  added  to  or  set  inferior  to  a  symbol,  reads  sub  i  or  sub  2,  and  is  used 
to  designate  corresponding  values  of  the  same  element,  as  h,  hi,  h2,  etc. 

2,  3,  4,  added  or  set  superior  to  a  number  or  symbol,  signify  that  that  num- 
ber, etc.,  is  to  be  squared,  cubed,  etc. ;  thus,  42  means  that  4  is  to  be  multi- 
plied by  4 ;  4^,  that  it  is  to  be  cubed,  as  43  =  4  X  4  X  4  =  64.     The  pmoer, 
or  number  of  times  a  number  is  to  be  multiplied  by  itself,  is  shown  by  the 
number  added,  as  2,  3,  *,  s,  etc. 


22  ALGEBRAIC   SYMBOLS   AND   FORMULAS. 

2,  *,  etc.,  aet  superior  to  a  number,  signify  square  or  cube  root,  etc.,  of  the 

i  J?     4     4.    j>. 

number;  as  2s*  signified  square,  root  of  2}  aiso  3,  *,  3,  3,  etc.,  set  superior 
to  a  number,  signify  two  thirds  power,  etc.,  or  cube  root  of  square,  or  square 

or  cube  root  of  4th  power,  or  cube  root  of  sixth  power  ;  as  83  =  V~&  or 


I-7,  3-6,  etc.,  set  superior  to  a  number,  signify  tenth  root  of  iyth  power,  etc. 

•02^  .<ot  set  superior  to  a  number,  signify  hundredth  root  of  2d  power,  or 
thousandth  root  of  59th  power,  the  numerator  indicating  power  to  which 
quantity  is  to  be  raised,  and  denominator  indicating  root  which  is  to  be  ex- 
tracted. 

oo  signifies  Infinite,  as  -  or  a  quantity  greater  than  any  assignable  quan- 

tity.   Thus,  -  =  oo  signifies  that  o  is  contained  in  any  finite  quantity  an  in- 

°  a  a 

finite  number  of  times  :  -  —  a.—  =  ioa,  etc. 
i        x.i 

oc  signifies  Varies  as.  Thus,  M  <x  D  x  V  signifies  that  mass  of  a  body  in- 
creases or  diminishes  in  same  ratio  as  product  of  its  density  and  volume,  or 
S  oc  t2,  signifies  S  varies  as  t2. 

£.  signifies  Angle,     -i-  Perpendicular.     A  Triangle,     n  Square,  as  Q 
inches  ;  and  E3  cube,  as  cube  inches. 
NOTES.—  Degrees  of  temperature  used  are  those  of  Fahrenheit. 
g  is  common  expression  for  gravity  =  32.  166,  2^  =  64.33,  V  2  g  =  8.02  feet. 

85  signifies  Dead  Flat,  denoting  dimensions  or  greatest  amidship  section 
of  hull  of  a  vessel. 


ALGEBRAIC   SYMBOLS   AND   FORMULAS. 

I  representing  length,  h1  representing  h  prime,  v  representing  versed  sine, 

b  breadth,  c  chord,  h.  h  sub, 

d  depth,  a  area,  sin.       "          sine, 

h  height,  r  radius,  g  u          gravity. 

—j-  =  sum  of  length  and  breadth  divided  by  depth. 
-j-  =  product  of  length  and  breadth  divided  by  depth. 

—  j-  =  difference  of  length  and  breadth  divided  by  depth. 
I2  b3  =  product  of  square  of  length  and  cube  of  breadth. 
Ty-r  =  square  root  of  length  divided  by  cube  root  of  breadth. 

V7-4-6 

—  --—  =  square  root  of  sum  of  length  and  breadth  divided  by  depth. 


L^  =  cube  root  of  difference  of  h  prime  and  k  sub,  divided  by 
2g 
square  root  of  23. 

V«+(c—  r)2  =  x.  Add  square  of  difference  between  the  chord  and  ra- 
dius to  the  area,  and  extract  the  square  root  ;  the  result  will  be  equal  to  x, 

NOTE.—  It  is  frequently  advantageous  to  begin  interpretation  of  a  formula  at  its 
right  hand,  as  in  the  above  case. 


ALGEBRAIC    SYMBOLS    AND    FOKMULAS. 


*\/—  —- — i  =  z.    Divide  square  of  sum  of  x  anoy  by  square  of  y; 

subtract  unity  from  quotient ;  extract  square  root  of  result ;  multiply  it  by 
length,  and  product  will  be  equal  to  z. 

2  (sin  75°)2 

f  •        oV2-    Divide  twice  square  of  sine  of  the  angle  of  75°  by  square 
i  "T"  vslHu  75  ) 
of  sine  of  the  angle  of  75°  added  to  unity. 

?=-  \  M~W  (Vh-Vh)+  2.303  c.  log.  &V*9^b  j.  =  tt    Multiply 
20)2  *  Sv  2gk'— b  J 


__ 

S  by  the  V  of  2#,  and  this  product  by  difference  between  square  roots  of  h 
and  h  prime ;  add  this  to  2.303  times  common  logarithm  of  quotient  arising 
from  dividing  product  of  S  into  V zgh  diminished  by  b  by  product  of  S 
into  Vzgh  prime  diminished  by  6,  and  multiply  this  sum  by  the  quotient 
of  2  a  divided  by  square  of  product  of  S  into  V '2/7,  which  will  be  equal  to  t. 

2  a  +  3  cos.  98°  =  2  a  —  3  cos.  82°  =  twice  a  diminished  by  three  times 

cosine  of  82°. 

Cosine  of  any  angle  greater  than  90°  and  less  than  270°  is  always  —  or  negative, 
but  is  numerically  equal  to  cosine  of  its  supplement,  t.  e.,  remainder  after  subtract- 
ing angle  from  180° 

39.127  —  .09982  cos.  2  L  =  I.  Assuming  L  less  than  45°,  as  42°,  this  equation 
becomes  39.127  —.09982  cos.  (2  X  42°  =  84°)  =  J;  and  also,  L  greater  than  45°,  as 
50°,  it  becomes  39. 127  +  .099  82  cos.  (180°  —  2  X  50°  =  80°)  =  I. 

L  —  10°  N  =  L  -p- 10°  S,  as  a  negative  result  furnished  by  a  formula  in- 
dicates a  positive  result  in  an  opposite  direction. 

—  i — — J  ^"T2 =  v.    Minus,  the  fraction  B  minus  6,  tunes  v,  plus 

D  -\-  b 
2  times  BV,  divided  by  B  plus  ft,  is  equal  to  y. 

Sin.  ~*  x,  tan.  ~x  x,  cos.  ~~z  x,  signifies  the  arc,  the  sine,  tangent  or  cosine 
of  which  is  x.  Thus,  if  #  =  -5,  this  is  30°,  as  30°  is  the  arc,  the  sine  of 
which  is  .5. 


Raise  r  to  nth  power,  »*.  e.,  multiply  r  by  itself  and  this  result  by  r,  and  so 
on,  until  r  appears  in  result  as  a  factor,  as  many  times  as  there  are  units 
in  n.  Multiply  this  result  by  /,  dimmish  this  by  / ;  divide  remainder  by  r 
raised  to  the  nth  power,  diminished  by  r  raised  to  a  power  whose  exponent 
is  n  diminished  by  i,  and  quotient  =  or  is  value  of  S. 

fl^:I/-==r.    Divide  I  by  a  and  extract  that  root  of  the  quotient,  index 

V  a 
of  which  is  n  diminished  by  i,  and  this  root  is  =  or  value  of  r. 

Logarithm  of  a  Number  is  exponent  of  the  power  to  which  a  particulal 
constant  quantity  must  be  raised  in  order  to  produce  that  number. 
Constant  Quantity  is  termed  the  base  of  the  system. 
Common  (or  Brigg's)  Log.  is  the  logarithm  the  base  of  which  is  10. 
Hyperbolic  Log.  is  the  logarithm  the  base  of  which  is  2.71828. 
Com.  I^og.  =  Hyp.  log.  X  .434  294* 
Hyp.  Log.  =  Com.  log.  x  2.302  585  052  994,  ordinarily  2.303  or  2.3026 


24  DIFFERENTIAL   AND   INTEGRAL    CALCULUS. 

ILLUSTRATION.— When  a  number,  hyp.  log.  of  which  =  a  given  figure  or  number, 
is  required. 

Multiply  figure  or  number  (hyp.  log.)  by  .434294  (modulus  of  com.  log.)  =  com.  log. 
of  figure. 

Thus,  Required  the  number,  hyp.  log.  of  which  =  .02.  .02  X  .434  294  =  .00868  588, 
com.  log.,  and  1.0202=  number. 

Log.  xoo'°S>=  059  x  log.  of  zoo  =  .059  X  2=r.n8;  the  number  corresponding  to 
log.  .118,  is  1.3122;  hence,  ioo-°59rr  1.3122.  That  is,  if  100  is  raised  to  sgth  power, 
and  the  loooth  root  is  extracted,  the  result  will  be  1.3122. 

Differential  and.  Integral  Calculus. — In  Equation,  u=.^xe — 
2  a?,  u  is  termed  a  function  of  x.  If  it  is  desired  to  indicate  the  fact  that  u 
thus  depends  for  its  value  upon  value  of  x,  without  expressing  exact  value 
of  u  in  terms  of  #,  following  notation  is  used : 

M  =/(#),         u=.F(x'),         or  M  =:</>(£). 

Each  of  these  notations  is  read,  u  is  a  function  of  x.  If  in  such  function 
of  x  value  of  x  is  assumed  to  commence  with  o  and  to  increase  uniformly,  the 
notation  indicating  rate  of  increase  is  dx,  and  is  read  "  the  differential  of  x" 

Differentiation,  d  is  its  symbol,  and  it  is  the  process  of  ascertaining  the 
ratio  existing  between  the  rate  of  increase  or  decrease  of  a  function  of  a 
variable  and  the  rate  of  increase  or  decrease  of  the  variable  itself.  If 
y  =  3  as8,  y  or  its  equal  3 x2  is  the  function  of  a?,  and  x  is  the  independent 
variable,  while  the  exponent  of  the  variable  or  the  primitive  exponent  is  2, 

By  the  operation  of  Calculus,  such  expressions  are  differentiated  by  di- 
minishing the  exponent  of  the  variable  by  unity,  multiplying  by  the  prim- 
itive exponent,  and  attaching  the  d  x. 

Hence,  dy=zz  Xsxdx  —  bxdx.  This  indicates  the  relation  between 
the  differential  of  y,  the  function  of  #,  and  the  differential  of  x  itself. 

Assume  that  x  increasing  at  rate  of  3  per  second  becomes  4 ;  that  is,  x  =  4, 
and  d  x=.  3;  hence  dy  =  6  X  4  X  3  =  72.  That  is,  if  x  is  increasing  at 
rate  of  3  per  second,  at  the  time  that  x  •=.  4,  the  function  itself  is  increasing 
at  rate  of  72  per  second. 

To  differentiate  an  expression  of  two  or  more  terms,  it  is  necessary  to 
differentiate  them  separately  and  connect  the  results  with  the  signs  with 
which  the  terms  are  connected. 

Thus,  differentiating  u  =  3  x2  —  2  #,  we  have  d  u  =  d  (3  x2  —  2  x)  =  6  x  d  x 

—  2dx=(6x  —  z)  dx. 

Assuming  x  =  4  and  d  x= 3,  we  have  c?  M  =  (6  x  4  —  2)  X  3  =  66.  This 
indicates  that  when  x  =  4,  and  is  increasing  at  rate  of  3  per  second,  the  func- 
tion M,  or  3  x2  —  2  a:,  is  at  same  instant  increasing  at  rate  of  66  per  second. 

Integration.  Its  symbol  /  was  originally  letter  S,  initial  of  sum,  the 
symbol  of  an  operation  the  reverse  of  differentiation ;  and  when  the  oper- 
ation of  integration  is  to  be  performed  twice,  thrice,  or  more  times,  it  is 
written  //,  ///,etc. 

By  the  operation  of  Calculus,  expressions  are  integrated  by  increasing  the 
exponent  of  the  variable  by  unity,  dividing  by  the  new  exponent,  and  de» 
taching  the  dx. 

Hence,  integrating  the  differential  6 x  d  a*,  we  have  /*  6  x  dx  =  3  or2.  This 
result  is  the  function,  the  differential  of  which  is  6xd'x. 

To  integrate  an  expression  of  two  or  more  terms,  it  is  necessary  to  inte- 
grate the  terms  separately  and  connect  the  results  with  the  signs  with  which 
the  terms  are  connected. 

Thus, integrating  (6 #—2)  d#,  we  have  f  (6x— 2)  dx  =  f(6xdx—zdx) 
=  3  a;2  —  2x.  This  result  is  the  function  the  differential  of  which  is  (6x 

—  2)  dx  or  (6x  —  2  a?°)  d  x. 

NOTE.—  A  quantity  with  the  exponent  °,  as  «°  or  3°,  is  equal  to  unity. 


NOTATION.  25 

The  operation  of  summation  may  also  be  illustrated  in  use  of  the  sym- 
bol / .  Assuming  x  =  4,  the  former  of  the  preceding  results  becomes 
/  6  x  d  x  =  3  x2  =.  48,  the  latter  /  (6  x  —  2)  d  x  =  3  x2  —  2  x  =  40. 

Here  x  is  assumed  to  commence  at  o  and  to  continue  to  increase  by  in- 
finitely small  increments  of  dx  until  it  becomes  4.  The  summation  is  the 
addition  of  all  these  values  of  x  from  o  to  4. 

Arithmetically. — The  first  formula  may  be  written 

6  (aj'  -f-  x"  -\-  x'"  -+-  etc.)  d  x.  If  then  x  is  to  advance  from  o  to  4  by  in- 
crements of  i,  we  have  6  (o +1  +  2  +  3-!-  4)  x  1=60,  which  exceeds  48. 
If  da;  is  assumed  to  be  .5,  the  result  is  54.  The  correct  result  is  obtained 
only  when  d  x  is  taken  infinitely  small.  By  Arithmetic  this  is  approximated, 
but  it  is  reached  by  the  operations  of  Calculus  alone. 

The  second  formula  may  be  written 

(6  [x'  +  x"  +  x"'  +  etc.]  —  2  O0'  +  x°"  +  x°"'  etc.] )  d  x.  Assuming  x  = 
4,  and  d  x  =  i,  we  have  (6  [i  +  2 +  3  +  4]  -2  [i  +  i  +  i  +  i])  x  i  =  52, 
which  exceeds  40.  If  dx=  .25,  the  result  would  be  43,  and  if  .125  it  would 
be  41.5,  ever  approaching  but  never  reaching  40,  so  long  as  a  finite  value  is 
assigned  to  dx. 

A,  Delta,  when  put  before  a  quantity,  signifies  an  absolute  and  finite  in- 
crement of  that  quantity,  and  not  simply  the  rate  of  increase. 

2,  Sigma,  signifies  the  summation  of  finite  differences  or  quantities.  Thus, 
Zy2  A#=(y'2  +  y"2+y"2  +  etc.)  A  x.  Assume  y  =  6,  y"  =  8,  y  "  =  4,  and 
A  x  the  common  increment  of  x  =  5,  then  2y2  A  x  =  (36  +  64+16)  x  5  = 
580. 


NOTATION. 

1  =  I.  2o  =  XX.  iooo  =  M,orCIO. 

2  =  11.  3o=XXX.  2ooo  =  MM. 

3  =  III.  40  =  XL.  5000  =  ^01100. 

4  =  IV.  so  =  L.  6ooo  =  VI. 

5  =  V.  6o=LX.  ioooo  =  X,orOCIOa 

6  =  VI.  70  =  LXX.  50  ooo  =  I,  or  1000. 

7  =  VII.  8o  =  LXXX.  6oooo  =  LX.  . 

8  =  VIII.  9o  =  XC.  iooooo  =  C,orCCCIOOO. 

9  =  IX.  100  =  C.  i  ooo  ooo  =  M,or  CCCCIOOOD. 
10  =  X.  500  =  D,  or  10.  2  ooo  ooo  =  MM. 

As  often  as  a  character  is  repeated,  so  many  times  is  its  value  repeated, 
its  CC  =  200. 

A  less  character  before  a  greater  diminishes  its  value,  as  IV  =  V  —  I. 
A  less  character  after  a  greater  increases  its  value,  as  XI  =  X  +  I. 
For  every  0  annexed  to  10  the  sum  as  500  is  increased  10  times. 

If  C  is  placed  on  left  side  of  I  as  many  times  as  0  is  on  the  right,  tht 
number  is  doubled. 

A  bar,  thus  ~~,  over  any  number,  increases  it  1000  times. 
Illustration  i.~i88o,  MDCCCLXXX.    18  560,  XVlUDLX. 
2.  — - 10  =  500.      CIO  =  500  x  2  =  looo.      100  =  500  x  10  =  5000. 
CCIOO  =  5000  x  2  =  10  ooo.    1000  =  500  x  10  x  10  =  50  ooo.    CCCIOOO 

=  SO  OOO  X  2  =  100  000. 


26     CHRONOLOGICAL    ERAS. MEASURES    AND    WEIGHTS. 

CHRONOLOGICAL  ERAS   AND    CYCLES  FOR  1906. 

The  year  1906,  or  the  1300/1  year  of  the  Independence  of  the  United  States  of  America, 
corresponds  to 

The  year  7414-15  of  the  Byzantine  Era; 

u       6619  of  the  Julian  Period; 

"        5666-67  of  the  Jewish  Era; 

u  2071  of  the  Olympiads,  or  the  second  year  of  the  67181  Olympiad,  com- 
mencing in  July  (1892),  the  era  of  the  Olympiads  being  placed  at 
775-5  years  before  Christ,  or  near  the  beginning  of  July  of  the 
3938th  year  of  the  Julian  Period; 

"        2659  since  the  foundation  of  Rome,  according  to  Varro; 

44        2218  of  the  Grecian  Era,  or  the  Era  of  the  Seleucidse; 

44        1622  of  the  Era  of  Diocletian. 

The  year  1323-24  of  the  Mohammedan  Era,  or  the  Era  of  the  Hegira,  begins  on 
the  26th  of  July,  1906. 

The  first  day  of  January  of  the  year  1906  is  the  2,412,115^  day  since  the  com- 
mencement of  the  Julian  Period. 

Dominical  Letter G  I  Lunar  Cycle  or  Golden  Number 7 

Epact 5  I  Solar  Cycle n 

Roman  Indiction  3.  was  a  period  of  15  years,  in  use  by  the  Romans.  The  precise 
time  of  its  adoption  is  not  known  beyond  the  fact  that  the  year  313  A.D.  was  a  first 
year  of  a  Cycle  of  Indiction. 

Julian  Period  is  a  cycle  of  7980  years,  product  of  the  Lunar  and  Solar  Cycles  and 
the  Indiction,  and  it  commences  at  4714  years  B.C. 

6513  +  (given  year  — 1800)  =  year  of  Julian  Period,  extending  to  3267. 


MEASURES    OF   LENGTH. 

Standard  of  measure  is  a  brass  scale  82  inches  in  length,  and  the 
yard  is  measured  between  the  27th  and  63d  inches  of  it,  which,  at  tem- 
perature of  62°,  is  standard  yard. 


Lineal. 


12    inches     =  i  foot. 

3    feet        =  i  yard. 

5.5  yards      =  i  rod. 
40    rods        =  i  furlong. 

8    furlongs  =  i  mile. 


Inches.  Feet.  Yarda.        Rods.    Furl. 

36=        3- 
198=      16.5=        5.5. 

7920=      660      =      220     =     40. 
63360=5280      =1760     =320  =  8. 


Inch  is  sometimes  divided  into  3  barleycorns,  or  12  lines. 
A  hair's  breadth  is  .02083  (48th  part)  of  an  inch, 
i  yard  =  .ooo  568,  and  i  inch  =  .00001 5  8  of  a  mile. 

Grianter's    Chain. 

7.92  inches  =  i  link.      |      100  links  =  i  chain,  4  rods,  or  22  yards. 
80  chains  =  i  mile. 

Ropes   and   Cables, 
x  fathom  =  6  feet.  j      i  cable's  length  =  120  fathoms. 

Greographical   and  ]N~au.tical. 

I  degree,  assuming  the  Equatorial  radius  at  6967  459.893  yards  (3958.784 
miles),  as  given  by  U.  S.  Coast  Survey,  =  69.094  Statute  miles, 
i  mile  =  2026. 7566  yards  or  6080.27  feet, 
i  league  =  3  Nautical  miles. 


MEASURES    AND   WEIGHTS.  2J 

Log  Lines. 

Estimating  a  mile  at  6080.27  feet,  and  using  a  30"  glass, 

i  knot = 50  feet  8.03  inches.        |          i  fathom  =  5  feet  .08  inch. 
If  a  28"  glass  is  used,  and  8  divisions,  then 

i  knot  =  47  feet  5  inches.  |       i  fathom  =  5  feet  1 1.25  inches. 

The  line  should  be  about  150  fathoms  long,  having  10  fathoms  between  chip  and 
first  knot  for  stray  line. 

NOTE.— This  estimate  of  a  mile  or  knot  is  that  of  U.  S.  Coast  Survey,  assuming 
Equatorial  radius  of  Earth  to  be  6967459.893  yards  and  a  Meter  to  be  39.370433 
inches  of  the  Troughton  scale  at  62°. 

Cloth. 
i  nail  =  2.25  inches.    |    i  quarter  =  4  nails.    |    5  quarters  =  i  ell. 

3?endiilnm. 

6  points  =  i  line.        |        12  lines  =  i  inch. 

Shoemakers'. 

No.  i  is  4.125  inches,  and  every  succeeding  number  is  .333  of  an  inch. 
There  are  28  numbers  or  divisions,  in  two  series  or  numbers— viz.,  from  i 
to  13,  and  i  to  15. 

Miscellaneous. 

12  lines  or  72  points  =  i  inch.  i  hand  =  4  inches, 

i  palm  =  3  inches.  i  span  =  9  niches, 

i  cubit  =  1 8  inches. 

"Vernier   Scale. 

Vernier  Scale  is  ^  divided  into  10  equal  parts ;  so  that  it  divides  a  scale 
of  loths  into  looths  when  two  lines  of  the  two  scales  meet. 


Metric,  "by  -A-ct   of  Congress   of  July  S8,  1866. 

Unit  of  Measurement  i$  the  METER,  which  by  this  Act  is  declared  to  be  39. 37  ins. 


Denominations. 

Meters. 

Inches. 

Feet. 

Yards. 

Miles. 

Millimeter  

.OOI 

.0394 

— 

.OI 

•3937 

Decimeter      

.  i 

7.  037 

.728083 

Meter     

I. 

39.37 

7.28083 

1.09361 

IO. 

393-7 

32.80833 

10.936  ii 



Hektamet^  

IOO. 

328.083  33 

109.361  ii 

I  OOO. 

3280.833  33 

1093.611  ii 

.621  77 

Mvriameter  .  .  . 

10  OCX). 



6.2177 

In  METRIC  system,  values  of  the  base  of  each  measure— viz. ,  Meter,  Liter,  Stere, 
Are,  and  Gramme— are  decreased  or  increased  by  following  prefix.    Thus, 


Milli,  loooth  part  or  .001. 
Centi,  looth       "      .ox. 


Deci,  loth  part  or  .x. 

Deka,  10  times  value. 

Myria,  10000  times  value. 


Hekto,  loo  times  value. 
Kilo,    looo          " 


NOTE.— The  Meter,  as  adopted  by  England,  France,  Belgium,  Prussia,  and  Russia, 
is  that  determined  by  Capt.  A.  R.  Clarke,  R.E.,  F.R.S.,  1866,  which  at  32°  in  terms 
of  Imperial  standard  at  62°  F.  is  39.370432  inches  or  1.09362311  yards,  its  legal 
equivalent  by  Metric  Act  of  1864  being  39.3708  inches,  the  same  as  adopted  in 
France. 

Captain  Rater's  comparison,  and  the  one  formerly  adopted  by  the  U.  S.  Ordnance 
Corps,  was  =  39. 370  797  i  inches,  or  3. 280  899  76  feet,  and  the  one  adopted  by  the 
U.  S.  Coast  Survey,  as  above  noted,  is  =  39.370  432  35  inches. 


28 


MEASURES    AND    WEIGHTS. 


Denominations. 

Value  in  Meters. 

Denominations. 

Values  in  Meters. 

Inch       •  ....... 

.025  4 

Rod  

5  029  2OQ  Q 

Foot  

.304  800  6 

Furlong  

201.  168  396 

Yard..., 

.014401  8 

Mile... 

1600.  ^47  168 

Approximate   Eqxii.valen.ts    of  Old.    and,    Metric    U.  S. 
.Measures   of  .Length. 


i  Chain =  20  meters. 

i  Furlong  . . .  =  200      " 

5  Furlongs  . . .  =     i  kilometer. 


i  Kilometer . . . .  =  .625  mile. 

i  Mile =  1.6  kilometers. 

i  Pole  or  Perch .  =  5  meters. 

i  Foot =3  decimeters  or  30  centimeters. 

i  Metre =3.280833^/66^  =  3^66^  3  ins.  and  3  eighths. 

ii  Meters =12  yards.     \     i  Decimeter ...  =4  inches. 

i  Millimeter  . .  =  i  thirty-second  of  an  inch. 

To  Convert  Meters  into  Inches. — Multiply  by  40;  and  to  Convert  Inches 
into  Meters. — Divide  by  40. 

Approximate  rule  for  Converting  Meters  or  parts,  into  Yards. — Add  one 
eleventh  or  .0909. 

Inches  Decimally  =  Millimeters. 


Inches. 

Milli- 
meters. 

Inches. 

Milli- 
meters. 

Inches. 

Milli- 
meters. 

Inches. 

Milli- 
meters. 

Inches. 

Milli- 
meters. 

.01 

•25 

.2 

5-08 

.48 

12.2 

.76 

19-3 

2 

50.8 

.02 

•51 

.22 

5-59 

•5 

12.7 

.78 

19.8 

3 

76.2 

•03 

.76 

.24 

6.1 

•52 

13.2 

.8 

20.3 

4 

101.6 

.04 

1.02 

.26 

6.6 

13-7 

.82 

20.8 

5 

127 

1.27 

.28 

7.11 

•  56 

14.2 

.84 

21.3 

6 

152.4 

.07 

1.52 
I.78 

•3 
•32 

7.62 
8.13 

:i8 

14.7 
15-2 

.86 

.88 

21.8 

22.4 

1 

177.8 

.08 

2.03 

8.64 

.62 

15-7 

•9 

22.9 

9 

228.6 

.09 

2.29 

•36 

9.14 

.64 

16.3 

.92 

23-4 

10 

254 

.1 

2-54 

.38 

9-65 

.66 

16.8 

•94 

23-9 

ii 

279-4 

.12 

3-05 

•4 

10.2 

.68 

17-3 

.96 

24.4 

12 

304-8 

.14 

3-66 

.42 

10.7 

•7 

17.8 

.98 

24.9 

=  i  foot 

!i8 

4.06 
4-57 

•$ 

II.  2 
II-7 

.72 
•74 

18.3 
18.8 

l' 

25-4 

Inches  in  Fractions  =  Millimeters. 


•79 
1-59 
2.38 
3-i7 
3-97 
4.76 
5-56 
6-35 


If 

H 

if 

be 

3 

(  •• 

33  g 

it 

J,  « 

1 

w 

9 

7.14 

17 

13-5 

5 

— 

7-94 

9 

— 

14-3 

ii 

8-73 

'9 

I5-1 

— 

— 

9-52 

5 

— 

— 

iS-9 

7 

13 

10.32 

21 

16.7 

7 

— 

ii.  ii 

ii 

— 

17.5 

15 

11.91 

23 

18.3 

— 

— 

12.7 

6 

— 

19 

8 

4      —     — 


29 


19.8 

20. 6 
21.4 

22.2 

23  8 
23.0 

24.6 
25-4 


By  means  of  preceding  tables  equivalent  values  of  inches  and  millimeters, 
equivalent  values  of  inches  in  centimeters,  decimeters,  and  meters,  may  be 
ascertained  by  altering  position  of  decimal  point. 

ILLUSTRATION. — Take  i  millimeter,  and  remove  decimal  point  successively  by  one 
figure  to  the  right;  the  values  of  a  centimeter,  decimeter,  and  meter  become 

i  millimeter 0394  I  i  decimeter 3-94|    .32  inch     =  8. 13  millimetera 

i  centimeter 394    (imeter... 39.4    (3.2 


MEASURES   AND   WEIGHTS. 


29 


MEASURES   OF   SURFACE. 

144  square  inches  =  i  square  foot.    |    9  square  feet  =  i  square  yard. 
Architect's  Measure,  100  square  feet  =  i  square. 


Land. 


30.25  square  yards 

=  i  square  rod. 

Yards. 

40 

square  rods 

=  i  square  rood. 

I2IO. 

4 

10 

square  roods 
square  chains 

>  =  i  acre. 

4840: 

640 

acres 

=  i  square  mile. 

3097600: 

Rood*. 


43  560  square  feet,  or  208.710326  feet  square,  or  220  x  198  feet  =  t  Acre. 

IPaper. 

84  sheets  =  i  quire.  |  20  quires  =  i  ream,  j  21.5  quires  =  i  printer's  ream. 
2  reams  =  i  bundle.  |  5  bundles  =  i  bale. 

Dra\ving. 

Columbier 23  x 

Atlas 26  x 

Theorem 28  x 

Doub.  Elephant.  27  X    40 
Antiquarian  ...  31  x    53 

Emperor 40  x    60 

Uncle  Sam 48  x  120      " 

Peerless 18  X    52      " 

Tracing. 

Grand  Royal 18  X  24  inches. 

Grand  Aigle 27  x  40      " 

Vellum  Writing,  18  to  28  ins.  in  width. 


Cap 13  X  17  inches. 

Universal 14  X  17      u 

Demy 15  X  20 

Medium 17  X  22, 

Royal 19  X  24 

Super-royal  ....  19  X  27 

Imperial 22  X  30 

Elephant 23  X  28 


34  inches. 

34 

34 


Double  Crown 20  x  30  inches. 

Double  D.  Crown  . .  30  X  40 
Double  D.  D.  Crown,  40  X  60 


Mounted  on  cloth,  38  ins.  in  width. 
^Miscellaneous. 


i  sheet    =  4  pages. 
I  quarto  =   8 
i  octavo  =  16     " 


i  duodecimo  =  24  pages, 
i  eighteenmo  =  36     " 
i  bundle         =  2  reams. 


i  piece  wall-paper,  20  ins.  by  12  yards, 
i     "        u        "     French,  4.5  sq.  yards. 
Roll  of  Parchment  =  60  sheets. 

Copying. 

too  Words  =  i  Folio. 

Metric,  by  A.ct   of  Congress   of  July   28,  1866. 

Unit  of  Surface  is  Are  or  Square  Dekameter. 

A  square  meter  (39.372)  =  1549.9969  sq.  ins.,  but  by  this  Act  is  declared  to  be 
1550  sq.  ins. 


Denominations. 

Sq.  Meters. 

Sq.  Inches. 

Sq.  Feet, 

Sq.  Yards. 

Acres. 

Centimeter  

.0001 

•  '55 

Decimeter 

I  £    CO 

107  638 

Centare  or        ) 
Square  Meter)  **"* 
Are  

I. 

too 

i55o. 

10.763888 
1076  388  88 

1.196 

- 

Hectare... 

10  000. 



11060. 

2.A7I 

MEASURES   AND   WEIGHTS. 


Equlval 

Denominations. 

ent  "Value 

Sq.  Meters. 

s  in  Metr 

Denominations. 

LC  Denom 
Sq.  Meters. 

illations 
Sq.  Hectares. 

of  XJ.  S. 

Sq.  Ares. 

Sq.  Inch  
"  Foot  
"  Yard  
"  Rod... 

.000645  16 
.09290323 
.83612907 

25.2Q2Q04 

Sq.  Chain  .  .  . 
"  Rood  
"  Acre  .... 
"  Mile... 

404.68647 
1011.716  175 
4046.864699 

.404686 
258.09934 

4.046865 
10.117  J62 
40.468647 
25899.934074 

Approximate    Equivalents    of  Old.   and.   Metric    U.  S. 
Square    M.easu.res. 

6. 5  square  centimeters  =  i  sq.  inch.     I    i  acre  —  1. 16  per  cent,  over  4000  sq.  meters 
i         "      meter  =  io.7ssq.feet.  \    i  square  mile  =  259  hectares. 


MEASURES   OF   VOLUME. 

Standard  gallon  measures  231  cube  ins.,  and  contains  8.3388822 
avoirdupois  pounds,  or  58  373  Troy  grains  of  distilled  water,  at  temper- 
ature of  its  maximum  density  (39.1°),  barometer  at  30  ins. 

Standard  bushel  is  the  WincJiester,  which  contains  2150.42  cube  ins., 
or  77.627  413  Ibs.  avoirdupois  of  distilled  water  at  its  maximum  density. 

Its  dimensions  are  18.5  ins.  diameter  inside,  19.5  ins.  outside,  and  8 
ins.  deep ;  and  when  heaped,  the  cone  must  not  be  less  than  6  ins.  high, 
equal  2747.715  cube  ins.  for  a  true  cone. 

A  struck  bushel  contains  1.24445  cube  feet. 


Liquid. 


Cube  Ins. 

28.875 

Gills.  Pints. 

57-75 

8. 

231. 

32  =  8. 

Dry. 

Cube  Ins. 

67.2006 

Pints.  Quarts.  Galls. 

268.8025 

8. 

537.605 

16  =   8. 

2150.42 

64  =  32  =  8. 

4  gills     =  i  pint. 
2  pints    =  i  quart. 
4  quarts  =  i  gallon. 


2  pints     =  i  quart. 
4  quarts  =  i  gallon. 
2  gallons  =  i  peck. 
4  pecks    =  i  bushel. 

Cube. 

1728  cube  inches  =  i  foot.  inches. 

27  cube  feet     =  i  yard.        I  46656 

NOTE. — A  cube  foot  contains  2200  cylindrical  inches,  or  3300  spherical  inches. 


IFTuid. 


60  minims  =  i  dram. 

8  drams   =  i  ounce. 
16  ounces  =  i  pint. 

8  pints     =  i  gallon. 


Minims.        Drams.      Ounces. 

480. 
7680=128. 

61  240  =  1024  =  128. 


!N~au.tical. 


i  ton  displacement  in  salt  water =35  cube  feet. 

i    "  registered  internal  capacity =40    "      " 

Dimensions   of  a   Barrel. 

Diameter  of  head,  17  ins. ;  bung,  19  ins. ;  length,  28  ins. ;  volume,  7689  cube  ina 
=  3-  5756  bushela 


MEASURES    AND    WEIGHTS. 

Miscellaneous. 

cube  foot 7.480  5  gallons. 

bushel 9-309 18  gallons. 

chaldron  =  36  bushels,  or 57.244  cube  feet. 

cord  of  wood 128  cube  feet. 

perch  of  stone 24.75  cube  feet. 


i  quarter  =  8  bushels. 

Galls. 

i  Barrel 32 

i  Tierce 42 

Butt  of  Sherry 35X50 108 

Pipe  of  Port 34X58....  115 

Pipe  of  Teneriffe 100 

Butt  of  Malaga 33X53 


i  load  hay  or  straw  =  36  trusses. 

Galls. 

Puncheon  of  Scotch  Whisky. .  no  to  130 
Puncheon  of  Brandy  34X52.  .no  to  120 

Puncheon  of  Rum 100  to  no 

Hogshead  of  Brandy  28X40. .  55  to   60 

Pipe  of  Madeira 92 

Hogshead  of  Claret 46 


A  Hogshead  is  one  half,  a  Quarter  cask  is  one  fourth,  and  an  Octave  is  one  eighth 
of  a  Pipe,  Butt,  or  Puncheon. 

Metric,  toy  Act  of  Congress  of  Jnly  28,  1866. 

Unit  or  Base  of  Measurement  is  a  cube  Decimeter  or  Liter,  which  is  declared  to  be 
61.022  cube  ins. 


Cu."be   M!ea 

Denominations.                            Values. 

snres. 

Cube  Inches.          Cube  Feet.       '  Cube  Yards. 

Cube  Centimete 
"     Decimetei 
"    Meter  .  .  . 

Denominations. 

r              .001  cube  re 

illiliter 

.o6l  022 

61.022             -035313657 

—        35-313657 

mres. 

QuarU.      Pecks.        Bushels. 

1.308 
Cube  Yards. 

*  i  cube  liter  . 

Kiloliter  or 
Dry 

Values. 

stere.. 
Meas 

Cube  Ins. 

Milliliter  

i  cube  centimeter. 

10                             " 

.1       decimeter.. 

.061 
.6102 

6.  1022 
6l.022 

.908* 
9.08 

•"35 
I-I35 
"•35 

•28375 
2-8375t 

28.375 

.l7o8 
1.308 

Centiliter  
Deciliter  
Liter  

Dekaliter  
Hektoliter.... 
Kiloliter  I 
or  Stere  j  ••« 

10                            " 

.1       meter  
i     "       "    

*  Or  .227  gallon.  \  3.531 365  7  cube  fed. 

NOTE.  —  In  practice,  term  cube  Centimeter,  abbreviated  to  cc,  is  used  instead  of 
Milliliter,  and  cube  Meter  instead  of  Kilometer. 

Equivalent   "Values    in.    Metric   Denominations   of  U.  S. 
Dry   Measures. 


Denominations. 

Centiliters. 

Deciliters. 

Liters. 

Inch  

Pint  

Quart  

no  is? 

Gallon  

** 

•> 

Peck  

0881 

88s 

8  81 

Bushel... 

.112A. 

1.  Z2A. 

1C.2A 

11.0125 

44-05 


Liquid.    Measures. 


Denominations. 

Liters. 

Drams. 

Ounces. 

Pints. 

Quarts. 

Gallon*. 

MPliliter  

27 

Centiliter  

.01 

2.  7 

o 

Deciliter  

27 

oR 

Liter  

_ 

00    8 

f- 

_/r       t 

Dekaliter  

10 

.204  I" 

Hektoliter........ 

IOO 

2.641  7 

Kiloliter    ) 
or  Stere  }  

IOOO 

— 

— 

— 

— 

264.17 

3  2  MEASURES   AND   WEIGHTS. 

Approximate    Equivalents    of  Old    and    Metric   TJ.  S. 
Pleasures    of  Volume. 

i  Gallon =  4-5  liters.       I      i  cube  meter =  1.33  cube  yards 

i  Liter =  .26  gallon.         i     "     yard =  .75    "    meter. 

i  cube  foot =  28. 3  liters.       \     i    "    kiloliter  ==  2240  Ibs.  nearly  of  water. 


MEASURES    OF   WEIGHT. 

Standard  avoirdupois  pound  is  weight  of  27.7015  cube  inches  of  dis< 
tilled  water  weighed  in  air,  at  (39.83°)  barometer  at  30  inches. 
A.  cube  inch  of  such  water  weighs  252.6937  grains. 


16  drams  =  i  ounce. 
16  ounces  =  i  pound. 
112  pounds  =  i  cwt. 
20  cwt.   =  i  ton. 


Ounces.        Pounda. 


Drams. 

256. 
28  672  =     I  792. 

573  440  =  35  840  =  2240. 
i  pound  =  14  oz.  ii  dwts.  i6grs.  Troy,  or  7000 grains. 
i  ounce  =  18  diets.  5.5  grains  Troy,  or  437.5  grains. 
i  dram  =  i  dwt.  3.343  75  grains  Troy,  or.  53.5  grains. 
i  stone   =  14  pounds. 


Dwt. 


20  grams 

3  scruples 

8  drams 

12  ounces 

45  drops 


Troy. 

Grains. 
480. 
5760  =  240. 

=     i  lb.  avoirdupois. 

=        IOZ. 

ins    =      i  dram      " 

=  144  Ibs. 

"     ounces  =  192  oz. 

"     ounce  =  480  grs.         " 

"     pound  =       .822  857  lb. 

avoirdupois  pound  =     1.215  27&  Ibs.  Troy. 

-A.poth.ecaries. 

=  i  scruple. 

=  i  dram. 

=  i  ounce.  480  =  24. 

5  760  =1288  =  96, 


24  grains  =  i  dwt. 
20  dwt.     =  i  ounce. 
12  ounces  =  i  pound. 
7000     Troy  grains 
437-5     "        " 
27-343  75  Troy  grai 
175     Troy  pounds 

175 

i 
i 
i 


Grains.  Scruples.  Dram. 
60. 


=  i  pound. 

=  i  teaspoonf  ul  or  a  fluid  dram. 
2  tablespoonfuls  =  i  ounce. 
The  pound,  ounce,  and  grain  are  the  same  as  in  Troy  weight 


Diamond. 


i  grain  =  16  parts. 
16  parts  =    .8   i  roy  grain. 


4  grains  =  3.2  Troy  grains, 
i  carat   =  4  grains. 


150  carats  =  i  Troy  ounce. 

Lead.. 

A  Fodder  of  lead  =  8  pigs. 

Sheet  lead  rolls  =  6.5  to  7.5  feet  in  width  and  from  30  to  35  feet  in  length, 

G-rain.. 

Standard    Weights   per    Bushel. 
Lbi.  I  Lbs.         I  Lbs.  I  Lbs.  I  Lbt. 

Wheat....  60  I  Corn....  56  and  58  I  Rye 56  I  Oats 32   I  Barley 48 


MEASURES    AND    WEIGHTS. 

Miscellaneous. 
jper    Cn"be    Foot    in    Bvilk    and    per    Ton. 

For  additional,  see  page  217. 


33 


MATERIALS. 

Per  Cube  Foot. 
In  Lbs. 

Cube  Feet. 
In  Tons. 

50  to  55 

44  "  50 

50-3 

42.2 

50 
52 

46.6 
18.5 

18 

80 

23  to  28 

97 
109 
80 
107 

21 
26 

41  to  45 
45  "  5i 
44-5 
53-8 
45 
43 
43 
48 

I2I.8 

124.4 

28 
80  to  97.4 

23 
20.5 
28 

21 

107 

86 

u    Cannel             

"    Welsh                           

"       flne                        , 

"      Southern.  .  .                              

NOTE.— These  weights  are  commercial,  not  computed  from  the  specific  gravity  of  the  material. 

Metric,  toy  Act  of  Congress  of  July  38,  1866. 

Unit  of  Weight  is  the  GRAM,  which  is  weight  of  one  cube  centimeter  of  pure  water 
weighed  in  vacuo  at  temperature  of  4°  C.,  or  39.2°  F.,  which  is  about  its  tem- 
perature of  maximum  density  =  15.432  grains. 


Denominations. 

Values. 

Grains. 

Ounces. 

Lba. 

Ton. 

Milligram 

i     cube  i 

[0           " 
.1      "      C 

i       " 

10           " 

i  deciliter 
i  liter.  .  . 

10      "... 

nillimeter 

14 

entimeter 

u 

•15432 
1-5432 
I5-432 

-03527 
•3527 
3-527 
35-27 

.22046 
2.2046 
22.046 

220.  46 
2204.6 

in. 
tions    of 

Grama.        K 

.098419 
.984  196 

TJ.  S. 

ilograms. 

Centigram    

Gram              .       .  . 

Pekagram  

Hektogram 

Kilogram  or  Kilo  .  . 



Quintal 

z  hektolit 
i  cube  me 
17  Ibs.  Troi 

ralues   i 

Grams. 

er.  

Millier  orTonneau. 
Kilogram  =  2.  679 

Equivalent  ~\> 

Denominations. 

>ter  —             — 
r,  or  2  Ibs.  8  oz.  3  dwts.  .  3072  gra 

n   Metric   Denomina 

Dekagrams.  II      Denominations. 

Grain  

.0648 
1.296 
1-5552 

J'SS  87 

3.  Boo 

S& 

28.3502 
31.104. 
453.6028 
373.2504 

.02835 
.03I  i 
•4536 
•37325 
IOI6.0S728 

Scruple      . 

"     Troy  
Pound  

Drachm        .          . 

„  "     Troy  
Ton  .  . 

"     (Apoth.)... 

Approximate    Eq.uivaleiats    of  Old    and    New    TJ.    S. 
IMeasxires    of  "Weight. 

The  ton  and  the  gram  are  at  nearly  equal  distances  above  and  below  the 
kilogram.    Thus, 

i  ton  . . . .  =  1 016057.28  grama.    |    i  kilogram =  1000  grams. 

i  gram  is  nearly  15.5  grains  (about  .5  per  cent.  less). 

i  kilogram  about  2*2  pounds  avoirdupois  (about  .25  per  cent.  more). 

looo  kilograms,  or  a  metric  ton,  nearly  i  Engl.  ton  (about  1.5  per  cent.  less). 


34 


MEASURES    AND    WEIGHTS. 


Absorption,    toy   Vegetation. 

Daily  Consumption  of  Water  by    Vegetation. 


Crop. 

Inches  o 

f  Water. 
Minimum. 

Crop. 

Inches  o 
Maximum. 

r  Water. 
Minimum, 

.14 
.11 

.02 
•134 
.122 

Oats  

•193 
•055 

.038 
.091 
.035 

Risler.) 

Corn,  Indian  
Fir-tree 

i-57 
.043 
.267 
.287 

Potato 

Rye  .   . 

Lucern  grass  
Meadow  grass  

Vineyard  

.031 
.11 
(M.  E 

Wheat 

Composition    of   Common    Food    Substances. 


Albu- 
min. 

Fat. 

Carbo- 
hydrate. 

Albu- 
u.in. 

Fat. 

Carbo- 
hydrate. 

Asparagus.  .  . 
Beans 

Per  cent. 
2 
IQ  "> 

Per  cent. 
•3 

Per  cent. 

2-5 
52 

Oysters  
Peas 

Per  cent. 
4-95 

Per  cent. 
•37 

Per  cent. 

Beer  

.  tj 

52C 

.  0 

Potatoes 

Butter 

Oatmeal 

c  oft 

fift  it 

Buttermilk.  . 
Cheese  

3 
33 

i-3 

91  • 

3 

ej 

Barleymeal.. 
Poultry 

12.5 
8.31 

.81 

75-19 

Egg   

12  ^ 

Milk  cows' 

Game  

23 

Wheat  bread 

5 

Kumyss  

3 

i-3 

3 

Rye        do. 

4-5 

46 

(Kcenig,  Munk,  &  Ujflemann.) 

"Weights    of  Q-rain    and.    Roots. 

Following  weights  have  been  fixed  by  statute  in  many  of  the  States ;  and 
these  weights  govern  in  buying  and  selling,  unless  a  specific  agreement  to 
the  contrary  has  been  made. 

founds   in    a    Bxasliel. 


ARTICLES. 

California. 

^ 
P 

CJ 

Delaware. 

1 

1 

•5 

c 

* 

Kentucky. 

1 
I 

j 
% 

j 

S 

Michigan. 

Minnesota. 

| 
1 

N.  Hampshire. 

1 
| 
Y< 

^ 

1 
1 

1 

1 

0 

4 

~. 

! 

•d 

1 

H 

+*    t* 

C      q 

1     1 
>     * 

Barley  

50 
40 

45 

- 

& 
14 

4* 
46 

60 

f 
60 

H 

5° 
46 
60 

fi 

H 

52 

t 
DO 

& 

i4 

52 

60 

32 

- 

46 
46 

48 
42 

48 
42 

f 
60 

14 

I* 

- 

48 
So 

t 

48 

48 

46 

42 

47 

4^ 

- 

46  45 
46  42 

Beans 

Blue  Grass  Seed. 
Buckwheat  
Castor  Beans  
Clover  Seed  
Dried  Apples  — 
Dried  Peaches  .  . 

- 

~ 

- 

60 

0° 

60 

oO 

60 

- 

64 

60 

60 

60 

nO 

- 

- 

—  60 
a8 

52 

56 

- 

33 
56 
44 
52 

S 

33 

56 
44 
56 
68 
50 

S^ 

56 

68 

56~ 
44 
56 

- 

~ 

- 

28 

28 

33 
56 

— 

55 

55 

56 

28 

_ 

- 

—  28 

Hemp  Seed  
Corn          

56 

56 

~ 

56 

56 

56 

52 

- 

56 

58 

56 

56 

56 

- 

5656 

Corn  in  ear  
Corn  Meal  
Coal 

So 

- 

5° 

50 

- 

~ 

°n 

- 

- 

- 

- 

- 

- 

5° 



Oats  

32 

28 

— 

32 
57 

% 

35 
57 

33* 
57 

32 

30 

30 
52 

32 

32 

35 
57 

30 

30 

32 

6n 

32 

34 

32 

5° 

32  36 
—  5° 

Onions  

Potatoes  

54 

60 
56 

60 
54 

60 

56 

60 
56 

60 
56 

60 

60 

56 

56 

56 

60 

56 

60 

60 
56 

60 
56 

56 

60 

56 

56 

60 

6060 
5656 

Rye  

Rye  Meal  
Salt 

- 

56 

- 

45 
60 
20 

50 

45 
60 

So 

45 
60 
20 

50 
45 
60 

20 

_ 

rfi 

Timothy  Seed.  .  . 
Wheat 

60 

60 

60 



60 

60 

60 

& 

20 



60 

00 

60 

60 
— 

60 

— 

- 

6060 

Wheat  Bran  

MEASURES    AND    WEIGHTS. 


35 


"Weight    of  IVIen   and.   "Women. 

Average  weight  of  20000  men  and  women,  weighed  in  Boston,  1864,  was 
— men,  141.5  Ibs. ;  women,  124.5  Ibs.  Average  of  men,  women,  and  chil- 
dren, 105.5  Ibs.  A  mass  of  people,  densely  packed,  weighs  85  Ibs.  per  sq.  foot, 
each  occupying  .8  of  one  sq.  foot  of  area  =  54  450  per  acre. 


Weight    of  Horses.— (TJ.  S.) 

Weight  of  horses  ranges  from  800  to  1200  Ibs. 


WEIGHT  OF  CATTLE. 
To   Compute  Dressed  "Weight   of  Cattle. 

RULE. — Measure  as  follows  in  feet: 

1.  Girth  close  behind  shoulders,  that  is,  over  crop  and  under  plate, 
immediately  behind  elbow. 

2.  Length  from  point  between  neck  and  body,  or  vertically  above 
junction  of  cervical  and  dorsal  processes  of  spine,  along  back  to  bone  at 
tail,  and  in  a  vertical  line  with  rump. 

Then  multiply  square  of  girth  in  feet  by  length,  and  multiply  product 
by  factors  in  following  table,  and  quotient  will  give  dressed  weight  of 
quarters. 


Condition. 

Heifer,  Steer, 
or  Bullock. 

Ball. 

Condition. 

Heifer,  Steer, 
or  Bullock. 

Ball. 

Half  fat  

•a.  1C 

3.36 

Very  prime  fat  .  .  . 

3.64 

3.85 

Moderate  fat 

•3    06 

3'  5 

Extra  fat  

0.78 

406 

Prime  fat..  . 

3.  5 

1.64, 

ILLUSTRATION.— Girth  of  a  prime  fat  bullock  is  7  feet  2  ins.,  and  length  measured 
as  above  4  feet  5  ins. 

7'  2"  =  7. 17,  and  7. 17*  =51. 4,  which  x  4'  5"  and  by  3.5  =  794.5  Ibs.      Exact 
weight  was  799  Ibs. 

NOTE.— i.  Quarters  of  a  beef  exceed  by  a  little,  half  weight  of  living  animal 
2.  Hide  weighs  about  eighteenth  part,  and  tallow  twelfth  part  of  animal. 

Comparative   "Weights   of*  Live   Beeves   and   of  Beef*. 


Lbs. 

Per  cent. 

Lbs. 

Per  cent. 

Bullocks  

2800 
2600 
2600 
2400 
2400 

2IOO 
2100 
I800 

72  to  78 
}  70  to  76 

|  66  to  70 

J  64  to  68 
63  to  66 

Bullocks 

1550 
1550 

I2OO 
1200 
1050 
1050 
980 
Q50 

|  61  to  64 
|  58  to  61 
}  57  to  58 
|  50  to  56 

Heifers  

Heifers 

Bullocks  

Bullocks 

Heifers  

Heifers 

Bullocks  

Bullocks  .     . 

Heifers  

Heifers 

Bullocks  

Bullocks 

Heifers  

Heifers  .  .  . 

"Weight  of*    Offal     in   a   Beef  and   Sheep. 


BEEF.  SHEEP. 

Lba.  Lbs. 

Hide  and  Hair ....  56  to   98  8  to  16* 

Tallow 42  "  140  5  "  14 

Head  and  Tongue  .  28  "    49  6  "  nf 

Feet 21  «    35  2  "    3 

*  Including  3  to  6  Iba.  for  fleece. 


BEEF. 
Lbs. 

Kidneys.  Heart.) 

Liver,  etc.... 7 3'  to  62 

Stomach,  En  trails,  etc.,  126  "  196 


Blood 


6  to  10 


42  "    56 
t  Including  2  to  5  Ibs.  for  horns. 


36  MEASURES,  WEIGHTS,  PRESSURES,  ETC. 

To    Compute    Equivalents    of  Old    and.    New   TJ.   S.  and 
of  Metric   t)enominations. 

By  Act  of  Congress,  July  28,  1866. 

RULE.  —  Divide  fourth  term  by  second,  multiply  quotient  by  first 
term,  and  divide  product  by  third  term. 

Or,  Ascertain  relative  ratio  of  first  and  second  terms,  and  multiply 
result  by  ratio  of  third  and  fourth  terms. 

NOTE.  —When  result  is  required  in  French  or  other  Metric  denominations  thai 
those  of  U.S.,  use  exact  denominations,  as,  61.025  387  for  61.022,  39.370432  for  39^ 
etc. 

EXAMPLE  i.— If  one  gallon  (ist),  per  sq.  foot,  yard,  acre,  etc.  (2d) ;  how  many  liters, 
(3d),  per  sq.  foot,  yard,  acre,  etc.  (4th)  ? 

—  X  231-:- 61.022 =  3-7851  liters  or  3.7848  Uteri. 

Or,  ^3  =  1.604, and  —^-  =  2.3598;  hence,  1.604  X  2.3598  =  3.7851  liters. 

144  OI.O22 

NOTE.— In  computing  ratios,  first  term  is  to  be  dividedby  second,  and  fourth  by  third 
EXAMPLE  2. — If  one  ton  per  cube  foot,  how  many  kilograms  per  cube  decimeter? 

Il02-X  2240-7-2.2046  =  35.881  liters,  or  35.882  litres. 


1728 


MEASURES. 

By  Act  of  Congress  of  U.  8.  By  Metric  Computation* 

i  Liter  per  sq.  foot,  etc.  =  .2642  Gallon  per  sq.foot,  or  .264  2  gallon. 
i  Liter  per  sq.  meter  .  =  .0245  Gallon  per  sq.foot,  or  .024  5  gallon. 
i  Gallon  per  sq.  foot  .  =40.746  Liters  per  sq.  meter,  or  40.745  4  litres. 
i  Sq.  foot  per  acre  . . .  =  .2296  Sq.  meters  per  hectare,  or  2.29609  metres. 


WEIGHTS    AND    PRESSURES. 

By  Act  of  Congress  of  U.  8.  By  Metric  Computation. 

Per  sq.  inch.  Per  sq.  inch. 

i  Centimeter =       .3937  In*>  or        -393  7°4  32  /«*. 

i  Atmosphere  . . . .  =     6.6679  Kilograms,  or      6.667 8  kilogrammes. 

i  Inch  mercury  . .  =     2.54     Centimeters,         or     2.54  centimetres. 

i  Pound =  453.6029  Grams,  or  453.592  6  grammes. 

i  Kilogram =  317.4624  Lbs.per  sq.foot,  or  317.465  Ibs. 

NOTE.— 30  ins.  of  mercury  at  62°  =  14.7  Ibs.  per  sq.  inch ;  hence,  i  Ib.  =  2.0408  in*., 
and  a  centimeter  of  mercury  =  30-4-  .3937  for  U.  S.  computation,  and  30---  .393  704  3* 
for  French  or  Metric. 

POWER   AND   WORK. 

i  Horse  -  power  =  Cheval  or  Cheval  -  vapeur  =  4500  k  X  m  =  33  ooo  4 
(4500  x  2.2046  X  39.37  -4-  12)  =  1.013  88  chevaux. 
i  Cheval  or  Cheval- vapeur  (75  Tcxm  per  second)  =  horse-power. 
(4500  x  2.2046  X  39.37  -4- 12)  -f-  33000  =  .9863  horse-power. 

By  A  ct  of  Congress  of  U.  8.  By  Metric  Computation. 

Kilograinmeter  k  x  w  =  7.233  jfroi-/6s. ;  hence, 

i -=-(2.2046x3.280  833)  =  .13826  Kilogr ammeter,  or  .13825  kilogrammetre. 
i  Cube  foot  per  IP  . . . .  =  .0279  Cube  meter  per  cheval,  or  .0279  cheval. 
I  Pound        "    "  . .  =  .447  38  Kilogram  per  cheval,  or  .447  38  kilogramme 
i,  Cube  meter  per  cheval  =  35.8038  Cube  feet  per  IP,  or  35.8058  IP. 


PRESSURES,   ETC. MEASURES    OF   TIME.  37 


TEMPER  ATURES. 

i  Caloric  or  French  unit  =  3.968  Heat-units,  and  i  heat-unit  =  i  -r-  3.968 
•=  .252  caloric. 

i  U.  S.  Mechanical  equivalent  (  772  foot-lbs. )  =  772  -r-  7.233  =  106.733 
Kilo gr ammeters  and  106.733  kilogrammetres. 

i  French  Mechanical  equivalent  (423.55  k  X  w)  =  3.280833  x  2.2046  X 
423.55  =  3063.505  foot-lbs.,  or  3063. 566 foot-lbs.  Metric. 

i  Heat-unit  per  pound  =  .5556  Kilogram,  or  .5556  kilogramme. 

i  Heat-unit  per  sq.  foot  =  .2715  Caloric  per  sq.  meter,  or  .zjizper  sq.  metre. 


VELOCITIES. 

i  Foot  per  second =  .3047  Meter  per  second,  or  .3047  metres. 

i  Mile  per  hour =  .447        "      "       "       or  .447 


MEASURES   OF   TIME. 


60  thirds     =  i  second. 
60  seconds  =  i  minute. 


60  minutes  =  i  degree. 
30  degrees  =  i  sign. 


360  degrees  =  i  circle. 

True  or  apparent  time  is  that  deduced  from  observations  of  the  Sun, 
and  is  same  as  that  shown  by  a  properly  adjusted  sun-dial. 

Mean  Solar  time  is  deduced  from  time  in  which  the  Earth  revolves 
on  its  axis,  as  compared  with  the  Sun ;  assumed  to  move  at  a  mean 
rate  in  its  orbit,  and  to  make  365.242  218  revolutions  in  a  mean  Solar 
or  Gregorian  year. 

Sidereal  time  is  period  which  elapses  between  time  of  a  fixed  star 
being  in  meridian  of  a  place  and  time  of  its  return  to  that  place. 

Standard  unit  of  time  is  the  sidereal  day. 

Sidereal  day  =  23  h.  56  m.  4.092  sec.  in  solar  or  mean  time. 

Sidereal  year,  or  revolution  of  the  earth,  365  d.  5  h.  48  m.  47.6  sec.  in  solar 
or  mean  time  =  365.242  218  solar  days. 

Solar  day,  mean  =  24  h.  3  m.  56.555  sec.  in  sidereal  time. 

Sol'ir  year  (Equinoctial,  Calendar,  Civil  or  Tropical)  =  365.242  218  solar 
days,  or  365  d.  5  h.  48  m.  47.6  sec. 

Civil  day  commences  at  midnight.  Astronomical  day  commences  at 
noon  of  the  civil  day,  having  same  designation,  that  is,  12  hours  later 
than  the  civil  day. 

Marine  or  sea  day  commences  12  hours  before  civil  time  or  i  day 
before  astronomical  time. 

New  Style  was  introduced  in  England  ii  \  1 752. 

NOTE.  — In  Russia  days  are  reckoned  by  Old  Style,  and  are  consequently  12  days 
behiud  Gregorian  record. 

J> 


MEASURES  OF  VALUE. 


MEASURES  OF  VALUE. 

10  mills  =  i  cent.  I  10  dimes   =  i  dollar. 

10  cents  =  i  dime.  10  dollars  =  i  eagle. 

Standard  of  gold  and  silver  is  900  parts  of  pure  metal  and  100  of 
alloy  in  1000  parts  of  coin. 

Fineness  expresses  quantity  of  pure  metal  in  1000  parts. 

Remedy  of  the  Mint  is  allowance  for  deviation  from  exact  standard 
fineness  and  weight  of  coins. 

Nickel  cent  (old)  contained  88  parts  of  copper  and  12  of  nickel. 

Bronze  cent  contains  95  parts  of  copper  and  5  of  tin  and  zinc. 

Pure  Gold   23.22   grains  =  $100.     Hence  value  of  an  ounce  is 
$20.67.1834-. 

Standard  Gold,  $18.60.465+  per  ounce. 


WEIGHT,  FINENESS,  ETC.,  OF  U.  S.  COINS. 
G-old. 


Denomination. 

Weigh 
of  Coin. 

t 
of  Pure 
Metal. 

Denomination. 

Weigh 
of  Coin. 

* 
of  Pare 
Metal. 

Dollar 

Oz. 
•053  75 
•134375 
.161  25 

•080375 
.16075 
.2009375 

Gra. 
25-8 
64-5 
77-4 

38.58 
77.16 
96-45 

Gra. 

23.22 
58-05 
69.66 

Sil 

34.722 
69.444 
86.805 

Half  Eagle  
Eagle  

Oz. 

.26875 

•5375 
1.075 

.401  875 
.875 
•859375 

Gn. 

1  29 
258 
5i6 

192.9 
420 
412-5 

Grs. 
116.1 
232.2 
464.4 

173.61 
378 
37I-25 

Quarter  Eagle.. 
Three  Dollar... 

Dime  

Double  Eagle.  .  . 

ver. 

|  Half  Dollar  
Trade  Dollar.  .  .  . 
1  Silver  Dollar  .  .  . 

20  Cent    

Quarter  Dollar  . 

Copper  and.   INTicltel. 


Weight. 

Copper. 

Tin  and  II 
Zinc.     || 

Weight. 

Copper. 

Tin  and 
Zinc. 

One  Cent  
Two  Cents  .  .  . 

Grains. 

48 
96 

Per  cent. 
95 
95 

Per  cent.  || 

5       N  Three  Cents. 
5       ||  Five  Cents.. 

Grains. 
77.16 

Per  cent. 
75 
75 

Per  cent. 
25 
25 

Tolerance.— Gold,  Dollar  to  Half  Eagle,  .25  grains.  Eagles,  .5  grains. 
— Silver,  1.5  grains  for  all  denominations.  —  Copper,  i  to  3  cents,  2  grains ; 
5  cents,  3  grains. 

Legal  Tenders.  —  Gold,  unlimited.  —  Silver.  Dollars  of  412.5  grains 
unlimited ;  for  subdivisions  of  dollar,  $10.  (Trade  dollars  [420  grains]  are 
not  legal  tender.)— Copper  or  cents,  25  cents. 

NOTE.— Weight  of  dollar  up  to  1837  was  416  grains,  thence  to  1873,  412.5.  Weight 
of  $1000,  @  412.5  gr.  =859.375  oz. 

BRITISH  standards  are :  Gold,  ||  of  a  pound,*  equal  to  u  parts  pure  gold 
and  i  of  alloy  ;  Silver,  fff  of  a  pound,  or  37  parts  pure  silver  and  3  of  alloy 
=  .925  fine. 

A  Troy  ounce  of  standard  gold  is  coined  into  £3  175.  lod.  2/,  and  an 
ounce  of  standard  silver  into  55. 6d.  i  Ib.  silver  is  coined  into  66  shillings. 

Copper  is  coined  in  proportion  of  2  shillings  to  pound  avoirdupois. 

£  Sterling  (1880)  $486.65;  hence  -^  of  this  =  value  of  i  penny  = 
2.027  708  33  cents. 

*  A  pound  is  assumed  to  be  divided  into  24  equal  parts  or  carats,  hence  the  pro 
portion  is  equal  to  22  carats. 


FOREIGN   MEASURES   OF   VALUE.  39 

To    Compute    "Valne   of  Coins. 

RULE.  —  Divide  product  of  weight  in  grains  and  fineness,  by  480 
(grains  in  an  ounce),  and  multiply  result  by  value  of  pure  metal  per 
ounce. 

Or,  Multiply  weight  in  ounces  by  fineness  and  by  value  of  pure  metal 
per  ounce. 

EXAMPLE  i.—  When  fine  gold  is  $20.67.183+  per  oz.,  what  is  value  of  a  British 
sovereign  ? 

By  following  tables,  p.  40,  Sovereign  weighs  .2567  oz.,  and  .2567  X  480  =  123.216 
grains,  and  has  a  fineness  of  .9165. 


Hence,  x.o.67..83+  =  $4.86.34. 

EXAMPLE  2.  —When  fine  silver  is  $i.  1  5.  5  per  oz.  ,  what  is  value  of  U.  S.  Trade  dollar  f 
By  table,  p.  38,  Dollar  weighs  .875  oz.  and  has  a  fineness  of  .900. 
Hence,  .87$x  -QooX  1.15.5  =  90.95625  cents. 


EXAMPLE  3.—  A  4-Florin  (Austrian)  weighs  49.92  grains  and  has  a  fineness  of  .900, 
What  is  its  value? 

4,^900  X2o67i8j+  = 

To    Convert    TJ.  S.  to    British    Currency    and   Contrari- 

•wise. 

RULE  i.  —  Divide  Cents  by  2.027  71—  (2.027  7°8  33),  or,  Multiply  by 
493  12—  (.493  118  26),  and  result  is  Pence. 

2.  Multiply  Pence  by  2.02771—  ,  or  divide  by  .49312—,  and  result 
is  Cents. 
EXAMPLE.—  What  are  100  cents  in  pence? 

loo  x  49312  —  ==49.312  —  pence  —  4*.  i.  3  1  2d 
a.  What  is  a  Pound  sterling  in  cents? 

20  X  12  =  240  pence,  which  x  2.02771—  =:  $486.65. 


FOREIGN   MEASURES    OP   VALUE. 

"Weight,  Fineness,    and.    Mint    "Values    of  Foreign 
Silver  and.    GJ-old   Coins. 

By  Laws  of  Congress,  Regulations  of  the  Mint,  and  Reports  of  its  Directors. 

Current  Value  of  silver  coins  is  necessarily  omitted,  as  the  value  of 
silver  is  a  variable  element.  Hence,  in  order  to  compute  current  value 
of  a  silver  coin,  the  price  of  fine  or  a  given  standard  of  silver  being 
known, 

Proceed  as  per  above  rule  to  compute  value  of  coins. 

The  price  of  silver  should  be  taken  as  that  of  the  London  market  for 
British  standard  (925  fine),  it  being  recognized  as  the  standard  value, 
and  governing  rates  in  all  countries. 

EXAMPLE.— If  it  is  required  to  determine  value  of  a  Mexican  dollar  in  cents. 

Weight  867.5  oz.  .go^Jine.  Value  of  Silver  in  London  52.75  pence  per  ounces 
106.96164-  cents. 

^  =  .846  867—  and  106.9616  X  .846  867  =  90.5822  cenfc. 
925 


3  FOREIGN   MEASURES    OF    VALUE. 

Weight    and    Mint    Values    of   Foreign    Coin. 

(Value  is  based  on  their  Value  on  April,  1901.) 
Countries  given  in  Italics  have  not  a  National  Coinage. 


Countr}'  and  Denomination. 

Weight. 

Fine- 
ness. 

Pure 

Silver 
or 
Gold. 

Current 
or 
Nominal. 

VALDB 
Gc 

U.S. 

Id. 
British. 

Arabia. 
Piastre  or  Mocha  Dollar.  .... 
Argentine  Republic. 
Dollar  =  100  Centisimos  .... 
Peso  

Oz. 

:l£s 
:i 

.104 

"hous'a. 

916 
916.5 

900 
900 

900 
986 
900 

870 

916.66 
918.5 
9*7-5 
914 

925 
925 

875 

853-1 
833 

900 
870 

901 

Grains. 

Cents. 
83.14 
50-69 

$    c. 

.965 

4-85-7 
5-32-37 

34-5 

1.93.49 
2.28.3 
6.75.4 

•451 
15-59-3 

•54-59 

10.90.6 
4.92 

i 
3-97-43 
3-99-97 

7es 

3.68.8 
14.96.39 

^65 
9-!5-4 
15-59-3 

.682 

X    6.     d. 

1911.5 
i     i  10.5 

.2 

7" 

9    4-6 
i    7    9.1 

•3' 
3    4    * 

26.9' 
2    4    9.8, 

I     O     2.6 

•o. 

•5 

•7. 

4    2 
16    4 

16    5.2 

15    i  8! 
3    i    5-9' 

•4. 

1  17    7-4. 
3    4    ' 

.0! 

3-32 

Australasia. 
Same  as  British. 
Australia. 
Sovereign  1855 

171.47 
257-47 

362.06 

12.67 
393-6 

66.6 
83.25 

"•34 

37^8 

•456 

Austria. 
Kreutzer  (copper)  

Dollar,    "    

•75 
•547 

.1 

1.014 

1.52 

•9 

.14 

6.75 

07.  C2 

Souverain  

.363 

.86~7 

.028.8 
.82 
•575 
.261 

•  *5 
•1875 

.027 

.209 
.869 

.492 
-867 

.087 

Belgium. 
Same  aa  France. 
Bolivia. 
Centena  

Boliviano  

Brazil. 
Rei  

Milreis  

Double  Milreis  

Moidore  4000  Reis   .... 

Canada. 

Cent     "      

20  Cent  currency  

Penny          u 

Shilling        "        

4    "  =20  shillings,  currency 
Pound 
Cape  of  Good  Hope. 

Same  as  British. 

Central  America. 
4  Reals  

Colon  

2  Escudos  

Doubloon  ante  1834  .  . 

Chili. 
Centaro  

Dollar,  new  

10  Pesos  .  ...  

Doubloon  

China. 
Cash,  Le  

10  Cents,  Leang  

Tael  Hankow  

Cochin  China. 
Mas,  60  Sapeks  

jo  Mas,  i  Quan... 

FOREIGN    MEASURES    OF   VALUE. 


"Weight   and   Mint   Values. 


Country  and  Denomination. 

Weight. 

Fine- 
ness. 

Pare 
Silver 
or 
Gold. 

Current 
or 
Nominal. 

V  A  L  U  ». 

Go 
'  U.S. 

Id. 
British. 

Cuba. 

Ox. 

:« 

.025 
.927 
.427 

.304 
.060.4 
.182.5 
.178 
•454-5 
.363-6 
.256.7 
.256.2 

.04 
•275 
•275 
i-  375 

.032 
.161 
.161 
.804 
.207.5 

.012.8 

.128 

•595 

.1X2 

.OIO.4 
'719 
185 

012.8 

»2.a 

844 
870 

900 
877 
895 

925 
924-5 
925 
925 
925 
916.5 
916.5 

755 
875 
875 
875 

900 

900 
900 
900 
986 

900 
900 
900 

900 
27  71  cent 

D* 

390.23 

26.82 
80.99 
79-  °3 

201.8 

161.44 
M-5 

69-55 
347-76 

257-04 
310.61 

8. 

I.  OX 

8.94 

1.  01 
2.02+* 

.2 
1.  01 

2.38 
.405 

.926 

.451 

7-55-5 
15-59-3 

26.8 
7.90 

•451 

4.86.65 
4.85.1 

4-9 
5-0-52 
4  94-3 
25.  2.6 

I!'3 
96-45 
3.85.8 

^3-8 
2.38.24 

2.28.38 

iy-3 
344.2 

5-  6.  ii 
23.8 

•5 

in    0.58 
3    4    i 

4-39 
13-22 

i  12    5.6 

•5 

..:* 

100 
100 

I     0     6.84 

i    o    5.3 

5     2  10.2 

.1 

-5 

15  10.26 

1.17 
11.74 

9    9-5 

9    4-63 

9-J 
H    i-75 

I     0     9.6 

11.74 

.2 

Colombia. 
Centaro  

Doubloon  old     

Costa  Rica. 

Same  as  Mexico. 

Denmark. 
Mark  i6Skilling          

Crown  

2  Rigsdaler  

jO  Thaler 

East  Indies. 

See  Hindustan  and  Japan. 

Ecuador. 
Centaro 

Sucre  

England. 

'  '       average      

Half  Crown  

Florin  

Sovereign  or  Pound,  new  .  .  . 
"                  "    average. 
Egypt 
Piastre  40  Paras  . 

Guinea,  Bedidlik  

Pound  

Purse  5  Guineas  

France. 

Sou  5  Centimes             . 

Franc  100  Centimes    

5  Francs  

20  Francs,  Napoleon,  new  .  .  . 
25  Francs  20  centimes  =£1  Stg. 
Germany. 
Groschen  10  Pfenning 

Mark  10  Groschen    

10  Marks  

Thaler  ,  

Ducat  

Greece  and  Ionian  Islands. 

Same  as  France. 
Drachma  100  Lepta  .   .  . 

5  Drachmas  

Pound      . 

Guatemala. 

Same  as  Mexico. 

Guiana,  British,  French,  and 
Dutch. 

Same  as  that  of  their  Countries. 

Hanse  Towns. 
Mark  

Holland. 
Cent  

FOREIGN   MEASURES    OF   VALUE. 


\Veight   and.   Mint  "Values. 


Country  and  Denomination. 

Weight. 

Fine- 
ness. 

Pure 
Silver 
or 
Gold. 

Current 
or 
Nominal. 

VALUI 
G 

U.S. 

>ld. 
British. 

Holland. 
Florin  or  Guilder,  100  cents. 
10  Guilders  

Ox. 
.021.6 

.215 

•374 

.16 
.864 

•375 
•375 

.279 
.866.7 
•053.6 
.289 
•  362 
1.072 

.861 
.867.5 
1.081 

.844 
.245 

.803 
.867 

Stg.,  nomi 

Thous". 
916.5 

835 
900 

916.5 
916.5 

890" 
900 
900 

572 
568 
900 

902.5 
870.5 
873 

83o 
996 

896 
858 

nal  value 

Grains. 
164-53 

65.12 

373-24 

16? 

119.19 
374-4 

372.98 
336-25 

=  2  shill 

Cents. 

•25 
3-03 

I 

21.63 

.083 
ings  sterlin 

*    c. 

40.2 
3-99-7 

19.3 

32.9 
4.86.65 
6.84.36 

75-3 

99-72 
3-57-6 
4.44 
19.94.4 

4.86.65 
49.0 

15.  6.1 
19-51-5 

5-  4-4 

iS-37-8 

.83 
g. 

£  s.    d. 

i    8 
16    5.1* 

x  10.5 

.13 

'•5 

I     O     O 

i    8    1.5 

.49* 

4~j.x8 
14    8.35 
18    2.96 
4    x  ii.  6 

X     O     0 

2 

3    4    1-88 
4    o    2.4 

x    o    8.75 

3    3    3-39 
xo.66 

Hindostan. 

Honduras. 
Sam*  M  Mexico. 
Italy. 
Same  as  France. 
Lira  100  Centimes       .  •  .  • 

Scudo  

Indian  Empire. 
Pic  nominal 

Anna     *  '      

Rupee,*  16  Annas  
10  Rupees,  and  4  Annas  
Mohur  15  Rupees  

Japan. 
Yen  

Itzebu  new    

Yen,  100  Sen  

Cobang  old  

*'      new  

Java. 

Same  as  Holland. 
Liberia. 
U.  S.  Currency. 
Malta. 
12  Scudi  =  i  Sovereign  
Mexico. 
Peso  new    

*'    Maximilian  

20  Pesos  Republic     .     .  • 

Morocco. 

10  Ounces  Mitkeel  

Naples. 
Scudo  

6  Ducati     

Netherlands. 

Same  as  Holland. 

New  Brunswick. 

Same  as  Canada. 
Newfoundland. 
Same  as  Canada. 
New  Granada. 
Dollar  1857  

Norway. 

Alike  to  Denmark. 

Mark  24  Skillingen  

Nova  Scotia. 
Same  as  Canada. 
Persia. 
Reran,  20  Shahis  

10  Keran  Toman  

Paraguay.    Foreign  coins. 

*  .092  76  of  a  £ 

FOREIGN   MEASURES    OF    VALUE. 


Weight    and    Mint   "Values. 


43 


Country  and  Denomination. 

Weight. 

Fine- 
ness. 

Para 
Silver 
or 
Gold. 

Current 
or 
Nominal. 

V  ALUB 
04 

U.S. 

British. 

Peru. 
Dollar  1858       

q*. 

766 

Thoas'». 

Grains. 

Cents. 

*      C- 

£   s.    d. 

gol            

,0- 

.867 

868 

I  e.  cc  7 

Portugal. 
Corda,  1838,  10000  Reis  

.308 
.005 

912 

912 

— 

- 

I0.8l.78 
10.8 

a    4    5-S 

Roumania. 
2  Lei       

•322 

81* 

129  06 

Russia. 

500 

77 

18 

ioo  Copek  Rouble         . 

66? 

Kin 

_j    - 

e  Roubles               .       ..... 

6  6 

2  OO 

Sandwich  Islands. 
U.  S.  Currency. 
Sardinia. 
Lira                  

.16 

8i< 

Spain. 

16 

81* 

6^  ii 

.193 

Dollar  5  Peseta 

3 

A 

I9.3 

ioo  Reals        .   ..   

268 

°rT 

1048 

10  Escudos  

270  8 

806 

r       re 

20  Reals  vellons=i  U.S.  Dollar. 
Sweden. 
Riksdaler,  ioo  Ore  

271 

98  28 

Rixdollar  

I  OQ2 

Carolin  10  Francs  

75 

Switzerland. 

Same  as  Franc*. 
St.  Domingo. 
Gomdes  ioo  Cents.  

900 

6  11 

Tunis. 
Piastre  16  Karubs  

ii  81 

c8i 

5  Piastre  

m  i 

808  <; 

220.  18 

.511 
.161 

2  OO  S 

12      17 

Turkey. 
Piastre  40  Paras  

20  Piastre  

8  TO 

>. 

88 

ioo  Piastre,  Medjidie  

211 

18    o 

Tuscany. 

.  112 

2  11   1 

96.  i 

Tripoli. 
20  Piastres  Mahbub  

?6 

i    o  So 

Uruguay. 
Dollar  ioo  Centimes.  

74.8 

West  Indies,  British. 
Same  as  England. 

Venezuela. 
Centaro  

j 

.5 

Bolivar,  i  Franc.  .  . 





10.  1 

O.4 

Memoranda. 

FRANCE.— Bronze  coins  9.5  copper,  4  tin,  and  i  zinc. 

HAXSE  TOWNS.— Monetary  system  same  as  that  of  German  Empire. 

SWITZERLAND. — The  Centime  is  termed  a  Rappe. 

SPAIN.— 25  Peseta  piece  is  195.  9.5^.  Stg. ;  Real  vellon  was  2.5^.  Stg. 

ITALY.— All  coins  same  weight  and  fineness  as  those  of  France. 

MALTA.— 7  Tari  and  4  Grani  =  i  Shilling  Sterling. 

EGYPT.— A  Para  =.  .061  5<J.  Sterling,  and  97.22  Piastres  —  i  Sovereign. 

INDIAN  EMPIRE.— i  Lac  Rupees=£ioooo  Sterling.    In  CEYLON,  Rupee  =  ioo  Cents. 


44 


ENGLISH  AND  FEENCH  MEASURES  AND  WEIGHTS. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS. 

MEASURES    OF   LENGTH. 

ENGLISH. — Imperial  standard  yard  is  referred  to  a  natural  standard, 
which  is  a  pendulum  39. 1393  ins.  in  length  vibrating  seconds  in  vacuo 
in  London,  at  level  of  sea ;  measured  between  two  marks  on  a  brass 
rod,  at  temperature  of  62°. 

NOTE.  — -  In  consequence  of  destruction  of  standard  by  fire  in  1834,  and  difficulty 
of  replacing  it  by  measurement  of  a  pendulum,  the  present  standard  is  held  to  be 
about  i  part  in  17  230  less  than  that  of  U.  S.,  equal  to  3  67  ins.  in  a  mile. 

Miscellaneous. 

Land. — Woodland  pole  or  perch  or  Fen =  18  feet. 

Forest  pole =  21    " 

Irish  mile  .......  =  2240  yards.    \    Scotch  mile =  1984  yards. 

Sea. — 10  cables,  or  1000  fathoms,  or  6080.27  feet,  or  1.1516  Statute  miles, 
i  A  dmiralty  or  Nautical  mile  or  knot  =  6080  feet. 

3  miles  =  i  league.  60  Nautical  or  69.094  Statute  miles  or  20  Leagues 
=  i  degree. 

Mean  length  of  a  minute  of  Latitude  at  mean  level  of  the  sea  =1.1451 
statute,  miles. 

Nautical  mile  is  taken  as  length  of  a  minute  at  the  Equator. 

Nautical  fathom  is  loooth  part  of  a  nautical  mile,  and  averages  about 
.0125  longer  than  the  common  fathom. 

FRENCH. — Standard  Metre  or  unit  of  measurement  is  defined  as  the 
ten  millionth  part  of  the  terrestrial  meridian,  or  the  distance  from  the 
Equator  to  the  Pole,  passing  through  Paris.  Actual  standard  is  a  plat- 
inum metre,  deposited  in  the  Palais  des  Archives,  Paris. 


Denomination. 

Metres. 

Inches. 

Feet. 

Yards. 

Miles. 

Millimetre  

OOI 

Centimetre  
Decimetre  

.01 

•3937 

- 

— 

- 

METRE  

I 

3  28087 

, 

Dekametre           . 

03  808  60 

°93 

Hektometre.  ...... 

100 

I  OOO 

- 

328.086  9 
3280  869 

109.36231 

621  38 

Mvriametre.  .  . 

IOOOO 

10016.271 

6.21177 

NOTE.— For  length  of  metre  see  p.  27. 

Old.   ]Vteasnre. 

i  Toise =  1.949  metres. 

i  Mille =  1.949  kilometres. 

i  Noaud  (knot).  =  1.855         " 


i  Terrestrial  league  =  4.444  kilometres, 
i  Nautical  league  .  =  5.555         " 
i  Arpent =  900  sq.  toises. 


MEASURES    OP   SURFACE. 
ENGLISH.— Same  as  that  of  United  States  of  America. 

Miscellaneous. 

Builders.      i  superficial  part =  i  square  inch. 

12  parts =  i  inch. 

12  inches =  square  foot. 

Boards. — Boards  7  inches  in  width  are  termed  battens,  9  inches  deals,  and 
12  inches  planks. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS.      45 

FRENCH. 
3Metrio   Surfaces   in   Sq.u.are   Iiich.es,  Feet,  etc* 


Denomination. 

Sq.  Inches. 

Sq.  Feet. 

Sq.  Yards. 

Sq.  Acre*. 

I  SqU£ 

M 

ire  millimetre  

.00155 
.155003 
15.500309 
1550.030916 

.  107  641 
10.764104 
1076.410358 

1.19601 
119.601  15 
II  960.11509 

.024711 
2.471098 
247.109816 

2A  710.081  6 

decimetre  

Metre  or  Centiare  .... 
dekametre  or  are  
hektometre  or  hectare 
kilometre  

Equal  38.610  90$  «q.  milti. 
Old   System. 

i  square  inch  =  1.13587  inches* 

i  toise  =  6.394  6  yeek 

i  arpent  (Paris)         =  900  square  toises  =  4089  square  yards. 

i  arpent  (woodland)  =  100  square  royal  perches  =  6108.24  square  yards. 


MEASURES    OF   VOLUME. 

Imperial  gallon  measures  277.123  cube  ins.,  but  by  Act  of  Parliament 
1825  its  volume  is  277.274  cube  ins.,  equal  to  10  Ibs.  avoirdupois  of 
distilled  water,  weighed  in  air,  at  temperature  of  62°,  barometer  at  30 
inches.  6.2355  gallons  in  a  cube  foot. 

Imperial  bushel,  18.5  ins.  internal  diameter,  19.5  external,  and  8.25 
in  depth,  contains  2218.192  cube  ins.,  and  when  heaped  in  form  of  a 
right  cone,  at  least  .75  depth  of  the  measure,  must  contain  2815.4872 
cube  ins.  or  1.6293  cube  feet. 

Grain. — i  quarter  =  8  bushels  or  10.2694  cube  feet. 

Vessels.  —  i  ton  displacement  =  35  cube  feet;  i  ton  freight  by  measure- 
ment =  40  cube  feet. 

i  ton  internal  capacity  =  100  cube  feet,  and  i  ton  ship  -  builders  =  94 
cube  feet. 

English  standard  No.  5  is  .008  grain  heavier  than  the  pound,  and  U.  S.  pound  it 
ooi  grain  lighter  than  English. 

Wine   and    Spirit   Measures. 

4  quarts  (231  cube  ins.) =  .8333  Imperial  gallon. 

=  i  anchor. 
=  i  runlet. 
=  i  barrel. 
==  i  tierce. 
=  i  hogshead. 
=  i  puncheon. 
=  i  pipe  or  butt. 

=  i  tun. 


10  gal 
18 

42 
63 
84 
126 

2  pip 

3PUE 

(15    in 
-       26.25 
35 
r        52.5 
70 

i<>5 
es  or     1 

« 

u 

u 

u 

cheonsj 

4  quails  (282  cube  ins.)  . .  =    1.017 

9  gallons  =  i  firkin =   9-I53 

2  firkins  =  i  kilderkin  . . .  =  18.306 


-Ale   and   Beer   Aleasvires. 

Imp'l  pall's. 


Imp'l  gall'B 

2  kilderkins  =  i  barrel  =  36.612 
54  gallons  =  i  hogshead  =   54.918 
108      "      =  i  butt =  109.836 


46      ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS. 


Apothecaries'    or  Fluid.   Measures. 

i  drop =  i  grain. 

60  drops =  i  drachm. 


4  drachms =  i  tablespoon, 

2  ounces  (875  grains)  =:  i  wineglass. 


Coal    Measures. 


50  pounds  . . . .  =  i  cube  foot. 

88      "       =  i  bushel. 

9  bushels  . . . .  =  i  vat. 


90  or  94      "      =i  Cornish         " 
93  pounds  . . . .  =  i  Welsh  bushel. 
3  heaped  bush.  =  i  sack. 
10  sacks =  i  ton. 


12  sacks = 

i  chaldron = 

5.25  chaldrons  . .  = 
i  London  chaldron: 
i  Newcastle  "  = 

i  ton = 

i  room = 

21  chaldrons = 

i  barge  or  keel . .  = 


:    i  chaldron. 
:  58.6548  cube  ft. 
:   i  room. 
:  26.5  CWts. 

'•53       " 
144.5  cube  feet. 
:    7  tons. 
:    i  score. 
:2i.2  tons. 


Miscellaneous. 


.  .  —  80  bushels. 

.  .  —  35.9  cube  j  eft  •. 

dicker  hides  .  . 

.  .  =  10  skins. 
.  .  —  :  20  dickers. 

x  *  —  26.5  gallons, 

6  bushels  wheat 

.  .=   i  sack  flour. 
.  .  =r   7  pounds. 

.  .  —  28.2    " 

truss  straw  .  .  . 

..=36      " 

35.9  cube  feet  =  i  ton  water. 


i  truss  old  hay =   50  noun  Is. 

i     "     new  "    =   60  u 

i  bushel  oats =  40  " 

i      "      barley  . . . .  =   47  " 

i      "      wheat =   60  " 

i  cube  yard  new  hay  =   84  " 

i     "       "    old     "    =126  " 

i  quintal =  100  " 

i  boll =  140  u 

i  sack  wool =  364  " 


LIQUID. 


i  wine  gallon  =  231  cube  ins. 
i  beer      "      =  282     "      " 

i  litre =       .220  09  gallon. 

i  gallon =     4-544  litres. 

i  cube  foot . .  =     6.2321  gallons. 
i  auker  . . . .  =     8.333        " 


i  hogshead  wine  . . : 
i  "  beer . . . : 
i  puncheon  wine  . . : 
i  pipe  or  butt  wine  : 
i  "  "  "  beer  : 
i  tun : 


:   52.5     gallons. 

•   54-9l8      " 

:    70 

:  105 

:  109.836       " 

:2io  " 


i  ton  water  62°  =  224  gallons. 


BUILDERS. 


i  solid  part =  12  cube  ins. 

12  u  parts =  i  "  inch." 

12  "  inches  " =  i  cube  foot. 

i  load  timber,  rough  =  40    "   feet. 

i    "        u       hewn  =  50    "      " 

i    "    lime =  32  bushels. 

z    "    sand =  36      " 


i  square 

i  bundle  laths  . . . 
i  rod  brickwork  . 
i  rood  masonry  . 
Batten,  in  section 
Deal,  *  u 
Plank,  "  " 


.  =  loo  sq.fect. 
.  =  120  laths. 
.  =  306  cube  feet, 
.  =  648     "       u 
.  =:    7  X  2.5  ins. 

•=   9X3       " 
.  =  11  X  3 


Volumes  in    Cube    Iiiolies,  Feet,  etc. 


Denominations. 

Litres. 

Gills. 

Pints. 

Quarts. 

Gallons. 

Bushels. 

Quarters. 

Centilitre  

•  OI 

.0704 

0176 

Decilitre  .  .  . 

1761 

Litre*  

i 

7  0420 

.lyui 
I  7607 

.8804 

22OI 

Dekalitre  

IO 

8.8036 

2.  2OOQ 

.  275  II 

Hectolitre  

IOO 

22  OOQI 

2.751  13 

O^-JQ 

Kilolitre... 

1000 







220.0008 

27.^11  •**> 

^.  4180 

*  Equal  61.025  24  cube  int. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS.       47 

"Wood    !Meas\ire. 

i  Stere  or  cube  metre  =  35.3150  cube  feet  or  1.308  cube  yards. 
i  Voie  de  bois  (Paris)  =  70.6312  cube  feet ;  i  voie  de  charbon  (charcoal) 
=  7.063  cube  feet ;  i  corde  =  4  cube  metres  =  141.26  cube  feet. 


MEASURES    OF   WEIGHT. 

BRITISH. — i  Troy  grain      =     .003  961  cube  inches  of  distilled  water, 
i  Troy  pound     =22.815  689  cube  inches  of  water, 
i  Avoir,  drachm  =  27.343  75  Troy  grains. 

16  drachms,  or) 

437-5  grains  j 
16  ounces,  or  1 

7000  grains  J 

20  hundredweights . 

The  grain,  of  which  there  are  7000  to  the  pound  avoirdupois,  is  same  as 
Troy  grain,  of  which  there  are  by  the  revised  table  7000  to  the  Troy  pound. 
Hence  Troy  pound  is  equal  with  the  Avoirdupois  pound. 
In  Wales,  the  iron  ton  is  20  cwt.  of  120  Ibs.  each. 


Avoirc 
—  i  ounce. 

lupois. 

8  pounds  .  .  =  i  stone  (for  meat). 
14      "        .  .  =  i  stone. 
28      "         .  .  =  i  quarter. 

112         "            .  .  =  I  CWt. 

.    .  .  .  —  i  pound. 

.  =  i  ton. 


Troy. 


16  ounces =  i  pound. 

25  pounds =  i  quarter. 

4quarters,  or  loo  pounds  =  i  cwt. 


24  grains =  i  dwt. 

20  pennyweights,  or  j 
437-5  grains        f"~ 

By  this  are  weighed  gold,  silver,  jewels,  and  such  liquors  as  are  sold  by 
weight. 

The  old  Troy  ounce  to  the  Avoirdupois  ounce  was  as  480  grains,  the 
weight  of  the  former,  to  437.5  grains,  weight  of  the  latter;  or,  as  i  to  .9115. 

Apothecaries.* 
437.5  grains  =  i  ounce.          |  16  ounces  =  i  pound. 

FRENCH. 

M.etrio    "Weights   in    Avoirdupois. 


Denominations. 

Grammes. 

Grains. 

Ounces. 

Pounds. 

Ton. 

Milligramme  

001 
.01 

.1 

,'  .-"'i  i  •  i 

10 
100 
1000 

IOOOO 
100  000 
I  OOOOOO 

ogramme  =  2  Ibs. 

01543 
•IS432 
i  543  23 
I5-43235 
154.32349 
1543.23487 
15432.34874 

3  oz.  4  drachms,  lo.t 

•3527 
3-5274 
35-2739 

734  pram*. 

.22046 
2.20462 
22.04621 

220.462  12 
2204.621  25 

.9842 

Centigramme  

Decigramme  

Dekagramme  

Kilogramme^  .   . 

Myriagramme  

Quintal 

Millier  or  Ton  

tKi 

NOTE.— For  the  values  of  the  prefixes,  as  Milli,  Centi,  etc. ,  see  p.  27. 

Old.    System. 

i  grain  . .  =    0.8188  grains  Troy.    I    i  ounce  =  1.0780  oz.  Avoirdupois. 
i  gross  . .  =  58.9548  |    i  livre   =  1.0780  Ibs. 


*  As  by  revised  Pharmacopoeia. 


48 


FOREIGN   MEASURES   AND   WEIGHTS. 


FOREIGN   MEASURES   AND   WEIGHTS. 

It  being  wholly  impracticable  to  give  all  the  denominations  of  measures 
and  weights  of  all  countries,  the  following  cases  are  selected  as  essential  and 
as  exponents. 

With  parent  countries,  as  England,  France,  etc.,  their  denominations  ex 
tend  to  their  colonies  and  dependencies.  Thus,  the  denominations  of  England 
extend  to  Canada,  a  large  portion  of  the  East  a,nd  West  Indies,  and  parts  of 
South  America,  and  those  of  France  to  a  part  of  the  West  Indies,  Algiers,  eta 

Abyssinia. 

Pic,  Stambouili 26.8    ins. 

"   geometrical 30.37   " 

Madega 3.466  bush, 

Ardeb 34.66      " 

"     Musuah 83.184    " 

Wakea 400  grains. 

Mocha i  Troy  oz. 

Rottolp 10    u      " 

Also,  same  as  in  Egypt  and  Cairo. 

Africa,  Alexandria,  Cairo, 
and.   Egypt. 

Cubit 20.65  ins- 

Derah 25.49   " 

Pic,  cloth 26.8     " 

"   geometrical 29.53   " 

Kassaba,  4.73  Pics 11.65  ft. 

Miie 2146  yds. 

Feddan  al-risach 552  48  acre. 

Roobak 1.684  galls. 

Ardeb 4.9  bush. 

Rottol 9821  Ib. 

Distances  are  measured  by  time. 
A  Maragha  —  15  Ddreghe*  or  i  hour. 

Aleppo   and.    Syria. 

Dra  Mesrour 21.845  ins. 

Pic 26.63     " 

Road  Measures  are  computed  by  time. 

Algeria. 

Rob,  Turkish 3.  n  ins. 

Pic,        "        24.92   u 

"   Arabic 18.89   " 

Also  Decimal  System. 

Alicante. 

Palmo 8.908  ins. 

Vara 35.632   " 

Amsterdam. 
Voet ii.  144  ins. 


El 


1.979 


Faden 5.57  ft. 

Lieue 6.383  yds. 

Maat 1.6728  acres. 

Morgen 2.0095     " 

Vat 40  cub.  ft 

Also  Decimal  System. 

Ant\verp. 

Fuss 1 1-  275  ins. 

Elle,  cloth 26.94     u 

Corde 24.494  cub.  ft. 

Bonnier 3-2507  acres. 

Also  Decimal  System. 


Arabia,    Bassora,  and 
IMoclia. 

Foot,  Arabic 1.0502  ft. 

Covid,  Mocha 19  ins. 

Guz,        "     25   " 

Kassaba 12. 3  ft. 

Mile,  6000  feet 2146  yds. 

Baryd,  4  farsakh 21 120   " 

Feddan 57  600  sq.  ft 

Noosfla,  Arabic 138  cub.  ins. 

Gudda 2  galls. 

Maund 3  Ibs. 

Tomand 168   " 

Other  Measures  like  those  of  Egypt. 

Argentine    Confederation, 
Paraguay,  and  Uruguay. 

Fanega 1.5  bush. 

Arroba, 25. 35  Ibs. 

Quintal 101,4     " 

Also  Decimal  System  in  Argentine  Con< 
federation  and  Paraguay. 

Australasia. 

Land  Section 80  acres. 

Other  Measures  same  as  English. 
Axistria. 

Zoll i-°37i  ins- 

Fuss i  0371  ft. 

Meile 24000  ft 

Klafter,  quadrat 35-854  sq.  yds. 

Jochart 6.884      " 

Cube  Fuss i-"55  cu^-  ft 

Achtel 1.692  galls. 

Eimer 12.774    u 

Viertel 3. 1143  " 

Metze i. 6918  bush. 

Unze 8642  grains. 

Pfund  (1853, 500  grammes),  1.2347  Ibs. 

Centner 123. 47       u 

Also  Decimal  System. 

Babylon. 

Pachys  Metrics 18. 205  ins 

Baden. 

Fuss 11.81  ins. 

Klafter 5.9055  ft. 

Ruthe 9.8427  " 

Stunden 4860  yds. 

Morgen 8896  acre. 

Stutze. 3-3014  galls. 

Malter o .......    4. 1268  bush. 

Pfund 1. 1023  Ibs. 

Also  Decimal  System. 


FOREIGN   MEASURES   AND   WEIGHTS. 


49 


Bagdad. 

Guz 31-665  ins. 

Barbary    States. 

Pic,  Tunis  linen. 18.62  ins. 

"        "     cloth 26.49  " 

"    Tripoli 21.75   " 

Batavia. 

Foot 12. 357  ins. 

Covid 27 

El... .....27.75     « 

Bavaria. 

Fuss 11.49  ins- 

Klafter 5-745  36  ft. 

Ruthe 3-  *9l8  yds- 

Meile 8060      " 

Ruthe,  quadrat 10. 1876  sq  yds. 

Morgen  or  Tagwerk 8416  acre. 

Klafter,  cube 4. 097  cub.  yds. 

Eimer 15-05856  galls. 

Scheffel 6.119 

Metze 1.0196  bush. 

Pfund 8642  grains. 

Also  Decimal  System. 

Belgiu.ni. 

Meile 2.132  yds. 

Also  Decimal  System. 

Benares. 
Yard,  Tailor's 33  ins. 

Bengal,  Bombay,  and.    Cal- 
cutta. 

Moot 3  ins- 

Span 9  " 

Ady,  Malabar 10.46  ins. 

Hath 18 

Guz,  Bombay 27 

"    Bengal 36 

Corah,  minimum 3.417  ft. 

Coss,  Bengal 1. 136  miles. 

"    Calcutta 1-2273    " 

Kutty. 9.8175  sq.  yds. 

Biggah,  Bengal 3306  acre. 

"      Bombay 8114    u 

Seer,  Factory 68  cub.  ins. 

Covit,  Bombay 12.704  cub.  ft. 

Seer,  Bombay 1-234  pints. 

Parah 4. 4802  galls. 

Mooda 112.0045     kt 

Liquids  and  Grain  measured  by  weight. 

Bohemia. 

Foot,  Prague u.88  ins. 

"     Imperial 12.45   " 

Also  same  as  Austria. 
Bolivia,  Chili,  and    Peru. 

Vara 33-333  ins- 

Fanegada 1.5888  acres. 

Gallon 74  gall. 

Fanega 1.572   " 

Libra 1.014  Ibs. 

Arroba 25.36     " 

Originally  as  in  Spain ;   now  Decimal 
System  in  Chili  and  Peru. 


Brazil. 

Palmo,  Bahia 8.5592  ins. 

Vara 3.566  ft 

Braca 7. 132  *  * 

Geira 1.448  acres. 

Also  same  as  Portugal,  and  sometimes 
as  in  England. 

Buenos   Ayres. 

Vara 2. 84  ft 

Legua 3.226  miles 

Suertes  de  Estancia ....  27  ooo  sq.  varas. 
Also  same  as  Spain. 

Burmah. 

Paulgat i  inch. 

Dain 4. 277  yds. 

Viss 3.6  Ibs. 

Taim 5.5    " 

Saading 22 

Also  same  as  England. 

Canary   Isles. 

Onza 927  inch. 

Pic,  Castiliaii 11.128  ins. 

Almude 0416  acre. 

Fanegada 5         " 

Libra 1.0148  Ibs. 

Also  same  as  Spain. 

Cape    of  Grood   Hope. 

Foot 11.616  ins. 

Morgen 2. 116  54  acres. 

Also  same  as  in  England. 

Ceylon. 

Seer i  quart 

Parrah 5.62  galls. 

Also  same  as  in  England. 

China. 

Li 486  inch. 

Chih,  Engineer's. 12. 71    ins. 

"     or  Covid 13-125   " 

"     legal 14.1 

Chang 131-25     ' 

"     legal 141          " 

Pu 4.05  ft 

Chang,  fathom 10.9375  ft 

Li 486  yds. 

Pu  or  Rung 3.32  sq.  yds 

King,  loo  Mau 16.485  acres. 

Tau 1. 13  galls. 

Tael 1.333  oz. 

Catty 1.333  Ibs. 

Cochin    China. 

Thuoc  or  Cubit 19.2  ins. 

Sao 64  sq.  yds. 

Mao 1.32  acres. 

Hao 6. 222  galls. 

Shita 12.444    " 

Nen 8594  Ib. 

Colombia  and  Venezuela. 

Libra 1. 102  lb& 

Oncha 25          " 

Also  Decimal  System. 


FOREIGN    MEASURES    AND    WEIGHTS. 


Denmarlt,*  Greenland,  Ice- 
land, and.   Norway. 

romme 1.0297  ins 

Fod 1.0297  ft. 

Favn,  3  Alen 6. 1783  " 

Mil 4-68°  55  miles. 

"   nautical 4.61072     ' 

Anker 8.0709  galls. 

Skeppe 478  DUstL 

Fjerdingkar 9558   " 

Fund 1. 1023  IDS. 

Lispund i7-367     ' 

Centner 110.23 

*  Also  Decimal  System. 

Ecuador. 

Decimal  System. 

Genoa,  Sardinia,  and 
Turin. 

Palmo 9.8076  ins. 

Piede,  Manual,    8  oncie. . .  13.488     ' 
"    Liprando,  12     "   ...20.23 

Trabuco  or  Tesa 10. 113  ft. 

Miglio 1.3835  miles. 

Starello 9804  acre. 

Giomaba 9394    * 

Germany. 

The  old  measures  of  the  different  States 
differ  very  materially  ;  generally,  how- 
ever, 

Foot,  Rhineland 12-357  iQS- 

Meile 4-603  miles. 

Decimal  System  made  compulsory  in  1872. 

Greece. 

Stadium 6155  mile. 

Also  Decimal  System. 

Guinea. 
Jachtan 12  ft. 

Hamburg. 

Fuss 11.2788  ins. 

Klafter 5.641311. 

Morgen 2.386  acres. 

Cube  Fuss 831 1  cub.  ft. 

Tehr 99.73 

Viertel i.  594  7  galls. 

Pfund  (500  grammes) ...    1. 102  32  Iba. 

Ton 2135.8  Ibs. 

Also  Decimal  System. 

Hanover. 

Fuss 11.5  ins. 

Morgen 6476  acre. 

Hindostan. 

Borrel 1.211  ins. 

Gerah 2.387   u 

Haut 19.08     " 

Kobe 29. 065   ' ' 

Coss 3.65  miles. 

Tuda 1. 184  cub.  ft 

Candy 14. 209      ' 


Hungary. 

Fuss 12.445  ins. 

lle 3°-67     ' 

Meile 9-  *39  yds- 

Also  as  in  Vienna. 

Indian   Empire. 

Guz 27.125  ins. 

Cowrie i  sq.  yd. 

Sen 61.025  39  cub.  ina. 

2.204 737  Ibs. 

Uniform  standard  of  multiples  of  the  Sen 
adopted  in  1871. 

Italy.          •-.> 
IVIilan   and   "Venice. 

Decimal  System. 
The  Metre  is  termed  Metra;  the  Are,  Ara; 
the  Stere,  Stero;  the  Litre,  Litro;  the 
Gramme,  Gramma,  and  the  Tonneau, 
Tonnelata  de  Mare. 

Naples   and  Two   Sicilies. 

Palmo 10.381  ins. 

Canna 6.921  ft. 

Miglio 1. 1506  miles. 

Migliago 7467  acre. 

Moggia 86 

Pezza,  Roman 6529 

Roman   States. 
Old  Measure. 

Foot ii. 592  ina 

"    Architect's 11.73     * 

Braccio 30.73     * 

Palmo 8.347  u 

Miglio 1628  yds. 

Quarta 1.1414  acres. 

Lucca  and   Tuscany. 

Pie 11.94  ins. 

Palmo "-49  " 

Braccio 22.98   " 

Passetto 3.829  ft 

Passo 5.74    " 

Miglio 1.0277  miles. 

Quadrato 8413  acre. 

Saccato i.  324      " 

Japan. 

Sun,        .303  03  Metre. .       1.193*  ins. 
Shaku,  3. 030 3  Metres..      11.9305*  ins. 
Jo,        30.303         "    ..       9.9421*  ft. 
Ken,      5.5  '    ..       5-9653*  " 

Ri,  11880  "    ..       2.4403  miles. 

Kai-ri 6o8ofeet.t 

Hiro 4-971*  feet- 

Momme 3.756  521  7  grammes  Fr. 

Hiyaku  me 828  17  Ibs. 

Kwam-me 8.28171 

Hiyak-kin 132.507  32 

Man's  load S7-972 

Koku 331.26831 

Hiyak-koku 33  126. 830  8 

*  These  are  as  equivalent  aa  they  are  practi- 
cable of  reduction, 
f  Admiralty  knot. 


FOKEIGN   MEASUEES   AND   WEIGHTS. 


Java. 

Duim 1.3  ins. 

Ell 27.08  " 

Djong 7-015  acres. 

Kan 328  galls. 

Tael 593.6  grains. 

Sach 61.034  Ibs. 

PecuL 122.068   " 

Catty 1.356  " 

Madras. 

Ady. 10.46  ins. 

Covid 18.6     " 

Guz 33         " 

Culy 20.92  ft. 

League 3472  yds. 

Puddy 338  galls. 

Marcal 2.704     " 

Tola 180  grains. 

Seer 625  Ibs. 

Viss 3.086   " 

Maund 24.686  " 

jNXalabar. 
Ady 10.46  ins. 

Malacca. 

Hasta  or  Covid 18. 125  ins. 

Depa 6ft. 

Orlong 80  yds. 

Malta. 

Palmo 10.3125  ins. 

Pie 11.167     " 

Canna 82. 5 

Salma 4-44  acres. 

Also  as  in  Sicily. 

Moldavia. 

Foot Sins. 

Kot,  silk 24.86  ins. 

Fathom 8  ft 

Molucca   Islands. 
Covid 18.333  ins. 

Morocco. 

Tomin 2. 810  25  ins. 

Cadee 20.34  ins. 

Cubit 21        " 

Muhd. 3.081  35  galls. 

Kula,  oil 3.356         " 

Rotal  or  Artal 1.12  Ibs. 

Liquids  other  than  oil  are  sold  by  weight. 

Mysore. 

Angle 2.12  ins. 

Haut 19.  i     ' 

Guz 38. 2     " 

Candy 500  Ibs. 

Netherlands. 

Elle 39.370432  ins. 

Decimal  System  since  1817. 

JPersIa. 

Gereh 2. 375  ins. 

Gueza,  common 25          " 

44     llonkelrer 37.5       " 


Archin,  Schah 31.55  ins. 

"      Arish 38.27   " 

Parasang 6076  yds. 

Chenica 80. 26  cub.  ins 

Artaba 1.809  bush. 

Mi  seal. 71  grains. 

Ratel 2.1136  Ibs. 

Batman  Maund 6.49       u 

Liquids  are  measured  by  weight. 

Polando 

Trewiee 14.03  ins. 

Precikow 17  ins. 

Pretow 4-7245  yds. 

Mile,  short 6075  yds. 

Morgen i.  3843  acres. 

^Portugal  and  ^dozana"biq-ue. 

Foot 13  ins. 

Milha 1.2788  miles 

Almude 3.7  galls. 

Fanga 1.488  bush. 

Alguieri 3.6        " 

Libra 1.012  Ibs. 

Also  Decimal  System. 

Prussia. 

Fuss 12.358  ins. 

Ruthe 4. 1 192  yds. 

Meile 24  ooo  feet 

Quadrat  Fuss 1.0603  sq.  ft. 

Morgen 631 03  acre. 

Cube  Fuss 1.092  cub.  ft 

Scheffel i.  5121  bush. 

Anker 7-559  galls. 

Pound 7217  grains. 

Zollpfund 1. 1023  IDS. 

Centner "3-43  Ibs. 

Russia. 

Vershok 1.75  ina 

Foot 12  ins. 

Arschine 28  " 

Rhein  Fuss 1.03(1. 

Sajene 7  ft. 

Verst 3500  " 

Mila 5. 5574  milea 

Dessatina 2.4954  acres. 

Vedro 2. 7049  galls. 

Tschel- werha 1-4424    " 

Pajak 1.4426  bush. 

Tschetwert 5-7704    " 

Pound 6317  grains. 

Funt 902  85  Iba 

Decimal  System  adopted  in  1872. 

Siam. 

K'up 9-75  ins. 

Covid 18  ins. 

Ken 39   " 

Jod 098  48  mile. 

Roeneng 2.462  miles. 

Silesia. 

Fuss 11.19  ins- 

Ruthe 4. 7238  yds. 

Meile 7086  yds. 

Morgen 1-3825  acrea 


FOREIGN   MEASURES   AND    WEIGHTS. 


Singapore. 

Hasta  or  Cubit 18  ins. 

Dessa. 6  ft. 

Orlong 80  yds. 

Smyrna. 

Pic 26. 48    ins. 

Indise 24.648   " 

Berri 1828  yds. 

Spain,  Cu"ba,  Malaga,  Ma- 
nilla, GJ-u.atemala,  tdondu.- 
ras,  and.  Mexico. 

Pie 11.128  ins. 

Vara 33-384  " 

Milla 865  mile. 

Legua,  8000  varas 4.2151  miles. 

Fanegada 1.6374  acres. 

Vara,  cubo 21. 531  cub.  ft. 

Cuartilla 888  gall. 

Arroba,  Castile 3. 554  galls. 

Fanega 1-5077  bush. 

Libra 1.0144  N>s. 

Tonelada 2028.2  Ibs. 

Also  Decimal  System. 

Stettin. 

Fuss ii.  12    ins. 

Foot,  Rhineland 12. 357    " 

Elle 25.6  ins. 

Morgen 1-5729  acres. 

Sumatra. 

Jankal  or  Span o  ins. 

Elle 18   « 

Hailoh 36   " 

Fathom 6  ft 

Tung 4  yds. 

Snrat. 

Tussoo,  cloth. 1.161  ins. 

Guz,         "    27.864   " 

Hath 20.  Q       " 

Covid 18.5       " 

Biggah 51  acre. 


Tunnland 1.2198  acres 

Anker - ....    8.641  galls. 

Spaun 1.962  bush. 

Centner 112.05  ^& 

Also  Decimal  System. 

S\vitzerland. 

Fuss,  Berne 11.52  ins. 

u     11.54   " 

Vaud 11.81    " 

Klafter 5.77  ft. 

Meile 4. 8568  miles 

Juchart,  Berne 85  acre. 

Haas 2.6412  pints. 

Eimer 8.918  galls. 

Malter 4. 1268  bush, 

Pfund 1. 1023  Ibs. 

Also  Decimal  System. 

Tripoli. 

Pik,  3  palmi 26.42  ins. 

Almud 319. 4  cub.  ins. 

Killow 2023       "      " 

Barile 14.267  galls. 

Temer 7383  bush 

Rottol 7680  grains. 

Oke 2.8286  Ibs. 

»    Tnrlzey. 

Pic,  great 27.9  ins. 

"    small 27.06" 

Berri 1.828  yds. 

Alma 1. 154  galls. 

Also  Decimal  System. 

Wiirtemtoerg. 

Fuss 11.29  ins. 

Elle 2.015(1. 

Meile 8146.25  yds. 

Morgen 7793  acre. 

Cube  Fuss 830  45  cub.  ft 

Eimer 64.721  galls. 

Scheffel 4. 878  bush. 

Pound 7217  grains. 

Zurich. 

Fuss ii. 812  ins. 

Elle 23.625    " 

Klafter 5.9062  ft. 

Meile , . .    4. 8568  miles 

Jachart 808  acre. 

Cube  Klafter 144  cub.  ft 


Sweden. 

Fot 11.6928  ins. 

Ref 32.4703  yds. 

Faden 5.845  ft. 

league 3. 3564  miles. 

Meile 6.6417 

Holland. 

Denominations  corresponding  to  the  French  are  as  follows: 

Length.  —  Millimetre,  Streep;  centimetre,  Duim;  decimetre.  Palm;  metre,  El; 
decametre,  Roede;  kilometre,  Mijle. 

Surface.— Square  millimetre,  Vierkante  Streep;  square  centimetre,  Vierkante 
Duim;  and  so  on.  Hectare,  Vierkante  Bunder. 

Cube  Measure.—  Millistere,  Kubicke  Streep,  and  so  on. 

Capacity.—  Centilitre,  Vingerhoed;  decilitre,  Maatje;  liquid  litre,  Kan;  dry  litre, 
Kop;  decalitre,  Schepel;  liquid  hectolitre,  Vat  or  Ton;  dry  hectolitre,  Mud  or  Zak; 
30  hectolitres  =  i  Lastr=  10.323  quarters. 

Weight.—  Decigramme,  Korrel;  gramme,  Wigteje;  decagramme,  Lood;  hecto- 
gramme, Onze;  kilogramme,  Pond. 

Belgium. 

Metric  system.— The  term  Livre  is  substituted  for  kilogramme,  Litron  for  lltre> 
and  Anne  for  metre. 


SCRIPTURE    MEASURES. — ANCIENT  WEIGHTS. 


53 


SCRIPTURE    AND    ANCIENT   LINEAR   MEASURES. 
Scripture. 

Digit 912  inch.  I  Span,  3  palms 10.944 

Palm,  40  digits 3-648  ins.     j  Cubit.  2  spans 21.888 

Fathom,  4  cubits 7  feet  3. 552  ins. 

Hebrew  and.   Egyptian. 


ins 


Nahud  cubit i-475      feet 

Royal      "    1.7216 

Egyptian  finger 06145 


Babylonian  foot 1. 140  feet 

Hebrew          u   1.212    " 

"      cubit 1.817    " 


Hebrew  sacred  cubit 2.002  feet. 


Digit 7554  inch. 

Pous  (foot). 1.0073  feet- 

Cubit i.i332    " 

Pythic  or  natural  foot 814  foot. 

Attic  or  Olympic     "    1.009  feet. 


Grecian. 

Ancient  Greek  foot     )  Q     .    . 

(16  Egyptian  fingers) f 9«4i  wot 

Arabian  foot 1-095    feet 

Stadium 604.0375    u 

Olympic  stadium 606.29       " 


Mile,  8  stadium  .........  4835  feet 

Alexandrian  or  Phileterian  stadium  (600  Phil,  feet)  =  708.65  feet 
Volume.—  Keramion  or  Metretes  ..............  8.488  gallons. 


Cubit.  .  .  ............  i.  824  feet 

Sabbath  day's  journey  ____  3648          " 


I  Mile,  4000  cubits  ........  7296  feet 

|  Day's  journey  ............  33-164  milea 


Digit...  ..........  72575  ins- 

Uncia  (inch)  ...............  967 

Pes  (foot)     .  .............  11.604       " 


Roman    Long   Measures. 

Cubit  ......................  -1.4505  feet 

Passus.  ...  .  .  ..............  .4-835 

Mile,  milliarmm  .........  4842 


ANCIENT   WEIGHTS. 


He"brew   and   Egyptian. 

Troy  grains, 

Atticobolus \  X'Ct        Denarius,  Roman {  |J;9* 

"       Nero 54$ 

Shekel 


151-9* 
"     drachma \  54-6t 

(69* 

Lesser  mina 3. 892 

Greater  mina 5.46 

Egyptian       mina 8. 326* 

Ptolemaic         "    8.985* 

Alexandrian     "     9-992* 

Obolus 4.63 


Ounce 


(431- 


Drachm 146. 5 

Libra 4086.  i 

Pound 12  Roman  ouncee. 

Talub c  581-71  ounces. 


Talent  (60  minae) 56  Ibs.  avoirdupois. 


GJ-recian. 


Mina. 


Troy  grains. 

Obolus,  ancient 8.33 

"       "-57 

Gramme 23. 15 

Drachma 50-01 

--'#<     Sreat 69-47 

Roman. 
Ounce 416.82  grains.    |    Pound 10.41  ouncea 


Troy  ounces. 

10.41 

"   great 14-47* 

Talent 625. 19 

u     Attic , 868.32 


t  Arbuthnot. 

E* 


t  Paucton. 


54 


GEOGRAPHIC   MEASURES   AND   DISTANCES. 


GEOGRAPHIC    MEASURES   AND    DISTANCES. 
To  Reduce  Ijongitncle  into  Time. 

RULE. — Multiply  degrees,  minutes,  and  seconds  by  4,  and  product  is 
the  time. 
EXAMPLE.— Required  time  corresponding  to  50°  31'.    50°  31'  X  4  =  3^-  22m.  45. 

To   Reduce    Time   into   Longitude. 

RULE. — Reduce  hours  to  minutes  and  seconds,  divide  by  4,  and  quo- 
tient is  the  longitude.     Or,  Multiply  them  by  15. 
EXAMPLE. — Required  longitude  corresponding  to  $h.  8m.  11.2$. 

5/i.  8m.  ii. 2$.  =  3o8m.  11.2$.,  which  -=-  4  =  77°  2'  8". 
Or,  multiplying  by  15:  5/1.  8m.  11.25.  X  15  =  77°  2/  8". 

Table   of  Departures   for  a  Distance   rnn   of*  1   Mile. 

Course.  Departure.  II  Course.  Departure.  ||  Course.  Departure. 


3.5  points.  .773  4.5  points.  .634       I     5.5  points.  .471 

4  -707      II     5  -556      II     6  .383 

Thus,  if  a  vessel  holds  a  course  of  4  points,  that  is  without  leeway,  for  distance 
of  i  mile,  she  will  make  .707  of  a  mile  to  windward. 

Or,  a  vessel  sailing  E.N.E.  upon  a  course  of  6  points  for  100  miles  will  make  38.3 
(100  X  .383)  miles  of  longitude. 

.Minutes,   and.    Seconds     of*    eacn    IPoint    of   the 
Compass   -with.    Meridian. 


Degrees, 


NOBTH. 

SOOTH. 

Points. 

o  i  n 

Sin.  A.* 

Cos.  A.* 

Tan.  A.* 

| 

•25 

2  48  45 

.0489 

.9988 

.0491 

N.  

.5 

c  07  OQ 

.008 

oo8< 

"t 

•75 

O  j/  y* 

8  26  15 

.uyo 
.1467 

•9952 
.9891 

•vy°5 
.1484 

( 

i 

11  J5 

.195 

.9808 

.1989 

N.by  E.... 
N.  by  W  

S.by  E  1 
S.by  W  1 

1.25 

'4  3  45 
16  52  30 

.2429 
•2903 

•9569 

.2504 
•3034 

I 

J-75 

19  4i  15 

•3368 

•9415 

•3578 

N.N.E..., 
N.N.W  

S.S.E  .. 

2 

2.25 
2-5 

22  30 

25  18  45 
27  7  30 

.3827 

•4275 
.4714 

•9239 
.904 
.8819 

.4142 
.4729 
•5345 

S.S.W.  1 

I 

2.75 

30  56  15 

•8577 

•5994 

N.E.  byN.  .. 
N.W.  byN... 

S.E.byS.  ... 
S.W.byS....  1 

3 
3-25 
3-5 

33  45 
36  33  45 
39  22  30 

•5556 

•5957 
•6344 

•8315 
•8032 
•773 

.6682 
.7416 
.8207 

I 

3-75 

42  ii  15 

•6715 

•7409 

.0063 

r 

4 

45 

.7071 

.7071 

i 

N.E  

S.E  I 

4-25 

47  48  45 

•7404 

1.103 

N.W  

S.  W  1 

A.  5 

4-75 

So  37  3° 
53  26  15 

•773 
.8032 

•UJ44 
•5957 

1.348 

N.E.  by  E.  .. 

S.E.  by  E... 

5 
5-25 

56  15 
59  3  45 

•8315 
•8577 

•5556 

1.497 
1.668 

N.W.byW... 

S.W.  by  W...  1 

5-5 

61  52  30 

.8819 

.4714 

1.871 

I 

5-75 

64  41  15 

.904 

•4275 

2.114 

E.N.E... 
W.N.W  

E.S.E.  .. 

w.s.w  1 

6 
6.25 
6-5 

67  30 
70  18  45 
73  7  30 

•9239 
•9569 

•3827 
•3368 
•2903 

2.414 

2-795 
3-296 

I 

6-75 

75  56  15 

•97 

.2429 

3-941 

E.byN... 
W.  by  N  

E.byS  J 
W.byS  1 

7 
7-25 
7-5 

78  45 
81  33  45 
84  22  30 

.9808 
.9891 
•9952 

•195 
.1467 
.098 

5.027 
6.741 

I 

7-75 

87  ii  15 

.9988 

.0489 

20-555 

East  or  West. 

East  or  West.  .  . 

8 

90 

i 

.0000 

00 

*  A,  representing  course  or  points  from  the  meridian. 

GEOGRAPHIC   LEVELLING. 


55 


GEOGRAPHIC    LEVELLING. 

Curvature   and.   Refraction.0 

Correction  for  Curvature  of  Earth,  to  be  subtracted  from  reading  of 
a  levelling-staff,  is  determined  as  follows : 

Divide  square  of  distance  in  feet  from  level  to  staff,  by  Earth's  Equa- 
torial diameter — viz.,  41  852  124  feet. 

Or,  Two  thirds  of  square  of  distance  in  statute  miles  equal  the  cur- 
vature in  feet. 

Correction  for  Refraction  is  to  be  subtracted  from  reading,  and  as  a  mean 
may  be  taken  at  about  one  sixth  of  that  for  curvature. 

Correction  for  Curvature  and  Refraction  combined,  is  to  be  added  to 
reading  on  staff. 

Formulas  of  Capt.  T.  J.  Lee,  U.  8.  Engineers. 

T)2  D2 

— —  =  correction  for  curvature,  -^  m  =  correction  for  refraction,  and 
2  R  K 

D2 
(i  —  2  m)  — ^  =  correction  for  curvature  and  refraction.     D  representing 

distance,  R  radius  of  earth,  and  m  a  coefficient  of  refraction  =  .075,  all 
in  feet. 

ILLUSTRATION.  —A  distance  is  3  statute  miles,  what  is  correction  for  curvature 
and  refraction? 

•';;....  (.-2  x. 075)  ^^=.85x5.996=5.097/0*. 

Approximately,  —  D2  =  curvature  in  feet. 


Hie  veiling  by  Boiling   IPoint   of  TVater. 
To   Compute  Height  A_"bove   or   Belo\v  Level   of  Sea* 

517  (212°  -  T)  +  (212°  -  T)2  =  Height. 
ILLUSTRATION.— What  is  height  of  an  elevation,  when  boiling  point  of  water  is  182°? 


517  X  212°  — 182°  +  212°  — 182°  =  517  X  30  +  so2  =  16  410  feet. 
Corrections  for  Temperature  to  be  made  in  Connection  with  Formula. 


Temp. 

Correc- 

Temp. 

Correc- 
tion. 

Temp. 

Correc- 
tion. 

Temp. 

Correc- 
tion. 

Temp. 

Correc- 
tion. 

Temp. 

Correc- 
tion. 

o 

.936 

18 

.972 

3°6 

008 

54 

.046 

72 

1.083 

90 

12 

2 

•94 

20 

.976 

38 

012 

56 

•05 

74 

1.087 

92 

124 

6 

8 

•944 
.948 
•952 

22 

It 

.98 

40 
42 

44 

016 

02 
024 

£ 

62 

•054 
.058 
.062 

76 

£ 

1.091 
1.096 
i.i 

91 
96 

98 

128 
132 
136 

10 

•956 

28 

.992 

46 

028 

64 

.066 

82 

1.104 

100 

.14 

12 

.96 

30 

.996 

48 

032 

66 

.071 

84 

1.108 

IO2 

.144 

14 

.964 

32 

I 

So 

036 

68 

•075 

86 

1.  112 

104 

.148 

16 

.968 

34 

1.004 

52 

.041    / 

70 

.079 

88 

1.  116 

106 

.152 

ILLUSTRATION.— Assume  temperature  in  preceding  illustration  to  have  been  80*. 
Then  i64ioX  i.i  =  18051  feet. 


GEOGRAPHIC   LEVELLING  A1TD   DISTANCES. 


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GEOGRAPHIC  LEVELLING. — MAGNETIC  VARIATION.      57 

ILLUSTRATION.  —  Curvature  of  Earth  independent  of  refraction  is  computed  at 
.667  foot  =  8. 004  ms.  for  i  geographical  mile,  and  as  refraction  on  land  is  taken  as 
.104  foot  or  1.248  ins.,  and  on  ocean  at  .099  foot  or  1.188  ins.,  relative  visible  dis- 
tances of  an  object,  including  curvature  and  refraction,  for  an  elevation  of 
.667  foot    is       1.09  miles  on  land,  and     i. 08  miles  at  sea. 

I  "  1.33        "        «         "          «  ^32        u        <(      « 

9        feet     "       4          «     «     "      «       3.08     «     »    « 
i        mile   "    104.03     "     "     "      "    103.54     "     "   " 

Difference  between  two  levels  in  feet  is  as  square  of  their  distance  in 
miles. 

ILLUSTRATION.— At  what  elevation  can  an  object  be  seen,  at  surface  of  ocean,  when 
it  is  2  miles  distant? 

i2  :  22  : :  . 667  —  .099  :  2. 272  feet  =  2  feet  3. 25  -f-  ins. 
Difference  between  two  distances  in  miles  is  as  square  root  of  their  heights 

ILLUSTRATION  i.  —At  an  elevation  of  9  feet  above  level  of  sea  at  what  distance 
can  an  object  be  seen  upon  its  surface? 

V-667  —.099  =  .754  :  i  ::  -y/g  :  3.98  miles. 

2VTIf  a  man,at  the  fore1-V)pgallant  mast-head  of  a  vessel,  100  feet  from  water,  sees 
another  and  a  large  vessel  "hull  to,"  how  far  are  the  vessels  apart? 
A  large  vessel's  bulwarks  are  at  least  20  feet  from  water 

Then,  by  table,  100  feet —  ,,  271 

5o    <*    _   5.03!  19-20  miles  distance. 

When  an  observation  for  distance  is  taken  from  elevation,  as  a  light-house 
a  vessel's  mast,  etc.,  of  an  object  that  intervenes  between  observer  and  hori^ 
zon,  or  contrariwise,  observer  being  at  a  horizon  to  elevated  object,  distance 
of  observer  from  intervening  object  is  determined  by  ascertaining  or  esti- 
mating its  elevation  from  horizon,  and  subtracting  its  distance  from  whole 
distance  between  observer  and  point  from  which  observation  is  taken  and 
remainder  is  distance  of  object  from  observer. 

ILLUSTRATION.— Top  of  smoke  pipe  of  a  steamer,  assumed  to  be  50  feet  above  sur- 
face of  water,  is  in  range  with  horizon  from  an  elevation  of  100  feet-  what  is  dis 
tance  to  steamer  from  elevation  ? 

100  feet ,..=1^27) 

50    ti  _  Q  3sj  3    9  mites  distance. 


MAGNETIC   VARIATION   OF  NEEDLE. 

A merica.—  Needle  reached  a  Westerly  maximum  in  1660,  and  then  varied  to  Fast 
until  1800,  when  it  reversed  to  West. 

London  (Eng.).—  From  1576  to  1815  variation  ranged  from  11°  15'  East  to  24°  27' 
West,  when  it  receded  gradually  to  21°  in  1865. 

Jamaica  (W.  I.).— No  variation  from  year  1660. 

Diurnal  Variation.— There  is  a  small  diurnal  variation,  being  greatest  in  sum- 
mer (15')  and  least  in  winter  (7'  30"),  added  to  which  a  change  of  temperature 
affects  a  needle. 

Variation  in  U.S.— Professor  Loomis  concludes  that  the  Westerly  variation  is 
increasing  and  Easterly  diminishing  in  every  part  of  United  States  ;  that  this 
change  occurred  between  i793  and  1819,  and  that  present  annual  change  is  about 
2  in  Southern  and  Western  States,  from  3'  to  4'  in  Middle  States  and  c'  to  7'  in 
Eastern  States. 

NOTB.— Rules  for  computation  of  variation  are  empirical,  except  in  each  par. 
ticular  locality,  as  annual  and  diurnal  variations  of  needle,  added  to  local  attrar 
tiou,  render  it  altogether  unreliable. 


MAGNETIC    VARIATION    OF   NEEDLE. 


Decennial  Magnetic  Variation  in  the  IT.  S.  and. 
some  Foreign  <Jou.ntries.  From  January,  1820,  to  January,  1900. 
U.  S.  Coast  and  Geodetic  Survey.  Chas.  A.  Scltott. 


LOCATION. 

1820. 

1830. 

1840. 

1850. 

1860. 

1870. 

1880. 

1890. 

1900. 

EAST. 

Acapulco   Mex  

o 

8  < 

O 

8  7 

o 

8  Q 

0 

8  88 

o 
8.75 

o 
8  5 

o 

8.12 

o 
7  64 

0 

7  i 

10  7 

IO  2 

9.  74 

927 

8.8 

8.34 

7.  9 

Charleston  S  C 

e 

i.  70 

45 

Chicago   111      ....           . 

4-°5 
6  12 

6  28 

6  25 

6  04 

5.67 

5  15 

4-  52 

3  81 

3'  * 

Cincinnati  Obs'y  0  

5 

4.82 

4-  51 

4.08 

3-57 

2.99 

2.39 

1.8 

i.  j 

i-43 

1.  1 

.66 

.16 

Denver   Col     

I5<  *4 

14.88 

14-52 

14.06 

Detroit   Mich  

2.84 

2  49 

2.04 

i 

•93 

•34 

Duluth  and  Superior,  Minn  
El  Paso  Tex     

9.8 

12   14 

10.02 
12.34 

10.  1  1 

12.38 

1  0.06 
12.23 

9-9 

II.  Q^ 

9-5 
11.5 

Erie  Marine  Hos'l,  Peun  

•39 

.09 

Fernandina  Fla    .  . 

.  c 

4  5 

3.8 

3  2 

2.5 

I  Q 

1.2 

Florence  Ala  

6.58 

6.54 

637 

6  ii 

5-74 

5*3 

4.81 

4.28 

3.8 

Galveston  Tex 

8s' 

8  Q 

8.8 

8.  <;<; 

8  16 

7  62 

6  9 

Havana,  Coll'e  de  Belen,  Cuba. 
Milwaukee,  N.  P't,  Wis  

6.2 

6.38 

5-77 

5-39 
7.4 

4-95 
6.9 

a 

6.2 

3-97 
5.4 

7-u^ 
3.46 

4  5 

!:« 

Mexico,  Ast'l  Obs'y,  Mex  

8? 

8.5 

86 

8.62 

8.55 

8.39 

8.13 

7.77 

17.4 

Mobile  Ala    

4.6 

7-  03 

7.  i 

6  QQ 

6.71 

6.i7 

5.71 

^  06 

3 

Monterey  Cal  

i^-91 

14.45 

14.  QI 

15.32 

15.65 

15.89 

i. 

16.04 

T6? 

Nashville,  near  Van't  U'y,  Tenn. 
Newbern    N.  C 

% 

if 

1.23 

6.9; 

.7 

6.7 
OQ 

6-3 

5-78 

5.i3 

4-4 

3-6 

7.96 

8.25 

8'T6 

8^ 

7.66 

7.18 

6.59 

5.91 

5.2 

Olympia,  Wash  

18.8 

19.4 

20.  i 

20.65 

21.17 

21.63 

22.01 

22.20 

22.5 

Omaha,  Neb  

12.64 

£56 

I2-  33 

ii.  06 

11.47 

10.80 

IO.23 

9-  5^ 

89 

Pensacola,  Fla  

7.42 

7-5 

7-4 

7-  *4 

6-73 

o.  19 

5-55 

1  *5 

4-  J4 

Portland,  C.  House,  Ore  

18.8 

19.5 

20.3 

21 

21.  5 

22 

22.3 

22.5 

Port  Townsend,  Wash  

19.04 

20.06 

20.67 

21.  22 

21.71 

22.15 

22-4 

22.58 

22.  7 

St.  Louis,  Mo  

8.9 

86 

8.2 

7-7 

7-  * 

6.} 

5.6 

5 

Salt  Lake  City,  T'mpleB'k,Utah 
San  Antonio,  Tex     

9.8 

IO.  I 

10.28 

15-8 
10.31 

10.27 
10.  17 

16.54 
9.87 

16.61 
9  44 

16.46 
8  9 

16.1 
8-3 

San  Diego,  C'y  Park  Ob's,  Cal.  . 
San  Francisco,  Presidio,  Cal  
Savannah,  Hutch'n  Is'd,  Ca.  .  .  . 

WEST. 

Albany  Obs'y,  N.  Y 

11.79 
14.6 
4-7 

12.27 
15-1 

4-5 

12.67 

15-43 
4.2 

12.99 
15-8 
3.78 

13.21 
16.  ii 

3-25 

8    A/I 

IIP 

2.65 

13-32 
16.57 

2.01 

n  87 

13.2 
16.64 
i-37 

i3-7 
16.7 
.8 

Baltimore,  F't  McH.,  Md  
Bangor  Tho's  Hill   Me 

•93 

1.29 

1.77 

7-73 
2-35 

y  A    A%. 

2-99 

3-65 

1  6  48 

4.89 
16  89 

5-4 

Boston,  Mass.  .  .  . 

1    8,4 

12.9 

0     .„ 

'06 

24.48 

TD    -38 

15.92 

Buffalo,  N.  Y  

9.0 

9  73 

2    8.1 

i  67 

~ 

T  63 

Burlington  U'y,  Vt  

7   78 

•79 

8  2Q 

J-35 
8  o 

o  c8 

6 

TJCR 

12.  5 

Cambridge,  Coll'e  Obs'y,  Mass. 
Charleston,  St.  M.  Ch.,S.  C  
Cleveland,  0  

8.12 

8.7 

9-32 

9-97 

10.6 

11.18 
n6 

11.68 

12.08 
.09 

12.4 
.05 

Detroit,  Mich  

•39 

.90 

I   2 

Erie,  Pa  

06 

i  6 

'(^ 

Halifax,  N.  S  

18  2 

•3° 

18   A 

•94 

20  69 

Harrisburg,  Pa 

3 

19.4 

^'46 

* 

r    £)A 

~67 

Hartford   C    Hill,  Conn. 

e   eg 

,' 

2-9 

8  62 

5-°4 
0  80 

Ithaca,  N.  Y  

2  8 

7-93 

A  88 

6  =;8 

7   "? 

Montreal  Can                   .   ... 

o7 

8    7 

1  1  6 

12  6 

IA   6 

New  Brunswick,  N.  S  

4  66 

r   /-vR 

New  Haven,  Conn  

7    28 

8  69 

Q  Q 

New  York,  C.  Hall,  N.  Y  

5-04 

A  08 

5-97 

8   AQ 

Philadelphia,  Gi'd  Coll'e,  Pa... 
Pittsburg,  Pa  

2-44 

2.9I 

5    « 

3-46 

18 

4.07 
68 

J-3 

5-44 
i  8? 

7'9 

6.2 

0.49 
6.97 
o  06 

7-7 
3-  5 

Portland,  Bram'l  Hill,  Me  
Portsmouth,  N.  H  

9.46 
8  •* 

10.  1 

8  Q 

10.82 

19.56 
10  28 

12.29 

12.97 

13-58 

14.08 

14.4 
J3  3 

Providence,  near  B.  U'y,  R.  I.. 
Quebec,  Can  ." 

o.  j 

6-95 

0.9 
7.67 

8.49 

TO    8 

9.06 

9.67 

10.23 
16  9 

10.85 

n.48 

12 

17  5 

Toronto,  Can  

5 

i  6 

2  66 

TTfU 

4  8 

Washington,  N.  Obs'y,   D.  C... 

.2 

•65 

1.17 

1.77 

2-43 

3i 

3-72 

4.28 

6.2 

MAGNETIC    VARIATION    OF   NEEDLE.  59 

Magnetic  ^Variation  of  USTeedle  at  Locations  in. 
United  States  and.  Canada,  19OO. 

U.   8.  Coast  and   Geodetic   Survey.       Cfias.    A.   Schott. 
For  many  other  locations,  see  them  on  page  58,  under  year  1900. 

EAST. 


LOCATION. 

Varia- 
tion. 

LOCATION. 

Varia- 
tion. 

LOCATION. 

Varia- 

Aiken    S  C 

0 

.7 

Grand  Haven,  Mich. 

0 

i 

Natchez,  M  iss  

0 

5.8 

Appalachicola,  Fla.. 
Astoria,  Ore  

Augusta    Ga 

3.6 

22.6 

1.8 

Green  Bay,  Wis  
Helena,  Ark  
Helena   Mont. 

3-3 
5-5 
19.  i 

Nebraska  City,  Neb. 
New  Orleans  P'k,  La. 
Oakland    Ore  .   .   . 

8.9 
4.8 

IQ    Q 

Baton  Rouge,  La.  ... 
Billings    Mont 

5.7 

17.  4 

Huntsville,  Ala  
Indianapolis,  Ind  

3-2 
1.6 

Olympia,  Wash  
Pensacola,  Fla  

22.5 

4*  2 

Bismarck,  C.H.,  N.D. 
Brunswick  Ga.    .   . 

I4.6 

I 

Iowa  City,  Iowa.  .... 
Jackson,  Miss.   .   . 

6.4 
5.6 

Sacramento  C.G.,  Cal. 
St.  Augustine,  Fla. 

16 

i 

Cairo,   111     .. 

4.4 

Jacksonville,  Fla.  .   . 

2.3 

St.  Paul,  Minn  

8.7 

Carson  City,  Nev  
Cheyenne,    Wis 

16.6 

14.  2 

Kalamazoo,  M  ich  .  .  . 
Keokuk,  Iowa  

i 

C.Q 

Salt  Lake  T.,  Utah  .  . 
San  Antonio  Ob.,  Tex. 

s!3 

Colorado  Sp'gs  Col 

JO     Q 

Lexington    Kan 

6 

San  Bias   Mex  

8 

Columbia  S  C. 

Lexington,  Mo  

7.5 

San  Diego,C  Pk.,Cal. 

13  .  7 

Columbus  Ga  

2.  3 

L.  Rock,  Ars'l,  Ark.. 

6.6 

Santa  Barbara,  Cal  .  . 

14.6 

Darien  Ga 

I    e 

Los  Angeles  Ob    Cal 

14.2 

SantaFe,F.  M'r'y,  N.  M 

12.4 

Denver  Col  

I3"  5 

Louisville,  Ky.    . 

Seattle,  Wash  

22.5 

Dodge  City  Kan 

II    I 

Madison  Ind 

I  •  Q 

Selma  Ala.   .... 

2  9 

Dubuque,  Iowa  
Duluth,  Minn 

5-4 

T>     1 

Magdalena  B.  ,  L.  Cal. 
Magnetic  St'n,  Idaho 

IO 

17.6 

Sheboygan,    Wis  
Shreveport,  La  

2.2 

6.6 

Fort  Bowie,  Ariz..  .. 
Fort  Garland,  Cal... 
Fort  Gibson,  Ind.  T 
Ft  Leavenw'th  Kan 

X>  OOOJ  OJ 

Marelsl'd,  N.Y.,  Cal. 
Mazatlan,  Mex  
Memphis,   Tenn  
Mexico  City  0    Mex 

Ta 

o.o 
5-3 
7  .4 

Sitka,  Par'e  G,  Alaska 
Springfield,  111  
Tallahassee,  Fla  
Tampa  Fla  

29.4 
4.2 

2 
2.2 

Fort  Lyon  Col 

13  .2 

Michigan  City,  Ind.  . 

T    8 

Tuscaloosa,  Ala  

4.6 

Ft.  McKiuney,  Wyo.. 
Gainesville,  Flo  
Galena  111 

16 

2.  I 

7  * 

Milledgeville,  Ga.... 
Minneapolis,  Minn.. 
Montgomery,   Ala.  .  . 

2.7 

*:l 

VicksburgC.H.,Miss. 
Vincennes,  Ind  
Yankton,  S.  Dak.... 

5.6 
ii 

Acapulco   Mex 

WEST. 
Ithaca  NY        .... 

7    "» 

Richmond   Va     .... 

3.7 

Alleghany,  Pa  
Atlantic  City,  N.J... 
Auburn,  N.  Y  

3-6 
7.2 
8.6 

Keeseville,  N.  Y.... 
Kittery,  Me  
Knoxville,  Tenn  .... 

12.4 

13.3 
.2 

Rochester,  N.  Y.  
Rockland,  Me  
Rome,  N.  Y  

£3 

9-4 

Bath    Me            .   .  . 

14.    7 

Little  Falls   N.  Y 

Rutland,  C.  P'k,  Vt.. 

12  .4 

Beaver  Pa. 

2    8 

Lowell     Mass  . 

Saginaw  Mich.... 

Belfast    Me 

Lynn   Mass  .   .  . 

12    2 

Sandusky  0  

•J 

Bellows  Falls,  Vt.  .  .  . 
Bridgeport   Conn 

12.4 

Mackinac,  Mich  
Madison  O           .... 

1.6 
3  2 

Saybrook,  Conn  
Schenectady   N  Y 

10.4 

jO 

Buffalo,  N.  Y  
Calais,  Me  

Cape  May  N  J 

r 

T5 

Marietta,  0  
Newark,  N.  J  
Newbern  N'l  C'y  N  C 

1:1 

2    6 

South  Bethlehem,  Pa. 
Springfield,  Mass  
Stamford   Conn  

7.2 

II.  2 

Carlisle,  Pa  
Chambersburg,  Pa.  . 
Cheboygan   Mich 

5-2 

5.03 

Newburyport,  Mass.. 
NewLondon,G.Pt.,Ct. 
Newport   R.  I  

12.8 

11.  1 

12 

Stonington,  Conn... 
Tappan&PTdes,N.Y. 
Toledo  M'n  Line,  O.  . 

II.  2 

9.2 
1.5 

Columbus  'O 

New  Rochelle    N   Y 

8  s 

7  *9 

Concord,  N.  H  
Daubury,  Conn  
Delaware  City   Del 

12.4 

12.6 

Norfolk  C.  H.,  Va... 
Norwalk,  Conn  
Machinac    Mich 

4 

IO 

i  6 

Troy,  N.  Y  
Union  town,  Pa  
Utica  N  Y  

TR 

Dunkirk,  N.  Y  
Geneva  N  Y 

4.6 

Oswego,  N.  Y  
Ottawa    Can 

8-5 

Wash'tonN.Ob.,D.C. 
Wheeling   Va 

3-9 

Gettysburg,  Pa  
Greenport,  N.  Y.  .  .  . 
Hackensack,  N.J...  . 
Hanover   N  H 

6.1 

10.8 
8.7 

12    8 

Owego,  N.  Y  
Penobscot,  Me  
Perth  Amboy,  N.J... 
Pittsburg  Pa    

7.8 
19 
8-5 
3-6 

Williamsburg,  Va  
Wilmington,  N.  C... 
Wilmington,  Del  
York    Pa        

3-9 
1.6 

Hudson  'N   Y 

IO.2 

Provincetown    Mass. 

12.9 

Zanesvilie.  0  

i  .  i 

Huntington,  Pa  

5.6 

Raleigh,  nr  Cap'l,N.C. 

1.8 

Ypsilanti,  Mich  

2.2 

GEOGRAPHIC  LEVELLING. — BASE  LINE. — SOUNDINGS. 


Dip  of  Horizon- 
Approximate,  57.4  VR=-dip  in  seconds,  varying  with  temperature 

air.  H  representing  height  of  observer's  eye  in  feet. 
.667w2=H:  «498s2=H:  1.42 -/H  =  «: 

n  representing  distance  in  geographical  miles  and  s  in  statute. 


Multi- 
plier. 

3V£ea 

Angle. 

sureir 

Multi- 
plier. 

lent  o 

Angle. 

C   Hei 

Multi- 
plier. 

gilts    \vith 

H  Multi- 
Angle.    II  piier. 

a   Sext 
Angle 

ant. 

Multi- 
plier. 

Angle. 

1-5 

2 

si 

63  26 

2.5 
3 
3-5 

68  II 
7i  34 

4 
4-5 

75  58        5-5 
77  29        6 

79  42 
8d  32 
81  52 

8 

9 
10 

82  52 
83  40 
84  17 

Operation.  —  Set  sextant  to  any  angle  in  table,  and  height  will  equal  distance 
multiplied  by  number  opposite  to  it 

ILLUSTRATION.  —  When  sextant  is  set  at  80°  32',  and  horizontal  distance  from  ob- 
ject in  a  vertical  line  is  100  feet,  what  is  its  height? 
100  X  6  =  600  feet 
By  Trigonometry:  1:100::  5.997  (tan.  angle)  ;  599.7  feet. 

To  Reduce   a   Sounding:   to   Low  Water. 

-  f  i  q:  cos.  -  --  J  =  h'.  h  representing  vertical  rise  of  tide,  and  h'  sound* 
ing  or  depth  at  low  water,  both  in  feet  ;  t  time  between  high  and  low  water,  and 
t  time  from  time  of  sounding  to  low  water,  in  hours.  —  cos.  when  -  <9Q°, 
and  +  cos.  when  >9o°. 

ILLUSTRATION.  —Low  water  occurring  at  3.45,  and  high  water  at  10.15  P.M.,  a 
sounding  taken  at  5.30  P.M.  was  18.25  feet;  what  was  depth  at  low  water,  vertical 
rise  being  10  feet? 

h  =  10  feet  ;  t'  =  5^.  ymi.  —  3^.  45W,.  =  ih.  45*11.  =  i.  75  hours. 

t  ==  loh.  ism.  —  3/1.  45  m.  =  6/1.  307/1.  =  6.  5  hours. 


Then 


i  ipcos. 


=5  (i-coa  48°  27'  4i")=5x  (1—663  i24)=i.68438/tf«fe 


Sounding  18.  25  feet  —  Reduction  1.68407  feet  =  16.565  93  feet 


Lengths  of  a  Degree  of  Longitude  on  parallels  of  Lati- 
tude, for  eaclx  of  its  Degrees  from  Equator  to  IPole. 


Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

i° 

59-99 

1  6° 

57-67 

3i° 

51-43 

460 

41.68 

61° 

29.09 

"76°" 

I4-52 

2 

59-  96 

17 

57-38 

32 

50.88 

47 

40.92 

62 

28.17 

77 

13-5 

3 

59-92 

18 

57.06 

33 

50-32 

48 

40.15 

63 

27.74 

78 

12.48 

4 

59-  85 

19 

56.73 

34 

49-74 

49 

39-36 

64 

26.3 

79 

"-45 

5 

59-77 

20 

56.38 

35 

49-  J  5 

50 

38.57 

65 

25-36 

80 

10.42 

6 

59-67 

21 

56-01 

36 

48.54 

5i 

37-76 

66 

24.4 

81 

9-38 

7 

59-55 

22 

55-63 

37 

47.92 

.52 

36.94 

67 

23-44 

82 

8-35 

8 

59-42 

23 

55-23 

38 

4728 

53 

36.11 

68 

22.48 

83 

7-3i 

9 

59-26 

24 

S4-8i 

39 

46-63 

54 

35-27 

69 

21.5 

84 

6.27 

10 

59-  °9 

25 

54-38 

40 

45-96 

55  i  34  4i 

7° 

20.52 

85 

5-23 

ii 

58-89 

26 

53-93 

4i 

45-28 

56 

33-45 

7i 

1953 

86 

4.18 

12 

58.69 

27 

53-46 

42 

44-59 

57   32-68 

72 

18.54 

87 

3-H 

13 

58.46 

28 

52.97 

43 

43-88 

58  '  31  79 

73 

17-54 

88 

2 

14 

58.22 

29 

52.48 

44 

43.16 

59  i  30-9 

74 

16.54 

89 

1.05 

IS 

57-95 

30 

51.96 

45 

42.43 

60   30 

75 

15-53 

90 

.00 

NOTE.  —  Degrees  of  longitude  are  to  each  other  in  length  as  Cosines  of  theii 
latitudes. 


FIGURE  OF  EARTH. — BOAED  AND  TIMBER  MEASURE.  6 1 

Klements   of  Figure   of  ttie   Earth. 

Capt.  A.  R.  Clarke,  1866. 

Feet.  Mile*. 

Major  semi-axis  of  Equator  (longitude  15°  34'  E. ) 2o  926  350  3  963. 324. 

Minor    "       '     "        "        (               105°  34' E.) 20919972  3962.115. 

Polar     "      "      20853429  3949.513. 

Equatorial  semi-axis 20  926  062  3  963. 269. 

Circumference,  mean 24  898. 562. 

Diameter,             "     « 79J6- 


BOARD  AND  TIMBER  MEASURE. 

BOARD   MEASURE. 

In  Board  Measure,  all  boards  are  assumed  to  be  i  inch  in  thickness 
To   Compute   Measure   or   Surface. 

When  all  Dimensions  are  in  Feet. 

RULE. — Multiply  length  by  breadth,  and  product  will  give  surface  in 
square  feet. 

When  either  of  Dimensions  are  in  Inches. 

EXAMPLE.  —  What  are  number  of  square  feet  in  a  board  15  feet  in  length  and  16 
inches  in  width? 

15  X  16  =  240,  and  240  -f- 12  =  20  sq.  feet. 

When  all  Dimensions  are  in  Inches. 
RULE.— Multiply  as  before,  and  divide  product  by  144. 


TIMBER   MEASURE. 
To   Compute  "Volume    of  Round  Timber. 

When  all  Dimensions  are  in  Feet. 

RULE. — Add  together  squares  of  diameters  of  greater  and  lesser  ends, 
and  product  of  the  two  diameters ;  multiply  sum  by  .7854,  and  product 
by  one  third  of  length. 

Or,  a  +  a'-\-a"  x  -  =  V,  and  c2  +  c'2  -f  c  x  c'  x  .07958  X  -  =  V.    a  and 

a'  representing  areas  of  ends,  a"  area  of  mean  proportional,  I  length,  and  c 
and  c'  circumference  of  ends. 

NOTE.— Mean  proportional  is  square  root  of  product  of  areas  of  both  ends. 

ILLUSTRATION.— Diameters  of  a  log  are  2  and  1.5  feet,  and  length  15  feet. 


(aa4-x.sa+2X  1.5)  =  9-25,  which  X  .7854  and  ——.36.3245  cube  feet. 

When  Length  in  Feet,  and  Areas  or  Circumferences  in  Inches. 
RULE. — Proceed  as  above,  and  divide  by  144. 

When  all  Dimensions  are  in  Inches. 
RULE. — Proceed  as  before,  and  divide  by  1728. 
NOTE.  —  Ordinary  rule  of  Hutton,  Ordnance  Manual  of  U.  S.,  and  Molesworth,  of 

I  X  c-f-  4,  giVv  s  a  result  of  about  .25  less  than  exact  volume,  or  what  it  would  be 
if  the  log  was  hewn  or  sawed  to  a  square,     c  representing  mean  circumferences. 

F 


62 


BOARD   AND   TIMBER   MEASURE. 


To   Compute  "Volume    of  Squared.  Tim"ber. 

When  all  Dimensions  are  in  Feet. 

RULE.—  Multiply  product  of  breadth  by  depth,  by  length,  and  product 
will  give  volume  in  cube  feet. 

When  either  Dimension  is  in  Inches. 
RULE.  —  Multiply  as  above,  and  divide  product  by  12. 
When  any  two  Dimensions  are  in  Inches. 
RULE.  —  Multiply  as  before,  and  divide  by  144. 

EXAMPLE.—  A  piece  of  timber  is  15  inches  square,  and  20  feet  in  length;  required 
its  volume  in  cube  feet. 

15  XX20 


lao  deals  .........  =  i  hundred. 


Allowance  is  to  be  made  for  bark,  by  deducting  from  each  girth  from 
.5  inch  in  logs  with  thin  bark,  to  2  inches  in  logs  with  thick  bark. 

3VIeasu.res   of  Timlber. — (English.) 

50  cube  feet  of  squared  > i     , 

timber  J  —     loaa* 

40  feet  of  unhewn  timber  =  i  load. 
600  superficial  feet  of  inch  planking  =  i  load. 

Deals. 

Deals.  —  Boards  exceeding  7  ins.  in  width,  and  if  less  than  6  feet  in 
length,  are  termed  deal  ends. 

Battens  are  similar  to  deals,  but  only  7  inches  in  width. 
Balk. — Roughly  squared  log  or  trunk  of  a  tree. 
Planks  are  boards  12  ins.  in  width. 


Country. 

Long. 

Broad. 

Loc 

Thick. 

al   St 

Volume. 

andards. 

Country. 

Long. 

Broad. 

Thick. 

Volume. 

Russia  and 
Prussia  .  . 
Sweden  .  .  . 

Ft. 

12 
14 

Ins. 
II 

9 

Ins. 

i-5 
3 

Cub.  ft. 

1-375 
2.625 

Norway  .  . 
Christiana 
Quebec.  .  . 

Ft. 
12 
II 
12 

Ins. 
9 
9 
II 

Ins. 
3 
1.25 

2-5 

Cub.  ft. 
2.25 

•859 
2.292 

ioo  Petersburgh  standard  deals  equal  60  Quebec  deals. 


SPARS    AND   POLES. 

Pine  and  Spruce  Spars,  from  10  to  4.5  inches  in  diameter  inclusive, 
are  to  be  measured  by  taking  their  diameter,  clear  of  bark,  at  one  third 
of  their  length  from  abut  or  large  end. 

Spars  are  usually  purchased  by  the  inch  diameter ;  all  under  4  inches 
are  termed  Poles. 

Spars  of  7  inches  and  less  should  have  5  feet  in  length  for  every 
inch  of  diameter,  and  those  above  7  inches  should  have  4  feet  in  length 
for  every  inch  of  diameter. 

IJOBS    or   Waste   in    Hewing   or    Sawing   of  Timber. 

(C.  Mackrow.) 


Oak,  English 200  per  cent. 

"    African ioo   " 

"    Dantzic 50  "      " 

"    American 10  "      " 


Yellow  Pine  from  planks. .    10  per  cent. 

Teak 15    "      " 

Elm,  English 200   " 

"    American 15  **      '* 


CISTERNS. — SHINGLES. 


CISTERNS. 

Capacity  of  Cisterns  in   Cu.be   Feet  and  GJ-allons. 

For  each  10  Indies  in  Depth. 

Diam. 

Cub.  ft. 

Gallons. 

Diara. 

Cub.  ft. 

Gallons. 

Diam.  |  Cub.  ft.   |  Gallons. 

Feet. 

Feet. 

Feet. 

2 

2.618 

19.58 

9-5 

59.068 

441.8    ! 

17 

189.15 

1414.94 

2-5 

4.091 

30.6 

10 

65449 

489.6 

17-5 

200.432 

J499-33 

3 

5.89 

44.07 

10.5 

72.158 

539.78 

18 

212.056 

1586.28 

3-5 

8.018 

59-97 

ii 

79.194 

592-4 

19 

236.274 

1  767.45 

4 

10.472 

78.33 

"•5 

86.558 

647-5 

20 

261.797 

1958.3 

4-5 

i3-254 

99.14 

12 

94.248 

705 

21 

288.632 

2159.11 

5 

16.362 

122.4 

12.5 

102.265 

764.99 

22 

316.776 

2369.64 

5-5 

19.798 

148.1 

13 

110.61 

827.4 

23 

346.23 

2589-97 

6 

23.562 

176.24 

13-5 

119.282 

892.29 

24 

376.992 

2820.09 

6.5 

27.652 

206.84 

14 

128.281 

959-6 

25 

409.062 

3059-8 

7 

32.07 

239-88 

14-5 

137.608 

1029.38 

26 

442.44 

3309-67 

7-5 

36.816 

275-4 

15 

147.262 

noi.6 

27 

47LI3 

3569-17 

8 

41.888 

3!3«33 

15.5 

I57-243 

1176.26 

28 

513.126 

3838.44 

8-5 
9 

47.288 
530H 

353-72 
39^-55 

16 
16.5 

rfj-SS2 
178.187 

1253-37 
1332.93 

29           550.432 
30         |  589.048 

4"7.5i 
4406.08 

Excavation    and   Lining   of  "Wells    or    Cisterns. 

For  each  10  Inches  in  Depth. 


1 

1 

Bricks. 

Masonry. 

1 

1 

Bricks. 

Masonry. 

9 

5 

Num- 

Laid 

8  inches 

i  foot 

1 

09 

Num- 

Laid 

8  inches 

i  foot 

5 

w 

ber. 

dry. 

thick. 

thick. 

g 

i 

ber. 

dry. 

thick. 

thick. 

Feet. 

Cub.  ft. 

Cub.  ft. 

Cub.  ft. 

Cub.  ft. 

Feet. 

Cub.  ft. 

Cub.  ft. 

Cub.  ft. 

Cub.ft. 

3 

12.29 

126 

5-24 

6.4 

10.47 

8.5 

63.29 

356 

14.83 

16 

24.87 

3-5 

15.29 

M7 

6.  ii 

7.27 

11.78 

9 

69.89 

377 

16.87 

26.18 

4 

18.62 

168 

6.98 

8.14 

13.09 

9-5 

76.81 

398 

16^58 

17-75 

27.49 

4-5 

22.27 

1  88 

7.85 

9.02 

14.4 

10 

84.07 

419 

17-45 

18.62 

28.8 

5 

26.25 

209 

8-73 

9.89 

10.5 

91.65 

440 

18.33 

19.49 

30.11 

V 

30-56 
35-2 

230 
251 

9.6 
10.47 

10.76 
11.64 

17.02 
18.33 

ii 

12 

99-56 
1  16.  36 

461 
503 

19.2 
20.94 

20.36 

22.11 

31.42 
34-03 

6.5 

40.16 

272 

"•34 

12.51 

19.63 

13 

134.46 

545 

22.69 

23.85 

36.65 

7 

45-45 

293 

12.22 

I3-38 

20.94 

14 

153-88 

586 

24-43 

25-6 

39-27 

7-5 

51.07 

13.09 

14.25 

22.25 

15 

174.61 

628 

26.18 

27-34 

41.89 

8 

57-02 

335 

13.96 

I5-I3 

23-56 

16 

196.64 

670 

27.92 

29.09 

44-51 

Number  of  bricks  and  width  of  curb  are  taken  at  dimensions  of  ordinary 
brick — viz.,  8  by  4  by  2.25  ins.  =  72  cube  ins. 

In  computing  number  of  bricks  required,  an  addition  of  5  per  cent,  should 
be  added  for  waste.  It  is  to  be  considered,  also,  that  diameter  of  excavation 
necessarily  exceeds  that  of  masonry. 


SHINGLES. 

Usually  of  white  Cedar  and  Cypress ;  27  inches  in  length  and  6  to  7 
inches  m  width,  dressed  to  light  .25  inch  at  point  and  .3125  inch  at 
abut. 

Laid  in  three  thicknesses  and  courses  of  about  8  inches,  so  that  less 
than  .33  of  a  shingle  is  exposed  to  air,  or  about  2.25  shingles  are  re- 
quired per  square  foot  of  roof. 

Shingles,  alike  to  Slates,  are  laid  upon  boards  or  battens. 


64 


SLATES   AND    SLATING. 


SLATES   AND   SLATING. 

A  Square  of  Slate  or  Slating  is  100  superficial  feet. 
-      Gauge  is  distance  between  the  courses  of  the  slates. 

Lap  is  distance  which  each  slate  overlaps  the  slate  lengthwise  next 
but  one  below  it,  and  it  varies  from  2  to  4  inches.  Standard  is  assumed 
to  be  3  inches. 

Margin  is  width  of  course  exposed  or  distance  between  tails  of  the 
slates. 
Pitch  of  a  slate  roof  should  not  be  less  than  i  in  height  to  4  of  length. 

Po    Compnte    Surface   of  a   Slate    when,    laid,  arid.    Num- 
ber   of  Sq.vi.ares    of  Slating. 

RULE.  —  Subtract  lap  from  length*  of  slate,  and  half  remainder  will 
give  length  of  surface  exposed,  which,  when  multiplied  by  width  of 
slate,  will  give  surface  required. 

Divide  14  400  (area  of  a  square  in  inches)  by  surface  thus  obtained, 
and  quotient  will  give  number  of  slates  required  for  a  square. 

EXAMPLE.  —A  slate  is  24  X  12  inches,  and  lap  is  3  inches;  what  will  be  number 
required  for  a  square? 

24  —  3  =  21,  and  21  -f- 2  =  10.5,  which  X  12  =  126  inches;  and  14 400 -=-  126  = 
114.29  slates. 

Dimensions   of*  Slates. 
[AMERICAN.] 


Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

14  X  7 
14  X  8 
14X9 

14  X  10 
i6x  8 
i6x  9 

16  X  10 
18  X  9 
18  X  10 

18  X  ii 
18  X  12 

20  X  10 

20  X  II 
20  X  12 
22  X  II 

22  X  12 
22  X  13 
24  X  12 

24  X  13 
24  X  14 
24  X  16 

ENGLISH. 


Ins. 

Ins 

Ins. 

Doubles  

13  X  10 

C 

I2X    8 

Marchioness  .  . 

22X22 

u 

11  V     7 

1 

I4X   8 

Duchess  

24  X  12 

Small  doubles  . 

iiX  6 
iox  5 

Ladies  •( 

14X12 
15  X   8 

Imperial  
Rags  

30X24 
36X24 

f 

12  X  IO 

i6x   8 

->6X  24 

Plantations  .  .  j 
Viscountess  .  .  . 

13X10 

18X10 

Countess  

16x10 
20X10 

Empress  
Princess  

26X15 
24X14 

Thickness  of  slates  ranges  from  .125  to  .3125  of  an  inch,  and  their  weight 
varies  from  2  to  4.53  Ibs.  per  sq.  foot. 


Weight  of  One   Square   Foot   of  Slating, 

125  in.  thick  on  laths 4.75  Ibs, 

"    "      "      "  i  in.  boards. .  6.75    " 
1875  in.  thick  on  laths 7         " 

"     t:      "      "  i  in.  boards.  9        " 


.25  in.  thick  on  laths 9.25  Ibs- 

"    "      "      "   i  in.  boards..  11.25    " 

.3125  in.  thick  on  laths 11.15    " 

"     "      "      u   i  in.  boards,  14. 10    " 


Slate  weighs  from  167  to  181  Ibs.  per  cube  foot,  and  in  consequence  of 
laps,  it  requires  an  average  of  nearly  2.5  square  feet  of  slate  to  make  one  of 
slating. 

Weights  per  1000  and  Number  Required  to  Cover  a  Square. 


Doubles  13  x  6 

Lbs. 
1680 

480 

Countess  .  .  .  20  x  10 

Lbs. 
6720 

No. 
171 

2800 

2A.O 

Duchess  .  .  .  24  x  12 

44.8o 

i2«: 

*  Length  of  a  slate  is  iaken  from  nail-hole  to  tail. 


SHOT   AND    SHELLS.  —  FRAUDULENT   BALANCES.        65 
PILING  OF  SHOT  AND  SHELLS. 

To    Compute    NnmlDer    of   Sh.ot. 

Triangular  Pile.  RULE.  —  Multiply  continually  together,  number  of  shot 
in  one  side  of  bottom  course,  and  that  number  increased  by  i,  and  again  by 
2,  and  one  sixth  of  product  will  give  number. 

EXAMPLE.  —  What  is  number  of  shot  in  a  triangular  pile,  each  side  of  base  contain- 
ing 30  shot? 

30X30+1X30  +  2  =  £976o  shQt 

6  6 

Square  Pile.  RULE.  —  Multiply  continually  together,  number  in  one  side 
of  bottom  course,  and  that  number  increased  by  i,  double  same  number  in- 
creased by  i,  and  one  sixth  of  product  will  give  number. 

EXAMPLE.—  How  many  shells  are  there  in  a  square  pile  of  30  courses? 


Oblong  Pile.  RULE.  —  From  3  times  number  in  length  of  base  course  sub- 
tract one  less  than  number  in  breadth  of  it  ;  multiply  remainder  by  number 
in  breadth,  and  again  by  breadth,  increased  by  i,  and  one  sixth  of  product 
will  give  number. 

EXAMPLE.—  Required  number  of  shells  in  an  oblong  pile,  numbers  in  base  course 
being  16  and  7  ? 

,6X3-7=7X7XJTI  =  *J£  = 


Incomplete  Pile.  RULE.  —  From  number  in  pile,  considered  as  complete, 
subtract  number  conceived  to  be  in  that  portion  of  pile  which  is  wanting, 
and  remainder  will  give  number. 


FRAUDULENT   BALANCES. 

To  Detect  Them. After  an  equilibrium  has  been  established  between 

weight  and  article  weighed,  transpose  them,  and  weight  will  preponder- 
ate if  article  weighed  is  lighter  than  weight,  and  contrariwise  if  it  is 
heavier. 

To  Ascertain  True  Weight.  RULE.— Ascertain  weight  which  will  produce 
equilibrium  after  article  to  be  weighed  and  weight  have  been  transposed; 
reduce  these  weights  to  same  denomination,  multiply  them  together,  and 
square  root  of  their  product  will  give  true  weight 

EXAMPLE.  -If  first  weight  is  32  Ibs.,  and  second,  or  weight  of  equilibrium  after 
transposition,  is  24  Ibs.  8  oz.,  what  is  true  weight? 

24  Ibs.  8  oz.  =24.5  Ibs. 

Then  32  X  24. 5  =  784,  and  v/784  =  a8  lbs- 

Or,  when  a  represents  longest  arm,         \         A  greatest  weight,  and 
b         "         shortest  arm,        \         B  least  weight. 

Then  Wa= Aft,  and  W6=:Ba;  multiplying  these  two  equations,  W2a6  =  ABa6, 
or  W2  =  AB,  and  W  =  VAB- 

ILLUSTRATION.  —  A  =  32 ;  B  =  24. 5 ;  W  =  28.    Assume  length  of  longest  arm  -.=  i* 
Then  32  :  28  ::  10  :  8.75. 

He^jce,  a  =  10,  6  =  8. 75,  or  282  =  32  X  24-  5,  and  ' 


66 


WEIGHING   WITHOUT   SCALES. — PAINTING. 


"Weighing  wi.th.oirt   Scales. 

To    Ascertain    AVeiglit   of  a    Bar,  Beam,  etc.,  t>y   Aid    of 

a    known    "VVeignt. 

OPERATION. — Balance  bar,  etc.,  over  a  fulcrum,  and  note  distance  between 
it  and  end  of  its  longest  arm.  Suspend  a  known  weight  from  longest  arm, 
and  move  bar,  etc.,  upon  fulcrum,  so  that  bar  with  attached  weight  will  be 
in  equilibrio ;  subtract  distance  between  the  two  positions  of  fulcrum  from 
longest  arm  first  obtained ;  multiply  this  remainder  by  weight  suspended, 
divide  product  by  distance  between  f  ulcrums,  and  quotient  will  give  weight. 

EXAMPLE. — A  piece  of  tapered  timber  24  feet  in  length  is  balanced  over  a  fulcrum 
when  13  feet  from  less  end;  but  when  the  body  of  a  man  weighing  210  Ibs.  is  sus- 
pendecTfrom  extreme  of  longest  arm,  the  piece  and  weight  are  balanced  when  ful- 
crum is  12  feet  from  this  end.  What  is  weight  of  the  timber? 

13  — 12  =  i,  and  13  —  i  =  12  feet.    Then  12  X  210-:- 1  =  2520  Ibs. 


PAINTING. 

i  pound  of  paint  will  cover  about  4  square  yards  for  a  first  coat  and  about 
6  yards  for  each  additional  coat. 

Proportions  of  Colors  for  ordinary  Paints. — By  "Weight. 


COLORS. 

-•3 

£J 

ti 

HJ3 

li 

i 

m 

i* 
« 

l! 

COLORS. 

II 

White  .  .  . 

IOO 

Lead  

08 

Black  
(ireen  

25 

IOO 

— 



75 

— 

Red  
Chocolate.  . 

These  are  the  colors  alone,  to  which  boiled  linseed  oil,  litharge,  Japan  varnish, 
and  spirits  turpentine  are  to  be  added  according  to  the  application  of  the  paint. 

Lamp-black  and  litharge  are  ground  separately  with  oil,  then  stirred  into  the 
lead  and  oil. 

Thus  for  black  paint:  Lamp-black  25  parts,  litharge  i,  Japan  varnish  i,  boiled  lin- 
seed oil  72,  and  spirits  turpentine  i. 

Tar  Paint.— Coal  tar  9  gallons,  slaked  lime  13  Ibs.,  turpentine  or  naphtha  2 
or  3  quarts. 


A  GALLON  OF  PAINT  WILL  COVER 

Superficial  1 
feet. 

A  GALLON  OF  PAINT  WILL  COVBR 

Superficial 
feet. 

On  stone  or  brick,  about  
On  composite,  etc.,  from  
On  wood,  from  .  .  , 

190  to  225 
300  "  375 
37*:  "  $2* 

On  well-painted  surface  or  iron 
One  gallon  tar,  first  coat  
"       "       "   second  coat  ... 

600 
90 
160 

Boiled   Oil.— Raw  linseed  oil  91  parts,  copperas  3,  and  litharge  6. 
Put  litharge  and  copperas  in  a  cloth  bag  and  suspend  in  middle  of  a  kettle.     Boil 
oil  four  hours  and  a  half  over  a  slow  fire,  then  let  it  stand  and  deposit  the  sediment. 

\Vhite    Paint. 

Inside  work.    Outside  work. 

White  lead,  in  oil . .    80     80  Raw  oil 

Boiled  oil 14. 5 9  Spirits  turpentine. 


New  wood- work  requires  i  Ib.  to  square  yard  for  three  coats. 

Coats  for  100  Square  Yards  New  White  Pine. 


Inside  work.    Outside  work. 
•       9 
4 


8 


INSIDE. 

White 
lead. 

Raw 

Turpen- 
tine. 

Drier. 

OUTSIDE. 

White 
lead. 

Raw 

oil. 

Boiled 
oil. 

Turpen- 
tine. 

Priming  
2d  coat  
3d    "    

Lba. 
16 
15 
13 

Pt3. 

3-5 
2-5 

Pt8. 

6 
i-5 
i-5 

Lbs. 
•25 
•25 
•25 

Priming  — 
2d  and  3d  ) 
coats     j 

Lbs. 

18.5 

15 

Pts. 

2 
2 

Pte. 

2 
2 

Pts. 
•5 

.1  Ib.  of  drier  with  priming  and  coating  for  outside. 


HYDROMETERS. 


HYDROMETERS. 

U.  S.  Hydrometer  (Tralle's)  ranges  from  o  (water)  to  100  (pure  spirit) ; 
it  has  not  any  subdivision  or  standard  termed  "Proof,"  but  50,  upon 
stem  of  instrument,  at  a  temperature  of  60°,  is  basis  upon  which  com- 
putations of  duties  are  made. 

In  connection  with  this  instrument,  a  Table  of  Corrections,  for  differences  in  tem- 
perature of  spirits,  becomes  necessary ;  and  one  is  furnished  by  the  Treasury  De- 
partment, from  which  all  computations  of  value  of  a  spirit  are  made. 

ILLUSTRATION.  —A  cask  contains  100  gallons  of  whiskey  at  70°,  and  hydrometer 
sinks  in  the  spirit  to  25  upon  its  stem. 

Then,  by  table,  under  70°,  and  opposite  to  25,  is  22.99,  showing  that  there  are  22.99 
gallons  of  pure  spirit  in  the  100. 

Commercial  Hydrometer  (Gendar's)  has  a  "  Proof "  at  60°,  which  is 
equal  to  50  upon  U.  S.  Instrument  and  its  gradations,  run  up  to  100 
with  it,  and  down  to  10  below  proof,  at  o  upon  U.  S.  Instrument ;  or  o 
of  the  Commercial  Instrument  is  at  50  upon  U.  S.  Instrument,  from 
which  it  progresses  numerically  each  way,  each  of  its  divisions  being 
equal  to  two  of  latter. 

In  testing  spirits,  Commercial  standard  of  value  is  fixed  at  proof; 
hence  any  difference,  whether  higher  or  lower,  is  added  or  subtracted, 
as  case  may  be,  to  or  from  value  assigned  to  proof. 

A  scale  of  Corrections  for  temperature  being  necessary,  one  is  fur- 
nished with  a  Thermometer. 

Application  of  Thermometer.—  Elevation  of  the  mercury  indicates  correction  to 
be  added  or  subtracted,  to  or  from  indication  upon  stem  of  hydrometer. 

When  elevation  is  above  60°,  subtract  correction ;  and  when  below,  add  it. 

ILLUSTRATION.— A  hydrometer  in  a  spirit  indicates  upon  its  stem  50  below  proof, 
and  thermometer  indicates  4  above  60°  in  appropriate  column. 
Then  50  —  4  =  46  =  strength  below  proof, 

To  Compute  Strength,  of  a  Spirit,  or  ^Volume  of*  its  Pure 
Spirit,  "by  Commercial  Hydrometer,  and.  Convert  it  to 
Indication  of  a  TJ.  S.  Hydrometer. 

When  Spirit  is  above  Proof.    RULE.— Add  100  to  indication,  and  divide  sum  by  2. 

When  Spirit  is  below  Proof.  RULE.  — Subtract  indication  from  100,  and  divide 
remainder  by  2. 

EXAMPLE.  —A  spirit  is  n  above  proof  by  a  Commercial  Hydrometer;  what  pro- 
portion of  pure  spirit  does  it  contain  ? 

n-f-ioo-f-2  =  55.5  per  cent. 

To   Compute    Strength,  etc.,  by   a   U.  S.  Hydrometer. 

When  Spirit  is  above  Proof    RULE.— Multiply  indication  by  2,  and  subtract  100. 

When  Spirit  is  below  Proof.  RULE.  —  Multiply  indication  by  2,  and  subtract  it 
from  ioo. 

EXAMPLE,— A  spirit  is  55.5;  what  is  its  per  centage  above  proof? 

55.5X2  —  ioo  =  n  per  cent. 

Commercial  practice  of  reducing  indications  of  a  hydrometer  is  as  follows: 
Multiply  number  of  gallons  of  spirit  by  per  centage  or  number  of  degrees  above 

or  below  proof,  divide  by  ioo,  and  quotient  will  give  number  of  gallons  to  be  added 

er  subtracted,  as  case  may  be. 

ILLUSTRATION.  — 50  gallons  of  whiskey  are  1 1  per  cent,  above  proof. 

Then  50  X  1 1  -=-  ioo  =  5. 5,  which  added  to  50  =  55. 5  gallons. 


68 


HYGROMETER. 


HYGROMETER. 

Dew-point. — When  air  is  gradually  lowered  in  its  temperature  at  a 
constant  pressure,  its  density  increases,  and  ratio  of  increase  is  sensibly 
same  for  the  vapor  as  for  the  air  with  which  it  is  combined,  until  a  point  is 
reached  at  which  the  density  of  the  vapor  becomes  equal  to  the  maximum 
density  corresponding  to  the  temperature. 

This  temperature  is  termed  dew-point  of  given  mass,  and  any  further  re- 
duction of  it  will  induce  the  condensation  of  a  portion  of  the  vapor  in  form 
of  dew,  rain,  snow,  or  frost,  according  as  temperature  of  surface  is  above  or 
below  freezing  point. 

Mason's    or   like    Hygrometer. 
To   Ascertain    Dew-point. 

RULE.  —  Subtract  absolute  dryness  from  temperature  of  air,  and  remainder  is 
dew-point. 
EXAMPLE.— Temperature  of  air  57°,  and  absolute  dryness  7°. 

Hence  57°  —  7°  =  50°  dew-point. 

To   Ascertain   Absolute   Existing   Dryness. 
RULE. — Subtract  temperature  of  wet  bulb  from  temperature  of  air,  as  indicated 
by  a  dry  bulb,  add  excess  of  dryness  from  following  table,  multiply  sum  by  2,  and 
product  will  give  absolute  dryness  in  degrees. 
EXAMPLE. — Temperature  of  air  57°,  wet  bulb  54°  • 

Then  57°  —  54°  =  3°,  and  3°  +  -5°  (from  table)  X  2  =  7°  absolute  dryness. 


Observed  j  Excess  of 
Dryness.  |  Dryness. 

Observed 
Dryness. 

Excess  of 
Dryness. 

Observed 
Dryness. 

Excess  of 
Dryness. 

Observed 
Dryness. 

Excess  of 
Dryness. 

Observed 
Dryness. 

Excess  of 
Dryness. 

•5 

I 

.083 
.166 

5 

5-5 

.8°33 
.9165 

9-5 

0 

1.583 
1.666 

14 
M-5 

8-3°33 
2.4165 

18.5 
'9 

3.0*83 
3.166 

i-5 

•2495 

6 

o-5 

1-7495 

15 

2-5 

19-5 

3-2495 

2 

•333 

6-5 

.083 

i 

1-833 

15-5 

2.583 

20 

3-333 

2-5 

.4165 

7 

.166 

i-5 

1-9165 

16 

2.666 

20.5 

3-4165 

3 

•5 

7-5 

•2495 

2 

2 

16.5 

2-7495 

21 

3-5 

3-5 

•583 

8 

•333 

2-5 

2.083 

17 

2-833 

2I-5 

3-583 

4 

.666 

8-5 

•4165 

3 

2.166 

*7oS 

2.9165 

22 

3.666 

4-5 

•7495 

9 

•5 

3-5 

2.2495 

18 

3 

22.5 

3-7495 

To    Coxnpiate   "Volume    of  "Vapor   in    Atmosphere. 

By  a  Hygrometer. 

When  temperature  of  atmosphere  in  shade,  and  of  dew-point  are  given. — If  temper- 
ature of  air  and  dew-point  correspond,  which  is  the  case  when  both  thermometers 
are  alike,  and  air  consequently  saturated  with  moisture,  then  in  table*  opposite  to 
temperature  will  be  found  corresponding  weight  of  a  cube  foot  of  vapor  in  grains. 

ILLUSTRATION.— Assume  temperature  of  air  and  dew-point  70°.  Then  opposite 
temperature  weight  of  a  cube  foot  of  vapor  =  8. 392  grains. 

But  if  temperature  of  air  is  different  from  dew-point,  a  correction  is  necessary  to 
obtain  exact  weight. 

ILLUSTRATION.— Assume  dew-point  70°  as  before,  but  temperature  of  air  in  shade 
80°,  then  the  vapor  has  suffered  an  expansion  due  to  an  excess  of  10°,  which  re- 
quires a  correction. 

In  table  of  corrections  for  10°  is  1.0208.  Then  divide  8.392  grains  at  dew-point— 
viz.,  70°  by  correction  corresponding  to  degrees  of  absolute  dryness— viz.,  10°. 

'392— 8.221  grains  of  existing  vapor,  which,  subtracted  from  weight  of  vapor 

I.O2OO 

corresponding  to  temperature  of  80°,  will  give  number  of  grains  required  for  satu- 
ration at  that  temperature. 

"•333  grains  at  temperature  of  80°  —  8.221  contained  in  the  air  =  3. 112  required 
for  saturation. 

*  For  table,  see  Mason's  as  published  by  Pike  &  Sons,  New  York,  and  compared  with  Sir  Jobs 
Leslie's  and  Professor  Daniel's. 


HYGROMETER. — SUN-DIAL. — CHAINING.  69 

To  ascertain  relations  of  these  conditions  on  natural  scale  of  humidity  (complete 
saturation  being  1000),  divide  weight  of  vapor  at  dew-point  by  weight  at  tempera- 
ture of  air,  and  quotient  will  give  degrees  of  saturation. 

ILLUSTRATION. — Dew-point  =  70°,  we ight  =  8.392. 
Then  8. 392 -7-11.333  (at  8o°)  =  -74°5  degrees  of  humidity;  saturation  =  1000. 

To  Compute,  Weight  of  Vapor  in  a  Cube  Foot  of  Air. 

See  Pressures,  Temperatures,  Volumes,  and  Density  of  Steam,  p.  708. 

Thus,  Required  weight  of  vapor  in  a  cube  foot  of  saturated  air  at  212°. 

At  a  temperature  of  212°  density  or  weight  of  i  cube  foot  of  air  =  .038  /&. 

If  density  is  required  for  any  temperatures  not  in  table,  see  rule,  p.  706. 

Humidity. — Condition  of  air  in  respect  to  its  moisture  involves  amount  of 
vapor  present  in  air  and  ratio  of  it  to  amount  which  would  saturate  it  at  its 
temperature,  and  it  is  this  element  which  is  denoted  by  term  humidity,  and 
it  is  expressed  as  a  per  centage;  thus,  if  weight  of  vapor  present  is  .7  of  that 
required  for  saturation,  the  humidity  is  70. 

Dry  Air  is  air,  humidity  of  which  is  below  zero,  but  it  is  customary  to 
term  it  dry  when  its  humidity  is  below  the  average  proportion. 

NOTE. — Air  in  a  highly  heated  space  contains  as  much  vapor  (when  weight  of  it 
is  equal)  as  a  like  volume  of  external  air,  but  it  is  drrv  as  its  capacity  for  vapor 
is  greater.  

SUN-DIAL. 

To    Set   a    Sun-dial.  ,tr . 

Set  column  on  which  dial  is  to  be  placed  perpendicular  to  horizon.  Ascertain  by 
spirit  level  that  upper  surface  is  perfectly  horizontal ;  screw  on  plate  loosely  by  means 
of  centre  screw,  and  bring  gnomon  as  nearly  as  practicable  to  its  proper  direction. 

On  a  bright  day  set  dial  at  9  A.M.  and  3  P.M.  exactly,  with  a  correctly  regulated 
watch;  observe  difference  between  them,  and  correct  dial  to  half  difference.  Pro- 
ceed in  same  manner  till  watch  and  dial  are  found  to  agree  perfectly.  Then  fix 
plate  firmly  in  that  situation,  and  dial  will  be  correctly  set. 

This  is  obvious;  for,  if  there  were  any  defects,  the  Sun's  shadow  would  not  agree 
with  time  indicated  by  watch,  both  before  and  after  he  passed  meridian.  Take 
care,  however,  to  allow  for  equation  of  time,  or  you  may  set  dial  wrong.  Best  day 
in  the  year  to  set  a  dial  is  isth  of  June,  as  there  is  no  equation  to  allow  for,  and  no 
error  can  arise  from  change  of  declination.  A  dial  may  be  set  without  a  watch,  by 
drawing  a  circle  around  centre,  and  marking  spot  where  top  of  shadow  of  an  upright 
pin  or  piece  of  wire,  placed  in  centre,  just  torches  circle  in  A.M.,  and  again  in  P.M. 
A  line  should  be  drawn  from  one  spot  to  the  other,  and  bisected  exactly;  then  a 
line  drawn  from  centre  of  dial  through  that  bisection  will  be  a  true  meridian  line, 
on  which  the  XII  hours'  mark  should  be  set. 


CHAINING   OVER  AN   ELEVATION. 

I C  ==  L,  and  C  =  cos.  angle. 

I  representing  length  of  line  chained,  C  cos.  angle  of  elevation  with  horizon, 
and  L  length  of  line  reduced  to  horizontal. 

ILLUSTRATION.— Length  of  an  elevation  at  an  angle  of  30°  17'  is  100  feet;  what  is 
horizontal  distance  ? 
By  Table  of  Cosines,  30°  17'  =  .  863  54.    Hence,  100  X  .  863  54  =  86. 354  feet. 

To  set  out  a  Riglxt  Angle  witla.  a  Chain,  Tape-line,  etc. 

Take  40  links  on  chain  or  feet  of  line  for  base,  30  links  or  feet  for  perpendicular, 
and  50  for  hypothenuse,  or  in  this  ratio  for  any  length  or  distance. 

USEFUL  NUMBERS  IN  SURVEYING. 


For  Converting 


Multiplier.  Converse.  II  For  Converting 


Multiplier.    Convert*. 


Feet    into  links.. 
Yardb    "      "    .. 


4-545 


.66       Square  feet    into  acres..  . 0000229     43560 


[I  Square  yards 


.0003066 


4840 


7<D  CHRONOLOGY. 

CHRONOLOGY. 

Solar  day  is  measured  by  rotation  of  the  Earth  upon  its  axis  with  respect 
to  the  Sun. 

Motion  of  the  Earth,  on  account  of  ellipticity  of  its  orbit,  and  of  perturba- 
tions produced  by  the  planets,  is  subject  to  an  acceleration  and  retardation. 
To  correct  this  fluctuation,  timepieces  are  adjusted  to  an  average  or  mean 
solar  day  (mean  time),  which  is  divided  into  hours,  minutes,  and  seconds. 

In  Civil  computations  day  commences  at  midnight,  or  A.M.,  and  is  divided  into 
two  portions  of  12  hours  each. 

In  Astronomical  computations  and  in  Nautical  time  day  commences  at  M.,  or 
12  hours  later  than  the  civil  day,  and  it  is  counted  throughout  the  24  hours. 

Solar  Year,  termed  also  Equinoctial,  Tropical,  Civil,  or  Calendar  Year,  is  the 
time  in  which  the  Sun  returns  from  one  Vernal  Equinox  to  another;  and  its  average 
time,  termed  a  Mean  Solar  Year,  is  365.242218  solar  days,  or  365  days,  5  hours,  48 
minutes,  and  47.6  seconds. 

Year  is  divided  into  12  Calendar  months,  varying  from  28  to  31  days. 

Mean  Lunar  Month,  or  lunation  of  the  Moon,  is  29  days,  12  hours,  44  minutes, 
2  seconds,  and  5.24  thirds.* 

Bissextile  or  Leap  Year  consists  of  366  days;  correction  of  one  year  in  four  is 
termed  Julian  ;  hence  a  mean  Julian  year  is  365.25  days. 

In  year  1582  error  of  Julian  computation  of  a  year  had  amounted  to  a  period  of 
10  days,  which,  by  order  of  Pope  Gregory  VIII. ,  was  suppressed  in  the  Calendar,  and 
5th  of  October  reckoned  as  isth. 

Error  of  Julian  computation,  .007  76  days,  is  about  i  day  in  128.79  years,  and  adop- 
tion of  this  period  as  a  basis  of  intercalation  is  termed  Gregorian  Calendar,  or  New 
StyleJ  Julian  Calendar  being  termed  Old  Style. 

Error  of  Gregorian  year  (365.2425  days)  amounts  to  i  day  in  3571.4286  years. 

New  Style  was  adopted  in  England  in  1752  by  reckoning  3d  of  September  as  i4th. 

By  an  English  law,  the  years  1900,  2100.  2200,  etc.,  and  any  other  iooth  year,  ex- 
cepting only  every  4ooth  year,  commencing  at  2000,  are  not  to  be  reckoned  bissex- 
tile years. 

Dominical  or  Sunday  Letter  is  one  of  the  first  seven  letters  of  alphabet,  and  is 
used  for  purpose  of  determining  day  of  week  corresponding  to  any  given  date.  In 
Ecclesiastical  Calendar  letter  A  is  placed  opposite  to  ist  day  of  year,  January  ist; 
B  to  second;  and  so  on  through  the  seven  letters;  then  the  letter  which  falls  oppo- 
site to  first  Sunday  in  year  will  also  fall  opposite  to  every  following  Sunday  in  that 
year.  See  table,  p.  73. 

NOTE.— In  bissextile  years  two  Dominical  letters  are  used,  one  before  and  the  other 
after  the  intercalary  day. 

In  Ecclesiastical  Year  the  intercalary  day  is  reckoned  upon  24th  of  February; 
hence  24th  and  25th  days  are  denoted  by  same  letter,  the  dominical  letter  being  set 
back  one  place. 

In  Civil  Year  the  intercalary  day  is  added  at  end  of  February,  the  change  of  letter 
taking  place  at  ist  of  March. 

Dominical  Cycle  is  a  period  of  400  years,  when  the  same  order  of  dominical  letters 
and  days  of  the  week  will  return. 

Cycle  of  the  Sun,  or  Sunday  Cycle,  is  the  28  years  before  same  order  of  Dominical 
letters  return  to  same  days  of  month,  and  it  is  considered  as  having  commenced  9 
years  before  the  era  of  Julian  Calendar. 

To    Compnte    Cycle   of  the    Snn. 

RULE.— Add  9  to  given  year;  divide  sum  by  28;  quotient  is  number  of  cycles  that 
have  elapsed,  and  remainder  is  number  or  years  of  cycle. 

NOTE.— Use  of  this  computation  is  determination  of  dominical  letter  for  any  given 
year  of  Julian  Calendar  for  each  of  the  28  years  of  a  cycle. 

*  Ferguson.  f  Now  adopted  in  every  Christiaa  country  except  Russia  and  Greece. 


CHRONOLOGY.  *]\ 

By  adoption  of  Gregorian  Calendar,  order  of  the  letters  is  necessarily  interrupted 
by  suppression  of  the  century  bissextile  years  in  1900,  2100,  etc.,  and  a  table  of  Do- 
minical letters  must  necessarily  be  reconstructed  for  following  century. 

Lunar  Cycle,  or  Golden  Number,  is  a  period  of  19  years,  after  which  the  new 
moons  fall  on  same  days  of  the  month  of  Julian  year,  within  1.5  hours. 

Year  of  birth  of  Jesus  Christ  is  reckoned  first  of  the  Lunar  Cycle. 

To    Compute    Lniiar    Cycle,  or    Grolden    Number. 

RULE.— Add  i  to  given  year;  divide  sum  by  19,  and  remainder  is  Golden  Number. 

NOTE. — If  o  remain,  it  is  19. 

EXAMPLE. — What  is  Golden  Number  for  1879? 

1879  -}- 1  -T-  19  =  98,  and  remainder  =  18  =  Golden  Number. 
Epact  for  any  year  is  a  number  designed  to  represent  age  of  the  moon  on  ist  day 
of  January  of  that  year.    See  table,  p.  73. 

To    Compute   the   Roman    Indiotion. 

RULE.— Add  3  to  given  year;  divide  sum  by  15,  and  remainder  is  Indiction. 
NOTE.— If  o  remain,  Indiction  is  15. 

Number  of  Direction  is  the  number  of  days  that  Easter-day  occurs  after  2ist  of 
March. 

Easter-day  is  first  Sunday  after  first  full  moon  which  occurs  upon  or  next  after 
zist  of  March;  and  if  full  moon  occurs  upon  a  Sunday,  then  Easter-day  is  Sunday 
after,  and  it  is  ascertained  by  adding  number  of  direction  to  2ist  of  March.  It  is 
therefore  March  N  -j-  21,  or  April  N  — 10. 

ILLUSTRATION.  —  If  Number  of  Direction  is  19,  then  for  March,  19-1-21  =  40,  and 
40  —  31  =  o  =  gth  of  April ; 
agai  n  for  A  pr  il,  1 9  —  i  o  =  9  =  gth  of  April. 

NOTE.— Moon  upon  which  Easter  immediately  depends  is  termed  Paschal  Moon 

Full  Moon  is  i4th  day  of  moon,  that  is,  13  days  after  preceding  day  of  new  moon. 

Days    of  the    Roman    Calendar. 

Calends  were  the  first  6  days  of  a  month,  Nones  following  9  days,  and  Ides  remain- 
ing days. 

In  March,  May,  July,  and  October,  Ides  fell  upon  isth  and  Nones  began  upon  7th. 
In  other  months  Ides  commenced  upon  i^th  and  Nones  upon  sth. 

For  Roman  Indiction  and  Julian  Period  see  p.  26. 

B  c  Chronology. 

4004.  Creation  of  World  (according  to  Julius  Africanus,  Sept.  i,  5508 ;  Samaritan 
Pentateuch,  4700;' Septuagint,  5872;  Josephus,  4658 ;  Talmudists,  5344 ;  Sca- 
liger,  3950;  Petavius,  3984 ;  Hales,  5411). 


2348  Deluge  (according  to  Hales,  3154). 

2247.  Bricks  made  and  Cement  first  used. 
Tower  of  Babel  finished. 

2203.  Chinese  Monarchy. 

2090.  First  Egyptian  Pyramid  and  Canal. 

1920.  Gold  and  Silver  Money  first  intro- 
duced. 

1891.  Letters  first  used  in  Egypt. 

1822.  Memnon  invents  the  Egyptian  Al- 
phabet. 

1490.  Crockery  introduced. 

1240.  Axe,  WTedge,  Wimble,  Lever,  Masts 
and  Sails  invented  by  Daedalus 
of  Athens. 

.1180.  Troy  destroyed. 

1 1 20.  Mariner's  Compass  discovered  in 

China. 

753.  Foundation  of  Rome. 
640.  Thales  asserts  Earth  to  be  spherical. 
605.  Geometry,  Maps,  etc.,  first   intro- 
duced. 


576.  Money  coined  at  Rome. 

562.  First  Comedy  performed  at  Athens. 

480.  First  recorded  Map  by  Aristagoras. 

420.  First -Theatre  built  at  Athens. 

336.  Calippus  calculates  the  revolution  of 
Eclipses. 

320.  Aristotle  writes  first  work  on  Me- 
chanics. 

310.  Aqueducts  and  Baths  introduced  in 
Rome. 

306.  First  Light-house  in  Alexandria, 

289.  First  Sun  dial. 

267.  Ptolemy  constructs  a  Canal  from  the 
Nile  to  the  Red  Sea. 

224.  Archimedes  demonstrates  the  Prop- 
erties of  Mechanical  Powers  and 
the  Art  of  measuring  Surfaces,  Sol- 
ids, and  Sections. 

219.  Hannibal  crossed  the  Alps. 

219.  Surveying  first  introduced. 

202.  Printing  introduced  in  China. 


CHRONOLOGY. 


198.  Books  with  leaves  of  vellum  first 

introduced  by  Attalus. 
170.  Paper  invented  in  China. 
168.  An  eclipse  of  the  Moon  which  was 

predicted  by  Q.  S.  Callus. 
162.  Hipparchus  locates  the  first  degree 

of  Longitude  and  the  Latitude  at 

Ferro. 

A.D. 

69.  Destruction  of  Jerusalem. 
79.  Destruction  of  Herculaneum  and 

Pompeii. 

214.  Grist-mills  introduced. 
622.  Year  of  Hegira,  commencing  i6th 
July;  Glazed  windows  first  intro- 
duced into  England  in  thiscent'y. 
667.  Glass  discovered. 
670.  Stone  buildings  introduced  into  Eng- 
land. 

842.  Lands  first  enclosed  in  England. 
933.  Printing  said  to  have  been  invented 

by  the  Chinese. 
991.  Arabic  Numerals  introduced. 

1066.  Battle  of  Hastings. 

i in.  Mariner's  Compass  discovered. 

1 1 80.  Mariner's  Compass  introduced  in 
Europe. 

1368.  Chimneys    first    introduced    into 
Rome  from  Padua. 

1383.  Cannon  introduced. 

1390.  Woollens  first  made. 

1434.  Printing  invented  at  Mayence. 

1460.  Wood-engraving  invented  and  First 
Almanac. 

1471.  Printing  in  England  by  Caxton. 

1477.  Watches  first  introduced  at  Nurem- 
berg. 

1492.  America  discovered. 

1497.  Vasco  de  Gama  discovers  passage 
to  India. 

1500.  Variation  of  Mariner's  Compass  ob- 
served. 

152?.  F.  de  Magellan  circumnavigates  the 
Globe. 

1530.  Incas  conquered  by  Pizarro. 

1545.  Needles  first  introduced. 

rs86.  Potato  introduced  into  Ireland  from 
America. 

1590.  Telescopes  invented  by  Jansen  and 
used  in  London  in  1608. 

1616.  Tobacco  first  introduced  into  Vir- 
ginia. 

1620.  Thermometer  invented  by  Drebel. 

1627.  Barometer  invented. 

1629.  First  Printing  press  in  America. 

1639.  First  Printing-office  in  America  at 
Cambridge. 

1647.  Otto  Van  Gueriche  constructed  first 
electric  machine. 

1650.  Railroads  with  wooden  rails  intro- 
duced near  Newcastle. 

1652.  First  Newspaper  Advertisement. 

1704.  First  Newspaper  in  America. 

1705.  Blankets  first  made  at  Bristol,  Eng- 

land. 


159.  Clepsydra,  or  Water -clock,  invent 

ed. 

146.  Carthage  destroyed. 
70.  First  Water-mill  described. 
51.  Caesar  invaded  Britain. 
45.  First  Julian  Year  by  Caesar. 
8.  Augustus  corrects  the  Calendar. 


1752.  Benjamin  Franklin  demonstrated 
identity  of  the  electric  spark  and 
lightning,  by  aid  of  a  kite. 

1752.  New  Style,  introduced  into  Britain; 

Sept.*3  reckoned  Sept.  14. 

1753.  First  Steam  engine  in  America. 

1769.  James  Watt — First  design  and  pat- 
ent of  a  Steam-engine  with  sepa- 
rate vessel  of  condensation. 

1772.  Oliver  Evans — Designed  the  Non- 
condensing  Engine.  1792.  Ap- 
plied for  a  patent  for  it.  1801. 
Constructed  and  operated  it. 

1774.  Spinning  jenny  invented  by  Robert 
Arkwright. 

1776.  Iron  Railway  at  Sheffield,  England. 

1783.  First  Balloon  ascension,and  Vessel's 
bottoms  coppered. 

1790.  Water-lines  first  introduced  in  mod- 
els of  Vessels  in  the  U.  S. 

1797.  John  Fitch— Propelled  a  yawl  boat 
by  application  of  Steam  to  side- 
wheels,  and  also  to  a  screw-propel- 
ler, upon  Collect  Pond,  New  York. 

1807.  Robert  Fulton  —  First  Passenger 
Steamboat. 

1824.  Compound  marine  steam-engines 

first  introduced  by  James  P.  Al- 
lan, New  York. 

1825.  Introduction  of  steam  towing  by 

Mowatt,  Bros.  &  Co.,  of  New  York, 
by  steam- boat "  Henry  Eckford," 
New  York  to  Albany.  * 

1826.  Voltaic  Battery  discovered  by  Alex. 

Volta,  and  First  Horse-railroad. 

1827.  First  Railroad  in  U.  S.,  from  Quincy 

to  Neponset. 

1829.  First  Lucifer  Match  and  first  Loco- 

motive in  America. 

1830.  Liverpool  and  Manchester  Railroad 

opened.  First  Steel  Pen  and  first 
Iron  Steamer. 

1832.  S.  F.  B.  Morse  invents  the  Magnetic 
Telegraph. 

1836.  Robert  L.  Stevens  first  burned  An- 
thracite Coal  in  furnace  of  boiler 
of  steamboat  "  Passaic." 

1840.  First  steam-boiler  constructed  for 
burning  AnthraciteCoal  in  steam- 
boat "North  America,"  N.  Y. 

1844.  Telegraph  line  from  Washington  to 
Baltimore,  Md. 

1846.  First  complete  Sewing-machine. 
Elias  Howe,  inventor. 

1866.  Submarine  Telegraph  laid  from 
Valencia  to  Newfoundland,  N.  S. 


Witnesaed  by  author. 


CHRONOLOGY. 


73 


Dates    of   Day    of    Week,    corresponding    to    Day    deter- 
mined,   "by    folio-wing    Table. 


February, 
March, 
November. 

February,* 
August. 

May. 

January, 
October. 

January,* 
April, 
July.' 

September, 
December. 

June. 

I 

2 

3 

4 

5 

6 

7 

8 

10 

ii 

12 

'3 

14 

IS 

17 

18 

xi 

20 

21 

22 

23 

24 

«5 

26 

27 

28 

29 

3<> 

3' 

Thus,  if  Monday  is  the  day  determined  by  the  year  given,  the  following  dates  are 
the  Mondays  in  that  year. 

Epacts,  Dominical  Letters,  and  an  .AJmanac,  from 
1834    to    1935. 

USK  or  TABLE.  —To  ascertain  day  of  the  week  on  which  any  given  day  of  the 
month  falls  in  any  year  from  1800  to  1901. 

ILLUSTRATION.— The  great  fire  occurred  in  New  York  on  i6th  of  December,  1835; 
what  was  day  of  the  week  ? 

Opposite  1835  is  Sunday;  and  by  preceding  table,  under  December,  it  is  ascertained 
that  1 3th  was  Sunday;  consequently,  i6th  was  Wednesday. 


Years. 

Days. 

Dom. 
Let- 
ten. 

*i 
tt. 

Yeara. 

Day*. 

Dom. 
Let- 
ten. 

I 

Yean. 

D«yi. 

Dom. 
Let- 
ten. 

1 

1834 

Saturday. 

B 

20 

1868 

Sunday.  * 

ED 

6 

1902 

Saturday. 

E 

22 

1835 

Sunday. 

D 

*!' 

1869 

Monday. 

C 

*7 

1903 

Sunday. 

D 

3 

1836 

Tuesday.* 

CB 

1? 

1870 

Tuesday. 

B 

28 

1904 

Tuesday.* 

CB 

'4 

1837 

Wednesd. 

A 

23 

1871 

Wednesd. 

A 

9 

'90S 

Wednesd. 

A 

2S 

1838 

Thursday. 

G 

'  4 

1872 

Friday.* 

GF 

20 

1906 

Thursday. 

G 

6 

1839 

Friday. 

F 

is 

1873 

Saturday. 

E 

I 

1907 

Friday. 

F 

If 

1840 

Sunday.* 

ED 

rf 

1874 

Sunday. 

D 

19 

1908 

Sunday.* 

ED 

ii 

1841 

Monday. 

C 

7 

1875 

Monday. 

C 

-'3 

1909 

Monday. 

C 

9 

1842 

Tuesday. 

B 

ii 

1876 

Wednesd.* 

BA 

4 

1910 

Tuesday. 

B 

20 

1843 

Wednesd. 

A 

29 

1877 

Thursday. 

G 

15 

1911 

Wednesd. 

A 

I 

1844 

Friday.* 

GF 

ii 

1878 

Friday. 

F 

20 

1912 

Friday.* 

~GF 

12 

1845 

Saturday. 

E 

22 

1879 

Saturday. 

E 

7 

I9f3 

Saturday. 

E 

23 

1846 

Sunday. 

D 

3 

1880 

Monday.  * 

DC 

18 

1914 

Sunday. 

D 

4 

1847 

Monday. 

C 

'4 

1881 

Tuesday. 

B 

20 

1915 

Monday. 

C 

IS 

1848 

Wednesd.* 

BA 

25 

1882 

Wednesd. 

A 

II 

1916 

Wednesd.  * 

BA 

26 

1849 

Thursday. 

G 

6 

1883 

Thursday. 

G 

22 

1917 

Thursday. 

G 

7 

1850 

Friday. 

F 

*z 

1884 

Saturday.* 

FE 

3 

1918 

Friday. 

F 

18 

1851 

Saturday. 

E 

28 

1885 

Sunday. 

D 

14 

1919 

Saturday. 

E 

29 

1852 

Monday.  * 

DC 

9 

1886 

Monday. 

C 

*3 

1920 

Monday.  * 

DC 

ii 

1853 

Tuesday. 

B 

20 

1887 

Tuesday. 

B 

e 

1921 

Tuesday.  T 

B 

22 

1854 

Wednesd. 

A 

I 

1888 

Thursday.* 

AG 

17 

1922 

Wednesd. 

A 

3 

1855 

Thursday. 

G 

12 

1889 

Friday. 

F 

28 

1923 

Thursday. 

G 

M 

1856 

Saturday.* 

FE 

»3 

1890 

Saturday. 

E 

9 

1924 

Saturday.* 

FE 

25 

1857 

Sunday. 

D 

4 

1891 

Sunday. 

D 

20 

1925 

Sunday. 

D 

6 

1858 

Monday. 

C 

15 

1892 

Tuesday.* 

CB 

I 

1926 

Monday. 

C 

'7 

1859 

Tuesday. 

B 

26 

1893 

Wednesd 

A 

12 

1927 

Tuesday. 

B 

28 

1860 

Thursday.  * 

AG 

7 

,8^4 

Thursday. 

G 

23 

1928 

Thursday.* 

AG 

9 

1861 

Friday. 

F 

iS 

1895 

Friday. 

F 

4 

1929 

Friday. 

F 

20 

1862 

Saturday. 

E 

29 

1896 

Sunday.* 

ED 

IS 

1930 

Saturday. 

E 

I 

1863 

Sunday. 

D 

ii 

1897 

Monday. 

C 

26 

1931 

Sunday. 

D 

12 

1864 

Tuesday.  * 

CB 

22 

1898 

Tuesday. 

B 

7 

1932 

Tuesday.  * 

CB 

23 

1865 

Wednesd. 

A 

3 

1899 

Wednesd. 

A 

,8 

1933 

Wednesd. 

A 

4 

1866 

Thursday. 

G 

M 

1900 

Thursday. 

G 

29 

1934 

Thursday. 

G 

15 

1867 

Friday. 

F 

25 

1901 

Friday. 

F 

ii 

1935 

Friday. 

F 

26 

*  In  leap-year,  January  and  February  must  be  taken  In  columns  marked  * 


74  CHRONOLOGY. — MOON'S   AGE. — TIDES. 

To  Ascertain  Year  or  Years   of  Coincidences   of  a  given 
Day   of  the    Week   with   a  given    Z>ay   of  a   M.oiith. 
Look  in  preceding  table  and  ascertain  day  of  week  opposite  to  year  of 
occurrence,  and  every  year  in  which  same  day  is  given  will  be  year  of  coin- 
cidences required. 

ILLUSTRATION.— If  a  child  was  born  on  Saturday,  igth  Sept.  1829,  when  could  and 
can  his  birthdays  be  celebrated,  that  occurred  or  are  to  occur  on  same  day  of  week 
and  date  of  month  ? 

Opposite  to  1829  is  Sunday,  and  in  preceding  table  the  Sundays  for  September  of 
that  year  were  6th,  i3th,  2oth;  hence,  if  2oth  was  Sunday,  the  igth  was  Saturday. 

Hence,  every  year  in  table  opposite  to  which  is  Sunday  are  the  years  of  the  coin- 
cidence required,  as  1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885,  etc. 



MOON'S  AGE. 

To   Compute   Moon's   ^Lge. 

RULE. — To  day  of  month  add  Epact  and  Number  of  month ;  from  sum 
subtract  29  days,  12  hours,  44  min.  and  2  sec.,  as  often  as  sum  exceeds  this 
period,  and  result  will  give  Moon's  age  approximately  at  6  o'clock  A.M.  in 


d.  h. 

.7  16 
.9  4 
•9  15 

EXAMPLE.— Required  age  of  Moon  on  25th  February,  1877  ? 

Given  day  25  -j-epact  i5-f-numl>er  of  month  1.22  =  41  d.  22  h.  — 29  d.  12  h.  44  m. 
2  sec.  =  12  d.  9  h.  15  min.  58  sec. 

In  Leap-years  add  i  day  to  result  after  28th  February. 

To   Compute  Age   of  3VEoon   at   Mean    Moon  at  any  other 
Location    than,    that   Given. 

RULE. — Ascertain  age,  and  add  or  subtract  difference  of  longitude  or  time, 
according  as  place  may  be  West  or  East  of  it,  to  or  from  time  given. 

0r,  when  time  of  new  Moon  is  ascertained  for  a  location,  and  it  is  required 
to  ascertain  it  for  any  other,  add  difference  of  longitude  or  time  of  the  place, 
if  East,  and  subtract  it  if  it  is  West  of  it. 

Moon's  Southing,  as  usually  given  in  United  States  Almanacs,  both  Civil  and  Nau- 
tical, is  computed  for  Washington. 


United  States, 

d 
January  
February  i 

east 

h. 

5 

22 

of  Mississippi  Riv 
Nurnlaers    of 
d.  h. 
April  i  21 
May  2    8 

er. 
the   !Mon 

July  
August  
September  . 

ths. 

d.  h. 

October  

November  
December..  . 

March.  .  . 

Q 

June.  .  .        .  .  •*  IQ 

To  Compute   Time  of  High--water  "by  Aid.   of  American 
Nautical   Almanac. 

RULE.— Ascertain  time  of  transit  of  Moon  for  Greenwich,  preceding  time 
of  the  high-water  required. 

For  any  other  location  (west  of  Greenwich),  multiply  the  time  in  column 
"  diff.  for  one  hour  "  by  longitude  of  location  west  of  fcrreenwich,  expressed 
in  hours,  and  add  product  to  time  of  transit. 

NOTE. — It  is  frequently  necessary  to  take  the  transit  for  preceding  astronomical 
day,  as  the  latter  does  not  end  until  noon  of  day  under  computation. 

EXAMPLE.— Required  time  of  high- water  at  New  York  on  2$th  of  August,  1864. 

Longitude  of  New  York  from  Greenwich  =  4  h.  56  m.  1.65  sec.,  which,  multiplied 
by  2. 17  min.,  the  difference  for  i  hour  — 10.71  min.  for  correction  to  be  added  to  time 
of  transit,  to  obtain  time  of  transit  at  New  York. 


TIDES. — MOON'S   SOUTHING. 


75 


Time  of  transit,  18  h.  38.8  m. ;  then  18  h.  38.8  m.  + 10.71  TO.  =  18  Aowr*  49.51  mm. 
Time  of  transit  at  New  York,    24  d.  18  h.  50  m. 

Establishment  of  the  Port,        8     13 

25  d.    3  A.    3  m.  =  time  of  high-water. 

NOTE.  —Time  of  2$th  at  3  A.  3  m.  Astronomical  computation  =  25th  at  3  h.  3  m. 
I. M.  Civil  Time. 

To   Compute  Time    of  Higli-water   at   Full   and  Change 
of  Moon. 

Time  of  High-water  and  Age  of  Moon  on  any  Day  being  given. 

RULE. — Note  age  of  Moon,  and  opposite  to  it,  in  last  column  of  following 
table,  take  time,  which  subtract  from  time  of  high-water  at  this  age  of 
Moon,  added  to  12  h.  26  m.,  or  24  h.  52  m.,  as  case  may  require  (when  sum  to 
be  subtracted  is  greatest),  and  remainder  is  time  required. 

EXAMPLE.— What  is  time  of  high-water  at  full  and  change  of  Moon  at  New  York? 

Time  of  high- water  at  Governor's  Island  on  25th  of  Jan.  1864,  was  9  h.  20  m.  A.M. 
civil  time.  Age  of  Moon  at  12  M.  on  that  day  was  16  d.  8  h.  59  m. 

Opposite  to  16  days,  in  following  table,  is  13  h.  28  m. ,  and  difference  between  16  d. 
and  16  d.  12  h.  =(16.5  — 16,  or  13.53  — 13.28)  is  25  m. :  hence,  if  12  h.  =25  w.,  16  d. 
8  h.  59  w.  — 16  d.  —  8  h.  59  m.  =  18.71  or  19  m.,  which,  added  to  13  h.  28  m.  =  13  h. 
47  m. 

Then  9  h.  20  m.  -f  12  h.  26  m.  (as  sum  to  be  subtracted  is  greater  than  time)  — 13  h. 
47  m.  =  21  h.  46  m.  — 13  h.  47  m.  =.  7  A.  50  m. 

This  is  a  difference  of  but  13  minutes  from  Establishment  of  Port. 

Time    after    apparent    Noon    "before    Moon    next 

passes   Meridian,  A-ge   at  Noon  being  given. 

(S.  H.  Wright,  A.M.,  Ph.D.) 


Age  of 
Moon. 

Moon  at 
Meridian. 

Age  of 
Moon. 

Moon  at 
Meridian. 

Age  of 
Moon. 

Moon  at 
Meridian. 

Are  of 
Moon. 

Moon  at 

Meridian. 

Age  of 
Moon. 

Moon  at 
Meridian. 

Days. 

B.  M. 

Days. 

H.  M. 

Days. 

H.  M. 

Days. 

H.  M. 

Days. 

H.  M. 

P.  M. 

P.M. 

.  M. 

A.  M. 

A.  M. 

.0 

o 

6 

5     3 

12 

o    6 

18 

15    8 

24 

20   II 

•5 

25 

6-5 

528 

12.5 

o  31 

18.5 

15  34 

24-5 

20  37 

I 

50 

7 

5  53 

13 

o  56 

19 

15  59 

25 

21       2 

i-5 

i  16 

7-5 

6  19 

13-5 

I    21 

i9-5 

16  24 

25-5 

21    27 

2 

i  41 

8 

644 

14 

i  47 

20 

16  49 

26 

21    52 

.  M. 

2-5 

2      6 

8.5 

7    9 

H-5 

2    12 

20.5 

17  i5 

26.5 

22    IJ 

3 

2    31 

9 

7  34 

15 

2  37 

21 

17  40 

27 

22  43 

3-5 

2  57 

9-5 

7  59 

15-5 

3     2 

21-5 

18    5 

27-5 

23    8 

4 

3   22 

10 

8  25 

16 

3  28 

22 

18  30 

28 

23  33 

4-5 

3  47 

10.5 

8  50 

16.5 

3  53 

22-5 

18  56 

28.5 

23  58 

5 

4  12 

ii 

9  '5 

i7 

4  28 

23 

19  21 

29 

24  24 

5-5 

438 

"•5 

9  40 

i7-5 

4  43 

23-5 

19  46 

29-5 

24  48 

Tidal   Phenomena. 

The  elevation  of  a  tidal  wave  towards  the  Moon  slightly  exceeds  that  of  the  op- 
posite  one,  and  the  intensity  of  it  diminishes  from  Equator  to  the  Poles. 

The  Sun  by  its  action  twice  elevates  and  depresses  the  sea  every  day,  following 
the  action  of  the  Moon,  but  with  less  effect. 

Spring  Tides  arise  from  the  combined  action  of  the  Sun  and  Moon  when  they  are 
on  both  sides  of  the  Earth. 

Neap  Tides  are  the  consequence  of  the  divided  action  of  the  Sun  and  Moon,  when 
they  are  on  opposite  sides  of  the  Earth,  and  the  greatest  elevations  and  depressions 
do  not  occur  until  the  2d  or  ^d  day  after  a  full  or  a  new  Moon. 

When  Sun  and  Moon  are  in  conjunction,  and  the  time  is  near  to  the  Equinoxes, 
the  tides  are  fullest.  The  mean  effect  of  the  Moon  on  the  tidal  wave  is  4.5  times 
that  of  the  Sun.  If,  therefore,  the  Moon  caused  a  tide  of  6  feet,  the  Sun  will  cause 
one  of  i. 33  feet;  hence  a  spring  tide  will  be  7.33  feet,  and  a  neap  tide  4.67  feet. 

Particular  locations  as  to  contour  of  shores,  straits,  capes,  and  rivers,  lengths  and 
depths  of  channels,  shoals,  etc.,  disturb  these  general  rules. 


LATITUDE   AND    LONGITUDE. 


LATITUDE   AND    LONGITUDE. 

Latitude   and.   Longitude    of  Principal  Locations 
and   Observatories. 

Compiled  from  Records  of  If.  S.  Coast  and  Geodetic  Survey  and  Topograph- 
ical Engineer  Corps,  Imperial  Gazetteer,  and  Bowditch's  Navigator. 

Longitude  computed  from  Meridian  of  Greenwich. 

A.,  represents  Academy;  Az.,  Azimuth;  A.  S.,  Astronomical  Station;  C.,  College; 
Cap. ,  Capitol ;  Ch. ,  Church ;  C.  H. ,  City  Hall ;  C.  S. ,  Coast  Survey ;  Ct. ,  Court-house ; 
Cy.,  Chimney;  F.S.,  Flagstaff;  G.S.,  Geodetic  Station;  Hos.,  Hospital;  L.  Light- 
house; Obs. , Observator y ;  S.H.,  State-house;  Sp.  Spire ;  Sq.,  Square;  S.S.,  Signal 
Station  ;  T. ,  Telegraph  ;  T. H. ,  Town  Hall ;  U. ,  University;  Un. ,  Union  ;  B. ,  Baptist ; 
Con.,  Congregational;  E.,  Episcopal;  P.,  Presby.  ;  and  M.Ch.,  JtfeM.  Churches. 


LOCATION. 

Latitude. 

Longitude. 

LOCATION. 

Latitude. 

Longitude. 

WORTH  AND   SOUTH 
AMERICA. 

Acapulco  Mex. 

N. 
o    /    // 

ID  50  IQ 

W. 

0        /      // 

99  49    9 

NORTH    AND   SOUTH 
AMERICA. 

Canandaigua  ...  .N.Y. 

N. 

0      1      tl 

42  54  9 

W. 

o     /    // 
77  17 

Albany,  P.  Ch....N.Y. 
Ann  Arbor  Mich 

42  39    3 
42  1  6  48 

73  45  24 
83  43    3 

Cape  Ann,  S.  L.  .Mass. 
Cape  Breton  ....    Va. 

42  38  ii 

AS.  C? 

70  34  10 

38  58  42 

76  29    6 

Cape  Canaveral.  .  .  Fla. 

28  27  30 

80  33 

Apalachicola,F.S.Fla. 
Astoria,  F.  S  Or. 

29  43  30 

46   II    IQ 

8459 
123  49  42 

Cape  Cod,  L.  P.  L...  Ms. 
Cape  Fear  N.C. 

42  2 

33  48 

70    9  48 

77    57 

Atlanta  C  H  ..  Ga. 

00    AA     C7 

84  23  22 

Cape  Flattery  L  W  T 

4.8  23  m 

Auburn  N.Y 

42  55 

76  28 

Cape  Florida,  L.  .  .  Fla. 

2C  QQ  54 

Augusta  Ga 

TO    28 

81  54 

Cape  Hancock  Colo  R 

f>    f> 

' 

Augusta,  B.Ch....  Me. 
Austin  Tex 

44  18  52 

69  46  37 

Cape  Hatteras,  L.  ,  N.  C. 
Caoe  Henlopen  L   Del 

35  15  2 
og  A.6  6 

75  3°  54 

Balize  La 

2Q      8      5 

Cape  Henry  L.   .'.Va 

^6  5^  3O 

75    4    7 

Baltimore,  Mon't  .  Md. 
Bangor  Tho's  Hill  Me 

y              o 

39  i7  48 

76  36  59 
68  46  59 

Cape  Horn,  S.  Pt.  ,  Her- 
mit's Island  .  . 

6V   33   Ju 
o. 

67  16 

Barbadoes,  S.Pt.  .  W.  I. 
Barnegat,  L  N.  J. 
Bath,  W  S  Ch....  Me 

13    3 

39  46 

4q     CA     CC 

5937 
74    6 

Cape  May,  L  N.J. 
Cape  Race  ....     N  S 

55  ft 
38  55  48 

46  3Q  24 

74  57  18 

Baton  Rouge  La 

QI  18 

Cape  Sable             N  S 

53    4    3 

Beaufort  Ct  .  .  N  C 

~ 

Cape  Sable  C  S     Fla 

43  24 

81  is 

Beaufort,  E.Ch...S.C. 
Belfast,  M.Ch  Me. 
Benicia,  Ch  Cal 

32  26      2 

44  25  29 

og      o       C 

80  40  27 
69        19 

Cape  St.  Roque,  Brazil 

"5  s.  53 

5N8 

35  17 

Benington  Vt. 
Bismarck,  S.  S  .  .  .Neb 

42  40 

4.6  AS 

73  18 

Carthagena  N.G. 
Castine  .   .  .            Me 

10  26 

7538 
68  AC 

Boston,  L  Mass. 
Boston,  S  H.  ...  a 

42  19    6 

7053    6 

Cedar  Keys,  Depot  Isl. 
Chagres                 N  G 

29  7  3° 

83    245 

80      I    21 

Brazos  Santiago.  .Tex. 
Bridgeport  Conn. 
Bristol  R.  I 

26    6 
41  10  30 

97  12 
7l\l    4 

Charleston,  C.Ch.,S.C. 
Charlestown,Mon.,Ms. 
Cheboygan  L      Mich 

32  46  44 

42  22  36 

79  55  39 
7i    3  18 

Brooklyn,  C.H...  N.Y. 
Brownsville,  S.S.  .Tex. 
Brunswick,  A  Ga. 
Brunswick,  C.Sp.  .Me. 
Buffalo  L  NY 

40  41  31 
26 
3i    8  51 
43  54  29 

73  59  27 
97  30 
81  29  26 
69  57  24 

?8    CO 

Chicago,  C.Ch  111. 
Chickasaw  Miss. 
Cincinnati,  Obs  O. 
Cleveland,  Hos  " 
Colorado  Springs  Col 

4i  53  48 
35  53  3° 
39  6  26 
41  30  25 
18  co 

87  37  47 
88    6  25 
84  29  45 
81  40  30 

Burlington  N.  J. 
Burlington,  C  Vt. 
Burlington,Pub.Sq.,Ia. 
Bushnell  Neb 

40    4  52 
44  28  52 

40  48   22 

74  52  37 
78  10 
91    6  25 

Columbia,  S.H....S.C. 
Columbus,  Cap  0. 
Concord,  S.H....N.H. 
Corpus  Christi      Tex 

33  59  58 
39  57  40 
43  12  29 

81    2    3 
82  59  40 
71  29 

Cairo  Ill 

Council  Bluffs    Neb  T 

Qe  ^8 

Calais,  C.S.  Obs...  Me. 
Callao,  F.S  Peru 
Cambridge,  Obs.  Mass 

45s!  5 

12  N4 

6716    5 
77  13 

71      7  4.3 

Crescent  City,  L...  Cal. 
Cumberland  Md. 
Darien,W.H  Ga. 
Davenport,  S.  S  la. 
Dayton  0. 

41  44  34 
39  39  H 
3i  21  54 
4i  32 

•3Q  A  A 

124   II   22 

78  45  25 
81  25  39 
90  38 
84  ii 

Camden  S  C 

80  33 

Deadwood  S  S      Dak 

Campeachy  .  .Yucatan 

19  49 

9033 

Decatur,  S.  S  Tex. 

33  10 

9730 

LATITUDE  AND  LONGITUDE. 


77 


LOCATION. 

Latitude. 

Longitude. 

LOCATION. 

Latitude. 

Longitude. 

NORTH   AND  SOUTH 
AMERICA. 

Denver,  S.  H.Sp..Col. 
Des  Moines,C.H...Ia. 
Detroit,  St.  P.  Ch.,  Mich. 
Dover  Del 

N. 

Q      1      II 

3945 
4i35 
42  19  46 

39  I0 
43  13 

42  29  55 
4648 
44  54  15 
36    3  24 
36  17  58 
42    843 
40  48  ii 
44  58  40 
30  40  i  8 
34  47  13 
35  47  35 

36  30  22 
42    12    10 

39  2I  14 

38  14 

39  24 

3818    6 

46    3 
29  18  17 

32  22      2 

3322    8 
42  30  46 

43    5 
44  39    4 

40  ID 

4i  45  59 
23    9 

25  5i    5 
41  27  13 
4214 
34  36 
39  55 
28  32  28 
32  23 

30  19  43 

19  30    8 
38  36 
40  43  28 
46  26 
40  23 
24  33  3i 
1758 
44    8 
35  59 
43  58  50 

40     2  36 

28  37  36 

39  29 
38    6 
S. 

12  N3 

344<> 

W. 

0         4       II 

104  59  33 
93  37  16 
83    223 
75  30 
?o  54 
9°  39  57 
92    8 

76  36  31 
76  13  23 
80    4  12 
124    9  41 
93  10  30 
81  27  47 
87  41  40 
95  15  10 
88    340 
104  47  43 
94  44 
8440 
77  '8 

77  27  38 
66  38  15 
94  47  26 
64  37    6 
79  l6  49 
70  39  59 

86  18 
6335 
76  50 
72  40  45 

82  21   23 

77  10    6 
70  35  59 
7346 
8657 
86    5 
9631    i 
90    8 

81  39  14 
96  54  30 
92    8 
74    2  24 
122  50  39 
9125 
81  48  31 
7646 
76  28  37 
8354 
91  14  40 
76  20  33 
96  37  21 
94  58 
84  18 

77    6 

92  12        1 

NORTH  AND  SOUTH 
AMERICA. 

Lockport  N.Y. 

N. 

O       1      II 

43  " 

U!5 

42  38  46 

4443     i 
32  50  25 
43    4  33 
42  30  14 
14  27 
28  41  29 
25  52  50 
23    3 
35    7 
»9  25  45 
43    2  24 
445838 

30  22  54 
30  41  26 

36  15" 
34  g. 

32  22  45 
45  3i 
37    4  47 
41  23  24 

4»  16  57 
36    933 

25      5      2 

3»  34 
4i    S    5 

41  38  10 
41  18  28 
41  21  16 
29  57  46 
40  42  44 

35    6  21 
41  30    6 
42  48  30 
39  39  36 
41  29  12 
36  50  47 
41     2  50 

41    33      « 

35    628 
44  45 
37         2 
47    3 
4i  15  43 
43  28  32 
45  23 
8  57    9 
39  l6    2 
30  20  42 
30  24  33 
37  *J  47' 

W. 

?84'6" 
118  14  33 
8530 

71  19     2 
67  27  21 

83  37  36 

8924    3 

7°  50  39 
6055 
95  57  56 
97  27  50 
81  40 
90    7 
99    5    6 
8754    4 
93  '4    8 

.89    i  57 
88    2  28 
121  52  59 

56  13 

86  18 
73  32  56 

89   12 
70     2  24 

70    557 
8649    3 
77  21    2 
91  24  42 

101  21  24 

70  55  36 
72  55  45 
72    5  29 
90    328 
74       24 

77    5 
74       33 
70  52  28 
75  33  48 
71  18  49 
76    7  22 
73  25  35 
72    7 
75  58  5« 
75  30 
7618    6 
122  55     f 
95  56  14 
7635    5 
75  42 
79  27  «7 
81  34  12 
88  32  45 
87  is  53 
77  24  16 

Los  Angeles  Cal. 
Louisville  Ky. 

Lowell,  St  Ann's  Ch., 
Mass. 
Machias  Th  Me. 

Dover  N.  H. 

Dubuque    .       .      la. 

Duluth,  S.S  Min. 
Eastpoii,  Con.Ch.  .  Me. 
Edenton,  C.H  N.C. 
Elizabeth  City,  Ct.   u 
Erie  L                Penn 

M  aeon,  Arsenal  Ga. 
Madison,  Dome...  Wis. 
Marblehead,  L.  ..Mass. 
Martinique,  S.  P't  .  W.  I. 
Matagorda,G.S...Tex. 
Matamoras  " 

Eureka,  M.Ch....Cal. 
Falls  St.  Anth'y..  Minn. 
Fernandina,  A.S.  .Fla. 
Florence  Ala. 
Fort  Gibson  Ind.  T. 
Fort  Henry  Tenn. 
Fort  Laramie.Wyo.  T. 
Fort  Leaven  worth,  Ks. 
Frankfort        .      Ky 

Matanzas              Cuba 

Memphis,  S.S.  ..Tenn. 
Mexico  Mex. 
Milwaukee  Mich. 
Minneapolis,U.C.,Min. 
Mississippi  City,  G.  S., 
Miss. 
Mobile,  E.Ch  Ala. 
Monterey,  Az.S...  Cal. 

Montevideo...  Rat  Is'  d 

Montgomery,  S.  H.,  Ala. 
Montreal  C  E 

Frederick  Md. 

Fredericksburg,  E.  Ch.  , 
Va. 
Fredericton  N  B 

Galveston,Cath'l.  .Tex. 
Georgetown  Ber. 
G  eorgeto  wn,  E.  Ch.,  S.C. 
Gloucester,  U.  Ch.  .  .  Ms. 
Grand   Haven,  S.  S., 
Mich. 
Halifax,  Obs  N.S. 
Harrisburg  Penn. 
Hartford,  S.H...  Conn. 
Havana,  Moro  .  .  .Cuba 
Hole  in  the  Wall,  L., 
Bahamas 
Holmes's  Hole,  Ch.,  Ms. 
Hudson  .   .        .NY 

Mound  City  111. 

Nantucket,L....Mass. 
Nantucket,  S.  Tower, 
Mass. 
Nashville,  U.  .  .  .Tenn. 
Nassau  L  N  P 

Natchez  Miss 

Nebraska,  Junction  of 
Forks  of  Platte  Riv. 
New  Bedford,  B.  Ch., 
Mass. 
New  Haven,  Col.,  Conn. 
New  London,  P.  Ch.  " 
New  Orleans,  Mint,  La. 

NEW  YORK,  C.H.,  N.Y. 

Newbern,  E.  Sp.  .  .N.C. 
Newburg,  A.  Sp.,N.Y. 
Newburyport,  L.  ,  Mass. 
New  Castle,  E.  Ch.  ,  Del. 
Newport,  Sp  R.  I. 
Norfolk  C  H  Va. 

Huntsville.  ..  .     Ala. 

Indianapolis  Ind. 
Indianola,G.S  Tex. 
Jackson  Miss 

Jacksonville,   M.  Ch., 
Fla! 
Jalapa  Mex 

Jefferson  City  .  .  .  .Mo. 
Jersey  City,  Gas  Ch'y. 
Kalama,M.Ch...WT. 
Keokuk  S  S            la 

Norwalk              Conn 

Key  West,  T.  Obs.,  Fla. 
Kingston  Jamaica 
Kingston,  C.  H.  .  .C.  W. 
Knoxville  Tenn 

Norwich.      ....     " 

Ocracoke,  L  N.C. 
Ogdensburg,  L.  .  .N.Y. 
Old  Point  Comfort,  Va. 
Olympia  Wash.T. 
Omaha,  P.  Ch  Neb. 
Oswego,  S.  S  N.Y. 
Ottawa  Can 

La  Crosse,  Ct.S.  .  .Wis. 
Lancaster  Penn. 
Lavaca,  A.  S  Tex. 
Leavenworth,  S.  S.  .  Ks. 

Panama,  Cath'l.  ..N.G. 
Parkersburg.  .  .  .  W.  Va. 
Pascagoula  Miss. 
Pensacola,  Sq're.  .Fla. 
Petersburg,  C.  H.  .  .  Va. 

Lima  Peru 

Little  Rock  Ark. 

78  LATITUDE    AND    LONGITUDE. 

Latitude   and.   Longitu.de— Continued. 


LOCATION. 

Latitude.  |  Longitude. 

LOCATION. 

Latitude. 

Longitude, 

NORTH   AND   SOUTH 
AMERICA. 

Philadelphia,  S.H.,  Pa. 
Pike's  Peak,  S.S..  Col. 
Pittsburg  Penn 

N. 

O      1      II 

39  56  53 
3848 
40  32 

44  41  57 
41  58  44 

48    7    3 
1833 

48    6  56 
43  39  28 
45  30 
9  34 

10  28 
43    4  16 
43    2 
40  20  40 
41  49  26 

42    3 

'9        i5 
46  49  12 

43    9 
35  46  50 
37  32  16 

22  56 
N. 
43    8  17 
44    6    6 
4355 
38  34  4i 
42  31  10 

40  46    4 

25   26  22 
29  25   22 

34  '5  46 

32  42    42 
3748 

37  19  58 
35  10  38 
33  43  20 
41  32  30 
40  27  40 

34  26  10 
37  20  49 
36  57  3i 
35  4i    6 
32    4  52 
42  48 
4i    7  So 
32  30 
33  54  58 

39  47  57 
42    6 

20  4.8  ^o 

W. 

0        /      II 

75    9    3 
104  59 
80    2 
73  26  54 
70  39  12 
122  44  33 
72  16    3 

122  44  58 
70  15     i 

122  27   30 
7940 

68    7 
70  42  34 
9i    835 
74  39  55 
71  24  19 

70  ii  18 

98      2   21 
71    12   15 

79    8 
7838    5 
77  26    4 

43    9 

69    6  52 

75  57 
121  27  44 
70  53  58 

i"  53  47 
101    4  45 
98  29  15 

119  15  56 
"7    9  43 

122   23    19 

121  53  39 
120  43  31 
118  16    3 
82  42  15 
74          9 

119  42  42 

121    26   56 
122      I    29 
IO6      I    22 

81    5  26 
73  55 
105  23  33 
93  45 
78    i    8 

89  39  20 
72  36 
.  81  K 

NORTH    AND   SOUTH 
AMERICA. 

St.  Augustine,  P.  Ch., 
Fla. 
St.    Bartholomew,    S. 
Point  W.  I. 

N. 

0       1      II 

29  53  20 

17  53  3° 

17  24 
i7  44  30 
18  29 
17  29 

19  58 
45  M    6 
23    3  i3 
38    8    3 
30    9    i 
18    5 
30  43  12 
44  52  46 

18  21 

13    9 
38    8  51 
30  50 
41  19  36 

42  27  18 

46   12 

43     3 
30  28 
27  36 

22   15   30 

41  54  ii 

II    20 

43  39  35 
40  13  10 
1039 
42  43 
3312 

43  s649 

33  N2 
3850 
19  ii  52 
32  23 

28  46  57 
3843 
45  20 

38  53  20 

42  21  41 
41  23  26 
40    7 

34  14     2 
39  44  27 
42  16  17 
42  45 
33     5 
39  58 

37    17 

W. 

o     /    u 

81  18  41 
62  56  54 

62  50 
64  40  42 
69  52 
63 

75  52 
66    330 
109  40  44 
90  12  17 

84    12    30 

63    3 

81  32  53 

95    4  54 

64  55  18 

61  14 

79    4  15 
102  50 

7i  54 
107  45  27 

60   12 

76    9  16 
8436 
82  45  15 
97  5i  Si 
7i     5  55 
60  27 
79  23  21 
74  45  50 
61  32 
73    2  16 
8742 
75  13 

71  41 

89    2 
96    8  36 
9°  54 
97    * 
87  25 
112     3 

77        36 

7i    9  45 
73  57     » 
80  42 

77  56  38 
75  33     * 
7i  48  13 
97  3° 
90  20 
76  40 

76    -3A 

Plattsburg,  Sp....N.Y. 
Plymouth,  Pier  ...Ms. 
Point  Hudson  W.T. 
Port  au  Prince.  ..W.  I. 
Port  Townshend,  A.S., 
Wash.  T. 
Portland,  C.H  Me. 
Portland  S  S  0. 

St.  Christopher,  N.  Pt., 
W.I. 
St.  Croix,  Obs  " 
St.  Domingo  u 
St.Eustatia,Town.    " 
St.  Jago  de  Cuba,  En- 

Porto  Bello  N  G 

St  John               N  B 

Porto  Cabello,  Mara- 
caibo 
Portsmouth,  L.  .  .N.  H. 
Prairie  du  Chien..Wis. 
Princeton,  S.  Cap.,  N.J. 
Providence,  U.Ch.,R.  I. 
Provincetown,  Sp., 
Mass. 
Puebla  de  los  Angelos, 
Mex. 
Quebec,  Citadel.  .Can'a 
Queenstown  ....     " 
Raleigh,  Square..  N.C. 
Richmond,  Cap.  .  .  .  Va. 

Rio  de  Janeiro,  S.  Loaf. 

Rochester,  R.H..N.Y. 
Rockland,E.Ch...Me. 
Sackett's  Harbor,  N.Y. 
Sacramento  Cal. 
Salem,  So  Mass. 
Salt  Lake  City,  Obs., 
Utah 
Saltillo  Mex 

St.  Joseph  L.  CaL 
St.  Louis,  W.U  Mo. 
St.  Mark's,  Fort..  Fla. 
St.  Martin's,  Fort,  W.  I. 
St.  Mary's,  M.  H.  ..Ga. 
St  Paul               Minn 

St.  Thomas,  Fort  Ch'n, 
W.  I. 
St.  Vincent's,  S.  Point, 
W.I. 
Staunton          .  .  .  .Va 

Stockton,  S.S  Tex. 
Stonington,  L.  .  .Conn. 
Sweetwater  River, 
Mouth  of.  ..Wyo.T. 
Sydney  S.S  N.S. 

Syracuse               N  Y 

Tallahassee  .   .  .     Fla 

Tampa  Bay,  E.  Key  " 
Tampico,  Bar.  .  .  .Mex. 
Taunton,T.  C.Ch.,  Mass. 
Tobago,  N.E.P'r.  W.I. 
Toronto  Can 

Trenton,  P.  Ch  ...N.J. 
Trinidad,  Fort...  W.  I. 
Troy,  D.Ch  N.Y. 
Tuscaloosa  Ala. 

San  Antonio  Tex. 
San    Buenaventura, 
G  S    Cal 

Utica,Dut.Ch....N.Y. 
Valparaiso,  Fort.  .Chili 
Vandalia  111. 

San  Diego,  B.C  ...  " 
San    Francisco,  C.  S. 
Station  Cal. 

San  Jose,  Sp  " 

San  Luis  Obispo..  " 

Vera  Cruz  Mex. 
Vicksburg,  S.S...  Miss. 
Victoria  Tex. 
Vincennes       ....hid 

Sandusky,  L  0. 
Sandy  Hook,  L...  N.J. 
Santa  Barbara,  M.Ch., 
Cal. 
Santa  Clara,  C.Ch..   " 
Santa  Cruz,  F.  S..   " 
Santa  F4  N.  Mex. 
Savannah,  Sp  Ga. 
Schenectady  N.Y. 
Sherman,  R.  R.  D.  ,  Wy. 
Shreveport,  S.  S  La. 
Smithville,  G.S.  ..N.C. 
Springfield  Mass. 
Springfield,  S.H....  111. 
Springfield,  S.S  " 
St.  Augustine  Fla. 

Virginia  City,S.S.,M.T. 
WASHINGTON.  .  .  Capitol 

Watertown,  Ars'l.  .Ms. 
West  Point  N.Y 

Wheeling                Va 

Wilmington,    E.    Ch., 

N.C: 

Wilmington,  T.H.  .Del. 
Worcester,  Ant.  H.  .  Ms. 
Yankton,  S.S  Dak. 
Yazoo  Miss 

York                    Penn 

Yorktown...      ...Va. 

LATITUDE    AND   LONGITUDE. 


79 


Latitude   and    Longitude— Continued. 


LOCATION. 

Latitude. 

Longitude. 

LOCATION. 

Latitude. 

Longitudf 

KUROPE,   ASIA,    AFRICA, 
AND  THE  OCEANS. 

N. 
o  ^   // 

E. 

0        /      // 

EUROPE,   ASIA,  AFRICA, 
AND  THE  OCEANS. 

N. 

O        y       // 

44  24 

E. 

°8  53  " 

Alexandria  L     .... 

31    12 

29  53 

W? 

Algiers  L 

06  4.7 

3    4 

Gibraltar.  

36    7 

^   22 

55  S^ 

4   IO 

Antwerp  

51  *3 

4  24 

CT     oR     -»R 

Archangel  

64  32 

4°  33 

"^ 

Athens 

37  58 

23  44 

E. 

53  33 

0  ">8 

41  s3 

40  20 

9    6 

Batavia  Obs 

6    8 

106  50 

49  I 

W. 

Bencoolen,  Fort,  Su'a. 
Berlin  Obs 

3i8 

102  19 

Hawaii  or  Owyhee  — 
Hongkong       

20  23 

tf 

22   I  6   30 

155  14 

114  14  45 

Bombay  F  S 

?!  S  l6 

21    l8   12 

IC7  ao  36 

Botany  Bay,  C.  Roads. 
Bremen 

s. 

34  N2 
53    5 

151  13 

8  4Q 

Hood  Isrd,Gallapagos. 
Hood's    Island,    Mar- 

'I3 

Q  26 

"w.  3 
^ 

138  57 

Bristol 

51  27 

w9 

2  3S 

Jeddo  or  Tokio  

9N. 

•3C    40 

139  40 

E 

31    48 

37  20 

Brussels  Obs  

50  5'  ii 

4  22 

43  32 

10  18 

Bussorah 

-}Q     3O 

48 

Leipsic  

51  20  20 

12  22 

4  V 

C.2      Q  28 

42Q   IS 

Cadiz  

36     32 

6  18 

W 
W. 

E 

Lisbon  

38  42 

9Q 

Cairo 

30    3 

IT  18 

Liverpool  Obs  

53  24  48 

•3 

Calais 

5°  58 

i  51 

E. 

Calcutta 

88  20 

Madras     

80  I  ^  4>t 

35  31 

25    8 

w5  4S 

Canton         .        . 

23      7 

113   14. 

Madrid  

40  25 

3  42 

Cape  Clear 

51  26 

V 

O  2O 

Majorca  Castle 

•5Q    -34 

3E. 

2   23 

Cape  of  G.  Hope  Obs 

& 

33  S^    3 

9E9 

18  28  45 

36  43 

W. 
4  26 

Cape  St.  Mary,  Mad'r.  . 

*53p 

45    7 

Malta  Valetta  

35  54 

E. 

14  30 

Ceylon  Port  Pedro  .   . 

940 

80  23 

Manila.  

14  36 

121      2 

Christiana 

CQ     CC 

Marseilles           .   .... 

43  *8 

5  32 

5955 

Messina  L  

38  12 

15  35 

Congo  River  

6    8 

12     Q 

13  20 

43  12 

N 

Moscow 

•je    o-a 

Constantinople  St  S 

41    i 

28   co 

Muscat    

23   37 

58  35 

Copenhagen  
Corinth  

55  4i 

37  54 

12  34 

22   52 

Naples,!,  

40  50 

14  16 
W. 

Cronstadt  

CQ     CQ 

2Q  47 

New  €astle 

c.4  «;8 

I   37 

Dover  

5I    g 

I    IQ 

New    Hebrides    Table 

S 

K 

w9 

Island  

ic.  28 

167    7 

Dublin      

co   23   12 

6  20  30 

Niphon    Cape    Idron 

N. 

Edinburgh  

cc    e,7 

5557 

3  12 

Japan    

34  3^    3 

138  50  35 

Falkland   Islands    St 

Odessa 

46  28 

30  44 

Helena  Obs 

1C  ee 

e  4c 

38    8 

13  22 

N 

Paris  Obs 

48  50  13 

2   2O 

Fayal  S  E  Point 

•38   3O 

28  42 

Pekin       

OQ    C.4. 

116  28 

Feejee  Group,  Ovolau, 
Obs 

3   s. 

E. 

I?8    C.3 

Plymouth  

50  21 

W. 

4      Q 

Florence  

17  K 

43  46 

ii  16 

Port  Jackson  .  .  N.  S.  W. 

S. 

•3C     Cl     -52 

4E9 
151  18 

Funchal  Madeira 

02  38 

W. 

16  c,^ 

Porto  Praya,  Cape  Verd 

35  N. 

14   "54 

W. 

23      3 

Geneva  .  .  . 

J.6    II    C.Q 

E5 
6    o  !=; 

Prince  of  Wales  Island. 

1s! 

10  46 

3E3 

142  13 

8o 


LATITUDE  AND   LONGITUDE. 


Latit 

LOCATION. 

rule 

Latitude. 

and.  TJ 

Longitude. 

ongitnde—  Contini 

LOCATION. 

ted. 
Latitude. 

Longitude, 

EUROPE,   ASIA,  AFRICA, 
AND  THE  OCEANS. 

Quct'iistowu 

N. 
o   /    // 
5'  47 

4i  54 
5  54 

28  28 

49  54 
16    i 

4437 
36  59 

14  Is 
8f 

i  17 

38  26 

51  s. 

15  55 
'vatoi 

Lo 

titude. 

w. 

V 

12  27 

4»? 

16  16 
621 
16^2 

33  30 
558 

100 

W. 

13  18 
E. 
103  50 

"V 

i  30 
5  45 
ries.—  JV 

ngitude  g 

Longitude. 

EUROPE,  ASIA,  AFRICA, 
AND  THE  OCEANS. 

St.  Petersburg  

N. 

0      ,      „ 

59  56 
29  59 

21    II 

S. 
33  33  4' 

17  ff 

35  47 
43    7 
3454 
36  47 
40  50 
48  13 

52  s3 

41  N4 

35  26 

6  28 

Table. 

itude. 

E. 

O       /     ft 

30  19 
32  34 
72  47 

"5w3 

M930 

5  54 
5  22 
13  " 
10    6 
14  26 
16  23 
5    21    2    9 

'7444 
13939 
3933 

Longitude. 

Rome,  St.  Peter's  

Suez  

Surat  Castle  

Sydney  N.J 

3.W. 

Santa  Cruz  Ten'fe 
Scilly,  St.  Agnes,  L.... 
Senegal  Fort  

Tahiti  or  Otaheite 
Tangier  

Sevastopol          •  •  •  • 

Seville  

Tripoli  

Siam 

Tunis  City  

Sierra  Leone 

Vienna    

Singapore  

Warsaw  Obs  

Wellington...  New  Z'd 

St.  Helena  »  .  . 

Zanzibar  Island,  Sp.  .  . 
ot  included  in  previous 
wen  in  Time. 

LOCATION.               La' 

Obser 

LOCATION.               La 

Albany,  Dudley  .  . 
Alleghany,  Fenn.  . 
Birr   Castle,  Earl 
of  Rosse 

N. 

O      1     II    in 

42  39  49.55 
40  27  36 

53    5  47 
42  22  52 

52   12  51.6 
S. 

33  V 

55  4<>  53 

4i  44  43 
53  23  13 
55  57  23-2 

43  46  41.4 
46  ii  59.4 

38  53  39 
32  47    7 
5i  28  38 
53  33    5 

51   20  20.1 

52    9  28.2 
53  24  47.8 
40  43  49 

W. 

h.  m.       ,. 
4  54  59-  52 
5  20    2.9 

31  40.9 

4  44  30-9 
22-75 
i  13  55 

50  19.8 
W. 

3  1  6  49.1 

25   22 

"E43'6 
45    3-6 

*V37-7 

5    8  12.5 
5  19  44-7 

E. 

39  54-i 

49  28-5 

'7w57'5 

12     O.  II 

4  55  57 

Madras  

N. 

0     t     ii    m 
13     4     8.1 
43  '7  So 

39    6  26 

55  45  19-8 
48    845 
38    644 

5048    3 
46  48  30 

4i  53  52.2 
4046    4 

374JM 

33  26  24.8 

i7  44  30 

59  56  29.7 
59  20  31 

33  5N4'" 

24  33  3i 

49  35  4° 

38  53  39 
41  23  26 

E. 
A.  m.      i. 

5  20  57.3 

21   29 

W. 

537E59 

2  30  16.96 
46  26.5 

5^.7 

4  23.9 
4  44  49-°2 

4^54-7 
7  27  35-i 
8    9  38.  i 
4  42  18.9 

4  1  8  42.8 
E. 

2      I    13-5 
I    12  24.8 

10    4  59.86 

5  27  14-1 
E. 

1  V0'1 

5    8  12.03 
45548 

Mitchell'  s,Cin.,0. 
Moscow    

Cambridge,  U.  S.  .  . 
Cambridge,  Eng.  .  . 
Cape  oft?.  Hope.. 

Copenhagen,  Un'y. 
Crescent  City,  A. 
S.,Cal  
Dublin  

Munich,  Bogenh'n 
Palermo  

Portsmouth  

Quebec  

Rome,  College  
Salt  Lake  City, 
Utah 
San  Francisco,  Sq., 
Cal  

Edinburgh      .... 

Florence 

Santiago  de  Chili. 
St.  Croix,  W.  I.... 

St.  Petersburg,  A.  . 
Stockholm      .... 

Geneva     

Georgetown,  U.S.  . 
Gibbes's,  Charles- 
ton U  S  

GREENWICH  

Sydney 

Hamburg  

Tifft's,  Key  West. 
Fla  

Leipsic  

Ley  den  

Unkrechtsberg,01- 

Liverpool 

Washington  
West  Point,  N.  Y.  . 

L.  M.  Rutherfurd, 
New  York  

DIFFERENCE    IN    TIME. 


8l 


DIFFERENCE    IN    TIME. 
Difference    in.    Time    at    following    Locations. 

Longitude  computed  both  from  New  York  and  Greenwich. 

Exact  Difference  of  Time  between  New  York  and  Greenwich  is  4  h.  56  m. 
1.6  sec.,  but  in  following  table  2  seconds  are  given  when  the  decimal  in  any 
reduction  exceeds  .5  seconds. 

F  representing  Fast,  and  S  Slow. 


LOCATION. 

New  York. 

Greenwich. 

LOCATION. 

New  York. 

Greenwich. 

Acapulco  

A.  m.   *. 

.43.58- 

lilt  ' 

5  16    5 
5U38 
4354S. 
3  19  '7 
41  32 
950 
3i  34 
i655F. 
i  34558. 
10  26 
2054F 
5734 
228. 
16  46  F. 
i    9  108. 
1038 
2639 

20     iF. 

312368. 

5  49  37  F. 
i  46  30  S. 
9  47  38  F. 
ii  47.6 
53n8 
3  17 

16  12 
29  56  S. 
5  13  3°  F. 
i    234 
19  54  S. 
i    824 

329 
1638 

i  5930 
4  305oF. 
7    i  14 
'       43§. 
2657F. 
10  49  22 

12  508. 

ii  30  F. 

12  28  58 
I      2  10  S. 

6    957F. 
26  58 
2568. 
i  23  46  F. 
6  308. 

21      2F. 

A.  m.   ». 
6  39  178. 
4  55    2 
i  59  32  F. 

12   IO 
I9  32 
17  36 

5  39  568. 
815  19 
5  37  33 
5    552 
527  36 
4  39    6 
6  3027 
5    6  28 
435    8 
35828 
4  56  24 
4  39  l6 
6    5  12 
5    6  39 
5  22  40 
4  36    i 
8    838 
5335F. 
6  42  32  S. 
45i  36  F. 
444  148. 
35  16  F. 
4  52  448. 
455  58 
4  39  50 
52558 
17  28  F 
353288. 
5  i5  56 
6    4  26 
4  59  3° 
5  12  40 
6  55  32 
25  12 
2    5  12  F. 
556458. 
4  29    4 
5  5320F 
5    8528. 
4  44  3i 
7  3256F. 
5  58  128 
i  '355F. 
4  29    48. 
45858 
3  32  16 
5    2  32 
4  35 

Cedar  Keys  

A.  m.   «. 
36    98. 
24    3 
234i 
n48F. 
41  378. 
54  30 
56  24 
4i  57 
3040 

2    3  '5 

28    7 

35  57 
10    6F. 
651  58 
5  46  18 
i  33  47  S. 
i  27  10 
3  2044 
2941 

I   12  30 
4042 
I   58  30 

2  3^ 

3558 

12  26  F. 

4  3?  42o 
i    6  388. 

I    12   10 

28    6F. 
10  248. 
443  M  F. 
8  528. 
24  i5 
32037 
i  ID  40 
29  50 
3  10  F. 
54458. 
i  2459 
56  13 
239 
i  22  54 
13  10 

«3  49 
29  29  F. 

3  48  22 

i  23    88. 
5  20  39  F. 
12  148. 
53i  34  F. 
37  33 
21    68. 
4  34  34  F. 

*.  m.   ,. 
532IIS. 
5  20    5 
5  1943 
4  44  '3 
53738 

s  5031 

5  52  26 
5  37  59 
5  26  42 
6  59  '7 
5  24    8 

5  31  59 
4  45  56 
i  55  56  P. 
50  16 
6  29  48  S. 
6  23  12 
8  1645 

5  2543 
6    2  32 

5  3644 
654  32 
65958 
5  32  10 

4  43  36 
2522 
6    2  40 
6    8  32 
4  27  S^ 
5    6  26 

12  48 

5    454 
52017 
8  1639 
6  12  42 
5  25  5' 
4  52  5' 
5  5047 

6  21      I 

5  52  15 
6  59  « 
6  18  56 
5    9  I2 
5    9  5i 
4  26  33 
i    7  40 
6  19  10 

slliS 

35  32  F. 
4  18  288. 
517    7 

21  38 

Albany 

Alexandria.  .Egypt 
Algiers     

Charleston  

Charlestown  

Amsterdam 

Cheboygan  

Antwerp 

Chicago.  

Chickasaw. 

Astoria  

Cincinnati  

Atlanta 

Colorado  Springs.  . 
Columbia  .... 

Augusta             Ga. 

Augusta             Me 

Columbus  

Austin 

Concord,  

Baltimore            .  . 

Constantinople.... 
Copenhagen  

Bangor   

Barbadoes,  8.  Ft... 

Corpus  Christ!  
Council  Bluffs  
Crescent  City  
Darien    

Bath       

Baton  Rouge  
Beaufort  N.C. 
Beaufort           S  C 

Davenport  

Dayton  

Belfast 

Deadwood  

Denver  

Berlin 

Detroit  

Bismarck 

Dover  Del. 

Bombay  F  S 

Dover     .  .      N  H 

Boston  S  H 

Dublin  

Bremen. 

Bridgeport 

Duluth  

Brooklyn,  N.  Yard. 
Brunswick  Me. 
Brunswick  Ga. 
Brussels 

Eastport    

Eden  ton  

Edinburgh  

Elizabeth  City,  N.C. 
Erie    ' 

Buenos  Ayres  
Buffalo  L 

Eureka  

Burlington  la. 
Burlington  N.  J. 
Burlington  Vt. 
Bushnell  Neb. 
Cadiz 

Falls  St.  Anthony.. 

Fire  Island,  L  
Florence  Ala 

Fort  Gibson  

Cairo     

Fort  Henry..Tenn. 
Fort  Laramie  
Fort  Leaven  worth  . 

Cairo   111. 

Calais  Me. 

Calcutta. 

Callao  

Fredericksb'g..Va. 
Fredericton.  .  .  N.  B. 
Funchal  

Cambridge.  .  .  Mass. 
Canton       

Cape  Gircrdeau.  .  .  „ 
Cape  of  Good  Hope. 
Cape  Horn  

Galveston-.  

Geneva  

Geneva  N.Y. 

Cape  May 

Cape  Race 

Georgetown...  Ber. 
Georgetown...  S.C. 
Gibraltar  

Carthagena  
Castine  .  .  . 

82 


DIFFERENCE   IN    TIME. 


Difference   in  Time—  Continued. 


LOCATION. 

New  York. 

Greenwich. 

LOCATION. 

New  York. 

Greenwich. 

Glasgow  

A43858F. 

13   22 

24      5S. 

49  I0 
4  56    i  .6 
4I  42  F. 
5  35  54 
ii  i8S. 
5  19  F. 
33248. 
4  56  26  F. 
5  27  34  S. 
12  27    i  F. 
15  27  30 

I    12 

5I46S. 

48  18 

I    30     2 
I      430 

30  35 
i  31  S6 
14  16    2F. 

I    12  308. 

8 
7  25  22  F. 

3  15  21  S. 
i    938 
3i  i3 
9  53 

II      2 

39  34 
i    8  58 
27  54  F. 

19  21  b. 
I   30  27 
I   23  40 

5  37  i4  F. 

41  10  S. 

12  22 

4  19  26  F. 

I    12  46  S. 

444    2F. 

19     2  o. 

2  56  16 

4558 

10  45  F. 

26   12 

38  28  S. 

1  1  35^ 

441  14  F. 
4  38  18 
5  54    2    . 
13        10 
9    2 

12  41 

5  17  20 

52   22 

27  50  S. 
33  So 

3038 

4  26 

40  19 

A.  m.    s. 
17     48. 
4  42  40 
5  20    7 
5  45  12 

4  14  40 
3952*. 
5    7  208. 
4  50  43 
5  29  26 
24  F. 
10  23  36  S. 
7  36  59  F. 
10  31  28 
4  54  40  S. 
547  48 
5  44  20 
6  26    4 
6    o  32 
5  26  37 
62738 
9  20       F. 
6    8  32  S. 
4  56  10 

2   29   20  F. 

8  ii  238. 
6    5  40 
5  27  14 
5    5  54 
5    7    4 
5  35  36 
6    4  59 
4  28    8 
.5  *5  22 
6  26  29 
6  19  52 
41  12  F. 
5  37  "S. 
5    824 
36  36  F. 
6    8  48  S. 

12 

5  i5    4 
7  52  18 
542 
4  45  16 
4  29  49 
5  34  30 
5  57  36 
1448 

i7  44 
58       F. 
848 
447       S. 
4  43  21 
21  28  F. 
4    3  40  8. 
6  23  52 
6  29  51 
5  26  40 
6    o  28 
6  36  20 

Milwaukee  

h.  m.    i. 

55  35  S. 
i  16  55 
i         6 

$1*. 

3  ii  308. 
i  ii  10  F. 

49  10  S. 
i  50  F. 
47  M 
7  18  14 
i        268. 
i538F. 
5  53    6 
5i  158. 
13  23 
1    9  37 
i  49  24 
12  19  F. 
4  19 
7  40 
i    4  128. 

12  18 
i 
12  32  F. 

449  34 
6  138. 
10  46  F. 
9    88. 
2  19  F. 
7  34  ^ 

6  58  58  F. 
5588. 
9  ii 
3  1538 
i  27  43 
10  19 
646 

58  22 

5  49  30  F. 

I    21   48  S. 

5    522F. 
30  i5  S. 
12  41  54  F. 
52  508. 
i3  35. 
4  346 
2    3  50 
24    6 
2  14  F. 
4  39  26 
13  25 

16  34 
3  I458S. 
15    2F. 
3  I348S. 
3  23  50  F. 
33  26 
13  " 

A.  m.    «. 
55i  36S, 
6  12  57 
556    8 
5  52    9 
4  37  40 
8    7  32 
344  52 
5  45  12 
4  54  '2 
4    848 

2   22   12  F. 

5  56  28  S. 
4  40  24 
57    4F. 
5  47  16  S. 
5    9  24 

6  45  26 
4  43  42 
4  5i  43 
44822 
6        14 
4  56    1.6 
5    8  20 
4  56    2 
44330 
6  28 
5    2  15 
4  45  15 
559 
4  53  42 
44828 
5    3  55^ 
2    2  56  F. 
5    2       S. 
5    5  12 
8  ii  40 
6  23  45 
5    6  20 
5    2  48 
5  54  24 
5328F. 

5  17  49  s- 
9  20  F. 
5  26  37  S. 
7  45  52  F. 
5  48  52  S. 
5    9  37 
5        36'* 
6  59  52 
5  20    8 
45348 
16  36 
4  42  37 

4  39  28 
8  ii 

4  4i 
8    9  50 

I    32   12 
4   22   36 

4  42  Si 

Gloucester  
Grafton          .     . 

Minneapolis  
Mississippi  City.. 
Mobile 

Grand  Haven  
GREENWICH  
Halifax  

Montauk  Point... 
Monterey  

Montevideo  
Montgomery  
Montreal 

Hamburg 

Harrisburg 

Hartford 

Montserrat 

Havana,  Morro  .  .  . 
Havre  

Moscow  
Mound  City  

Hawaii  or  Owyhee 
Hongkong       .... 

Nantucket 

Naples  

Honolulu 

Nashville 

Hudson    

Nassau       

Huntsville  

Natchez  

Indianapolis  
Indianola  

Nebraska 

New  Bedford  
New  Haven  
New  London  
New  Orleans  

NEW  YORK    .  . 

Jacksonville  
Jalapa  .'.... 

Jeddo  or  Tokio.  .. 
Jefferson  City.... 
Jersey  City 

Newbern 

Jerusalem  
Kalama  
Keokuk  

Newburg 

Newburyport  
New  Castle  
New  Castle.  .  .  Del. 
Newport  

Key  West 

Kingston  Can. 
Kingston  Jam. 
Knoxville        .... 

Norfolk 

Norwalk     

La  Crosse  

Norwich  

La  Guayra 

Ocracoke 

Lancaster  

Odessa        

Lavaca  
Leaven  worth  
Leghorn         

Ogdensburg  

Old  Point  Comfort 
Olympia  

Lexington 

Omaha 

Lima            .       . 

Oswego     

Lisbon    

Ottawa  

Little  Rock        . 

Paducah 

Liverpool  .... 

Palermo         

Lockport 

Panama 

Los  Angeles  
Louisville  

Paris 

Parkersburg  

Lowell  
Machias  Bay  
Macon 

Pekin  
Pensacola  

Petersburg 

Madison 

Philadelphia  
Pike's  Peak  
Pittsburg 

Madrid     

Malaga 

Malta 

Plattsburg    

Manila  

Plymouth  

Maracaibo 

Plvmouth...Mass. 
Port  Au  Prince, 
St.  Domingo 
Port  Townshend.  . 
Portland  
Portland  Or. 
Porto  Praya  
Porto  Rico     .  .  . 

Marblehead,  L  
Marseilles  

Martinique  
Matagorda  
Matamoras  
Matanzas  

Memphis    . 

Mexico 

Portsmouth  

DIFFERENCE    IN    TIME. 


Differe 

LOCATION.           j  New  York. 

Mice   in. 

Greenwich. 

Time—  Contin 

LOCATION. 

ued. 

New  York. 

Greenwich. 

Prairie  du  Chien  . 
Princeton        .... 

h.  m.  9. 

i    8  33  S. 
238 
10  24  F. 
15  16 
ii  13 
442  46 
18  31  S. 

13  43 
2    3  26  F. 
15  228. 
19  34  F. 

5  45  50 
5  1358 
7468. 

3    9  49 
12  26  F. 
2  31  34  S. 
148  17 

i  37  55 
3     i     2 
2  52  37 

3  13  32 
3  13  So 
3  ii  33 

344?F. 
3    2  49  S. 
3    9  46 
3  12    4 

3  So  58  F. 

2      8      48. 
28  20 
22  F. 

432  10 

2      533S. 

i  18  58 

II    36      2F. 

4    2  50  S. 

II   51   22  F. 

16    38. 
6  44  30  F. 
4  50    2 
i    2368. 
539F- 
29  13  S. 
37I9F. 
4  33    2 
7  268. 
3i48F. 

2    22    AT  S. 

A.  m.    ». 
6    4  34  S. 
4  58  40 
4  45  37 
4  4°  45 
4  44  49 
33  16 
5  H  32 
5    9  44 
2  52  36 
5  ii  24 
4  36  27 
49  48  F. 
17  56 
5    348S. 
8    55i 
4  43  36 
7  27  35 
6  44  i9 
633  57 
7  57    4 
7  47  39 

8    933 
8    95i 
8    7  35 
5  30  49 
4  56    i 
7  58  50 
8    548 
886 
i    5    4 
745 
5  24  22 
4  55  40 
23  52 
7    134 
6  15 
6  40       F. 
53  12  S. 
6  55  20  F. 
5  12    58- 
i  48  28  F. 
6       S. 
5  5837 
4  5°  24 
5  25  15 
4  18  43 

5    328 
4  24  14 

7    l8    A1 

St  Louis 

A.  m.    t. 
t     4478. 
4048 
30  10 
ii  24  18 
6  57  18  F. 

36  20 
20   I5  S. 

6    8  26  F. 
8  26 

7    6  18 

2    15          S. 

8    2F. 

»5    i  34 
8358. 
14  44    2  F. 

42  22  S. 

34  59 
1  35  25 
ii  38  F. 

21   32  S. 

5  17  30  F. 
328. 
5  48  36  F. 

S^ 

II    22 

54  46  S. 
4  50 
9  18 

5  53  46  F. 

i  28  33  S. 
i     7  34 

I    32      2 

6    i  34  F. 
53  38  S. 
23210 

6  2O   II  -r. 

12    loS. 

14  F. 

26  46  S. 
6  ii 
15  45 
849F. 
i  33588. 
i    5  18 
14  14  42  F. 

H  14  43 

10  38  S. 

IO   14. 

A.  m.   t. 
6         488. 
5  36  50 
5  26  12 

6  20  20 

2    i  16  F. 
4  19  4i  S. 
5  16  17 
i  12  25  F. 
4  37  36  S. 
2  10  16  F. 

7  ii    28. 
448 
10    5  32  F. 
5    4  37  S. 
9  48       F. 
538243. 
5  3i     ' 
6  31  27 
4  44  24 
5  17  33 
21  28  F. 
4  59    38. 
52  44  F. 
4  52    98. 
40  24  F. 
4  44  40  S. 
5  5048 
5        52 
4  46  44 
556    8 
57  44  F. 
624348. 
6    336 
6  28    4 
i    5  32  F. 
5  49  40  S. 
7  28  12 
i  24    9  F. 
5    8  128. 
45548 
5  22  48 

5      2   12 

5  "  47 
4  47  13 
6  30 

6      I   20 

9  18  40  F. 
9  18  41 
5    6408. 

s    6  16 

St.  Mark's  

Providence  
Province  town  .... 
Quebec                .  . 

St.  Mary's  

St.  Paul  

St.  Petersburg  
St.  Thomas,  Fort.. 
St  aunt  «>n  

Queenstown,L.... 
Raleigh  

Richmond 

Stockholm  

Rio  de  Janeiro... 
Rochester 

Stonington  

Suez 

Rockland  

Sweetwater  River, 
Mouth  of.  

Rome  

Rotterdam 

Sydney  N.S 

Sackett's  Harbor. 
Sacramento 

Sydney...  N.S.W. 
Syracuse 

Salem  

Tahiti  or  Otaheite. 
Tallahassee  
Tampa  Bay  
Tampico  Bar  
Taunton  . 

Salt  Lake  City... 
Saltillo 

San  Antonio  
San  Buenaventura 
San  Diego 

Toronto  

San  Francisco, 

c.  s.  s. 

San  Francisco,  P. 
San  Jose 

Toulon  

Trenton 

Tripoli 

Troy  

Sandusky 

Tunis  

Sandy  Hook 

Turk's  Island  
Tuscaloosa  

Santa  Barbara.  .  .  . 
Santa  Clara 

Utica 

Santa  Cruz 

Valparaiso 

Santa  Cruz.  Ten'  fe 
Santa  Fe* 

Vandalia  

Venice  

Savannah 

Schenectady  
Seville  

Vicksburg 

Victoria  Tex. 
Vienna   

Sherman 

Shreveport 

Vincennes  

Siam       

Virginia  City  
Warsaw  
WASHINGTON,  Obs.. 
West  Point  

Sierra  Leone  
Singapore  
Smithville 

Smyrna 

Southampton  
Springfield  111. 
Springfield.  .Mass. 
St.  Augustine  
St.  Croix,0bs  
St  Helena 

Wilmington..  Del. 
Wilmington.  .N.C. 
Worcester  

Yank  ton 

Yazoo 

Yeddo  

St.  Jago  de  Cuba.  . 
St.  John  
St.  Joseoh..L.Cal 

Yokohama  

York  
Yorktown.  .  . 

To  Compute  Difference  of  Time  "between  !N~ew  Y'orlc  and 
Greenwich,    and.    any    Location  not   given    in    Table. 

RULE.  —  Reduce  longitude  of  location  to  time,  and  if  it  is  W.  of  as- 
sumed meridian  it  is  Sloio  ;  if  E.,  it  is  Fast. 

If  difference  for  New  York  is  required,  and  it  exceeds  4  h.  56  m. 
2  sec.,  subtract  this  time,  and  remainder  will  give  difference  of  time,  S. ; 
and  if  it  (4  h.  56  m.  2  sec.)  does  not  exceed  it,  iubtract  difference  from  it, 
and  remainder  will  give  difference  of  time,  F. 


TIDES. 


TIDES. 
Tide-Table   for    Coast   of  United    States, 

Showing  Time  of  High-water  at  Full  and  New  Moon,  termed  Establish- 
ment of  the  Port,  being  Mean  Interval  between  Time  of  Moon'1  s  Transit 
and  Time  of  High-water.  (U.  S.  Coast  and  Geodetic  Survey.) 


LOCATIONS  AND  TIME. 

| 

I 

LOCATIONS  AND  TIME. 

si 

a 

1 

1 

COAST  FROM  BASTPORT 
TO  NEW  YORK. 

Eastport  Me. 

A.      TO. 

3° 

25 
30 
23 
22 

13 
3° 
12 
27 
2    24 
2    l6 

s;i 

7  So 
7  57 
8  13 
3  3° 

7  45 
7  32 

8   20 

8  25 
9    7 
9  38 
9  28 
16 
ii 
7 
13 

22 

20 

9  35 
7  32 

S3 

8  19 

9  34 

8 
833 
9    4 
"  53 
i3  44 
o  r  se  am 

Feet. 
15 
25 
9.9 

9-9 

10.6 
6 
10.9 
10.3 
3-6 
2-5 
1.8 

2.8 

4-7 
4.6 

5-4 

i 

4.6 

3-7 
2.4 

3-i 
5 
3-2 
2.9 

8 
9.2 

8.6 

r 

r 

5 

tf 

L 

6.8 
fall  of 

Feet. 
7.6 

U 
7.6 

8.1 

8-5 

2.6 

1.6 
13 

r.8 
0  I 
2.8 

3-4 

3-i 

i.'  8 
2.4 

2.2 

2-3 
2.1 
5-2 

4-7 

6.6 
6.1 

3.6 
4 

4-3 

3 
3-9 

11 

tide  a 

CHESAPEAKE  BAY  AND 
RIVERS. 

OldPt.  Comfort§..Va. 
Cape  Henry*  u 
Point  Lookout.  .  .  Md. 
Annapolis  " 

h.    m. 
8  17 

12    58 

'A  4s 

18  59 
H  37 
16  58 

7     4 

9     \ 
7  26 

7  19 
7  26 

IT3 

8    21 

8  84 

9   22 
II    21 

13  *5 

9  38 
9  39 
9  25 
o    8 

0   22 

o  37 

2       6 

3  40 
4  10 

2    36 
I    17 
2       2 
2    42 
2    33 

3  49 

2 

230 

p.  85. 

Feet. 
3 
6 
1.9 
i 
i  3 
i-5 
3 
3-4 

2.2 

5 
3-3 
5-5 
6 

8 
7.6 

4-9 
1.8 
1.6 
1.8 
3-2 

5 
4-7 

4-3 

4-4 
4-3 
5-2 
5-i 
7-3 
4-7 
5-5 
7-4 
7-4 
5-5 

60 

12 

30 

1      2 

7-5 
1.5 
1.6 

Feet 

2 

:1 

.8 

•9 

2-5 
2-3 

1.8 

2.2 

3-S 
4.1 

5-S 

1.2 

I 
I 
1.6 

2-3 

2,2 
2.8 

2.J 

2.  == 

9.1 

2.8 

4-  ] 

3-7 

2.7 

.  t 

Campo  Bello*....  " 
Portland     u 

Cape  Ann*               " 

Portsmouth  N.H. 
Newburyport.  .  .Mass. 
Salem             .        " 

Bodkin  Light.  ...    " 
Baltimore  .     .       " 

James  R.  (City  Pt.  ),  Va 
Richmond  u 

Cape  Cod*              '  ' 

Boston  Light  ...    " 
Bostonf     " 

COASTS    OF    N.    AND    S. 
CAROLINA,  GEORGIA, 
AND   FLORIDA. 

Hatteras  Inlet  ..  N.C 
Cape  Hat  teras  ...    " 
Beaufort  " 

Nantucket  " 

Edgartown  " 
Holmes'  s  Hole  ..    u 
Tarpaulin  Cove  .    " 
Wood's  Hole,  n.  side. 
N.  Bedford  (Dump-) 
ing  Rock)           J 
New  York*  N.Y. 
Albany*  " 

Smithv'le(C.Fear)  " 
Charlestonll   (C.  H.  ) 
Wharf  S  C.  j 

FortPulaski  Ga, 
Savannah  " 

LONG  ISLAND   SOUND. 

Newport               R  I 

St.  Augustine  Fla. 
Cape  Florida  " 
Key  West  " 

Point  Judith  " 
Montauk  Point.  .  N.Y. 
Watch  Hill     .  .     R  I 

Tampa  Bay  " 
Cedar  Keys  " 

Providence*  " 
Stonington              Ct 

WESTERN  COAST. 

San  Diego  Cal. 
San  Pedro               " 

Little  Gull  Isl'd.  N.Y. 
New  London           Ct 

New  Haven  ...      " 

Cuyler's  Harbor  .    " 
San  Luis  Obispo..    " 
Monterey  " 
South  Farallone  .    " 
San  Francisco.  .  .    u 
Mare  Island  " 
Benicia          .   .  .    " 

Bridgeport  " 

Oyster  Bay  N.Y. 
Sand's  Point....     " 
New  Rochelle.  .  .     " 
Throg's  Neck...     " 
Hell  Gate*      ...     " 

COAST  OF  NEW  JERSEY. 

Cold  Spring  Inlet,  N.J. 
Sandy  Hook  N.J. 
Amboy  " 

Ravenswood  " 
Bodega                     " 

Humboldt  Bay.  .  .    " 
Astoria                    Or 

Nee-  ah  Harbor,  Wash. 
Port  Townshend    " 

MISCELLANEOUS. 

Bay  of  Fundy*..N.S. 
Blue  Hill  Bay*..    " 
St.  John's*  " 
Kingston*  Jam. 
Halifax*  N.  S. 

Cape  May  Landing  " 
Egg  Harbor*  u 

DELAWARE  BAY  AND 
RIVER. 

Delaware  Breakwater 
Higbee's  (Cape  May).  . 
Egg  Isl'd  Light..  N.J. 
New  Castle  Del. 
Philadelphia.  .  .  .Penn. 

*  Refers  t 

Pensacola*            Fla 

Galveston*  Tex. 

one.                     t  t  §  II  see 

in.  (half  a  mean  lunar  day)  for  som«  ports  in  Del- 
aware River  and  Chesapeake  Bay.  to  give  succession  of  times  from  the  mouth  ;  hence  12  A.  26  min.  is. 
to  be  «ubtr»cUd  from  the  Establishments  which  are  greater  than  that,  to  give  the  interval  required, 


TIDES. 


Bench   Maries   referred,   to   in   preceding   Table. 

t  BOSTON.  —Top  of  wall  or  quay,  at  entrance  to  dry- dock  in  Charlestown  navy- 
j-ard,  14.76  feet  above  mean  low- water. 

i  NEW  YORK. — Lower  edge  of  a  straight  line,  cut  in  a  stone  wall,  at  head  of  wooden 
wharf  on  Governor's  Island,  14.56  feet  above  mean  low- water. 

§  OLD  POINT  COMFORT,  Va. — A  line  cut  in  wall  of  light-house,  one  foot  from  ground, 
on  southwest  side,  n  feet  above  mean  low- water. 

II  CHARLESTON,  S.  C.  —  Outer  and  lower  edge  of  embrasure  of  gun  No.  3,  at  Castle 
Pinckney,  10.13  feet  above  mean  low- water. 

Establishment   of  the   Fort  for   several    Locations   in 
Europe,  etc. 


PORT. 

TIME. 

PORT. 

TIME. 

PORT. 

TIME. 

Am  sterdaw 

A.    m. 

Chatham 

h.     TO. 

ii  16 

Antwerp  
Beachy  Head    Eng 

4  25 

Cherbourg  
Clear  Cape 

7  49 

London  Bridge  
Newcastle    

2    7 

I   22 

Belfast  

10 

Cowes  

10  46 

Portsmouth  D.-yard, 

Bordeaux    .... 

6     1O 

Dover  Pier.  

II    12 

Eng. 

II  41 

Bremen 

6 

Dublin  Bar 

Quebec  

8 

Brest  Harbor      . 

Funchal          ... 

II    3O 

Ramsgate  Pier  

10  27 

Bristol  
Bristol  Quay 

7  21 

6    27 

Gravesend  Eng. 
Greenock 

I    14 

g 

Rye  Bay  Eng. 
Sheerness    

II  20 

57 

Cadiz    

Holyhead     

8  15 

Calais  

ii  40 

Hull  Eng. 

6  20 

Southampton.  .Eng. 

ii  40 

Calf  of  Man 

Land's  End 

Thames  R  mo'th  " 

12 

Caoe  St.  Vincent... 

2    ^O 

Lisbon..  . 

2    10 

Woolwich  ...    .  .  " 

2   IS 

Rise   and   Fall   of  Tides    in  Q-ulf  of  Mexico. 


1 

1 

1 

LOCATIONS. 

1 

I 

m 

i" 

Feet. 

Feet, 
i  8 

Fee 

Isle  Derniere  La. 

Feet. 
i  4 

Feet. 

1.2 

Feet. 
.7 

I 

i-5 

•4 

Entrance  to  Lake  Cal-) 
casieu                     La  \ 

i-5 

!•* 

.6 

, 

Aransas  Pass  " 

1.  1 

T  8 

6 

i.i 

1.4 

.5 

Brazos  Santiago  " 

•9 

1.2 

•  5 

St.  George's  Island.. .  .Fla. 
Fort    Morgan    ( Mobile ) 

Bay) Ala.  j 

Cat  Island Miss. 

Southwest  Pass La. 

Tides   of  Q-nlf  of  Mexico. 

On  Coast  of  Florida,  from  Cape  Florida  to  St.  George's  Island,  near  Cape  San  Bias, 
the  tides  are  of  the  ordinary  kind,  but  with  a  large  daily  inequality.  From  St. 
George's  Island,  Apalachicola  entrance,  to  Derniere  Isle,  the  tides  are  usually  of  the 
single-day  class,  ebbing  and  flowing  but  once  in  24  (lunar)  hours.  At  Calcasieu  en- 
trance, double  tides  reappear,  and  except  for  some  days  about  the  period  of  Moon's 
greatest  declination,  tides  are  double  at  Galveston,  Texas.  At  Aransas  and  Brazos 
Santiago  the  single-day  tides  are  as  perfectly  well  marked  as  at  St.  George's,  Pensa- 
cola,  Fort  Morgan,  Cat  Island,  and  the  mouths  of  the  Mississippi.  For  some  3,  to  5 
days,  however,  about  the  time  when  the  Moon's  declination  is  nothing,  there  are 
generally  two  tides  at  all  these  places  in  24  hours,  the  rise  and  fall  being  quite  small. 

Highest  high  and  lowest  low  waters  occur  when  greatest  declination  of  Moon 
happens  at  full  or  change.  Least  tides  when  Moon's  declination  is  nothing  at  first 
or  last  quarter. 

Tides   of  Pacific   Coast. 

On  Pacific  coast  there  is,  as  a  general  rule,  one  large  and  one  small  tide  during 
each  day,  heights  of  two  successive  high- waters  occurring,  one  A.M.,  and  other 
P.M.  of  same  24  hours,  and  intervals  from  next  preceding  transit  of  Moon  are  very 
different.  These  inequalities  depend  upon  Moon's  declination.  When  Moon's  de- 
clination is  nothing,  they  disappear,  and  when  it  is  greatest,  either  North  or  South, 
they  are  greatest.  The  inequalities  for  low  water  are  not  same  as  for  high,  though 
they  disappear,  and  have  greatest  value  at  nearly  same  time. 

When  Moon's  declination  is  North,  highest  of  two  high  tides  of  the  24  hours  oc- 
curs at  San  Francisco,  about  11.5  hours  after  Moon's  southing  (transit);  and  when 
declination  is  South,  lowest  of  the  two  high  tides  occurs  about  this  interval. 

Lowest  of  two  low- waters  of  the  day  is  the  one  which  follows  next  highest  high- 
water. 

H 


86 


STEAMING   DISTANCES. 


STEAMING   DISTANCES. 
Distances  "between  various  Ports  of  "United.  States 

and.    Canada. 
By    Lake,  River,  arid    Canal. 


LOCATIONS. 

Lake 
and 
River. 

Canal. 

Total. 

LOCATIONS. 

Lake 
and 
River. 

Canal. 

Total. 

Duluth  to  Buffalo... 
Chicago  to  Buffalo  .  . 

Miles. 
1024 
Q2S 

Miles. 

I 

Miles. 
1025 
925 

Chicago  to  New  York, 
via  Oswego  

Miles. 
IJ95 

Miles. 
232 

Miles. 
1427 

Duluth  to  Oswego.  .  . 
Chicago  to  Oswego.. 
Duluth  to  New  York, 
via,  Buffalo  
via  Oswego  
Duluth  to  Montreal 

133 
034 

166 

294 
280 

27 
26 

353 
233 

72 

1160 
1060 

1519 
1527 
1361 

Chicago  to  Montreal. 
Buffalo  to  Colborne, 
via  Welland  Canal. 
Buffalo  to  New  York. 
Welland  Canal  to 
Montreal  

1190 

142 

304.  5 

7i 

26.77 
352 

70.5 

1261 

26.77 
494 

•aye 

Chicago  to  New 
York,  via  Buffalo  . 

1067 

352 

1419 

Montreal  to  Kingston 
Ottawa  to  Kingston  . 

126.25 

1  20 
126.25 

246.25 
126.25 

Distances  between  varioias  IPorts  and  ISTew  York 
and   London. 


Not  included  in  preceding  Table. 


PORTS. 


Miles.   Miles. 


Alexandria. . . 
Amsterdam . . 
Barbadoes . . . 

Batavia 

Bermudas  . . . 

Bombay 

Boston 

Bremen 

Bristol 

Buenos  Ayres  6oio 

Cadiz 

Calcutta 


N.Y. 

4893 
3291 

1855 
8972 

682 
8522 

356 
3428 
2979 


3125 

9350 


Lond. 

3  102 
262 

3812 
11492 

3M2 
10703 

3030 
408 
50i 

6280 

i  "5 


Cape  Race 

Cowes 

Funchal 

Galway 

Gibraltar 

Glasgow 

Halifax 

Havana 

Hobart  Town . . 
Kingston,  Jam. 

Lima 

Madras 


Miles.    Miles. 


N.Y. 

1  004 
3092 

2  760 
2720 
3260 
2913 

590 

1 161 
9187 
M56 
10050 
8707 


Lond. 

2249 

200 

1303 
721 

1325 
765 
2  706 
41 
113 
4305 


PORTS. 


New  Orleans 

Norfolk 

Pensacola . . . 
Philadelphia. 

Quebec  

Queenstown . 
Rio  Janeiro. . 

St.  Johns 

Southampton 
Swan  River. . 

Tortugas 

Washington . 


N.Y. 

1790 
308 
1623 
262 
1360 
2780 
4970 
1064 
3103 
8480 
U5I 
461 


Lond. 
4730 
3447 
4654 
3404 
3080 
55i 
5200 
2214 

211 

I066I 
4182 


Distances    between    various   Ports   of  England, 
Canada,  TJnited    States,  etc. 

Not  included  in  preceding  Table. 


PORTS. 

Miles. 

PORTS. 

Miles. 

PORTS. 

Miles. 

Halifax  to 
Liverpool      

2  563 

Liverpool  to 
Havana       . 

Panama  to 
San  Diego 

St  Thomas  

I  563 

Portland  

* 

Monterey 

St  Johns  N.  F  .  . 

520 

Baltimore  .... 

San  Francisco 

Quebec  to  Glasgow  . 
Liverpool  to 
Boston  

2563 

2  955 

N  Orleans  to  Havana 
Cape  Race  to 
Fastnet  

570 
1711 

San  Francisco  to 
San  Juan  del  Sud  . 
Acapulco 

2685 

184.1 

Quebec  

2  855 

Halifax  

Manzanilla 

Philadelphia  

3  *47 

Boston  

goc 

San  Diego 

J543 

Callao 

II  37Q 

St  Johns  N  F    to 

Monterey 

Fastnet  

283 

Quebec  

80  1 

Humboldt 

Cape  Race     .   .  . 

I  QQ2 

Boston 

890 

Columbia  R  Bar 

Aspinwall  

4  650 

Greenock  

1848 

Vancouver       . 

638 

Port  Said  

3  290 

Bermudas  to  Nassau. 

804 

Portland  

Melbourne  

13  290 

Panama  to 

Port  Townshend 

r 

Rio  Janeiro  

5  I25 

San  Juan  del  Sud  . 

57° 

Victoria  

San  Francisco.  .  . 

13800 

Gulf  of  Fonseca.  .  . 

739 

Yokohama  . 

via  Panama.  .  .  . 
tnaTehuantepec 

7378 
6400 

Acapulco  
Manzanilla  

1416 
1724 

Honolulu  
Honolulu  to  Callao.. 

2080 
5HS 

STEAMING    DISTANCES. 


P*   £3  ~    CJ    — •    rt  ^«   ^  —   E3    ® 

fe.  c5         "SS^pce^s3 
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88 


STEAMING   DISTANCES. 

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FBACTIONS.  89 


FRACTIONS. 

A  FRACTION,  or  broken  number,  is  one  or  more  parts  of  a  UNIT. 
ILLUSTRATION. — 12  inches  are  i  foot.    Here,  ifoot  is  unit,  and  12  inches  its  parts; 
3  inches  therefore,  are  one  fourth  of  a  foot,  for  3  is  fourth  or  quarter  of  12. 

A  Vulgar  Fraction  is  a  fraction  expressed  by  two  numbers  placed  one 
above  the  other,  with  a  line  between  them ;  as,  50  cents  is  the  £  of  a  dollar. 

Upper  number  is  termed  Numerator,  the  lower  Denominator.  Terms  of  a  frac- 
tion express  numerator  and  denominator;  as,  6  and  9  are  terms  of  |. 

A  Proper  fraction  has  numerator  equal  to,  or  less  than  denominator;  as,  ^,  etc. 

An  Improper  fraction  is  reverse  of  a  proper  one;  as,  ^,  etc. 

A  Mixed  fraction  is  a  compound  of  a  whole  number  and  a  fraction;  as,  5?,  etc. 

A  Compound  fraction  is  fraction  of  a  fraction ;  as,  £  of  | ,  etc. 

A  Complex  fraction  is  one  that  has  a  fraction  for  its  numerator  or  denominator, 

or  both ;  as,  JL,  or  JL,  or  JL,  or  %.,  etc. 

6436 

NOTE.— A  Fraction  denotes  division,  and  its  value  is  equal  to  quotient,  obtained  by 
dividing  numerator  by  denominator;  thus,  ^  is  equal  to  3,  and  ^  is  equal  to  4^. 


Red.iaction   of  Fractions. 

To  Compute  Common.  Measure  or  greatest  Num"ber 
tliat  will  divide  Two  or  more  Numbers  without  a 
Remainder. 

RULE.—  Divide  greater  number  by  less;  then  divide  divisor  by  remainder;  and  so 
on,  dividing  always  last  divisor  by  last  remainder,  until  there  is  no  remainder,  and 
last  divisor  is  greatest  common  measure  required. 

EXAMPLE  i.— What  is  greatest  common      936)  1908  (2 
measure  of  1908  and  936  ?  1872 

"36)  936  (*6 

7£_ 

216.    Hence  36. 

2.— -How  many  squares  can  there  be  obtained  in  an  area  of  90  by  160  feet? 
Here  10  is  greatest  common  measure. 

Hence,  1^  =  16,  and  f$  =  9;  16  X  9  =  144,  and  9°Xl6°=  nxx 

To   Compute   least   Common   Multiple  of  Two   or  more 
Numbers. 

RULE.— Divide  given  numbers  by  any  number  that  will  divide  the  greatest  num- 
ber of  them  without  a  remainder,  and  set  quotients  with  undivided  numbers  in  a 
line  beneath. 

Divide  second  line  in  same  manner,  and  so  on,  until  there  are  no  two  numbers 
that  can  be  divided ;  then  the  continued  product  of  divisors  and  last  quotients  will 
give  common  multiple  required. 

EXAMPLE.  —  What  is  least  5)  40  .  50  .  25 
common  multiple  of  40,  50,  5)  g  .  10  .  5 
and25?  2)  8.  2.  r 


4  .    i  .    i.   Then  5X5X2X4X1X1  =  200. 
To   Reduce   a   Fraction   to   its   .Lowest   Term. 

RULE.— Divide  terms  by  any  number  or  series  of  numbers  that  will  divide  them 
without  a  remainder,  or  by  their  greatest  common  measure. 
EXAMPLE.— Reduce  Jf$  of  a  foot  to  its  lowest  terms. 
Ji«-'°  =  H-8  =  iV-3  =  !, 
H« 


9o 


FRACTIONS. 


To  Reduce  a  Mixed.  Fraction  to  its  Equivalent,  an.  Im- 
proper   Fraction . 

RULE.  —  Multiply  whole  number  by  denominator  of  fraction  and  to  product  add 
numerator;  then  set  that  sum  above  denominator. 

EXAMPLE  i.— Reduce  23!  to  a  fraction.     23X6  +  2  =  l-^.  =  1—. 

2.— Reduce  1J!L  inches  to  its  value  in  feet.     123  -f-  6  =  2of  =  i  foot  8^  int. 

To    Reduce    a    Complex    Fraction   to    a    Simple   one. 

RULE. — Reduce  the  two  parts  both  to  a  simple  fraction,  multiply  numerator  of  re- 
duced fraction  by  denominator  of  reduced  denominator,  and  denominator  of  numer- 
ator fraction  by  numerator  of  denominator  fraction. 

EXAMPLE.— Simplify  complex  fraction  -I.  ~  2\          3  x  2   ~ ~  =  — 

To  Reduce  a  "Whole  ;N"xim"ber  to  an  Equivalent  Fraction 
having    a   given    Denominator. 

RULE.— Multiply  whole  number  by  given  denominator,  and  set  product  over  said 
denominator. 

EXAMPLE. — Reduce  8  to  a  fraction,  denominator  of  which  shall  be  9. 
8  X  9  =  72 ;  then  ^-  result  required. 

To    Reduce     a    Compound    Fraction     to     an    Equivalent 
Simple    one. 

RULE.— Multiply  all  numerators  together  for  a  numerator,  and  all  denominators 
together  for  a  denominator. 

NOTE. — When  there  are  terms  that  are  common,  they  may  be  cancelled. 
EXAMPLE. — Reduce  ^  of  ^  of  ^  to  a  simple  fraction. 

i  x  i  x  f  =  ?x  =  i  •    Or>  i  x  t x  i  —  4 » by  cancellin9  2's  and  3's- 

To    Reduce   Fractions    of  different    Denominations    to 
Equivalents    having   a   Common    Denominator. 

RULE.— Multiply  each  numerator  by  all  denominators  except  its  own  for  new  nu- 
merators; and  multiply  all  denominators  together  for  a  common  denominator. 

NOTE.  —In  this,  as  in  all  other  operations,  whole  numbers,  mixed  or  compound 
fractions,  must  first  be  reduced  to  form  of  simple  fractious. 

2.  When  many  of  denominators  are  same,  or  are  multiples  of  each  other,  ascertain 
their  least  common  multiple,  and  then  multiply  the  terms  of  each  fraction  by  quo- 
tient of  least  common  multiple  divided  by  its  denominator. 

EXAMPLE.  —  Reduce  ^,  J,  and  J  to  a          1X3X4  =  12) 
common  denominator.  2X2X4  =  l6  (  —  27  =  ^.r  =  ^T> 

3X2X3^,8)  6      8        d    ^ 

2X3X4  =  24 


Addition.. 

RULE.— If  fractions  have  a  common  denominator,  add  all  numerators  together, 
and  place  sum  over  denominator. 

NOTE. — If  fractions  have  not  a  common  denominator,  they  must  be  reduced  to 
one.  Also,  compound  and  complex  must  be  reduced  to  simple  fractions. 

EXAMPLE  i.— Add  ^  and  J  together.    ^  +  f  =  f  =  !• 
2.— Add  i  of  |  of  T%  to  2J  of  f . 

ixfxA^if.  2tor|  =  vx*  =  » 

Tben,  J J  +  f  |  =  f  f  | o  +  ^^  =  j^  reduced  to  equivalent  fractions  having 
a  common  denominator  and  thence  to  its  lowest  terms. 


FRACTIONS.  91 

Subtraction. 

RULB.— Prepare  fractions  same  as  for  other  operations,  when  necessary;  then 
subtract  one  numerator  from  the  other,  and  set  remainder  over  common  denom- 
inator. 

EXAMPLE. -What  is  difference  6*i  =  54)       64      84      so      15       B 

between*  and  f?  3X8  =  24    =ff-?t  =  ff  =  if  =  ff- 

8X9  =  72) 


Multiplication. 

RULE.— Prepare  fractions  as  previously  required;  multiply  all  numerators  to- 
gether for  a  new  numerator,  and  all  denominators  together  for  a  new  denominator. 

EXAMPLE  i.  —What  is  product  of  f  and  f?      f  x  f  =  ^  =  J. 

2.— What  is  product  of  6  and  f  of  5?       *  X  f  of  5  =  *  x  ^  =  ¥  =  2a 


Division. 

RULE.— Prepare  fractions  as  before;  then  divide  numerator  by  the  numerator, 
and  denominator  by  the  denominator,  if  they  will  exactly  divide;  but  if  not,  invert 
the  terms  of  divisor,  and  multiply  dividend  by  it,  as  in  multiplication. 

EXAMPLE  i. —Divide  *-f  by  f .        2¥5  -r-  f  =  f  =  if. 
2.  -Dividef  by  ^.       f-&  =  fx^ 


.Application   of  Reduction   of  Fractions. 

To  Compute  Value   of  a   Fraction   in    IParts  of  a  "Whole 
Nvimloer. 

RULE.  —  Multiply  whole  number  by  numerator,  and  divide  by  denominator;  then, 
if  anything  remains,  multiply  it  by  the  parts  in  next  inferior  denomination,  and 
divide  by  denominator,  as  before,  and  so  on  as  far  as  necessary;  so  shall  the  quo- 
tients placed  in  order  be  value  of  fraction  required. 

EXAMPLE  i.—  What  is  value  of  J  of  f  of  9? 


a.—  Reduce  4  of  a  pound  to  an  avoirdupois  ounce.         4)  3  (°  &*• 

j.6  ounces  in  a  Ib. 

4)  48  (12  ounces. 

To    Reduce    a    Fraction    from    one    Denomination    to 
another. 

RULE.—  Multiply  number  of  required  denomination  contained  in  given  denomina- 
tion by  numerator  if  reduction  is  to  be  to  a  less  name,  but  by  denominator  if  to  a 
greater. 

EXAMPLE  i.—  Reduce  £  of  a  dollar  to  fraction  of  a  cent 
JXIOO  =  i^  =  ^. 

2.  —  Reduce  ^  of  an  avoirdupois  pound  to  fraction  of  an  ounce. 

iXi6  =  ¥  =  !  =  2|- 

3.  —  Reduce  f  of  %  of  a  mile  to  the  fraction  of  a  foot 

2of*=&Xsrto=*l£p- 
For  Rule  of  Three  in  Vulgar  Fractions,  see  Decimals,  page  94. 


92  DECIMALS. 

DECIMALS. 

A  DECIMAL  is  a  fraction,  having  for  its  denominator  a  UNIT  with 
as  many  ciphers  annexed  as  the  numerator  has  places ;  it  is  usually  ex- 
pressed by  writing  the  numerator  only,  with  a  point  at  the  left  of  it.  Thus, 
T%  is  -4;  TO  is  -85;  AVoV  is  -ooys ;  and  T^Vrnnr is  -00125.  When  there  is 
a  deficiency  of  figures  in  the  numerator,  prefix  ciphers  to  make  up  as  many 
places  as  there  are  ciphers  in  denominator. 

Mixed  numbers  consist  of  a  whole  number  and  a  fraction;  as,  3.25,  which  is  the 
same  as  3  T8^,  or  M§. 

Ciphers  on  right  hand  make  no  alteration  in  their  value;  for  .4,  .40,  .400  are  deci- 
mals of  same  value,  each  being  T^,  or  J. 


.A-clclition. 

RULE.  —  Set  numbers  under  each  other  according  to  value  of  their  places,  as  in 
whole  numbers,  in  which  position  the  decimal  points  will  stand  directly  under  each 
other;  then  begin  at  right  hand,  add  up  all  the  columns  of  numbers  as  in  integers, 
and  place  the  point  directly  below  all  the  other  points. 
EXAMPLE.— Add  together  25. 125  and  293.7325.         25. 125 

293-7325 
318.8575  sum. 


Subtraction. 

RULE.—  Set  numbers  under  each  other  as  in  addition;  then  subtract  as  in  whole 
numbers,  and  point  off  decimals  as  iu  last  rule. 
EXAMPLE.—  Subtract  15.  15  from  89.  1759.        89.  1759 


74.0259  remainder. 


Multiplication. 

RULE.— Set  the  factors,  and  multiply  them  together  same  as  if  they  were  whole 
numbers;  then  point  off  in  product  just  as  many  places  of  decimals  as  there  are 
decimals  in  both  factors.     But  if  there  are  not  so  many  figures  in  product,  supply 
deficiency  by  prefixing  ciphers. 
EXAMPLE.— Multiply  1.56  by  .75.  1.56 

•75 
780 
1092 
1. 1700  product. 


By   Contraction. 

To  Contract  the  Operation   so  as  to  retain  only  as  many 
Decimal   places   in   Ir»rod.vict   as  may  "be    required. 

RULE.  — Set  unit's  place  of  multiplier  under  figure  of  multiplicand,  the  place  of 
which  is  same  as  is  to  be  retained  for  the  last  in  product,  and  dispose  of  the  rest  of 
figures  in  contrary  order  to  which  they  are  usually  placed. 

In  multiplying,  reject  all  figures  that  are  more  to  right  hand  than  each  multiply- 
ing figure,  and  set  down  the  products,  so  that  their  right-hand  figures  may  fall  in  a 
column  directly  below  each  other,  and  increase  first  figure 
in  every  line  with  what  would  have  arisen  from  figures 
omitted;  thus,  add  i  for  every  result  from  5  to  14,  2  from 
15  to  24,  3  from  25  to  34, 4  from  35  to  44,  etc.,  and  the  sum  -6 
of  all  the  lines  will  be  the  product  as  required.  8  J44  go-}- 2  for  18 

EXAMPLE.— Multiply  13.57493  by  46. 2051,  and  retain  only 
four  places  of  decimals  in  the  product. 

627.23  ii 

NOTE. — When  exact  result  is  required,  increase  last  figure  with  what  would  have  arisen  from  all  the 
figures  omitted. 


DECIMALS.  93 

Division. 

RULB.— Divide  as  in  whole  numbers,  and  point  off  in  quotient  as  many  places  for 
decimals  as  decimal  places  in  dividend  exceed  those  in  divisor;  but  if  there  are  not 
so  many  places,  supply  deficiency  by  prefixing  ciphers. 

EXAMPLE.  Divide  53  by  6.75.      6.75)  53.00000  (=7.851+. 

Here  5  ciphers  are  annexed  to  dividend  to  extend  division. 
By   Contraction. 

RULE.— Take  only  as  many  figures  of  divisor  as  will  be  equal  to  number  of  figures, 
both  integers  and  decimals,  to  be  in  quotient,  and  ascertain  how  many  times  they 
may  be  contained  in  first  figures  of  dividend,  as  usual. 

Let  each  remainder  be  a  new  dividend;  and  for  every  such  dividend  leave  out 
one  figure  more  on  right-hand  side  of  divisor,  carrying  for  figures  cut  off  as  in  Con- 
traction of  Multiplication. 

NOTE. — When  there  are  not  so  many  figures  in  divisor  as  there  are  required  to  be  in  quotient,  con- 
tinue first  operation  until  number  of  figures  in  divisor  are  equal  to  those  remaining  to  be  found  in  quo- 
tient, after  which  begin  the  contraction. 

EXAMPLE.— Divide  2508.92806    92.410315)2508.928106(27.1498  13.849  912 

by  92.41035,  so  as  to  have  only                     1848207-}-!  9*41  832  +  4 

four  places  of  decimals  in  quo-                      660721  ~T6o8  ~8o~ 

646872  +  2  3696  J4  +  2 

13  849  912  6 


Reduction,   of  Decimals. 
To  Reduce  a  Vulgar  Fraction  to  its  Equivalent  Decimal. 

RULE.— Divide  numerator  by  denominator,  annexing  ciphers  to  numerator  to  ex- 
tent that  may  be  necessary. 

EXAMPLE.— Reduce  £  to  a  decimal.        5)  4.0 

~8 

To  Compute  Value  of  a  Decimal  in  Terms  of  an  Inferior 
Denomination. 

RULE.— Multiply  decimal  by  number  of  parts  in  next  lower  denomination,  and 
cut  off  as  many  places  for  a  remainder,  to  right  hand,  as  there  are  places  in  given 
decimal. 

Multiply  that  remainder  by  the  parts  in  next  lower  denomination,  again  cutting 
off  for  a  remainder,  and  so  on  through  all  the  parts  of  integer. 

EXAMPLE  i.— What  is  value  of  .875  dollars?  .875 

100 

Cents,  87.500 


Mills,     5.000 = 87  cent*  5  milk. 
2.  —What  is  volume  of  .140  cube  feet  in  inches? 
.140 

1728  cube  inches  in  a  cube  foot. 
2 4 1. 920  cube  ins. 
3.— What  is  value  of  .00129  of  a  foot?  .01548  ins. 

To   Reduce   a   Decimal    to    an   Equivalent  Decimal   of  a 
Higher   Denomination. 

RULE.— Divide  by  number  of  parts  in  next  higher  denomination,  continuing  op* 
eration  as  far  as  required. 
EXAMPLE  i.— Reduce  i  inch  to  decimal  of  a  foot.       i2|i.ooooo 

I  •o8333+/oot 
•.—Reduce  14"  12'"  to  decimal  of  a  minute.  14"  12'" 


.236  66'+ minute. 


94 


DECIMALS. — DUODECIMALS. — MEAN   PROPORTION. 


When  there  are  several  numbers,  to  be  reduced  all  to  decimal  of  highest. 
RULE. —  Reduce  them  all  to  lowest  denomination,  and  proceed  as  for  one  denomi- 
nation. Feet.     Ins.      Be. 
EXAMPLE.— Reduce  5  feet  10  inches  and  3  5        10       3 
barleycorns  to  decimal  of  a  yard.                               I2 

70 
3 


5-9166 


1.9722-}-  ycuds. 


Rule   of  Three. 

RULE.  —  Prepare  the  terms  by  reducing  vulgar  fractions  to  decimals,  compound 
numbers  to  decimals  of  the  highest  denomination,  first  and  third  terms  to  same 
denomination,  then  proceed  as  in  whole  numbers. 

EXAMPLE. — If  .5  of  a  ton  of  iron  cost  .75  of  a  dollar,  .5  :  .75  ::  .625 

what  will  .625  of  a  ton  cost?  .625 

.5) -468  75 

.9375,  dollar. 


DUODECIMALS. 

In  Duodecimals,  or  Cross  Multiplication,  the  dimensions  are  taken  in  feet, 
inches,  and  twelfths  of  an  inch. 

RULE.— Set  dimensions  to  be  multiplied  together  one  under  the  other,  feet  under 
feet,  inches  under  inches,  etc. 

Multiply  each  term  of  multiplicand,  beginning  at  lowest,  by  feet  in  multiplier,  and 
set  result  of  each  immediately  under  its  corresponding  term,  carrying  i  for  every 
12  from  one  term  to  the  other.  In  like  manner,  multiply  all  multiplicand  by  inches 
of  multiplier,  and  then  by  twelfth  parts,  setting  result  of  each  term  one  place  farther 
to  right  hand  for  every  multiplier.  And  sum  of  products  will  give  result. 

EXAMPLE.  —  How  many  square  inches  are       Feet.      Ins.   Twelfths. 
there  in  a  board  35  feet  4.5  inches  long  and  12         35         4         6 
feet  3^  inches  wide?  I2         3         4 

424         6         o 
8        10         i        6 

ii         960 
434          3        ii        o        o 

"Value   of  Duodecimals   in    Sq.uare   Feet   and    Inches. 

Sq.  Ft.          Sq.  Ins. 


y*£  of  i  twelfth  =  yy1^  or  .083  333,  etc. 
"  .006  944,  etc. 


Sq.  Ft.       Sq.  1 

i  Foot =    i     or  144. 

i  Inch 

i  Twelfth 

ILLUSTRATION.  —  What  number  of  square  inches  are  there  in  a  floor  100  feet 
6  inches  long  and  25  feet  6  inches  and  6  twelfths  broad? 

2566  feet  ii  ins.  3  twelfths  =  2566  feet  135  ins. 


MEAN   PROPORTION. 

MEAN  PROPORTION  is  proportion  to  two  given  numbers  or  terms. 
RULE.— Multiply  two  numbers  or  terms  together,  and  extract  square  root  of  their 
product. 

EXAMPLE.— What  is  mean  proportionate  velocity  to  16  and  81  ? 
1 6  X  8 1  =  1296,  and  -^1296  —  36  mean  velocity. 


RULE  OF  THREE. COMPOUND  PROPORTION.     95 

RULE  OF  THREE. 

RULE  OF  THREE.  —  It  is  so  termed  because  three  terms  or  numbers  are 
given  to  ascertain  a  fourth. 

It  is  either  DIRECT  or  INVERSE. 

It  is  Direct  when  more  requires  more,  or  less  requires  less ;  thus,  if  3  bar- 
rels of  flour  cost  $18,  what  will  10  barrels  cost? 

In  this  case  Proportion  is  Direct,  and  stating  must  be, 
As  3  :  10  : :  18  •  60. 

It  is  Inverse  when  more  requires  less,  or  less  requires  more;  thus,  if  6  men  build 
a  certain  quantity  of  wall  in  10  days,  in  how  many  days  will  8  men  build  like  quan- 
tity? Or,  if  3  men  dig  100  feet  of  trench  in  7  days,  in  how  many  days  will  2  men 
perform  same  work  ? 

Here  the  Proportion  is  Inverse,  and  stating  must  be, 

As  8  :  6  : :  10  :  7. 5,  and  2  :  3:17:  10. 5. 

The  fourth  term  is  always  ascertained  by  multiplying  2d  and  3d  terms  together, 
and  dividing  their  product  by  ist  term. 

Of  the  three  given  numbers  necessary  for  the  stating,  two  of  them  contain  the 
supposition,  and  the  third  a  demand. 

RULE. — State  question  by  setting  down  in  a  straight  line  the  three  necessary 
numbers  in  following  manner : 

Let  third  term  be  that  of  supposition,  of  same  denomination  as  the  result,  or  4th 
term  is  to  be,  making  demanding  number  2d  term,  and  the  other  number  ist  term 
when  question  is  in  Direct  Proportion,  but  contrariwise  if  in  Inverse  Proportion; 
that  is,  let  demanding  number  be  ist  term. 

Multiply  2d  and  3d  terms  together,  and  divide  by  ist,  and  product  will  give  re- 
sult, or  4th  term  sought,  of  same  denomination  as  2d  term. 

NOTE. — If  first  and  third  terms  are  of  different  denominations,  reduce  them  to  same.  If,  after  divis- 
ion, there  is  any  remainder,  reduce  it  to  next  lower  denomination,  divide  by  divisor  as  before,  and 
quotient  will  be  of  this  last  denomination. 

Sometimes  two  or  more  statings  are  necessary,  which  may  always  be  known  by 
nature  of  question. 

EXAMPLE  i.—  If  20  tons  of  iron  cost  $225,  what  will         Tons.  Tons.  Dolls. 
500  tons  cost?  20  :  500  ::  225 

500 


2|0)  II  250|0 

''    v  5625  dollars. 

2.— A  wall  that  is  to  be  built  to  height  of  36  feet,  was  raised  9  feet  by  16  men  in 
6  days;  how  many  men  could  finish  it  in  4  days  at  same  rate  of  working? 

Days.  Days.     Men.  Men. 
4   :    6  '.'.   16  :   24 

Then,  if  9  feet  requires  24  men,  what  will  27  feet  require? 
9  :  27  : :  24  :  72  men. 


COMPOUND   PROPORTION. 

COMPOUND  PROPORTION  is  rule  by  means  of  which  such  questions  as 
would  require  two  or  more  statings  in  simple  proportion  (Rule  of  Three) 
can  be  resolved  in  one. 

As  rule,  however,  is  but  little  used,  and  not  easily  acquired,  it  is  deemed  prefer- 
able to  omit  it  here,  and  to  show  the  operation  by  two  or  more  statings  in  Simple 
Proportion. 

ILLUSTRATION  i.— How  many  men  can  dig  a  trench  135  feet  long  in  8  days,  when 
16  men  can  dig  54  feet  in  6  days? 

Feet,  jrtei.    Men.  Men. 

First As    54  :  135  : :  16  :  40 

Days.  Days.    Men.  Men. 

Second As    8   :   6    ::   40  :  30 


gO  COMPOUND  PROPORTION. — INVOLUTION. — EVOLUTION. 

2.— If  a  man  travel  130  miles  in  3  days  of  12  hours  each,  how  many  days  of  10 
hours  each  would  he  require  to  travel  360  miles? 

Miles.  Miles.   Days.  Days. 
First As    130  :  360  ::  3  :  8.307+ 

Hours.  Hours.    Days.       Days. 

Second As    10  :    12  ::  8.307  :  9.9684 

3. — If  12  men  in  15  days  of  12  hours  build  a  wall  30  feet  long,  6  wide,  and  3  deep, 
in  how  many  days  of  8  hours  will  60  men  build  a  wall  300  feet  long,  8  wide,  and 
6  deep?  120  days. 

By    Cancellation. 

RULE. — On  right  of  a  vertical  line  put  the  number  of  same  denomination  as  that 
of  required  answer. 

Examine  each  simple  proportion  separately,  and  if  its  terms  demand  a  greater 
result  than  $d  term,  put  larger  number  on  right  and  lesser  on  left  of  line;  but  if  its 
terras  demand  a  less  result  than  $d  term,  put  smaller  number  on  right  and  larger 
on  left  of  line. 

Then  Cancel  the  numbers  divisible  by  a  common  divisor,  and  evolve  the  4th  term 
or  result  required. 

Take  Illustration  i,  page  95 :  3d  term,  or  term  of  supposition  of  same  denomination 
as  required  result,  16  men. 

Statement.  135  feet  require   more  men  than  54  feet.         Result  by  Cancellation. 


54 

8 


16      and  8  days  less  men  than  6  days. 
135 

6  2  X  5  X  3  =  30  men. 


2 

m  5 

0     3 


ILLUSTRATION  3. — 3d  term,  15  days. 

Statement.  60  men  require  less  days  than  12  men,          Result  by  Cancellation. 

15         8  hours  more  days  than  12  hours,  300  feet 


60 
8 

30 
6 
3 


1  2         more  days  than  30  feet,  8  feet  more  days 
than  6  feet,  and  6  feet  more  days  than 


300 
8 


3  feet. 


6  3X4X10  =  120  days. 


3 
4 

10 


INVOLUTION. 


INVOLUTION  is  multiplying  any  number  into  itself  a  certain  number  of 
times.  Products  obtained  are  termed  Powers.  The  number  is  termed  the 
Root,  or  first  power. 

When  a  number  is  multiplied  bv  itself  once,  product  is  square  of  that 
number  ;  twice,  cube  ;  three  times,  biquadrate  ;  etc.  Thus,  of  the  number  5. 

5  is  the  Root,  or  ist  power. 

5  X  5  =  25     "     Square,  or  2d  power,  and  is  expressed  5*. 
5  x  5  X  5  =  125     "     Cube,  or  3d  power,  and  is  expressed  53. 
5X5X5X5  =  625     "     Biquadrate,  or  4th  power,  and  is  expressed  54. 
The  lesser  figure  set  superior  to  number  denotes  the  power,  and  is  termed  the 
Index  or  Exponent. 

ILLUSTRATION  i.—  What  is  cube  of  9  ?  729. 

2.  —What  is  cube  of  j  ?  ff  . 

3.—  What  is  4th  power  of  i.  5  ?  5.0625. 


EVOLUTION. 

EVOLUTION  is  ascertaining  Root  of  any  number. 

Sign  ^J  placed  before  any  number  indicates  that  square  root  of  that  number  it  re- 
quired or  shown. 
Same  character  expresses  any  other  root  by  placing  the  index  above  it. 


7  =  3,  and  ^64  =      4. 
Roots  which  only  approximate  are  termed  Surd  Roots. 


EVOLUTION.  97 

To   Extract    Square   Root. 

BULB.— Point  off  given  number  from  units'  place,  into  periods  of  two  figures  each. 

Ascertain  greatest  square  in  left-hand  period,  and  place  its  root  in  quotient;  sub- 
tract square  number  from  this  period,  and  to  remainder  bring  down  next  period 
for  a  dividend. 

Double  this  root  for  a  divisor;  ascertain  how  many  times  it  is  contained  in  divi- 
dend, exclusive  of  right-hand  figure,  which,  when  multiplied  by  number  to  be  put 
to  right  hand  of  this  divisor,  product  will  be  equal  to,  or  next  less  than  dividend; 
place  result  in  quotient,  and  also  at  right  hand  of  divisor. 

Multiply  divisor  by  last  quotient  figure,  and  subtract  product  from  dividend; 
bring  down  next  period,  and  proceed  as  before. 

NOTE.— Mixed  decimals  must  be  pointed  off  both  ways  from  units. 

EXAMPLE  i. — What  is  square  root  of  2? 

2.000000  (MI+.  2-  What  is  S(luare  root  of  '44' 

144  (12 


24  ioo 

4l  96  22(044 

981!  400  I  44 

i    281 


Square   Roots   of  Fractions. 

RULE.— Reduce  fractions  to  their  lowest  terms,  and  that  fraction  to  a  decimal, 
and  proceed  as  in  whole  numbers  and  decimals. 

NOTE.— When  terms  of  fractions  are  squares,  take  root  of  each  and  set  one  above  the  other ;  aa 
§.  is  square  root  of  |~|. 

EXAMPLE.  —What  is  square  root  of  ^  ?  .  866  025  4. 

To   Compute   4th.   or   8th.    Root   of  a   Number,  etc. 

RULE.  —For  the  4th  root  extract  square  root  twice,  and  for  8th  root  thrice,  etc. 


To   Extract   Cube  Root. 

RULE.— From  table  of  roots  (page  272)  take  nearest  cube  to  given  number,  and 
term  it  the  assumed  cube. 

Then,  as  given  number  added  to  twice  assumed  cube,  is  to  assumed  cube  added 
to  twice  given  number,  so  is  root  of  assumed  cube  to  required  root,  nearly ;  and  by 
using  in  like  manner  the  root  thus  found  as  an  assumed  cube,  and  proceeding  in 
like  manner,  another  root  will  be  found  still  nearer;  and  in  like  manner  as  far  as 
may  be  deemed  necessary. 

EXAMPLE. — What  is  cube  root  of  10517.9? 

Nearest  cube,  page  272;  10648,  root  22.  10648.      10517.9 


21296        2I035-8 
1       10517-9     10648. 

31813.9  :  3l683-8  '•  22  :  21.9+- 

To  Ascertain  or  to  Compute  the  Square  or  Cube  Roots  of 
Roots,  \Vhole  Numbers,  and  of  Integers  and  Decimals, 
see  Table  of  Squares  and  Cubes,  and  Rules,  pp.  272,  300. 

To   Extract   any    Root  whatever. 
Let  P  represent  number.  I    Let  A  represent  assumed  power,  r  ita  root 

n         "         index  of  the  power.     |  R         "         required  root  of  P. 

Then,  as  sum  of  w+i  x  A  and  n—i  x  P  is  to  sum  of  n-f-i  X  P  and  n  —  i  X  A 
so  is  assumed  root  r  to  required  root  R. 
ILLUSTRATION. — What  is  cube  root  of  1500? 
Nearest  cube,  page  272,  is  1331,  root  u. 

then,  w-f-i  x  A  =  5324,  n-f-i  X  P  =  6ooo 
n  —  i  X  P  =  3000,  n  —  i  X  A  —  2662 

8324  8662 ::  u  :  11.446-!-. 


98    EVOLUTION. — PROPERTIES  OP  NUMBERS. — POSITION. 

To   Compute  the   Root  of  an    Even  IPower   greater   than 
any   given    in    Table   of  Square    and.    Cube    Roots. 

RULE. — Extract  square  or  cube  root  of  it,  which  will  reduce  it  to  half  the  given 
power;  then  square  or  cube  root  of  that  power;  and  so  on  until  required  root  is  ob- 
tained. 

EXAMPLE  i.— Suppose  a  i2th  power  is  given;  the  square  root  of  that  reduces  it  to 
a  6th  power,  and  the  square  root  of  6th  power  to  a  cube. 

2. — What  is  biquadrate,  or  4th  root,  of  2560000? 

-^/2  560  ooo  =  1600,  and  -1/1600=40. 

NOTB.— For  other  rules  for  extraction  of  roots  sae  pp.  301-4. 


PROPERTIES    OF    NUMBERS. 

1.  A  Prime  Number  is  that  which  can  only  be  measured  (divided  without  a  re- 
mainder) by  i  or  unity. 

2.  A  Composite  Number  is  that  which  can  be  measured  by  some  number  greater 
than  unity. 

3.  A  Perfect  Number  is  that  which  is  equal  to  the  sum  of  all  its  divisors  or  ali- 
quot parts ;  as  6  =  |-,  ^ ,  |r. 

4.  If  sum  of  the  digits  constituting  any  number  be  divisible  by  3  or  9,  the  whole 
Is  divisible  by  them. 

5.  A  square  number  cannot  terminate  with  an  odd  number  of  ciphers. 

6.  No  square  number  can  terminate  with  two  equal  digits,  except  two  ciphers  or 
two  fours. 

7.  No  number,  the  last  digit  of  which  is  2,  3,  7,  or  8,  is  a  square  number. 


IPowers  of  tlie  first  Nine  Numbers. 


ISt. 

2d. 

3d. 

4th. 

5th. 

6th. 

7th. 

8th. 

9th. 

I 

I 

i 

i 

i 

i 

i 

i 

i 

2 

3 

4 

8 

16 

32 

64 

128 

256 

512 

9 

27 

81 

243 

729 

2187 

6561 

19083 

4 

16 

64 

256 

1024 

4096 

16384 

65536 

262144 

5 
6 

25 

125 

625 

3125 

15625 

78125 

390625 

1953125 

36 

216 

1296 

7776 

46656 

279936 

i  679616 

10077  696 

7 

49 

343 

2401 

16807 

117649 

823  543 

5  764  801 

40  353  607 

8 
9 

64 

512 

4096 

32768 

262  144 

2097152 

16777216 

-134217728 

81 

729 

6561 

59049 

531  441 

4  782  969 

43046721 

387420489 

POSITION. 

POSITION  is  of  two  kinds,  SINGLE  and  DOUBLE,  and  it  is  determined  by 
number  of  SUPPOSITIONS. 

Single    Position. 

RULE.— Take  any  number,  and  proceed  with  it  as  if  it  were  the  correct  one;  then, 
as  result  is  to  given  sum,  so  is  supposed  number  to  number  required. 

EXAMPLE  i.  — A  commander  of  a  vessel,  after  sending  away  in  boats  A,  J,  and  J 
of  his  crew,  had  left  300;  what  number  had  he  in  command? 

Suppose  he  had 600. 

i  of  600  is  200 
•^  of  600  is  100 
£•  of  600  is  150  450 

150  :  300  : :  600  :  1200  men. 


POSITION. — FELLOWSHIP.  99 

2.  —  A  person  asked  his  age,  replied,  if  ^  of  my  age  be  multiplied  by  2,  and  that 
product  added  to  half  the  years  I  have  lived,  the  sum  will  be  75.  How  old  was  he  ? 

37. 5  year*. 

Double   ^Position. 

RULE.— Assume  any  two  numbers,  and  proceed  with  each  according  to  conditions 
of  question ;  multiply  results  or  errors  by  contrary  supposition ;  that  is,  first  posi- 
tion by  last  error,  and  last  position  by  first  error. 

If  errors  are  too  great,  mark  them  -f ;  and  if  too  little,  — . 

Then,  if  errors  are  alike,  divide  difference  of  products  by  difference  of  errors;  but 
if  they  are  unlike,  divide  sum  of  the  products  by  sum  of  errors. 

EXAMPLE  i.— A  asked  B  how  much  his  boat  cost;  he  replied,  that  if  it  cost  him  6 
times  as  much  as  it  did,  and  $30  more,  it  would  have  cost  him  $300.  What  was 
price  of  the  boat? 

Suppose  it  cost  him. .  60 30 

6  times.  6  timet. 

360                            180 
and    30  more           and    30  more 
390                           210 
300                           300 
90-}-                            90— 
30  2d  position.    60  ist  position. 


90    2700  5400 

9°    54oo 
180)  8100    (45  dollars. 


. 

2.  —What  is  length  of  a  fish  when  the  head  is  9  inches  long,  tail  as  long  as  its  head 
and  half  its  body,  and  body  as  long  as  both  head  and  tail  ?  6  feet. 


FELLOWSHIP. 

FELLOWSHIP  is  a  method  of  ascertaining  gains  or  losses  of  individuals 
engaged  in  joint  operations. 

Single    Fellowship. 

RULE.— As  the  whole  stock  is  to  the  whole  gain  or  loss,  so  is  each  share  to  the 
gain  or  loss  on  that  share. 

EXAMPLE. — Two  men  drew  a  prize  in  a  lottery  of  $9500.  A  paid  $3,  and  B  $2  for 
the  ticket;  how  much  is  each  share? 

5  :  9500  1:3:  5700,  A's  share. 
5  :  9500  ::  2  :  3800,  B's  share. 

Double    Fellowship, 
Or  Fellowship  with  Time. 

RULE. — Multiply  each  share  by  time  of  its  interest ;  then,  as  sum  of  products  is  to 
product  of  each  interest,  so  is  whole  gain  or  loss  to  each  share  of  gain  or  loss. 

EXAMPLE. — A  cutter's  company  take  a  prize  of  $10000,  which  is  to  be  divided  ac- 
cording to  their  rate  of  pay  and  time  of  service  on  board.  The  oflScers  have  been 
on  board  6  months,  and  the  crew  3  months;  pay  of  lieutenants  is  $100,  ensigns  $50, 
and  crew  $10  per  month;  and  there  are  2  lieutenants,  4  ensigns,  and  50  men;  what 
is  each  one's  share  ? 

2  lieutenants $100  =  200  x  6  =  1200 

4  ensigns 50  =  200  X  6  =  1200 

50  men 10  =  500  x  3  =  1500 

3900 

Lieutenants 3900  :  1200  : :  10000  :  3076.92 -f-  2  =  1538.46  dolls. 

Ensigns 3900  :  1200  ::  10000  :  3076.92-:-  4=  769.23     ' 

Men 3900  :  1500  ::  10 ooo  :  3846.16-7-50=     76.92     " 


100 


PERMUTATION. 


PERMUTATION. 

PERMUTATION  is  a  rule  for  ascertaining  how  many  difl  arent  ways  auy 
given  number  of  numbers  of  things  may  be  varied  in  their  position. 

Permutation  of  the  three  letters  abc,  taken  all  together,  are  6 ;  taken  twe 
and  two,  are  6 ;  and  taken  singly,  are  3. 

RULE.— Multiply  all  the  terms  continually  together,  and  last  product  will  give 
result. 

EXAMPLE  i.— How  many  variations  will  the  nine  digits  admit  of? 
1X2X3X4X5X6X7X8X9  =  362  880. 

2. — How  many  years  would  there  be  required  to  elapse  before  10  persons  could 
be  seated  in  a  varied  position  collectively,  each  day  at  dinner,  including  one  day  in 
every  4  years  for  a  leap  year?  9935  years,  42  days. 


When  only  part  of  the  Numbers  or  Elements  are  taken  at  once.  RULE. 
Take  a  series  of  numoers, 'beginning  with  number  of  things  given,  decreasing  by  i, 
until  number  of  terms  equals  number  of  things  or  quantities  to  be  taken  at  a  time, 
and  product  of  all  the  terms  will  give  sum  required. 

EXAMPLE  i. — How  many  changes  can  be  made  with  2  events  in  5? 

5  —  1  =  4,  and  4X5  =  2  terms.     Hence,  5  x  4  =  20  changes. 
2.— How  many  changes  of  2  will  3  playing  cards  admit  of? 

3—1  =  2,  and  2X3  =  2  terms.    Hence,  2X3  =  6  changes. 
3.— How  many  changes  can  be  rung  with  4  bells  (taken  4  and  4  together)  out  of  6  ? 
4  — 1  =  3,  and  3X4X5X6  =  4  terms  or  changes. 


[ence,  3X4X5X6  =  360  changes. 

When  several  of  the  Elements  are  alike.  RULE.—  Ascertain  the  permutations 
of  all  the  numbers  or  things,  and  of  all  that  can  be  made  of  each  separate  kind  or 
division;  divide  number  of  permutations  of  whole  by  product  of  the  several  partial 
permutations,  and  quotient  will  give  number  of  permutations. 

EXAMPLE.  —How  many  permutations  can  be  made  out  of  the  letters  of  the  word 
persevere  (9  letters,  having  4  e's  and  2  r's)? 

i  X2X  3X4X  5X6X7X8X9  =  362880; 

i  X  2  X  3  X  4  =  24  for  the  e's  ;  1X2  =  2  for  the  r's,  and  24  X  2  =  48. 
Hence,  362  880  -4-  48  =  7560. 

Or,  Add  logarithms  of  all  the  terms  together,  and  number  for  the  sum  will  give 
result. 

EXAMPLE  i.—  How  many  permutations  can  be  made  with  three  letters  or  figures? 
Log.  i  =  .oo 

2  =  .3oio3 

3  =  .  4771213 

.7781513  =  log.  of  number  6. 

a.—  How  many  variations  will  15  numbers  in  16  places  admit  of? 
Add  logarithms  of  numbers  i  to  16  and  take  logarithm  of  their  sum  — 

viz.  ,  13.  320  661  97  =  20  922  789  888  ooo. 
Number  of  positions  of  the  blocks  in  the  "  15  puzzle  "  is  as  above  for  their  16  permutations. 

IPermutatioxis, 

Whereby  any  questions  of  Permutation  may  be  solved  by  Inspection,  number  of 
terms  not  exceeding  20. 


I 

5 

120 

2 

6 

720 

6 

7 

5040 

24 

8 

40320 

362880 

3628800 

39916800 

479001600 


6227020800 

87178291200 

1307674368000 

20922789888000 


355687428096000 

6402373705728000 

121645100408832000 

2432902008176640000 


ARITHMETICAL    PROGRESSION.  IOI 

ARITHMETICAL   PROGRESSION. 

ARITHMETICAL  PROGRESSION  io  a  series  of  numbers  increasing  or  de- 
creasing by  a  constant  number  ov  difference ;  'Is,  i,<$,  3,  7/1.2,  9, 0,  3.  The 
numbers  which  form  the  series  are  designated  Terms ;  the  first  and  last 
are  termed  Extremes,  and  the  others  Means. 

When  any  three  of  following  elements  are  given,  the  remaining  two  can  be  ascer- 
tained—viz., First  term,  Last  term,  Number  of  terms,  Common  Difference,  and  Sum 
of  all  the  terms. 

To    Compute   First   Term. 

When  Last  term,  Number  of  terms,  and  Sum  of  series  are  given.  RULE.  —From 
quotient  of  twice  sum  of  series,  divided  by  number  of  terms,  subtract  last  term. 

I  —  d          S        dn  —  i  ,     , 

Or, ; ;       and  y  (I  + . 5 d)2 — zdS±-5d  =  a.    a  represent- 

n  —  i         n  2 

ing  ist,  I  last,  n  number  of,  and  S  sum  of  all  terms,  and  d  common  difference. 

ILLUSTRATION.— A  man  travelled  390  miles  in  12  days,  travelling  60  miles  last  day. 
How  far  did  he  travel  first  day  ? 

-  =  65,  and  65  —  60  =  5  first  term. 


To   Compute   Last   Term. 

When  First  term,  Common  Difference,  and  Number  of  terms  are  given.  RULE.  — 
Multiply  the  number  of  terms  less  i,  by  common  difference,  and  to  product  add  first 
term. 

EXAMPLE.—  A  man  travelled  for  12  days,  at  the  rate  of  5  miles  first  day,  10  second, 
and  so  on  ;  how  far  did  he  travel  the  last  day  ? 

12  —  i  X  5  =  55>  and  55  +  5  =  60  miles. 

When  First  term,  Number  of  terms,  and  Sum  of  series  are  given.  RULE.  —  Divide 
twice  sum  of  series  by  number  of  terms,  and  from  quotient  subtract  first  term. 

Or,  —  -a;         V2dS  +  (a-.5d)2±.5  d  ;          and  -  +  li!Lnl>  =  i 
n  n  2 

ILLUSTRATION.  —  A  man  travelled  360  miles  in  12  days,  commencing  with  5  miles 
first  day;  how  far  did  he  travel  last  day? 

-  =  65,  and  65  -  5  =  60  miles. 


To   Compute    Number   of*  Terras. 

When  Common  Difference  and  Extremes,  or  First  and  Last  term,  are  given. 
RULE.—  Divide  difference  of  extremes  by  common  difference,  and  add  i  to  quotient. 

EXAMPLE.—  A  man  travelled  3  miles  first  day,  5  second,  7  third,  and  so  on,  till  he 
went  57  miles  in  one  day  ;  how  many  days  had  he  travelled  at  close  of  last  day  ? 
57  —  3-7-2  =  27,  and  27+  i  =28  days. 

When  Sum  of  series  and  Extremes  are  given.    RULE.—  Divide  twice  sum  of  series 
by  sum  of  first  and  last  terms. 


r  l~a  i  /2 

r>— +'>       v~ 


S        /2  a  —  d\*      d  —  za 


ILLUSTRATION.  —  A  man  travelled  840  miles,  walking  3  miles  first  day  and  57  last 
day;  how  many  days  was  he  travelling? 


IO2  ARITHMETICAL    PROGRESSION. 

To    Compiate    Common    Difference. 

When  Number  of  terms  and  Extremes  are  given.     RULE. — Divide  difference  of 
extremes  by  i  less  than  number  of  terms. 

•         28  —  2  an         '    l-^a/Xl —  a.  2  nl  —  2  S 

r'"w(n~ir;  TtT-T^-a   '  ~n~(n ^TjT  ~~ 

ILLUSTRATION.— Extremes  are  3  and  15,  and  number  of  terms  7 ;  what  is  common 
difference  ? 

*5  —  3-M7  — i)  =  j,  and  ^  =  2  com-  dif- 

To    Compute    Sum    of*  the    Series    or   of*  all    Terms. 

When  Extremes  and  Number  of  terms  are  given.    RULE. — Multiply  number  of 
terms  by  half  sum  of  extremes. 


l+ax(l-a)  .  l  +  a 


Or,  2  a  +  d  (n-i)  X  .5  n;  M — ^ -  +  - 

and  2  J —  (d  X  n— i )  X  .  5  n  =  S. 
.  ILLUSTRATION.— How  many  times  does  hammer  of  a  clock  strike  in  12  hours? 


12  X  12+  J  —  JS^,  and  156-7-2  =  78  fo'wes. 

To  Comp-ute    any    N"um"ber   of*  Arithmetical    Means   or 
Terms   between    t\vo    Extremes. 

RULE.  —  Subtract  less  extreme  from  greater,  and  divide  difference  by  i  more 
than  number  of  means  or  terms  required  to  be  ascertained,  and  then  proceed  as 
in  rule. 

To  Compute  T\vo  Arithmetical  IMeans  or  Terms  "bet-ween 
two   given    Extremes. 

RULE. — Subtract  less  extreme  from  greater,  and  divide  difference  by  3,  quotient 
will  be  common  difference,  which  being  added  to  less  extreme,  or  taken  from  great- 
er, will  give  means. 

EXAMPLE  i. — Compute  two  arithmetical  means  between  4  and  16. 
16  —  4-7-31=  4  com.  dif. 
4  -}-  4  =   8  one  mean. 
1 6  —  4  =  12  second  mean. 

2.— Compute  four  arithmetical  means  between  5  and  30. 

30  —  5  =  25,  and  25 ^-44-1=5=: com.  dif. 
5  4-  5  =  10=  i  st  mean.  15-}-  5  =  20  =  3*2  mean. 

io-j-5  =  i5  —  2d      "  20  -j-  5  =  25  =  ^th    " 

!M!iscellaneoi*s    Illustrations,, 

1.  A  steamer  having  been  purchased  upon  following  terms  —  viz.:  $5000  upon 
transfer  of  bill  of  sale  and  balance  in  monthly  instalments,  commencing  at  $4500 
for  first  month,  and  decreasing  $500  in  each  month,  until  whole  sum  is  paid. 

ist.  How  many  months  must  elapse  before  final  payment? 
2d.  What  was  amount  of  purchase  money,  or  sum  of  series? 
Here  are  first  and  last  terms  —  viz.,  500  and  5000,  and  common  difference,  500. 
Hence,  To  compute  number  of  terms  and  amount  of  purchase, 
5000  —  500  -r-  500  =  9,  and  9  -f- 1  =  10  =  number  of  terms  or  months,  and  10  X 

'  '  5°°  =  10  X  2750  =  $  27  500,  amount  of  purchase. 

2.  If  TOO  stones  are  placed  in  a  right  line,  one  yard  apart;  how  many  yards  must 
a  person  walk,  to  take  them  up  one  at  a  time  and  put  them  into  a  basket,  one  yard 
from  first  stone? 

JFirst  term  2,  last  term  200,  and  number  of  terms  100. 

Hence,  100  x  —    —  =  10 100  yards. 


GEOMETRICAL   PROGRESSION.  IO3 

3.  If  in  the  sinking  of  curb  of  a  well,  $3  is  to  be  given  for  first  foot  in  depth,  $5 
for  second,  $7  for  third,  and  increasing  in  like  manner  to  a  depth  of  20  feet,  what 
would  it  cost? 

First  term  3,  common  difference  2,  and  number  of  terms  20. 
Hence,  20  —  1X2-1-3  =  41,  last  term. 

Then,  3  -|-  41  X  —  =  440,  sum  of  all  terms,  or  cost  of  curb. 

4.  If  a  contractor  engaged  to  sink  a  curb  to  depth  of  20  feet  for  $400,  and  the 
contract  was  annulled  when  he  had  reached  a  depth  of  8  feet;  how  much  had  he 
earned? 

400  -r-  20  =:  number  of  terms.  But  inasmuch  as  400  may  be  divided  into  20  terms 
in  arithmetical  proportion  in  many  different  ways,  according  to  value  of  ist  term, 
it  becomes  necessary  to  assume  the  value  of  the  first  foot  as  value  of  ist  term. 

Assuming  it  at  $5,  the  required  proportion  will  be,  ist  term  5,  number  of  terms  20, 
rum  of  series  400. 

Hence,  400-^X^X2  =  &»      1 11  common  difference. 

_20X(20— •  l)  380  ld' 

Then,  5  -f-  i4-J  x  7  =  16^  =  ist  term  +  product  of  common  difference  and  Sth 
term  less  i,  which  added  to  5  —  21-^,  and  X  4  =  half  number  of  terms  for  which 
cost  is  sought  =  84^  dollars,  sum  earned. 


GEOMETRICAL  PROGRESSION. 

GEOMETRICAL  PROGRESSION  is  any  series  of  numbers  continually  in- 
creasing by  a  constant  multiplier,  or  decreasing  by  a  constant  divisor,  as 
i,  2,  4,  8,  16,  etc.,  and  15,  7.5,  3.75,  etc. 

The  constant  multiplier  or  divisor  is  the  Ratio. 

When  any  three  of  following  elements  are  given,  remaining  two  can  be  computed, 
viz.  :  first  term,  Last  term,  Number  of  Terms,  Ratio,  and  Sum  of  all  Terms. 

To    Compute   First   Terxn. 

When  Ratio,  Last  Term,  and  Number  of  Terms  are  given.  RULE.  —  Divide  last 
term  by  ratio  raised  to  a  power  denoted  by  number  of  terms  less  i. 

Or,  K~  and  rl  —  S  (r  —  i)  =  a.  a  representing  ist  term,  I  last,  n  number  of, 
S  sum  of  all  terms,  and  r  ratio. 

ILLUSTRATION.  —Last  term  is  4374,  number  of  terms  8,  and  ratio  3;  what  is  first 
term? 


To  Compute   Last   Term. 

When  First  Term  and  Ratio  are  Equal.  RULE.—  Write  a  few  of  leading  terms 
of  series  and  place  their  indices  over  them,  beginning  with  a  unit.  Add  together 
the  most  convenient  and  least  number  of  indices  to  make  the  index  to  term  required. 

Multiply  terms  of  the  series  of  these  indices  together,  and  product  will  give  term 
required. 

Or,  Multiply  first  term  by  ratio  raised  to  a  power,  denoted  by  number  of  terms 
less  i. 

EXAMPLE  i.  —  First  term  is  2,  ratio  2,  and  number  of  terms  13;  what  is  last  term? 

Indices,  12345 

Terms,    2,  4,  8,  16,  32. 
Then,  5-j-s-j-3=:i3z=  sum  of  indices,  and  32  X  32  X  8  =  8192  =  last  term. 

Or,  2  X  2  J3—  x  =  8192.     Also  by  inspection  of  table,  page  105,  isth  term  =  8192. 


IO4  GEOMETRICAL    PROGRESSION. 

2.— The  price  of  12  horses  being  4  cents  for  first,  16  for  second,  and  64  for  third, 
and  so  on;  what  is  price  of  last  horse? 

Indices,  1234 

Terms,   4,  16,  64,  256. 
Then,  4  +  4  +  4  =  i2=swm  of  indices,  and  256  X  256X256  =  2563  =  1167772.16. 

When  First  Term  and  Ratio  are  Different.  RULE.— Write  a. few  of  leading  terms 
of  series,  and  place  their  indices  over  them,  beginning  with  a  cipher.  Add  together 
the  most  convenient  indices  to  make  an  index  less  by  i  than  term  sought. 

Multiply  terms  of  these  series  belonging  to  these  indices  together,  and  take 
product  for  a  dividend. 

Or,  Raise  first  term  to  a  power,  index  of  which  is  i  less  than  number  of  terms 
multiplied;  take  result  for  a  divisor;  proceed  with  their  division,  and  quotient  will 
give  term  required. 

EXAMPLE  i.— First  term  is  i,  ratio  2,  and  number  of  terms  23;  what  is  the  last 
term? 

Indices,  01234    5 
Terms,   i,  2,  4,  8,  16,  32. 

Then,  5  +  5  +  5  +  5-1-2  =  22  =  sum  of  indices,  and  32  X  32  X  32  X  32  X  4  = 
4 194  304,  and  4 194  304  -r-  the  sth  power  (6  —  i)  of  i  =  i  =  4 194  304. 

Or,  i  X  2  23— I  =  4 194  304.    By  inspection  of  table,  page  105,  23d  term  =  4  194  304. 

2.— If  i  cent  had  been  put  out  at  interest  in  1630,  what  would  it  have  amounted 
to  in  1834,  if  it  had  doubled  its  value  every  12  years? 

1834  — 1630  =  204,  which  -r- 12 ,=  17,  and  17  +  1  =  18  =  number  of  terms. 
Indices,  01234     7 
Terms,    i,  2,  4,  8,  16,  128. 

16  X  8  X  4  X  2  X  i  =  131  °72>  and  J3i  072 


Then,  7  +  4  +  3  +  2  +  1  =  17,  and  128  X  16 
-r- 1,  the  4th  power  (5  —  i)  of  i  —  $  1310.72. 


When  First  Term,  Ratio,  and  Sum  of  the  series  are  given.  RULE.— From  sum  of 
series  subtract  quotient  of  first  term  subtracted  from  sum  of  series,  divided  by 
ratio.  Oraxrn~I=Z 

EXAMPLE.— First  term  is  2,  ratio  3,  and  sum  of  series  2186;  what  is  last  term? 
ai86— — — —  =  2186  —728  =  1458,  last  term. 

To   Compute   !N~umlber   of  Terms. 

When  Ratio,  First,  and  Last  Terms  are  given.  RULE. — Divide  logarithm  of  quo- 
tient of  product  of  ratio  and  last  term,  divided  by  first  term,  by  logarithm  of  ratio. 

ft    log-  (q  +  Sr  — i)  — log,  q  log.  I  —  log,  q 

log.  r  log.  (S  -  q)  -  log.  (S  - 1)  "*" 

and  — '• ^ 1-  i  =  n. 

log.  r 

EXAMPLE.  — Ratio  is  2,  and  first  and  last  terms  are  i  and  131072;  what  is  num- 
ber of  terms? 

log.  2X  I3I°72  _  i0g  262  I44  =  5. 4i8  54,  and  5. 418  54 -r- log.  of  2  =  5'41   54  =  18. 
i  .30103 

To   Compute    Sunn   of  Series. 

When  First  Term,  Ratio,  and  Number  of  Terms  are  given.  RULE. — Raise  ratio  to 
a  power  index  of  which  is  equal  to  number  of  terms,  from  which  subtract  i ;  thei 
divide  remainder  by  ratio  less  i,  and  multiply  quotient  by  first  term. 


GEOMETRICAL   PROGRESSION. 


105 


ILLUSTRATION  i. — First  term  is  2,  ratio  2,  and  number  of  terms  13;  what  is  sum 
of  series  ? 

2*3  — 1  =  8192  — 1  =  8191,  and  8191-^(2  — 1)  =  8191,  and  8191  x  2  =  16382. 
2.  — If  a  man  were  to  buy  12  horses,  giving  2  cents  for  first  horse,  6  cents  for 
second,  and  so  on,  what  would  they  cost  him  ?  $5314.40. 

To    Compute    Ratio. 

When  First  Term,  Last  Term,  and  Numbers  of  Terms  are  given.     RULE. — Divide 
last  term  by  first,  and  quotient  will  be  equal  to  ratio  raised  to  power  denoted  by  i 
less  than  number  of  terms;  then  extract  root  of  this  quotient 
^    S  — o 


ILLUSTRATION.— First  term  is  2,  last  term  4374,  and  number  of  terms  8;  what  is 
ratio  ? 

4074  8—i, 

—  =  2i87,and    -^2187  =  3,  ratio. 

Miscellaneous    Illustrations. 

1.  What  is  gth  term  in  geometrical  progression  3,  9,  27,  81,  etc.?  and  what  is 
sum  of  terms? 

ist  term  =  3,  number  of  terms  9,  and  ratio  3. 

Hence,  by  rule  to  compute  last  term,  ist  term  and  ratio  being  equal- 
Indices,  1234 
Terms,    3,  9,  27,  81. 

Then,  2-1-3-1-4  =  9  =  sum  of  indices,  and  9  X  27  X  81  =  19  683  =  last  term. 

By  rule  to  compute  sum  of  terms — 

3   ~*  X  3  =  I9a  2  =  9841  X  3  =  29  523,  sum  of  terms. 

2.  First  term  is  i,  ratio  2,  and  last  term  131072;  what  is  sum  of  series? 

131 072  X  2  —  i  =  262  143,  and  262  143  -r-  2  —  i  =  262 143. 

3.  What  are  the  proportional  terms  between  2  and  2048  ? 

4  -f-  2  =  6,  and  6  — 1  =  5,  and  A/ =  4. 

Hence,  2  :  8  :  32  :  128  :  512  :  2048. 

4.  Sum  of  series  is  6560,  ratio  3,  and  number  of  terms  8;  what  is  first  term? 

6^x~f^: 


Greoxnetrical   ^Progressions, 

Whereby  any  questions  of  Geometrical  Progression  and  of  Double  Ratio  may 
solved  by  Inspection,  number  of  terms  not  exceeding  56. 


I 

1  'I 

16384 

29 

268  435  456 

43 

4398046511104 

2 

2    16 

32768 

30 

536870912 

44 

8  796  093  022  208 

3 

4 

17 

65536 

3i 

1  073  741  824 

45 

17  592  186044416 

4 

8 

18 

131  072 

32 

2147483648 

46 

35184372088838 

5 

16 

J9 

262  144 

33 

4  294  967  296 

47 

70368744177664 

6 

•32 

20 

524288 

34 

8589934592 

48 

140737488355328 

7 

64 

21 

I  048  576 

35 

17179869184 

49 

281474976710656 

8 

128 

22 

2097152 

36 

34359738368 

5° 

562949953421  312 

9 

256 

23 

4  194  304 

37 

68719476736 

51 

1  125  899  906  842  624 

10 

512 

24 

8  388  608 

38 

i37438953472 

52 

2251799813685248 

ii 

1024 

25 

16777216 

39 

274  877  906  944 

53 

4503599627370496 

12 

2048 

26 

33554432 

40 

549755813888 

54 

9007199254740992 

13 

4096 

27 

67  108  864 

41 

1099511627776 

55 

18  014  398  509  481  9^4 

14 

8192 

28 

134217728 

42 

2199023255552 

56 

36  028  797  oi  8  963  968 

ILLUSTRATIONS.  — i2th  power  of  2  =  4096,  and  7th  root  of  128  =  2. 


IO6  ALLIGATION. 


ALLIGATION. 

ALLIGATION  is  a  method  of  finding  mean  rate  or  quality  of  different  ma- 
terials when  mixed  together. 

To    Compute    UVIean    JPrice    of  a,    Mixture. 

When  Prices  and  Quantities  are  known.  RULE.  —  Multiply  each  quantity  by  its 
mte,  divide  sum  of  products  by  sum  of  quantities,  and  quotient  will  give  rate  of  the 
composition. 

EXAMPLE. —  If  10  Ibs.  of  copper  at  20  cents  per  lb.,  i  Ib.  of  tin  at  5  cents,  and  i  Ib. 
of  lead  at  4  cents,  be  mixed  together,  what  is  value  of  composition? 

10  X  20  =  200 

iX  5=  5 
_iX  4=  4 
12  )  209  (17. 416  cents. 

To    Compute    Quantity    of  each,    Article. 

When  Prices  and  Mean  Price  are  given.  RULE. — Write  prices  of  ingredients,  one 
under  the  other  in  order  of  their  values,  beginning  with  least,  and  set  mean  price 
at  left.  Connect  with  a  line  each  price  that  is  less  than  mean  rate  with  one  or  more 
that  is  greater. 

Write  difference  between  mixture  rate  and  that  of  each  of  simples  opposite  price 
with  which  it  is  connected;  then  sum  of  differences  against  any  price  will  express 
quantity  to  be  taken  of  that  price. 

EXAMPLE.— How  much  gunpowder,  at  72,  54,  and  48  cents  per  pound,  will  compose 
a  mixture  worth  60  cents  a  pound? 


(48  \  12,  at  48  cents. 

60  ]  54\/  12,  at  54  cents. 

(727  12 -f  6  =  18,  at  72  cents. 


Here,  72  — 60  =  12  at  48,  72  — 60  =  12  at  54,  60— 48  =  12,  and  60  —  54  =  6  = 
12  +  6  =  18  at  72. 

Then  12  X  48-4-12  X  54 -f  18  X  72  =  2520,  and  2520^-12  +  12+  12 -f 6  —  60  cents. 

NOTE.  —  Should  it  be  required  to  mix  a  definite  quantity  of  any  one  article,  the  quantities  of  each, 
determined  by  above  rule,  must  be  increased  or  decreased  in  proportion  they  bear  to  defined  quantity. 

Thus,  had  it  been  required  to  mix  18  pounds  r\t  48  cents,  result  would  be  18  at  48, 
18  at  54,  and  27  at  72  cents  per  pound. 

When  the  whole  Composition  is  limited.  RULE  —As  sum  of  relative  quantities, 
as  ascertained  by  above  rule,  is  to  whole  quantity  required,  so  is  each  quantity  so 
ascertained  to  required  quantity  of  each. 

EXAMPLE.— Required  100  pounds  of  abore  mixture 

Then,  12 -f  12+  18  =  42.    Then,  42  :  100  ::  12  :  28.571  pounds. 

42  :  loo  ::  12  :  28.571  pounds. 

42  :  loo  ::  18  :  42.857  pounds. 

When  Price  of  Several  Articles  and  Quantity  of  one  of  them  is  given.  RULE. — As- 
certain proportionate  quantities  of  ingredients  by  previous  rule. 

Then,  as  number  opposite  ingredients,  quantity  of  which  is  given,  is  to  given 
quantity;  so  is  number  opposite  to  each  ingredient  to  quantity  required  of  that  in- 
gredient. 

EXAMPLE.  —  Having  35  Ibs.  of  tobacco,  worth  60  cents  per  pound,  how  much  of 
other  qualities,  worth  65,  70,  and  75  cents  per  pound,  must  be  mixed  with  it,  so  as  to 
sell  mixture  at  68  cents  per  pound? 

By  previous  rule,  it  is  ascertained  there  must  be  7  Ibs.  at  60,  2  at  65,  3  at  70,  and 
8  at  75  cents;  but  as  there  are  35  Ibs.  at  60  cents  to  be  taken,  other  quantities  and 
kinds  must  be  increased  in  like  manner. 

Hence,  7  :  35  :  :  2  :  10  =  10  at  65  cents. 
7  :  35  :  :  3  :  i5  —  15  "  7°  cents. 
7  :  35  :  :  8  :  40  =  40  "  75  cents. 


SIMPLE   INTEREST.  IO/ 

SIMPLE   INTEREST. 

To  Compvite   Interest  on.   any  Griven    Sum    fbr   a   Period 
of  One   or   more    Years. 

RULE.— Multiply  given  sum  or  principal  by  rate  per  cent,  and  number  of  years; 
point  off  two  figures  to  right  of  product,  and  result  will  give  interest  in  dollars  and 
cents  for  the  period. 

EXAMPLE.— What  is  interest  upon  $  1050  for  5  years  at  7  per  cent.  ? 
1050  X  7  X  5  =  36  750,  and  367-  5o  —  $  367-5<>-*  ^  ^ 

When  Time  is  less  than  One  Year.  RULE.— Proceed  as  before,  multiplying  by 
number  of  months  or  days,  and  dividing  by  following  units— viz.,  12  for  months, 
and  365  or  366,  as  the  case  may  be,  for  days. 

EXAMPLE.— What  is  interest  upon  $1050  for  5  months  and  30  days  at  7  per  cent.? 
5  months  and  30  days  =  183  days.    *°5°  XJ  X  183  =  ^  ftn 

The  operation  of  computing  interest  may  be  performed  thus : 

Assuming  interest  upon  any  sum  at  6  per  cent.—  i  per  cent  for  2  months. 

Interest  at  5  per  cent,  is  ^th  less  than  at  6  per  cent. 

Interest  at  7  per  cent,  is  ^th  greater  than  at  6  per  cent. 

Taking  preceding  example — 2  months  =  i  per  cent=  10.50 

2      "      =i        "  10.50 

«      "      =i       "  5-25 

30  days      =  i  month    =  5.25 

31-50 
Add  J  for  7  per  cent.=  5.25 

$36-  75 

NOTB.— Difference  between  this  amonnt  and  preceding  arises  from  183  days  being  taken  in  one  case, 
and  half  a  year,  or  182.5  days,  in  the  other. 

In  every  computation  of  interest  there  are  four  elements— viz. ,  Principal,  Time, 
Rate,  and  Interest  or  Amount,  any  three  of  which  being  given,  remaining  one  can 
be  ascertained. 

To    Compute    Principal. 

When  Time,  Rate  per  Cent.,  and  Interest  are  given.  RULE.— Divide  given  interest 
by  interest  of  $i,  etc.,  for  given  rate  and  time. 

EXAMPLE.— What  sum  of  money  at  6  per  cent,  will  in  14  months  produce  $  14? 
14  -T-  .07  =  200  dollars. 

To   Compute   Rate  per  Cent. 

When  Principal,  Interest,  and  Time  are  given.  RULE.— Divide  given  interest  by 
interest  of  given  sum,  for  time,  at  i  per  cent. 

EXAMPLE.  —If  $  32.66  was  discounted  from  a  note  of  $400  for  14  months,  whaf 
was  that  per  cent.  ? 
Interest  on  400  for  14  months  at  i  per  cent.  =  4. 66. 

Then  32.66  -=-  4.66  =  7  per  cent. 

To    Compute    Time. 

When  Principal,  Rate  per  Cent.,  and  Interest  are  given.  RULE. — Divide  given  in 
I  erest  by  interest  of  sum,  at  rate  per  cent,  for  one  year. 

EXAMPLE.— In  what  time  will  $  108  produce  $  11.34,  at  7  Per  cent.  ? 
Interest  on  108  for  one  year  is  7.56. 

1 1. 34  ^-7. 56  =1.5  years. 

ILLUSTRATION  i.  —If  an  amount  of  $ 2175  is  returned  for  a  period  of  15  months 
rate  of  interest  having  been  7  per  cent.,  what  was  principal  invested?  $2000. 

».— If  $  looo  in  1 8  months  will  produce  $  1090,  what  is  rate  ?  6  ver  cent. 


io8 


COMPOUND    INTEREST. 


COMPOUND  INTEREST. 

If  any  Rrincipal  be  multiplied  by  number  (in  following  table)  opposite 
years,  and  under  rate  per  cent.,  sum  will  be  amount  of  that  principal  at  com- 
pound interest  for  time  and  rate  taken. 
EXAMPLE.— What  is  amount  of  $500  for  10  years  at  6  per  cent.  ? 

Tabular  number. . . .  i.  790  84,  and  i. 790  84  x  500  =  895.42  dollars. 


i 

3 

Per  Cent. 

4 
Per  Cent. 

5 
Per  Cent. 

6 
Per  Cent. 

1 

3 

Per  Cent. 

4 

Per  Cent. 

5 
Per  Cent. 

6 
Per  Cent, 

i 

•°3 

.04 

•05 

i.  06 

13 

1.46853 

1.66507 

1.88564 

2.1329* 

2 

.0609 

.0816 

.1025 

1.1236 

14 

1-51529 

1.73167 

1-97993 

2.2609 

3 

.09273 

.12486 

.15762 

1.191  01 

15 

1-55797 

1.80095 

2.07892 

2.39655 

4 

•12551 

.16986 

•2155 

1.26247 

16 

1.60471 

1.87298 

2.18287 

2-54035 

•IS927 

.21668 

•  276  28 

1.338  22 

17 

1.65285 

1.94799 

2.29201 

2.69277 

6 

•  194  05 

•26532 

•34 

1.41851 

18 

1.70244 

2.025  8  1 

2.40661 

2-85433 

7 

.22987 

•31593 

.4071 

L50363 

J9 

1-7535 

2.10684 

2.52695 

3-02559 

8 

.26677 

.36857 

•47745 

I-59384 

20 

i.  806  1  1 

2.191  13 

2.65329 

3.207  13 

9 

•30477 

•42331 

•55132 

1.68947 

21 

1.86029 

2.27876 

2.  785  96 

3-39956 

10 

•34392 

.48024 

.62889 

1.79084 

22 

1.916  i 

2.36992 

2.925  26 

3-603  53 

ii 

.38424 

•53945 

•7I033 

1.89829 

23 

I-9736 

2.464  21 

3.07152 

3.81974 

12 

•42576. 

.  601  03 

•79585 

2.OI2  19 

24 

2.03279 

2-5633 

3.22509 

4.04873 

For  any  other  Rate  or  Period. — Multiply  logarithm  of  rate  + 1  by  period,  and 
number  for  logarithm  will  give  tabular  amount  as  above. 
ILLUSTRATION.— What  is  tabular  number  for  4  per  cent,  for  10  years? 
Log.  of  i. 04  =  .017  033  3,  which  x  io  =  .  170  333,  and  number  for  log.  =  1.48024. 

Time    in    "years    in    "which    a    Sum    of   ]Vtoiiey    "will    "foe 
doubled    at    Several    Rates   of  Interest. 


Rate. 

Time. 

Rate. 

Time. 

Rate. 

Time. 

|      Rate. 

Time. 

Per  cent. 

i 

2 

3 

69.68 
35 
23-44 

Per  cent. 
4 

I 

17.67 
14.21 
11.88 

Per  cent. 
9 

10.34 
9.01 
8.04 

Per  cent. 
JO 
20 
30 

7-27 

3-8 
2.64 

"Value  of  $1,  etc.,  Computed  Semi-annually  for  a  Period 
of  13    Years. 


Years. 

Per  Cent. 

4 
Per  Cent. 

5 

Per  Cent. 

6 
Per  Cent. 

Years. 

3 

Per  Cent. 

Per  Cent. 

5 

Per  Cent. 

6 
Per  Cent, 

•5 

.015 

i.  02 

.025 

1.03 

6-5 

.2134 

•2936 

•3785 

1.4684 

i 

.0302 

1.0404 

.0506 

1.0609 

7 

•2317 

•3195 

•413 

i  5102 

i-5 

•0457 

i.  0612 

.0769 

1.0927 

7-5 

•3459 

-4483 

1.558 

2 

.0614 

1.0824 

.1038 

1-1255 

8 

.3728 

•4845 

1.6047 

2-5 

•0773 

1.  1041 

•1314 

I-I593 

8-5 

.4002 

.5216 

1.6528 

3 

•0934 

1.1262 

•1597 

1.1941 

9 

•3073 

.4282 

•5597 

1.7024 

3-5 

.1098 

1.1487 

.1887 

1.2299 

9-5 

.3269 

.4568 

•5987 

1-7535 

4 

.1265 

1.1717 

.2184 

1.2668 

o 

•3469 

.486 

.6386 

i.  8061 

4-5 

•1434 

1.1951 

.2489 

1.3048 

0-5 

•3671 

•5157 

6796 

1.8603 

5 

.1604 

1.219 

.2801 

1-3439 

i 

.3876 

546 

.7216 

1.9161 

5-5 

.178 

1.2434 

.3121 

1.3842 

I-5 

.4084 

•5769 

.7606 

I-9736 

6 

.1956 

1.2689 

•3449 

1-4258 

2 

•4295 

.6084 

.8087 

2.0356 

ILLUSTRATION. — What  is  amount  of  $500  at  semi-annual  interest  of  5  per  cent 
compounded  for  10  years  ? 

Tabular  number  i. 6386.    Then,  500  X  i. 628  89  =  $  814. 44. 5. 

To    Compute    Interest   on    any    GHven    Sum. 

'A  »/A 


For  a  Period  of  Years.    P  (i  +  r) n  =  A  ; 


log.  A  —  log.  P 


(i-f-r)« 


= n.     P  representing  principal,  r  rate  per  cent.-^-ioo  per  annum, 


log.  (1-r-r) 
n  number  of  years,  and  A  amount  of  principal  and  interest. 


DISCOUNT  OK  REBATE. — EQUATION  OF  PAYMENTS.       1 09 

ILLUSTRATION. —Assume  as  preceding,  $500  at  5  per  cent,  for  10  years. 

500  x  i. 05 10  =  500  X  i. 628  89  =  $814. 44. 5,  amount.     (^'4^JO  =  590,  principal 

10  7814.44.5  log.  814. 44. 5  —  log.  500 

~ '  =  -°5, '<*•  =  I0' num 


For  any  Period.  —  Assume  elements  of  preceding  case,  interest  payable  semi- 
annually.    10  X  2  =  20,  number  of  payments ;    —  =  .025,  rate. 

Then,-5oox  i. 0252°=  500  X  1.63862  =  $819.31. 
When  term  of  payments  and  rate  are  not  given  in  table. 
[log.  (^  +  i)  x  n  P]  =  log.  A. 
ILLUSTRATION.— Assume  $1000  for  30  years,  at  7  per  cent,  half-yearly. 

log.  —  -f  i  =  .014  940  3,  and  log.  .0149403  X  30  X  1000  =  $  28o6.7& 


DISCOUNT  OR  REBATE. 
DISCOUNT  or  REBATE  is  a  deduction  upon  money  paid  before  it  is  due. 

To   Compute    R,e"bate   upon    any    Sum. 

RULE.— Multiply  amount  by  rate  per  cent,  and  by  time,  and  divide  product  by 
sum  of  product  of  rate  per  cent,  and  time,  added  to  100. 

EXAMPLK  i. — What  is  discount  upon  $  12075  for  3  years,  5  months,  and  15  days 
at  6  per  cent.  ? 

3  years  5  months  and  15  days  =  3. 4574  years. 
I2075X6X  3-4574      250488.63 
,oo+,6x  3-4574)  =-T^^-  =  20"t-« 

2.— What  is  present  value  of  a  note  for  $963.75,  payable  in  7  months,  at  6  per 
cent.  ? 

6rate.    7  months=  T7^of  i  year  =  6  X  7  4-12  =  3.5,  and  3  5  + 100  =  103. 5  -r- 100  = 
i-035       963  75  4"  i  035  =  $931.16 

To  Compute  tlie  Sum  for  a  given  Time  and  Rate,  to  yield 
a   Certain.    Sum. 

RULE.— Divide  given  sum  by  proceeds  of  $  i  for  given  time  and  rate. 

EXAMPLE.— For  what  sum  should  a  note  be  drawn  at  90  days,  that  when  dis- 
counted at  6  per  cent,  it  will  net  $  200  ? 

Discount  on  $  i  for  90+3  days  at  6  per  cent.  =  $  .0155. 
Hence,  $i— . 0155  =  .9845, proceeds,  and  $200--- .9845  =  $203. 14.9. 


EQUATION    OF    PAYMENTS. 

RULE.— Multiply  each  sum  by  its  time  of  payment  in  days,  and  divide  sum  of 
products  by  sum  of  payments. 

EXAMPLE.— A  owes  B  $300  in  15  days,  $60  in  12  days,  and  $350  in  20  days;  when 
is  the  whole  due  ? 

300  X  15  =  4500 
60X1*=    720 
350  X  20  =  7000 
710          )  izaao  (ij  +  dayi. 
Jf 


IIO 


ANNUITIES. 


ANNUITIES. 
To    Compute    Amount   of  Annuity. 

When  Time  and  Ratio  of  Interest  are  Given.  RULE.— Raise  the  ratio  to  a  power 
denoted  by  time,  from  which  subtract  i ;  divide  remainder  by  ratio  less  i,  and  quo- 
tient, multiplied  by  annuity,  will  give  amount. 

NOTE.— $  i  added  to  given  rate  per  cent,  is  ratio,  and  preceding  table  in  Compound  Interest  is  a 
table  of  ratios. 

EXAMPLE.— What  is  amount  of  an  annual  pension  of  $100,  interest  5  percent., 
which  has  remained  unpaid  for  four  years? 

1.05  ratio;  then  1.054 — 1  =  1.21550625  —  i  =.21550625,  and  .215 506 25-7- (1.05 
—  i). 05  =  4. 310 125,  which  x  zoo  =  $431. 01. 25. 

To    Compute    ^Present   ^Worth.    of  an.    Aiinuity., 

When  Time  and  Rate  of  Interest  are  Given.  RULE.— Ascertain  amount  of  it  for 
whole  time;  divide  by  ratio,  involved  to  time,  and  result  will  give  worth. 

EXAMPLE. — What  is  present  worth  of  a  pension  or  salary  of  $500,  to  continue  10 
years  at  6  per  cent,  compound  interest  ? 

$  500,  by  last  rule,  is  worth  $6590.3975,  which,  divided  by  i.o610  (by  table,  page 
108,  is  1.79084)  =  $  3680.05. 

Or,  Multiply  tabular  amount  in  following  table  by  given  annuity,  and  product 
will  give  present  worth. 

ILLUSTRATION  i.— As  above;  10  years  at  6  per  cent. =j. 360 08,  and  7.36008  x  50° 
=  3680.04  dollars. 

2.  What  is  present  worth  of  $150  due  in  one  year  at  6  per  cent,  interest  per  annum  ? 
. 943  39  X  150  —  $141.50.85. 

Present  "Worth,  of*  an  Annuity  of  $1,  at  4,  £>,  and.  €> 
3?er  Cent.  Compound.  Interest  for  ^Periods  under  25 
Years. 


Years. 

4  Per  Cent. 

5  Per  Cent. 

6  Per  Cent. 

Years. 

4  Per  Cent. 

5  Per  Cent. 

6  Per  Cent. 

i 

.961  54 

.95238 

•94339 

13 

9.98562 

9-39357 

8.85268 

2 

1.88609 

1.85941 

I-83339 

H 

0.56307 

9.  898  64 

9.29498 

3 

2-7751 

2.72325 

2.67301 

15 

1.11843 

10.37966 

9.71225 

4 

3.6299 

3-54595 

3-465I 

16 

1.651  28 

10.83778 

o.  105  89 

M52Q3 

4.32948 

4.21236 

17 

2.16626 

11.27407 

0.47726 

6 

5.24215 

5-07569 

4.91732 

18 

2.659  26 

11.68958 

0.8276 

- 

6.00203 

5-78637 

5-58238 

J9 

3.13388 

12.08532 

1.158  ii 

8 

6.73176 

6.46321 

6.20979 

20 

3-59029 

12.46221 

1.46992 

9 

7-4364 

7.  107  82 

6.80169 

21 

4.029  12 

12.821  15 

1.76407 

10 

8.11085 

7.72173 

7.36008 

22 

4.451  12 

13.163 

2.041  58 

ii 

8.76044 

8.30641 

7.88687 

23 

4.85682 

13.48807 

2.30338 

12 

9-38S05 

8.86325 

8.38384 

24 

5-24695 

13.79864 

2.55035 

For  a  Rate  of  Interest  and  Term  of  Years  not  given  in  either  Table. 
—    i  —      '         =  A.    Notation  as  preceding. 

ILLUSTRATION.—  Take  $  i  at  4  per  cent,  for  24  years. 

Log.  1.04  =  .017033,  which  x  24  =  .408  799.    log.  .408  799  =  2.5633  =  ratio  raised 
to  power  of  24. 

Then,  —  X  (i  --  ^—  )  =  25  X  i—  390122  =  $  15.  24.695. 
.04      \       2.56337 

To  Compute  Yearly  Amount  tliat  -will  Liquidate  a  Delot 
in  a  Q-iven  NunVber  of  Years  at  Compound  Interest. 

n 

•  —  A.     ILLUSTRATION.  —  What  is  amount  of  an  annual  payment  that 


x  _1_  Y 

— 


\rill  liquidate  a  debt  of  $100  in  6  years  at  5  per  cent,  compound  interest? 


ANNUITIES. 


Ill 


(i  -f  .os)6  per  table,  page  108, 
=  1.34. 


4_-7  =  |        6 
-34 


(i+.o5)6-i  i-34- 1 

When  Annuities  do  not  commence  till  a  certain  period  of  time,  they  are  said  to  be 
in  Reversion. 

To  Compute  Present  \Vorth  of  an  Annuity  in  Reversion. 

RULE.— Take  two  amounts  under  rate  in  above  table— viz.,  that  opposite  sum  of 
two  given  times  and  that  of  time  of  reversion;  multiply  their  difference  by  an- 
nuity, and  product  will  give  present  worth. 

EXAMPLE.— What  is  present  worth  of  the  reversion  of  a  lease  of  $40  per  annum, 
to  continue  for  6  years,  but  not  to  commence  until  end  of  2  years,  at  rate  of  6  per 
cent.  ? 

6  +  2  =  8  years 6.209  79 

2    *       1-83339 

4.37640X40  =  1175-05-6. 

Amount   of  Annuity   of  $1,  etc.,  Compound    Interest, 
from.    1    to    SO    Years. 


E 

* 

4 

5 

6 

7 

i 

4 

5 

6 

7 

> 

Per  Cent. 

Per  Cent. 

Per  Cent. 

Per  Cent. 

£ 

Per  Cent. 

Per  Cent. 

Per  Cent. 

Per  Cent. 

:     I 

I. 

i. 

I. 

i. 

ii 

I3-48635 

14.20679 

14.97164 

15.7836 

,,  3 

2.04 

2.05 

2.06 

2.07 

12 

15.0258 

15.91713 

16.86994 

17.88845 

3 

3.1216 

3-I525 

3-1836 

3.2149 

13 

16.62684 

17.71298 

18.88214 

20.14064 

4 

4.24646 

4.310  12 

4.37462 

4-439  94 

14 

18.291  91 

19.59863 

21.01507 

22.55049 

5.41632 

5-52563 

5-63709 

5-75074 

15 

20.023  59 

21.57856 

23-27597 

25.12902 

7 

6.63297 
7.89829 

6.801  91 
8.14201 

6-97532 
8.39384 

7-15329 
8.65402 

16 
17 

21.824  53 
23-69751 

23-65749 
25.84037 

25-67253 
28.21288 

27.88805 
30.  840  22 

8 

9.21423 

9.54911 

9.89747 

10.259  8 

18 

25.64541 

28.13238 

30.90565 

33.99903 

9 

10 

10.  582  79 
12.006  ii 

11.02656 
12.57789 

11.49132 

13-18079 

11.97799 
13.81645 

*9 

20 

27.671  23 
29.77808 

30-539 
33-o6595 

33-75999 
36.78559 

37-37896 
40.99549 

ILLUSTRATION.—  What  is  amount  of  $  1000  for  20  years  at  5  per  cent? 

5  per  cent,  for  20  years  =  33.065  95 ;  hence,  1000  x  33.065  95  =  $  33.06.595. 

To    Compute    Amount    of    an    Annuity    for    any    Period 
and    Rate. 

RULE.— From  table  for  Compound  Interest,  page  108,  take  value  for  rate  per  cent, 
for  i  year,  and  raise  it  to  a  power  determined  by  time  in  years,  from  which  subtract 
i,  divide  remainder  by  rate,  and  quotient  multiplied  by  annuity  will  give  amount 
required. 

EXAMPLE.— What  will  an  annuity  of  $  50,  payable  yearly,  amount  to  in  4  years,  at 
5  per  cent.  ? 

By  table,  page  108, 1.054  =  1.2155. 

i.  2155  —  i  -f-  (1-05  —  i)  =  4-31.  and  4. 31  X  50  =  $ 215.50. 
For   Half-yearly   and    Quarterly   Payments. 

Multiply  annuity  for  given  time  by  amount  in  following  table: 


ite  per  cent. 

Half-yearly. 

Quarterly. 

Rate  per  cent. 

Half-yearly. 

Quarterly. 

3 
3-5 
4 
45 
5 

.007445 
.008675 
.009902 
.on  126 
.012  348 

.on  181 
.013031 
.014877 
.016729 
.018559 

5-5 
6 

6-5 
7 
7-5 

.013567 
.014781 

•015993 
.017  204 
.018414 

.020395 
.022227 
.024055 
.02588 
.027704 

ILLUSTRATION  i.— Annuity  as  determined  in  previous  case  =  $21 5. 50. 
Hence,  215. 50  X  1.012348  from  above  table  =  $21 8.16  for  half  yearly  payments. 
2.  A  person  30  years  of  age  has  an  annuity  for  10  years,  present  worth  of  it  being 
$1000,  provided  he  may  live  for  10  years.    What  is  annuity  worth,  assuming  that 
60  persons  out  of  every  3550,  between  the  ages  of  30  and  40,  die  annually? 
3550 —  600  (60  X  10)  =  2950  would  therefore  be  living. 
And,  3550  :  2950  ::  1000  =  $830. 98. 


1 1 2  PERPETUITIES. — COMBINATION. 

PERPETUITIES. 
PERPETUITIES  are  such  Annuities  as  continue  forever. 

To    Compute  Value   of  a  I?erpetu.al  Annuity. 

RULE.— Divide  annuity  by  rate  per  cent.,  and  multiply  quotient  by  unit  in  pre- 
ceding table. 

EXAMPLE.— What  is  present  worth  of  an  annuity  for  $  100,  payable  semi-annually, 
at  5  per  cent.  ? 

100^-  .05  =  2,  and  2  X  i.  01  *  348,  from  preceding  table  =  2.024.70. 

To  Compute  "Value   of*  a   Perpetuity   in.   Reversion. 

RULE.— Subtract  present  worth  of  annuity  for  time  of  reversion  from  worth  of 
annuity,  to  commence  immediately. 

EXAMPLE. — What  is  present  worth  of  an  estate  of  $50  per  annum,  at  5  per  cent., 
to  commence  in  4  years  ? 

50  -T-  .05 =  1000 

$50,  for  4  years,  at  5  per  cent.  =  3. 545  95  (from  table,  page  no)  X  50=  177.2975 

822.7025 
which  in  4  years,  at  5  per  cent,  compound  interest,  would  produce  $1000. 


COMBINATION. 

COMBINATION  is  a  rule  for  ascertaining  how  often  a  less  number  of  num- 
bers or  things  can  be  chosen  varied  from  a  greater,  or  how  many  different 
collections  may  be  formed  without  regard  to  order  of  each  collection. 

Combinations  of  any  number  of  things  signify  the  different  collections 
which  may  be  formed  of  their  quantities,  without  regard  to  the  order  of  their 
arrangement. 

Thus,  3  letters,  a,  6,  c,  taken  all  together,  form  but  one  combination,  abc. 
Taken  two  and  tivo,  they  form  3  combinations,  as  ab,  ac,  be. 

NOTE.— Class  of  the  combination  is  determined  by  number  of  elements  or  things  to  be  taken  ;  if  two 
are  taken,  the  combination  is  of  ad  class,  and  so  on. 

RULE.— Multiply  together  natural  series  i,  2,  3,  etc.,  up  to  the  number  to  be  taken 
at  a  time.  Take  a  series  of  as  many  terms,  decreasing  by  i,  from  number  out  of 
which  combination  is  to  be  made,  ascertain  their  continued  product,  and  divide 
this  last  product  by  former. 

EXAMPLE  i.— How  many  single  combinations,  as  a&,  ac,  may  be  made  of  2  letters 
out  of  3?  i  X  2  _  2  _  6  _ 

3~x~^  ~~  ~6~  ~~  V  ~ 3' 

2. — How  many  combinations  may  be  made  of  7  letters  out  of  12? 
iX  2X  3X4X5X6X7  =  5040  and  399i68o  _ 
12X11X10X9X8X7X6  3991680'  5040 

3.— How  many  different  hands  of  cards  may  be  held,  as  at  whist,  combinations 
13  out  of  52?  635013559600. 

"When    t\vo    Nu.xn'bers    or    Tilings    are    Combined. 

RULE.— Multiply  together  natural  series  i,  2,  3,  etc.,  to  one  less  term  than  numbei 
of  combinations ;  ascertain  their  continued  product,  and  proceed  as  before. 

EXAMPLE. — There  are  3  cards  in  a  box,  out  of  which  two  are  to  be  drawn  in  a  re 
quired  order.  How  many  combinations  are  there? 

Here  there  are  2  terms ;  hence,  2  —  1  =  1,  and  — - —  =  —  =6-7-1  =  6. 

3X2       6 

To  Compxite  Number  of  "Ways  in  -which,  any  Number  of 
Distinct  Objects  can  "be  Divided,  among  any  Number. 

RULE.— Multiply  together  numbers  equal  to  number  given,  as  often  as  objects 
are  to  be  divided  among  them. 

EXAMPLE.— In  how  many  different  ways  can  10  different  cards  be  divided  amon/ 
3  persons?  3X3X3X^*3X3X3X3X3X30r3IO  =  59°49- 


COMBINATION.  —  CIRCULAR   MEASURE.  113 

Combinations   -with    Repetitions. 

In  this  case  the  repetition  of  a  term  is  considered  a  new  combination.  Thus, 
i,  2,  admits  of  but  one  combination,  if  not  repeated;  if  repeated,  however,  it  admits 
of  three  combinations,  as  i,  i  ;  i,  2;  2,  2. 

RULE.—  To  number  of  terms  of  series  add  number  of  class  of  combination,  less  i  ; 
multiply  sum  by  successive  decreasing  terms  of  series,  down  to  last  term  of  series; 
then  divide  this  product  by  number  of  permutations  of  the  terms,  denoted  by  class 
of  combination. 

EXAMPLE.—  How  many  different  combinations  of  numbers  of  6  figures  can  be 
made  out  of  n? 

„  -i.  (6  _  j)  -  ,6  =  sum  of  number  of  terms,  and  number  of  class,  Uss  i. 

16  X  15  X  14  X  13  X  12  X  ii  =  5  765  7f>°=product  of  sum,  and  successive  terms  to 
last  term. 

1X2X3X4X5X6  =  720  permutations  of  class  of  combination. 


Then,  5-7  =  8008. 

720 

Variations   with.   Repetitions. 

Every  different  arrangement  of  individual  number  or  things,  including  repeti- 
tions, is  termed  a  Variation. 
Class  of  Variation  is  denoted  by  number  of  individual  things  taken  at  a  time. 

RULE.—  Raise  number  denoting  the  individual  things  to  a  power,  the  exponent 
of  which  is  number  expressing  class  of  variation. 

EXAMPLE  i.—  How  many  variations  with  4  repetitions  can  be  made  out  of  5  fig- 
ures ?  54  =  625. 

2.—  How  many  different  combinations  of  4  places  of  figures  can  be  made  out  of 
the  9  digits  ? 

12  X  ii  X  10X9       Il88° 

=—  =  495- 


Combination   witho-ut    Repetitions. 

RULE.—  From  number  of  terms  of  series  subtract  number  of  class  of  combination, 
less  i  ;  multiply  this  remainder  by  successive  increasing  terms  of  series,  up  to  last 
term  of  series;  then  divide  this  product  by  number  of  permutations  of  the  terms, 
denoted  by  class  of  combination. 

EXAMPLE  i.  —  How  many  combinations  can  be  made  of  4  letters  out  of  10,  exclud- 
ing any  repetition  of  them  in  any  second  combination? 
10  —  (4  —  i)  =  7  =  number  of  terms  —  number  of  class,  less  i. 
7X8X9X10  =  5040  =  prod,  of  remainder  7,  and  successive  terms  up  to  last  term. 
1X2X3X4  =  24  =  permutations  of  class  of  combination. 

Then,  5°i£  =  2I0. 

24 

2.  —How  many  combinations  of  the  sth  class,  without  repetitions,  can  be  made 
of  12  different  articles? 

/          x      a        .8X9X10X11X12       85040 
12  —  (5  —  i)  =  8,  and  -  -  —  ---  =  =  702. 

1X2X3X4X5  120        /y 


CIRCULAR   MEASURE. 

Unit  of  Circular  Measure  is  an  angle  which  is  subtended  at  centre  of  a  circle 
by  an  arc  equal  to  radius  of  that  circle,  being  equal  to 

1  80° 


Circular  measure  of  an  angle  is  equal  to  a  fraction  which  has  for  its  numerator 
the  arc  subtended  by  that  angle  at  centre  of  any  circle,  and  for  its  denominator  the 
radius  of  that  circle. 

K* 


114  CIRCULAR   MEASURE.  -  PROBABILITY. 

To   Compnte    Circular    ^Measure    of  an    Angle. 
RULE.  —  Multiply  measure  of  angle  in  degrees  by  3.1416,  and  divide  by  180. 
EXAMPLE.—  What  is  circular  measure  of  24°  10'  8"? 
24°  10'  8"  X  3-1416  _  87008  x  3-1416 

180  ~  - 


To  Compute  ]Measu.re  of  an  Angle,  its  Circular  Measure 
being   Q-iven. 

RULE.—  Multiply  circular  measure  of  angle  by  180,  and  divide  by  3.1416. 


PROBABILITY. 

Probability  of  any  event  is  the  ratio  of  the  favorable  cases,  to  all  the 
cases  which  are  similarly  circumstanced  with  regard  to  the  occurrence.  If 
an  event  have  3  chances  for  occurring  and  2  for  failing,  sum  of  chances 
being  5,  the  fraction  f  will  represent  probability  of  its  occurring  and  is  taken 
as  measure  of  it.  Thus,  from  a  receptacle  containing  i  white  and  2  black 
balls,  the  probability  of  drawing  a  white  ball,  by  abstraction  of  i,  is  £ ;  prob- 
ability of  throwing  ace  with  a  die  is  ^ :  in  other  words,  the  odds  are  2  to  i 
against  first,  and  5  to  i  against  second. 

If  m  -f-  n  =  whole  number  of  chances,  m  representing  number  which  are  favorable, 

and  n  unfavorable.     Therefore =  probability  of  event. 

m-f-w     ' 

Probabilities  of  two  or  more  single  events  being  known,  probability  of  their  oc- 
curring in  succession  may  be  determined  by  multiplying  together  the  probabilities 
of  their  events,  considered  singly. 

Thus,  probability  of  one  event  in  two  is  expressed  by  ^;  of  its  occurring  twice  in 
succession,  ^  X  J,  or  £;  of  thrice  in  succession,  J  x  ^  X  J?  or  J,  etc. 

ILLUSTRATION  i.— If  a  cent  is  thrown  twice  into  the  air,  the  probability  of  its  fall- 
ing with  its  head  up,  twice  in  succession,  is  as  i  to  4.  Thus,  it  may  fall: 

1.  Head  up  twice  in  succession.  \ 

2.  Head  up  ist  time  and  wreath  2d  time,  f  i  i 

3.  Wreath  up  ist  time  and  bead  2d  time,  f  Hence,  jq—  _  .25  =  —  =  4  times. 

4.  Wreath  up  twice  in  succession. 

These  are  the  only  results  possible,  and  being  all  similarly  circumstanced  as  to 
probability,  the  probability  of  each  case  is  as  i  to  4,  or  odds  are  as  3  to  i. 

Probability  of  either  head  or  wreath  being  up  twice  in  succession  is  as  i  to  i,  or 
chances  are  even,  because  ist  and  4th  cases  favor  such  a  result;  probability  of  head 
once  and  wreath  once  in  any  order  is  as  i  to  2,  because  2d  and  3d  cases  favor  such  a 
result;  and  probability  of  head  or  wreath  once  is  as  3  to  4,  or  odds  are  as  3  to  i,  be- 
cause ist,  ad,  and  3d,  or  2d,  3d,  and  4th  cases  favor  such  a  result. 

NOTE. — i  to  2  is  an  equal  chance,  for  i  out  of  2  chances  =  i  to  i,  being  an  equal  chance  ;  again,  i  to 
5  is  4  to  i,  for  i  out  of  5  chances  is  i  to  4. 

2. — If  there  are  4  white  balls  and  6  black  in  a  bag,  what  is  the  chance  of  a  person 
drawing  out  2  black  at  two  successive  trials? 
This  is  a  combination  without  repetition.     Hence,  6  —  (2  —  i)  =  5, 

and —  —  —  — ,  which  x  2  for  successive  trials  =  —  or  — . 

1X221'  ,  2       15 

3. — Suppose  with  two  bags,  one  containing  5  white  balls  and  2  black,  and  the  other 
7  white  and  3  black. 

Number  of  cases  possible  in  one  drawing  from  each  bag  is  (5  -4-  2)  x  (7  -f-  3)  =  7 
X  10  =  70,  because  every  ball  in  one  bag  may  be  drawn  alike  to  one  in  the  other. 


PROBABILITY.  115 

Number  of  cases  which  favor  drawing  of  a  white  ball  from  both  bags  is  5  X  7  =  35, 
for  every  one  of  the  5  white  balls  in  one  bag  may  be  drawn  in  combination  with  every 
one  of  the  7  in  the  other.  For  a  like  cause,  number  of  cases  which  favor  drawing  of 
a  white  ball  from  ist  bag  and  a  black  one  from  2d  is  5  x  3  —  15  ;  a  black  ball  from  ist 
bag  and  a  white  ball  from  2d  is  7  X  2  =  14  ;  and  a  black  ball  from  both  is  3  X  2  =6. 

Probability,  therefore,  of  drawing  is  as 

£2l!-3i  =  J.  =  x  to  i,  a  white  ball  from  both  bags.    1*J  =  1*  =  A  =  3  to  n, 

70  70          2  70  70         14 

a  white  ball  from  ist,  and  a  black  from  2d.       1^2.  —  11  =  —  =  x  to  4,  a  black 

7°        7°       5 

ball  from  ist.  and  a  white  from  zd.     -  —  -  =  —  =  —  =  3  to  32,  a  black  ball  from 

70        70      35 
j  a  white  ianfrom  one,  and  a  black  from  other, 


for  both  2d  and  3d  cases  favor  this  result  :  hence,  —  -f  -    =     . 

5       14      7°  7° 

=  —  =  —  =  32  to  3,  at  least  one  white  ball,  for  the  ist,  2d,  and  3d  cases  favor  this 
7«>      35 

result  ;  hence,  —  +  A  +  _L  =  3£. 
2       14  ^  5       35 

Again,  if  number  of  white  and  black  balls  in  each  bag  are  same,  say  5  white  and 
2  black,  5  +  2  X  5  +  2  =  49,  then  probability  of  drawing  is  as 

i-*-5.  =  ?S  _  25  to  24  a  white  bau  from  both.    5211  =  —  —  I0  to  39,  a  wAife  ball 

49         49  jo  49         49 

from  ist,  and  a  black  from  zd.    -   —  =  —  =  10  to  39,  a  black  ball  from  ist,  and  a 

white  from  id.    ^2H  =  A  —  4  to  45,  a  fcZacfc  ball  from  both. 
49        49 

4.  —When  two  dice  are  thrown,  probability  that  sum  of  numbers  on  upper  sides 
is  any  given  number,  say  7,  is  as  follows: 

As  every  one  of  the  six  numbers  on  one  die  may  come  up  alike  to,  or  in  combi- 
nation with  the  other,  number  of  throws  is  6  X  6  =  36. 

!  i  and  61 
2    "   5}  ;  and  as  these  numbers  may  be 
3    "   4) 
upon  either  die,  there  are  3  x  2  =  6  throws  in  favor  of  the  combination  of  7  ;  hence 

6        i 
probability  of  throwing  7  is  —  ?  =  —  ,  or  as  i  to  5. 

5.  —Probability  of  a  player's  partner  at  Whist  holding  a  given  card  is  as  follows: 
Number  of  cards  held  by  the  other  3  players  is  3  x  13  =  39;  probability,  there- 

fore, that  it  is  held  by  partner  is  —  ,  but  it  may  be  one  of  the  13  cards  which  he 

holds;  hence  probability  is  —  X  13  =  —  =  —  ,  or  as  i  to  2. 
39  39       3 

6.  —  Probability  of  a  player's  partner  at  Whist  holding  two  given  cards  is  as  follows: 

Number  of  combinations  of  39  things,  taken  2  and  2  together,  is  —  —  —  =  741  ; 


therefore,  probability  that  these  2  cards  are  in  partner's  hand  is  39  x  38  = 

~7^7~     39Xl9 
—  -^-  =  i  to  740;  but  they  may  be  any  2  cards  in  partner's  hand;  therefore,  since 

number  of  combinations  of  13  cards,  taken  2  and  2  together,  is  — — —  =  —  =  78, 

78       2 
probability  required  is  -f—  =  — ,  or  as  2  to  17. 

Similarly,  probability  that  he  holds  any  3  given  cards  is  as  — ,  or  as  22  to  681. 


Il6  PROBABILITY. 

Probabilities  at  a  game  of  Whist  upon  following  points  are  : 

9  to  7,  that  one  hand  has  two  honors,  and  two  hands  one; 

g  to  55,  that  two  hands  have  each  two  honors  ; 

3  to  29,  that  each  hand  holds  an  honor; 

3  to  13,  that  one  hand  has  three  honors,  and  one  hand  one; 

i  to  63,  that  four  honors  are  held  by  one  hand. 

7.—  If  3  half-dollars  are  thrown  into  the  air,  probability  of  any  of  the  possible  com 
binations  of  their  falling  is  determined  as  follows  : 


Hence,     (^-)3=  .125  =  i  to  7  in  favor  of  3  heads. 

JL  /Z.  j  —  .  375  =  3  to  5      "      "     2  heads  and  i  tail. 
3X2  (—  )3=  -375  =  3  to  5      "      "      i  head  and  2  tails. 

3x2x1/^3  u    u 

IX2X  3\2/ 

And  in  like  manner,  if  5  were  thrown  up,  probability  of  any  of  their  possible 
combinations  would  be  determined  as  follows  : 

\*\    5X4/'\5      5X4X3/i\5,    5  X  4  X  3  X  2  /  i  \5 

/+o^W~i~ix  2x3  W  +  1X2X3  X4W 

,   5X4X3X2X  i  /_i_\S 

"""1X2X3X4X5    \2/ 


5-'I5-  5 


1X2X3X4X5 

Hence,  (—  )  =  .031  25  —  i  to  31  in  favor  of  5  heads  ; 

—  (—  )  =  .15625  =  5  to  27      "      "      4  heads  and  i  tail  ; 
-  —  -  (  —  )  =  .3125   =  10  to  22    "      "      3  heads  and  2  tails; 

I  X  2   \  2  / 

-5=  .312  5   =  10  to  22    u      "      2  heads  and  3  tails; 


All  Wagers  are  founded  upon  the  principle  of  product  of  the  event, 
and  contingent  gain,  being  equal  to  amount  at  stake. 

ILLUSTRATION  i.—  Suppose  3  horses,  A,  B,  and  C,  are  entered  for  a  race,  and  X 
wagers  12  to  5  against  A,  n  to  6  against  B,  and  10  to  7  against  C. 

If  A  wins,  X  wins  6  4-  7  —  12  =  i. 
"         «     X 
X 


, 

B     «     X     «    s--n^L 
«     5-1-6-10=1. 


Hence,  X  wins  i,  whichever  horse  wins,  from  having  taken  field  against  each 
horse  at  odds  named. 

.     (  A  are  5  to  12  )  (  -^  in  favor  of  A, 

Odds  given  m  fa-  J           ^           (  ;  corresponding  probabil-  )  V 

vorof                1  B        6       «  f  ity  is                                 1  ff                     B' 

(  C    »   7  "  10  )  (  A          "          °» 

*nd  x"^i"^==i"  =  I'o6==Xi°6  <0  J  infavor  of  taker  of  odds. 


PROBABILITY. 


2. Odds  given  upon  first  seven  favorite  horses  for  Oaks  Stakes  of  1828  were  so 

great,  that  probability  in  favor  of  taker  of  the  odds  when  reduced  was  as  follows : 

ist,  5  to  2 ;  2d,  5  to  2 ;  3d,  4  to  i ;  4th,  7  to  i ;  sth,  14  to  i ;  6th,  14  to  i ;  7th,  15  to  i 

(  4X3  X  161=192 
_2  2  ,  £  •  £  •  _L  _i_  _L  i  _L  _  1  _i_  _l-j_  -3_==!_i_£-i-JL:_  J  1X7X16  =  112 

7  7  (  7  X  3  X  16  "3^6 

_  367  _:_  336  =  LQQ2  —  ,[.092  to  i,  in  favor  of  taker  of  odds,  yet  neither  of  the  horses 
upon  which  these  odds  were  given  won. 

3. if  odds  are  3  to  i  against  a  horse  in  a  race,  and  6  to  i  against  another  horse 

in  a  second  race,  probability  of  ist  horse  winning  is  £,  and  of  other  ^.  Therefore 
probability  of  both  races  being  won  is  ^,  and  odds  against  it  27  to  i ,  or  1000  to  37.037. 
Odds  upon  such  an  event  were  given  in  1828  at  1000  to  60,  or  16.67  to  i. 

4._Two  persons  play  for  a  certain  stake,  to  be  won  by  winner  of  three  games  or 
results.  One  having  won  one  and  the  other  two,  they  decide  to  divide  the  sum, 
proportioaate  to  their  interest.  How  much  of  it  should  each  one  receive? 

OPERATION.— If  winner  of  two  games  should  win  game  to  be  played,  he  would  be 
entitled  to  the  whole  sum;  if  he  lost,  he  would  be  entitled  to  half  of  it.  Now  as 

one  event  is  as  probable  as  the  other,  —  +  —  =  —,  half  of  which  =  — ,  or  sliare 

122  4 

of  winner  of  two  games. 

When  events  are  wholly  independent,  so  that  occurrence  of  one  does  not 
affect  that  of  the  other,  probability  that  both  will  occur  is  product  of  proba- 
bilities that  each  will  occur. 

NOTE.— It  is  indifferent  whether  events  are  to  occur  together  or  consecutively. 

ILLUSTRATION  i.—  Assume  three  boxes,  each  containing  white  and  black  balls  as 
follows : 

6  white,  5  black ;  7  white,  2  black ;  8  white,  10  black.  What  is  chance  of  drawing 
from  them  a  white,  black,  and  a  white  ball  ? 


Probabilities  are  — ,  — ,  and  — ,  product  of  which  = 
ii    9  10 


+  2-1-8 
297 


=  17.625  to  i. 


2.— A  gives  an  answer  correctly  3  times  out  of  4,  B  4  times  out  of  5,  and  C  6  out 

of  7.    What  is  probability  of  an  event  which  A  and  B  declare  correct  and  C  denies? 

OPERATION.— Compound  probability  that  A  and  B  answer  correctly  and  C  denies 

(all  i  of  which  are  in  favor  of  event)  is—  X  —  X—  =  —  =  — . 

4        5         7        140      35 
Compound  probability  that  A  and  B  deny  and  C  is  correct  (all  3  of  which  are 

against  event)  is  —  X  —  X  —  =  —  =  — . 
4        5        7        140      70 


Then  correct,  divided  by  sum 
of  correct  and  incorrect, 


.8714 


35 


\35 


"•85714 


. =.68  or-. 

428  57  3 


Odds  "between  "Results  or  Chances,  and  bet-ween  any 
Number  and  "Whole  Num'ber,  at  various  Odds  against 
each.,  also  "Value  of  each  Chance  in  parts  of  1OO. 


Odds  against 
each. 

Value  of 
Chance. 

Odds  against 
each. 

Value  of 
Chance. 

Odds  against  ;  Value  of 
each.         i   Chance. 

Odds  against 
each. 

Value  of 
Chance. 

Even 

50 

2      tO 

33-33 

6.5  tc 

13-33 

15  tc 

6.25 

IX       tO  10 

47.62 

2-5' 

28.57 

7 

12-5 

18 

5-26 

6 

5 

45-45 

3     ' 

25 

7-5 

11.76 

20 

4.76 

5 

4 

44.44 

3-5  ' 

22.22 

8 

;     ii.  ii 

25 

3-84 

5-5 

4 

42.1 

4     ' 

2O 

8.5 

10.52 

3° 

3-22 

6 

4 

40 

4-5' 

18.18 

9 

i    I0 

40 

2.44 

6-5 

4 

38.1 

5     ' 

16.66 

9-5 

9-52 

5° 

1.96 

7 

4 

36.36 

5-5' 

15-38 

10 

9.09 

60 

1.64 

7-5 

4 

34-78 

6     ' 

14.28 

12 

i      7-7 

IOO 

•99 

OPERATION.  —  Divide  100,  or  unit,  as  case  may  be,  by  sum  of  odds,  and  multiply 
quotient  by  lesser  chance  or  odds. 
ILLUSTRATION.— 6  to  4.     6  +  4  =  10,  and  100  -r- 10  x  4  =  40,  value  of  chance. 


Il8     WEIGHTS   OF  IRON/ STEEL,  COPPER,  AND  BRASS. 


WEIGHTS   OF  IRON,  STEEL,  COPPER,  AND   BRASS. 
"Wrought    Iron,   Steel,  Copper,  and    Brass    T*latest 


u 

Stand 

No.  of 
Gauge. 

S.  Law, 
ard  Gau 

THI 

Approxi- 
mate 
Fractions. 

March  $d,  3 
ge.     Iron  ai 

CKNBSB. 

Approxi- 
mate 
Decimals. 

893- 
id  Steel 
WEIGHT. 

Wro't  Iron 
Per  Sq.  Ft. 

No.  of 
Gauge. 

American 

THICKNESS. 

Approximate 
Decimals. 

Gauge. 
WEI 
PER  Squ 

Copper. 

GHT. 
ARE   FOO* 

Brass 

Inch. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Lbs. 

DOOOOOO 

1-2 

•5 

20. 

oooo 

.46  or  %  f. 

20.838 

19.688 

000000 

15-32 

.46875 

iS-75 

ooo 

.40964 

18.5567 

I7-5326 

00000 

7-16 

•4375 

17-5 

00 

.3648or%  1. 

16.5254 

I5-6I34 

oooo 

13-32 

.40625 

16.25 

0 

.324  86  or  %1. 

14.7162 

13.904 

000 

3-8 

•375 

15- 

i 

.2893 

13-1053 

12.382 

oo 

11-32 

•34375 

13-75 

2 

•257  63  or  X  f- 

11.6706 

11.0266 

o 

5-16 

•3125 

12.5 

3 

.229-42 

10.3927 

9.819  2 

X 

9-32 

.28125 

11.25 

4 

.20431  or'/5f. 

9-2552 

8-7445 

2 

17-64 

.265625 

10.625 

5 

.18194  orjl^  1. 

8.2419 

7.787 

3 

1-4 

•25 

10. 

6 

.  162  02 

7-3395 

6.934  5 

4 

15-64 

•234375 

9-375 

7 

.14428 

6-5359 

6.1751 

5 

7-32 

.21875 

8-75 

8 

.1284901  >£  f. 

5.8206 

5-4994 

6 

13-64 

.203125 

8.125 

9 

•"443 

5-1837 

4.8976 

1 

11-64 

-1875 
.171875 

6-875 

10 

ii 

.  101  89  or  Vio  f. 

.090742 

4.6156 
4.1106 

4.  360  9 

9 

•15625 

6.25 

12 

.080808 

3.6606 

3-4586 

xo 

9-64 

.140625 

5-625 

13 

.071  961 

3-2598 

3.0799 

II 

1-8 

.125 

5- 

14 

.064084 

2-903 

2.7428 

12 

7-64 

•109375 

4-375 

15 

.057068 

2.5852 

2.4425 

»3 

3-32 

•°9375 

3-75 

16 

.050  82  or  Vaof. 

2.  302  I 

2.1751 

14 

5-64 

.078  125 

3-125 

17 

•045257 

2.050  I 

1-937 

X5 

9-128 

.0703125 

2.8125 

18 

.040303 

1.8257 

1-725 

16 

1-16 

.062  5 

2-5 

'9 

•03589 

1.6258 

1.5361 

17 

9-160 

-05625 

2.25 

20 

.031  961 

1.4478 

1-3679 

18 

1-20 

•°5 

2. 

21 

.028462 

1.2893 

1.2182 

19 

7-160 

•04375 

i-75 

22 

•025347 

1.1482 

1.0849 

20 

3-8o 

•°375 

23 

.022571 

1.0225 

.96604 

21 

x  1-320 

•034375 

i-375 

24 

.0201 

.910  53 

.86028 

22 

1-32 

•03125 

1.25 

25 

.0179 

.81087 

.766  12 

23 

9-320 

.028  125 

1.125 

26 

-01594 

.72208 

.68223 

24 

1-40 

.025 

i. 

27 

.014195 

•64303 

•60755 

25 

7-320 

.021875 

•875 

28 

.012641 

•57264 

.541  03 

26 

3-160 

.01875 

•75 

29 

.011257 

.50994 

.4818 

27 

i  1-640 

.0171875 

•6875 

3° 

.010025 

•454  J3 

42907 

28 

1-64 

.015625 

.625 

31 

.008928 

•404  44 

.382  12 

29 

9-640 

.014062  5 

•5625 

32 

.00795 

.36014 

.34026 

30 

1-80 

.012  5 

•  5 

33 

.00708 

.32072 

.30302 

31 

7-640 

.0109375 

•4375 

34 

.006304 

•28557 

.26981 

32 

13-1280 

.010  156  25 

.40625 

35 

.005614 

•25431 

.24028 

33 

3-320 

•009375 

•375 

36 

.005 

.226  5 

.214 

34 

11-1280 

•00859375 

•34375 

37 

•004453 

.201  72 

.19059 

35 

5-640 

.0078125 

•3125 

38 

.003965 

.17961 

.1697 

36 

9-1280 

.007031  25 

.28125 

39 

.003531 

•15995 

•I5H3 

37 

17-2560 

.006640625 

.265  625 

40 

.003144 

.14242 

•13456 

38 

1-160 

.00625 

•25 

In  the  practical  use  and  application  of  the  U.  S.  Gauge,  a  variation  of 
two  and  one-half  per  cent,  either  way  may  be  allowed. 


Wr't  Iron. 

Specific  Gravities 7.704 

Weights  of  a  Cube  Foot 481.75 

"         Inch 2787 


Steel. 
7.806 


Copper. 


.2823 


Brass. 

8.218 


543-6 

.3146  i 


WEIGHTS   OF   IRON,  STEEL,  COPPEE,  ETC. 


Iron,  Steel,  Copper,  and.   Brass 

(Birmingham  Gauge.) 


IPlates. 


No.  of 
Gauge. 

Thickness. 

Iron. 

PER  SQCAI 
Steel. 

IE  FOOT. 
Copper. 

Brass. 

Inch. 

Lbs. 

Lb8. 

Lbs. 

Lbs. 

0000 

.454  or  ^  f  uU 

18.2167 

18.4596 

20.5662      |      19.4312 

000 

•425 

17.0531 

17.2805 

19.2525      i      18.19 

00 

.38    or  f  full 

15.2475 

15.4508 

17.214         |      16.264 

0 

•34    or*.    " 

13-6425 

13.8244 

15402 

14.552 

i 

•3 

12.0375 

12.198 

13-59 

12.84 

2 

.284 

11-3955 

n-5474 

12.8652 

12.1552 

3 

.259  or  i  full 

10.3924 

10.5309 

H.7327 

11.0852 

4 

.238 

9-5497 

9.6771 

10.7814            10.1864 

5 

.22 

8.8275 

8.9452 

9.966                 9.416 

6 

.203  or  i  full 

8.1454 

8.254 

9.1959              8.6884 

7 

.18    or  T\  light 

7.2225 

7.3188 

8.154 

7.704 

8 

.165  or  J       " 

6.6206 

6.7089 

7-4745 

7.062 

9 

.148  or  i  full 

5.9385 

6.0177 

6.7044 

6.3344 

JO 

.134 

5.3707 

5.4484 

6.0702 

5-7352 

ii 

.12    or  ^  light 

4-815 

4.8792 

5.436 

5>I36 

12 

.109 

4.3736 

4.4319 

4.9377 

4.6652 

13 

.095  or  ^  light 

3.809 

3-8627 

4.3035 

4.066 

14 

.083 

3.3304 

3.3748 

3-7599 

3.5524 

15 

.072 

2.889 

2.9275 

3.2616 

3.0816 

16 

.065 

2.6081 

2.6429 

2-9445 

2.782 

i? 

.058 

2.3272 

2.3583 

2.6274 

2.4824 

18 

•049  or  ihj  light 

1.9661 

1.9923 

2.2197 

2.0972 

19 

.042 

1.6852 

1.7077 

1.9026 

1.7976 

20  :  .035 

1.4044 

1.4231 

1.5855 

1.498 

21        .032 

1.284 

1.3011 

1.4496 

1.3696 

22 

.028 

I-i235 

1.1385 

1.2684 

1.1984 

23 

•025  or  ^ 

1.0031 

1.0165 

1.1325 

1.07 

24 

.022 

.8827 

.8945 

.9966 

.9416 

25      .02    or^ 

.8025 

.8132 

.906 

.856 

36    i    .018 

.7222 

.7319 

.8i54 

.7704 

27    :    .Ol6 

.642 

.6506 

.7248 

.6848 

28 

.014 

•5617 

.5692 

-6342 

.5992 

29 

.013 

•5216 

.5286 

•5889 

•5564 

30 

.012 

.4815 

.4879 

•5436 

•5136 

31 

.01    or^ 

.4012 

.4066 

•453 

.428 

32 

.009 

.3611 

•3659 

.4077 

.3852 

33 

.008 

.321 

•3253 

•3624 

-3424 

34 

.007 

.2809 

.2846 

.3171 

.2996 

35 

.005  or  2W 

.2006 

.2033 

.2265 

.214 

36  ;  .004  or  ^ 

.1605 

.1626 

.1812 

.1712 

Thickness    of   Sheet    Silver,  Grold,  etc. 
By  Birmingham  Gauge  for  these  Metals. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

i 

2 

3 
4 
5 
6 

.004 
.005 
.008 

.01 

.013 
.013 

7 
8 

9 

10 

ii 

12 

.015 
.016 
.019 
.024 
.029 
-034 

13 
T4 
15 
16 

J7 
18 

.036 
.041 
.047 
.051 

-057 
.061 

19 

20 
21 
22 

23 
24 

.064 
.067 
.072 
.074 

-077 
.082  j 

25 
26 

27 
28 

29 
30 

•095 
.103 

•"3 

.12 
.124 
.126 

31 
32 
33 
34 
35 
36 

•133 
•143 
.145 
.I48 

.158 
.167 

I2O  WEIGHTS    OF    IKON,   STEEL,   COPPER,  ETC. 

Wrought  Iron,  Steel,  Copper,  and  Brass  Wire. 

American  Gauge,     f.  full,  1.  light. 


No.  of 

Gauge. 

Diameter. 

Iron. 

PER  LINE.A 

Steel. 

L  FOOT. 
Copper. 

Brass. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lba. 

oooo 

.46  or  T7^  f. 

.56074 

.56603 

.640513 

.605  176 

ooo  .400.  64*" 

.444683 

.448  879 

.507  946 

.479  908 

00 

.364  8  or  1  1. 

.352  659 

.355986 

.402  83 

.380666 

o  .324  86  orfg  f. 

.279665 

.282  303 

.319451 

.301  816 

i 

.2893 

.221  789 

.223  891 

•253  342 

•239  353 

2 

.257  63  or  i  f. 

.175888 

.177  548 

.200911 

.189818 

3  .229  42 

.13948 

.140  796 

.159323 

.150522 

4  .204  31  or  \  f. 

.H06l6 

.11166 

.126353 

.119376 

5  .  18194  or  A  I- 

.087  72 

.088  548 

.1002 

.094666 

6 

.16202 

.069565 

.070  221 

.079  462 

•075  075 

7 

.14428 

•055  165 

.055  685    -063  013 

.059  545 

8  .128  49  or  \  f. 

•043  751 

.044  l64    .049  97° 

.047219 

9 

.11443 

.034699 

.035  026    .039  636 

•037  437 

10 

.101  89  or  ^  f. 

.027  512 

.027  772 

.031  426 

.029  687 

ii 

.090  742 

.021  82 

.022  026     .024  924 

.023  549 

12 

.080808 

.017304 

.017468     .019766 

.018  676 

J3 

.071  961 

.013  722 

.013  851      .015  674 

.014809 

14 

.064084 

.010  886 

.010989     .OI2435 

.on  746 

15 

.057068 

.008631 

.008712      .009859 

.009315 

16 
i? 

.050  82  or  ^j.  f  . 
.045  257 

.006845 
.005  427 

.006  009     .O07  819 
.005  478      .006  199 

.007  587 
.005  857 

18 

.040  303 

.004  304 

.004  344    .004  916 

.004  645 

19 

•035  89 

.003  413 

.003  445    .003  899 

.003684 

20 

.031  961 

.002  708 

.002  734 

.003094 

.00292 

21 

.028  462 

.002  147 

.002  167     j  .002  452 

.002317 

22 

•025  347 

.001  703 

.001  719   !  .001  945 

.001  838 

23 

.022  571 

•00135 

.001  363   ,  .001  542 

.001  457 

24 

.020  i  or  ^  f  . 

.OOI  071 

.001  081   !  .001  223    .001  155 

25 

.0179 

.000  849  i 

.000  857  i   .000^69  9 

.000  916  3 

26 

.015  94 

.0006734 

.000-679  7   .000  769  2 

.000  726  7 

27 

.014  195 

.000534 

.000  539  i   .000  609  9   .000  576  3 

28 

.012  641 

.000  423  5   .000  427  5 

.000  483  7   .000  457 

29 

.on  257 

.000  335  8   .000  338  9 

.0003835 

.000  362  4 

30 

.010  025  or  i  f. 

.000  266  3   .000  268  8 

.000  304  2 

.000  287  4 

31 

.008  928 

.000211  3 

.0002132 

.000  241  3 

.000228 

32 

.00795 

.000  167  5 

.000  169  i 

.000  191  3 

.0001808 

33 

•00708 

.000  132  8   .000  134  i   .000  151  7 

.000  143  4 

34 

.006304 

.0001053   .0001063   .0001204   .0001137 

35 

.005  614 

.000  083  66  .000  084  45  .000  095  6  ;  .000  090  i 

36 

•005  or  ^ 

.00006625  .00006687  .0000757   .0000715 

37 

.004  453 

.000  052  55  .000  053  04  .000  060  03  .000  056  7 

38 

.003965 

.000  041  66  .000  042  05  .000  047  58  .000  044  9 

39 

•°03  53i 

.000  033  05  .000  033  36  .000  037  75  .000  035  6 

40 

.003144 

.000  026  2   .000  026  44  .000  029  92  .000  028  2 

8.88      8.^86 

Weights  of  a  Cube  Foot  .  .  485.87 

490.45 

554.988 

524.16 

"      "    Inch..   .2812 

.2838 

.3212 

•3033 

Specific  Gravities  to  determine  the  computations  of  these  weights  were  made  by 
author  for  Messrs.  J.  R.  Browne  &  Sharpe,  Providence,  R.  I. 


WEIGHTS    OF   IBOX,  STEEL,  COPPER,  ETC. 


121 


Iron,  Steel,  Copper,  and.   Brass   AV^ire. 
Birmingham  Wire  Gauge,     f.  full,  1.  light. 


T        » 

PER 

LINEAL  FOOT 

auge. 

Diameter. 

Iron.    |   Steel. 

Copper. 

Brass. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

xxx)  .454  or  ^  f. 

.546  207 

•551  36 

.623  913 

.589  286 

ooo  .425 

.478  656 

.483  172 

.546  752 

.516407 

oo  .38  or  |  f. 

.38266 

.38627 

•437099 

.41284 

o  .34  or  J  f. 

•30634 

.30923 

.349921 

.3305 

i  -3 

•2385 

.240  75 

.272  43 

•257  31 

2  '  .284 

.213738 

•215  755 

.244  146 

.230  596 

3  .259  or  *  f. 

.177765 

.179442 

.203  054 

.191  785 

4 

.238 

.150  107 

.151523    .171461 

.161  945 

5 

.22 

.12826 

.12947 

.146507 

•138376 

6 

.203  or  i  f  . 

.109204 

.110234    -12474 

.117817 

7 

.18  or&L 

.08586 

.086  667    .098  075 

.092  632 

8 

.165  or  J  1. 

.072  146 

.072  827 

.08241 

.077  836 

9 

.148  or  iL  f. 

.058  046 

•058  593 

.066303 

.062  624 

io 

.134 

.047  583 

.048  032    .054  353 

.051  336 

n 

.12  or  ^  1. 

.038  1  6 

.038  52 

.043  589 

.041  17 

12 

.109 

.031  485 

.031  782    .035  964 

.033968 

13 

.095  or  Tiff  1. 

.023  916    .024  142 

.027  319 

.025  802 

14 

.083 

.018  256 

.018428 

.020  853 

.019696 

15 

.072 

.CU3  728 

.013  867 

.015692   j  .014821 

16 

.065 

.on  196 

.on  302 

.012  789      .012  079 

J7 

.058 

.008915 

.008999 

.010  183 

.009618 

18 

.049  or  -^  1. 

.006363 

.006423 

.007268 

.006864 

19 

.042 

.004675 

.004  719    .005  34 

.005  043 

20 

•035 

.003  246 

.003  277 

.003708 

.003  502 

21 

.032 

.002  714 

.002  739 

.003  I 

.002928 

22 

.028 

.OO2  078 

.002  097    .002  373 

.002  241 

23 

•025  or  ^ 

.001  656 

.001  672 

.001  892 

.001  787 

24 

.022 

.001  283 

.001  295 

.001  465 

.001  384 

25 

.02  or  -I* 

.001  06 

.OOI  070     .OOI  211 

.001  144 

26 

.018 

.000  858  6 

.000  866  7   .000  980  7 

.000  926  3 

27  1  .Ol6 

.0006784 

.000  684  8   .000  774  9 

.000  731  9 

28 

.014 

.000  5194 

.000  524  3   .000  593  3   . 

0005604 

29 

.013 

.0004479 

.0004521  1.0005116 

0004832 

30 

.012 

.000  381  6 

.000  385  2  !  .000  435  9   .000  4117 

31 

.01  or  ^j 

.000265 

.000  267  5   .000  302  7  1  . 

0002859 

32 

.009 

.0002147 

.000  216  7 

.000  245  2  j  .000  231  6 

33 

.008 

.000  169  6   .000  1712   .000  193  7 

ooo  183 

34 

.007 

.000  129  9   .000  131  i   -ooo  148  3 

ooo  140  i 

35 

.005  or  YOU 

.00006625  1  .00006688  .00007568  1  .00007148 

36 

.004  or  ^ 

.000  042  4  j  .000  042  8  !  .000  048  43  .000  045  74 

Thickness  of  IPlates. 

No. 

Inch.     No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

I 

•312  5      9 

.156  25 

17 

.056  25 

25 

.02344 

2 

.281  25     io 

.140625 

18 

•05 

26 

.021  875 

3 

.25       n 

.125 

19 

•043  75 

27 

.020312 

4 

•234  375    I2 

.1125 

20 

•0375 

28 

.018  75 

5 

.218  75    13 

.1 

21 

•034  375 

29 

.017  19 

6 

.203  125    14 

.0875 

22 

•031  25 

30 

.015  625 

7 

-1875     15 

•075 

23 

.028  125 

31 

.014  06 

8 

.171  875    16 

.0625 

24 

.025 

32 

.0125 

122 


WIKE    GAUGES. 


WIKE    GAUGES.    (English.) 
Warrington  (Rylands  Brothers). 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

•053 
.047 
.041 
.036 

•0315 
.028 

7/o 
6/0 
5/o 

4/o 
3/0 

2/0 

i 

0 

i 

2 

3 
4 
5 

.326 
•3 
.274 

•25 
.229 
.209 

6 

8 
9 

10 

10.5 

.191 
.174 

•159 
.146 

•133 
.125 

ii 

12 
13 
H 
15 

16 

.117 
.1 
.09 
.079 

*£ 

.0625 

i? 
18 

J9 

20 
21 
22 

Sir  Joseph  Whitworth  &  Co.'s. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

i 

.001 

14 

.014 

34 

•034 

85 

•085 

240 

.24 

2 

.002 

15 

.015 

36 

.036 

90 

.09 

200 

.26 

3 

.003 

16 

.016 

38 

.038 

95 

.09 

280 

.28 

4 

.004 

17 

.017 

40 

.04 

100 

.1 

300 

•3 

5 

.005 

18 

.018 

45 

•045 

no 

.11 

325 

•325 

6 

.006 

19 

.019 

50 

•05 

120 

.12 

350 

•35 

7 

.007 

20 

.02 

55 

•055 

135 

•135 

375 

•375 

8 

.008 

22 

.022 

60 

.06 

150 

•15 

400 

•4 

9 

.009 

24 

.024 

65 

-065 

165 

.165 

425 

•425 

10 

.01 

26 

.026 

70 

.07 

180 

.18 

450 

•45 

ii 

.on 

28 

.028 

75 

•075 

200 

.2 

475 

•475 

12 

.012 

30 

•03 

80 

.08 

22O 

.22 

500 

•5 

13 

.013 

32 

.032 

Sir  Joseph  Whitworth,  in  1857,  introduced  a  Standard  Wire-Gauge,  rang- 
ing from  half  an  inch  to  a  thousandth,  and  comprising  62  measurements. 

It  commences  with  least  thickness,  and  increases  by  thousandths  of  an  inch 

up  to  half  an  inch.  Smallest  thickness,  YirVff  °^  an  mcni  ^s  No.  i ;  No.  2 
is  YJJ%^,  and  so  on,  increasing  up  to  No.  20  by  intervals  of  y<^ '  ^rom 
No.  20  to  No.  40  by  YI&TT ;  and  from  No.  40  to  No.  100  by  y^^.  The 
thicknesses  are  designated  or  marked  by  their  respective  numbers  in  thou- 
sandths of  an  inch. 
This  gauge  is  entering  into  general  use  in  England. 

!N"ew    Standard   Wire    Grange    of  Grreat   Britain, 

1884. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

7/o 

.5 

8 

.100 

22 

.028 

36 

.0076 

6/0 

.464 

9 

.144 

23 

.024 

37 

.0068 

5/0 

•432 

10 

.128 

24 

.022 

38 

.006 

4/° 

•4 

ii 

.Il6 

25 

.02 

39 

.0052 

3/0 

•372 

12 

.104 

26 

.018 

40 

.0048 

2/0 

•348 

13 

.092 

27 

.0164 

4i 

.0044 

O 

•324 

14 

.08 

28 

.0148 

42 

.004 

I 

.3 

15 

.072 

29 

.0136 

43 

.0036 

2 

.276 

16 

.064 

30 

.OI24 

44 

.0032 

3 

.252 

J7 

.056 

31 

.OIl6 

45 

.0028 

4 

.232 

18 

.048 

32 

.0108 

46 

.0024 

5 

.212 

!9 

•04 

33 

.OI 

47 

.002 

6 

.192 

20 

.036 

34 

.0092 

48 

.OOl6 

7 

.I76 

21 

.032 

35 

.0084 

49 

.0012 

No.  50,  .001  inch. 

WIKE  GAUGES. — GAS  PIPES  AND  WIRE  COKD.   123 

French  (Jauges  de  Fits  de  Fer). 

French  wire-gauges,  alike  to  the  English,  have  been  subjected  to  variation, 
Following  table  contains  diameters  of  the  numbers  of  the  Limoges  gauge. 

'Wire-Q-aiage  (Jauge  de  Limoges). 


Number.  ;  Millimetre. 

Inch. 

Number. 

Millimetre. 

Inch. 

Number.  Millimetre. 

Inch. 

0 

•39 

.0154 

9 

1.35 

•0532 

18 

3-4 

•134 

I 

•45 

.0177 

10 

1.46 

•0575 

19 

3-95 

.156 

2 

•56 

.0221 

ii 

1.68 

.0661 

20 

4-5 

.177 

3 

.67 

.0264 

12 

1.8 

.0706 

21 

.201 

4 

•79 

.0311 

13 

1.91 

.0752 

22 

5^5 

.222 

5 

•9 

•0354 

14 

2.02 

•0795 

23 

6.2 

.244 

6 

I.OI 

.0398 

15 

2.14 

.0843 

24 

6.8 

.268 

7 

1.  12 

.0441 

16 

2.25 

.0886 

8 

1.24 

.0488 

17 

2.84 

.112 

Number. 

Millimetre. 

For 

Inch. 

(3-alva 

Number. 

mized 

Millimetre. 

Iron   "V 

Inch. 

Vire. 

Number. 

Millimetre. 

•     Inch. 

I 

.6 

.0236 

9 

1.4 

•0551 

17 

3- 

!  .us 

2 

•7 

.0276 

10 

i-5 

.0591 

18 

3-4 

•134 

3 

.8 

•0315 

ii 

1.6 

.063 

19 

39 

•154 

4 

•9 

•0354 

12 

1.8 

.0709 

20 

44 

•173 

5 

i. 

•°394 

13 

2. 

.0787 

21 

4.9 

•193 

6 

i.i 

•0433 

!4 

2.2 

.0866 

22 

5-4 

.213 

7 

1.2 

•0473 

15 

2.4 

•0945 

23 

5-9 

.232 

8 

i-3 

.0512 

16 

2-7 

.106 

For   TVire    and    Bars. 


Mark. 

Millimetre. 

Mark.  {Millimetre. 

Mark. 

Millimetre. 

Mark. 

Millimetre. 

Mark. 

Millimetre. 

~~P~ 

5 

7 

12 

13 

20 

19 

39 

25 

70 

I 

6 

8 

13 

14 

22 

20 

44 

26 

76 

2 

7 

9 

14 

15 

24 

21 

49 

27 

82 

3    i         8 

10 

15 

16 

27 

22 

54 

28 

88 

4 

9 

ii 

16 

17 

30 

23 

59 

29 

94 

5 

10 

12 

18 

18 

34 

24 

64 

30 

100 

6 

ii 

Diameter. 

Thickness   of 

Thickness.  II         Diameter. 

G-as   !> 

Thickness. 

Lpes. 

Diameter. 

Thickneu. 

i-5  to  3 
4      "6 

•25 

•375     (I 

8  to  10 

12    "    13 

^625 

14  to  15 
16  "  48 

•75 
.875 

Copper  "Wire   Cord.. 
Circumference   and    Safe   Load. 

Inch.  Inch.   Inch.  Inch.   Inch.   Inch.  Ins.    Ina 

Circumference 25    .375      .5    .625     .75        i     1.125    1-25 

Safe  load  in  Lbs 34       50     75     112    168    224       336     448 

Zinc— sheets. 
Thickness   and   'Weight   per    Square   Foot. 

Inch.  i  Inch.  i  inch. 

.031 1  =  10  oz.  .0534  =  14  oz.  .0686  =  18  oz. 

.0457  =  12  OZ.  .O6l  I  =  l6  OZ.  .0761  =  20  OZ. 


124    WEIGHT   AND    STRENGTH    OF   WIRE,  IRON,   ETC. 

WEIGHT   AND    STRENGTH    OF   WIRE,  IRON,  ETC. 
"Weight   arid.    Strength,    of  "Warring-ton    Iron    "Wire, 

Manufactured  by  Rylands  Brothers.    (England.) 
Weight  per  100  Lineal  Feet. 


No. 

Diame- 
ter. 

Weight 

Breaking 
An- 
nealed. 

Weight. 
Bright. 

No. 

Diameter. 

Weight. 

Breaking 
An- 
nealed. 

Weight 
Bright. 

Gauge. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Gauge. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

7/0 

X 

64.46 

3490 

5233 

9 

.146 

5-5 

208 

447 

6/0 

%Z 

56.66 

3066 

4603 

10 

•133 

4-43 

247      |     370 

5/o 

%6 

49-36 

2673 

4OOO 

10.5 

.125 

4-03 

218 

327 

4/0 

/Is 

42-53 

2303 

3457 

ii 

.117 

3-53 

191 

288 

3/o 

% 

36.26 

1963 

2945 

12 

.1 

2.66 

145 

2I7 

2/O 

/&, 

30.46 

1653 

2473 

13 

.09 

2.1 

H3 

169 

O 

.326 

27.36 

I486 

2226 

14 

.079 

1.6 

87 

130 

I 

•3 

23-3 

1257 

1885 

15 

.069 

1.23 

66 

99 

2 

.274 

19.36 

1046 

1572 

16 

.0625 

.96 

53 

77 

3 

•25 

16.13 

873 

1309 

17 

•053 

•73 

39 

59 

4 

.229 

13-53 

732 

1098 

18 

.047 

.56 

3i 

46 

5 

.209 

11.26 

610 

913 

19 

.041 

•43 

23 

35 

6 

.191 

94 

509 

763 

20 

.036 

•33 

18 

27 

7 

.174 

7.8 

422 

633 

21 

.031  25 

.26 

14 

21 

8 

•159 

6-53 

353 

519 

22 

.028 

.2 

ii 

16 

To  Compute  Length  of  1OO  JPoxiiids  of  Wire  of  a  Griveii 
Diameter. 

RULE. — Divide  following  numbers  by  square  of  diameter,  in  parts  of  an 
inch,  and  quotient  is  length  in  feet. 

37.68  for  wrought  iron.      I        33.42  for  copper.        I       28    for  silver. 

37.45  for  steel.  34-4*  for  brass.         |       15.3  for  gold. 

13.64  for  platinum. 


WINDOW  GLASS. 

Thiolrness   and   "Weight   per   Sqnare   Foot. 


No. 

Thickness. 

Weight. 

No. 

Thickness. 

Weight. 

No. 

Thickness. 

Weight. 

12 

13 
15 

16 

Inch. 
•°59 
.063 
.071 
.077 

Oz. 
12 

13 
15 

17 

«9 

21 
24 

Inch. 
.083 
.091 
.1 
.III 

Oz. 
I? 
19 
21 
24 

26 
32 
36 
42 

Inch. 
.125 
•154 
.167 
.2 

Oz. 
26 

$ 

42 

Terne   Plates. 

Teme  Plates — Are  of  iron  covered  with  an  amalgam  of  lead. 

Thickness  and  'Weight  of  Gralvanized.  Sheet  Iron. 

Sheet  2  Feet  in  Width  by  from  6  tog  Feet  in  Length  (M.  Le/erts). 


.?! 

&5 

Weight 
per 

Sq.  Foot. 

t* 

t>  * 

£o 

Weight  ||  „& 
per         J:  § 
Sq.  Foot.     £a 

Weight 
per 
Sq   Foot. 

«& 

Si 

Weight 
per 
Sq/Foot. 

hi 
t§£ 

No. 

17 

16 
1  J5 

Weight 
per 
Sq.  Foot. 

1  & 

II 

Weight 
per 

Sq.  Foot. 

No. 
29 
28 
27 

Oz, 
12 
13 

J4 

No. 
26 
25 
24 

Oz.      H  No. 
15          23 
l6           22 

18     H  21 

Oz. 
20 
22 
24 

No. 
20 
19 

18 

Oz. 
27 
30 

35 

Oz. 
36 
42 
46 

No. 
14 
13 
12  j 

Oz. 

53 
61 
70 

WEIGHTS    OF    METALS. 


WROUGHT  IKON  AND  STEEL. 


125 


Weights    of    Square    Rolled.    Iron    and.    Steel, 

From  .125  to  10  Inches.    ONE  FOOT  IN  LENGTH. 
Iron,  485  Ibs.      Steel,  489.6  Ibs.      PER  CUBE  FOOT. 


SIDE. 

IRON. 

STEEL. 

SIDE. 

IRON. 

STEEL. 

SIDE. 

IROX. 

STEEL. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

.125 

•053 

•053 

2-75 

25-47 

25.71 

6.25 

I3I.6 

132.8 

•1875 

.118 

.119 

•875 

27.84 

28.1 

•375 

137 

138.2 

•25 

.21 

.212 

3 

30.3  1 

30.6 

-5 

142.3 

143.6 

•3125 

•329 

•333 

.125 

32.89 

33-2 

.625 

147-9 

149.2 

-375 

•474 

.478 

•25 

35-57 

35-92 

•75 

153-5 

154.9 

•4375 

.645 

.651 

•375 

38.57 

38.73 

•875 

159.2      160.8 

•5 

.812 

•85 

•5 

41.26 

41.65 

7 

165 

166.6 

•5625 

1.066 

1.076 

.625 

44.26 

44-68 

.125 

171 

172.6 

.625 

1.316 

1.328 

•75 

47-37 

47.82 

•25 

177 

178.7 

.6875 

1.592 

1.608 

•875 

50.37 

51-05 

•175 

183.2 

184.9 

•75 

1.895 

I-9I3 

4 

53.89 

54-4 

•5 

189.5 

191.3 

.8125 

2.223 

2.245 

.125 

57.85 

.625 

195.8 

197.7 

•875 

2-579 

2.608 

•25 

60.84 

61.41 

•75 

202.3 

204.2 

•9375 

2.96 

2.989 

•375 

64.17 

65.08 

.875 

208.9 

210.8 

i 

3-368 

3-4  ' 

•5 

68.2 

68.85 

8 

215.6 

217.6 

.125 

4.263 

4-303 

-625 

72.05 

72.73 

.125 

222.4 

224.5 

•25 

5.263 

5-312 

-75 

75-99 

76-71 

.25 

229.3 

231.4 

•375 

6.368 

6.428 

•875 

80.05 

80.8  1 

•375 

236 

238.5 

•5 

7.578 

7-65 

5 

84.20 

85 

•5 

2434 

245.6 

•5625 

8.893 

8.978 

.125 

88.47 

89-3 

.625 

250.6 

252.9 

*75 

10.31 

10.41 

•25 

92.83 

93.72 

•75 

257.9 

260.3 

•875 

11.84 

"•95 

•375 

98.23 

•875 

265.3 

267.9 

2 

13.37 

13-6 

•5 

101.9 

102.8 

9 

272.8 

275.4 

.125 

15.21 

15.35 

.625 

106.6 

107.6 

•25 

288.2 

290.9 

.25 

17.08 

17.22 

•75 

111.4 

112.4 

•5 

304 

306.8 

•375 

19 

19.18 

•875 

116.3 

117.4 

•75 

320.2 

323.2 

•5 

21.05 

21.25 

6 

121.3 

122.4 

•875 

328.6 

331.6 

.625 

23.21 

23-43 

.125 

— 

127.6 

10 

336.8 

340 

Weight   of  .A^ngle   Iron, 

From 

1.25  to  4.5  Inches.    ONE  FOOT  IN  LENGTH. 

Thickness  measured  in  Middle  of  each  Side. 

L  EQUAL  SIDES 

'         1 

-L  UNEQUAL  SIDES. 

L  UNEQUAL  SIDES. 

Thick- 

Thick- 

Thick- 

Sides. 

ness. 

Weight 

Side..         *£ 

Weight. 

Sides. 

ness. 

Weight 

Ins. 

Inch. 

Lbs. 

Ins.            Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

I.25XI.25 

.1875 

3    X2.5    .375 

6.25 

6    X3-5 

.625 

18 

1.5   Xi.5 

•1875 

2 

3-5X3       -4375 

7-75 

6    X4-5 

-625 

20 

I.75XI.75 

.25 

3 

3.5X3       -4375 

9.6 

2         X2 

.25 

3-5 

4    X3       -5 

ii 

TT 

2.25X2.25 

.3125 

4-5 

4    X3-5    -5 

"•5 

2      X  2.375* 

•375 

5-5 

2.5   X2.5 

•3125 

5 

4    X3-5    -5 

"•75 

2.5X2.875 

•375 

6-5 

3      X3 

•375 

7 

4.5X3       -5 

"•75 

3-5X3.5 

•4375 

10.5 

3-5   X3-5 

•4375 

9 

5    X3       -5 

12.65 

V,                                             f 

•4375 

11 

4      X4 

•5 

12.5 

5    X3       -5625 

13.7 

4         3o     "^ 

•75 

AO 

4-5   X4-5 

•5 

14 

5-5X3.5    -5 

14-5 

4    X3-5 

.75 

13.5 

4-5    X4-5 

16 

5-5X3.5    .5625 

15-6 

*  This  column  gives  depth  of  web  added  to  the  thickness  of  base  or  flange. 

L* 


126 


WEIGHTS    OF    METALS. 


WROUGHT  IRON  AND  STEEL. 
"Weights    of  Round    Rolled    Iron,    and    Steel, 

From  .125  to  10  Inches.    ONE  FOOT  IN  LENGTH. 
Iron,  485  Ibs.       Steel,  489.6  Ibs.       PER  CUBE  FOOT. 


Diameter. 

IRON. 

STEEL. 

Diameter 

IRON. 

STEEL. 

Diameter 

IRON. 

STEEL. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

.125 

.041 

.042 

2-75 

20.  01 

20.2 

6.25 

103-3 

104.3 

-1875 

•093 

.094 

.875 

21.87 

22.07 

•  375 

107.7 

108.5 

•25 

•  165 

.167 

3 

23.81 

24.03 

•  5 

in.  8 

II2.8 

•3125 

•  258 

.261 

.125 

25-83 

26.08 

.625 

116.4 

117.2 

•375 

•372 

-375 

•25 

27.94 

28.2 

•75 

120.5 

121.7 

•4375 

•  506 

.511 

•375 

30-13 

30.42 

•875 

124.9 

126.2 

•  5 

.661 

.667 

•5 

32.41 

32.71 

7 

129.6 

130.9 

•837 

.845 

.625 

34-76 

35-09 

.125 

134.2 

135-6 

.625 

1-033 

1.043 

•75 

37-2 

37-56 

•25 

139 

140.4 

.6875 

1-25 

1.262 

.875 

39-72 

40.1 

-375 

143-8 

145-3 

•75 

1.488 

1.502 

4 

42.33 

42.73 

.5 

148.8 

150.2 

.8125 

1.746 

'.763 

.125 

45.01 

45-44 

.625 

153-8 

155-2 

.875 

2.025 

2.044 

•25 

47.78 

48.24 

•75 

158.9 

160.3 

•9375 

2-325 

2-347 

•375 

50.63 

51.11 

•875 

164.1 

165.6 

i 

2.645 

2.67 

•  5 

53-57 

54-07 

8 

169.3 

171 

.125 

3-348 

3-379 

.625 

56.59 

57-12 

.125 

174.6 

'76.3 

•25 

4'*33 

4-I73 

•75 

59-69 

60.25 

-25 

1  80.  i 

181.8 

-375 

5 

5-049 

•875 

62.87 

63-46 

•375 

185-5 

187-3 

5.952 

6.008 

5 

66.13 

66.76 

•  5 

191.1 

.625 

6.985 

7.051 

.125 

69.48 

70.14 

.625 

196.6 

198.7 

•75 

8.104 

8.178 

•25 

72.91 

73-6 

•75 

202.5 

204.4 

.875 

9-3 

9-388 

-375 

76.43 

77.16 

.875 

208.1 

210.3 

2 

.125 

10.58 
"•95 

10.68 
12.06 

:LS 

80.02 
83-7 

80.77 
84.49 

9 

•  25 

214.3 
226.3 

216.3 
228.5 

•25 

13-39 

I3-52 

•75 

87.46 

88.29 

•  5 

238.7 

241 

•375 

14.92 

15.07 

•875 

91.31 

92.17 

-75 

251-5 

253-9 

16.53 

16.69 

6 

95-23 

96.14 

.875 

259.5 

260.4 

•625 

18.23 

18.4 

.125 

103.3 

100.2 

10 

264.5 

267 

"Weight    of  Steel    Angles. 

From  .75  to  7  X  3.5  Inches.     ONE  FOOT  IN  LENGTH. 
Thickness  measured  in  middle  of  each  side. 


SIDE. 

EQUAL 

Thick- 
ness. 

SIDES 
Area. 

Weight. 

SIDES. 

Thick- 
ness. 

U 

Area. 

NEQUAL 

Weight. 

SIDES. 
SIDES. 

Thick- 

Area. 

Weight. 

Ins. 

Ins. 

Sq.Ins 

Lbs. 

Ins. 

Ins. 

Sq.  Ins 

Lbs. 

Ins. 

Ins. 

Sq.Ins 

Lbs. 

•75 

.125 

•17 

.6 

i.375Xi 

•  25 

•53 

1.8 

5X3-5 

•  5 

4 

13.6 

.875 

.125 

.31 

•7 

2  Xi-375 

•25 

.78 

2-7 

5X3-5 

.625 

4.92 

16.8 

i 

.125 

.24 

.8 

2.25X1.5 

-25 

.88 

3 

5X3-5 

.75 

19.8 

1.25 

•I25 

•30 

i 

2.25X1.5 

•5 

1.63 

5-5 

5X3-5 

.875 

6.'  67 

22.7 

1.5 

•25 

.69 

2.4 

2.5     X2 

•25 

i.  06 

3-7 

5X4 

•5 

4-25 

14-5 

1.75 

.25 

.8l 

2.8 

2-5      X2 

•5 

2 

6.8 

5X4 

•  625 

5-23 

17.8 

2 

•25 

•94 

3-2 

3       X2 

•25 

I.I9 

4 

5X4 

•75 

6.19 

21.  I 

2.25 

•25 

1.06 

3-7 

3       X2 

•5 

2.25 

7-7 

5X4 

•875 

7.11 

24.2 

2-5 

•25 

1.19 

4.1 

3.25X2 

•25 

1.25 

4-3 

6X3-5 

•5 

4-5 

15-3 

2-75 

•25 

4-5 

3.25X2 

•5 

2.38 

8.1 

6X3.5 

.625 

5-55 

18.9 

3 

•5 

2-75 

9-4 

3-5    X2.5 

•25 

1.44 

4-9 

6X3-5 

•75 

6.56 

22.3 

3-5 

•5 

3-25 

ii.  i 

3-5    X3 

•75 

4-31 

14.7 

6X3-5 

i 

8-5 

28.9 

4 

•5 

3-75 

12.8 

4       X3 

•5 

3-25 

ii 

6X4 

•  5 

4-75 

16.2 

4 

•75 

5-44 

18.5 

4       X3 

•75 

4.69 

16 

6X4 

•  625 

5-86 

20 

5 

•5 

4-75 

16.2 

4       X3-5 

•5 

3-5 

11.9 

6X4 

•75 

6-94 

23.6 

5 

•75 

6-94 

23.6 

4       X3-5 

•75 

5-o6 

17.2 

6X4 

i 

9 

30.6 

5 

i 

9 

30.6 

4-5    X3 

5 

3-5 

11.9 

7X3-5 

•5 

5 

17 

6 

•5 

5-75 

19.6 

4-5    X3 

•75 

5-o6 

17.2 

7X3-5 

.625 

6.17 

21 

6 

•75 

8-44 

28.7 

5       X3 

•5 

3-75 

12.8 

7X3-5 

•75 

24.9 

$ 

ii 

37-4 

5       X3 

•75 

5-44 

18.5 

7X3-5 

i 

9-5 

32-3 

WEIGHTS    OF    METALS. 


127 


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WEIGHTS    OF    METALS. 


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GO 

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WEIGHT    OF    SHEET   AND    HOOP    IRON.  129 

Weight    of   Sheet    Iron.     (English.     D.  K.  Clark.) 

PER  SQUARE  FOOT  (at  480  Ibs.per  Cube  Foot). 
As  by  Wire-gauge  used  in  South  Staffordshire,  England. 


Thickness. 

Weight. 

Square 
Feet 
in  i  ton. 

Thickness. 

Weight. 

Square 
Feet 
in  i  ton. 

Thickness. 

Weight. 

Square 
JFeet 
in  i  ton. 

No. 

Inch. 

Lbs. 

No. 

No. 

Inch. 

Lbs. 

No. 

No. 

Inch. 

Lbs. 

No 

V 

.0125 

•5 

4480 

21 

•0344 

1.38 

1623 

IO 

.1406 

5.63 

398 

31 

.0141 

3986 

2O 

•0375 

i-5 

1493 

9 

•I563 

6.25 

358 

30 

.0156 

.625 

3584 

19 

.0438 

i-75 

1280 

8 

.1719 

6.88 

326 

29 

.0172 

.688 

3256 

18 

•05 

2 

II2O 

7 

•1875 

7-5 

299 

28 

.0188 

•75 

2987 

17 

•0563 

2.25 

996 

6 

.2031 

8.13 

276 

*T 

.0203 

.813 

2755 

16 

.0625 

25 

896 

5 

.2188 

8-75 

256 

26 

.0219       .875 

2560 

15 

•075 

3 

747 

4 

•2344 

908 

239 

25 

.0234       .938 

2388 

H 

.0875 

35 

640 

3 

•25 

10 

224 

24 

•025 

i 

2240 

13 

.1 

4 

56o 

2 

.2813 

11.25 

199 

*3 

.0281 

1.13 

1982 

12 

.1125 

4-5 

498   1    i 

•3125 

12.5 

I79 

22 

.0313 

1.25 

1792 

II 

.125 

5 

448 

Weiglit  of  Hoop   Iron.    (English.) 
PER  LINEAL  FOOT. 


Width. 

W.G. 

Weight. 

Width. 

W.G. 

Weight. 

Width. 

W.G. 

Weight. 

Ins. 
.625 

•75 
.875 

i 

No. 
21 
20 
19 

18 

Lbs. 
.067 

.0875 
.I2l6 
.1636 

Ins. 
I.I25 
1.25 
1-375 
«•$ 

No. 
17 

16 
15 
i5 

Lbs. 
.21 
.27 

Ins. 

1-75 
2 
2.25 
2-5 

No. 
14 
13 
13 
12 

Lbs. 
.484 

•634 
.714 
.91 

"W eight   of  Black  and  G-alvanized.   Sheet   Iron. 

(Morton's  Table,  founded  upon  Sir  Joseph  Whit  worth  fy  Co.'s  Standard  Bir- 
mingham Wire-Gauge.)     (D.  K.  Clark.) 

NOTE. — Numbers  on  Holtzapflel's  wire-gauge  are  applied  to  thicknesses  on  Whit- 
worth  gauge. 


Gauge  and  Weight  of 
Black  Sheets. 

Approxii 
o/Sq.F 
Black. 

iate  number 
t.  in  i  ton. 
Galvanized. 

Gauge  and  Weight  of 
Black  Sheets. 

A3SS 

Black. 

ute  number 
t.  in  i  ton 
Galvanized 

No. 

Inch. 

Lbs. 

Sq.  Ft. 

Sq.Ft. 

No. 

Inch. 

Lbs. 

Sq.Ft. 

Sq.  Ft. 

I 

•3 

12 

I87 

I8S 

I? 

.06 

2.4 

933 

876 

2 

.28 

II.  2 

200 

197 

18 

•05 

2 

II2O 

1038 

3 

.26 

IO.4 

215 

212 

19 

.04 

1.6 

1400 

1274 

4 

.24 

9-6 

233 

229 

20 

.036 

1-4 

1556 

1403 

5 

.22 

8.8 

254 

250 

21 

.032 

1.28 

1750 

1558 

6 

.2 

8 

280 

275 

22 

.028 

1.  12 

2000 

1753 

7 

.18 

7.2 

3" 

304 

23 

.024 

.96 

2333 

2004 

8 

.I65 

6.6 

339 

331 

24 

.022 

.88 

2545 

2159 

9 

•15 

6 

373 

363 

25 

.02 

.8 

2800 

2339 

IO 

•135 

5-4 

4i5 

403 

26 

.Ol8 

.72 

3III 

2553 

ii 

.12 

4.8 

467 

452 

27 

.Ol6 

.64 

3500 

2808 

12 

.11 

4.4 

509 

491 

28 

.014 

•56 

4000 

3122 

13 

•095 

3-8 

589 

566 

29 

.013 

•52 

4308 

3306 

14 

.085 

3-4 

659 

630 

30 

.012 

.48 

4667 

3513 

15 

.07 

2.8 

800 

757 

31 

.01 

•4 

5000 

4017 

16 

.065 

2.6 

862 

813 

32 

.009 

.36 

6222 

4327 

130 


WEIGHT    OF    ANGLE    AND    T   IRON. 


Weight    of  English   .A^ngle    and   T   Iron.    (D.  K.  Clark.  \ 

ONE   FOOT    IN    LENGTH. 

NOTE.— When  base  or  web  tapers  in  section,  mean  thickness  is  to  be  measured. 


SUM  OF  WIDTH  AND  DEPTH  IN  INCHES. 

1-5 

1.625     "-75 

1.875 

2 

2.125 

2-25      2.375 

2.5 

2.625 

2-75 

Lbs. 

Lbs.    |    Lbs.        Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•57 

.62 

.68       .73 

•78 

•S3 

.88 

-94 

-99 

1.04 

1.09 

.81 

.89 

•97     1.05 

LI3 

1.  21 

1.29 

I-37 

i-45 

1-52 

1.6 

1.04 

I.I5 

1.25 

1.36 

1.46 

1.56 

1.67 

I.77 

1.88 

I.98 

2.08 

1.24 

M 

1.63 

I.76 

1.89 

2.O2 

2.15 

2.28 

2.4I 

2.54 

2.875 

3 

3-125 

3.25 

3-375 

3-5 

3.625 

3-75 

3-875 

4 

4-25 

I.I4 

1.2 

1.25 

1.3 

1-45 

1.41 

1.46 

1.51 

1.56 

1.62 

1.72 

1.68 

I.76 

1.84 

1.91    1.99 

2.07 

2.15 

2.23 

2.3 

2.38 

254 

2.19 

2.29 

2.4 

2.5      2.6 

2.71 

2.81 

2.92 

3.02 

3.13 

3.33 

2.67 

2.8 

2.93 

3.06  '    3.19 

3-32 

3-45 

3.58 

3.71 

3-84 

4.1 

3.13 

3.28 

3-44 

3-59 

3«75 

3.91 

4.06 

4.22 

4-38 

4-53 

4.84 

3-57 

3-75 

3-93 

4.II 

4.29 

4.48 

4.66 

4.84 

5-02 

5-2 

5.56 

4-5 

4-75 

5 

5-25 

5-5 

5-75 

6 

6.25 

6-5 

6.75 

7 

2-7 

2.85 

3.01 

3-16 

3.32 

3.48 

3-63 

3-79 

3-95 

4.1 

4.26 

3-54     3-75 

3.96 

4.17 

4.38 

4-58 

4-79 

5 

5-21 

5.42 

5.63 

4.36'  4-62 

4.88 

5.14 

5-4 

5-66 

5-92 

6.18 

6-45 

6.71 

6.97 

5-i6    5.47 

5.78 

6.09 

6.41 

6.72 

7-03 

7-34 

7.66 

7-97 

8.28 

5.921  6.29 

6.65 

7.02 

7-38 

7-75 

8.11 

8.48 

8.84 

9.21 

9-57 

6.67!   7.08 

7-5 

7.92 

8-33 

8-75 

9.17 

9-58 

10 

10.42 

10.83 

7.38 

7.85 

8.32 

8.79 

9.26 

9-73 

10.2 

10.66 

11.13 

12.6 

12.07 

7-25 

7-5 

7-75 

8 

8.25 

8.5 

8-75 

9 

9-25 

9-5 

9-75 

5-83 

6.04 

6.25 

6.46 

6.67 

6.88 

7.08 

7.29 

7-5 

7.71 

7-92 

7-23 

7-49 

7-75 

8.01 

8.27 

8-53 

8-79 

9-°5 

9-3i 

957 

9-83 

8.59 

8.91 

9.22 

9-53 

9.84 

10.16 

10.47 

10.78 

11.09 

11.41 

11.72 

9-93 

10.3 

10.66 

11.03 

"•39 

11.76 

12.12 

12.49 

12.85 

13.22 

13.58 

11.25 

11.67 

12.08 

12.5 

12.92 

13.33 

13-75 

14.17 

14.58 

15 

12.54 

13.01 

13.48 

13.94 

14.41 

14.88 

15-35 

15.82 

16.29 

16.76 

17.23 

13-8 

14.32 

14.84 

15-36 

15.89 

16.41 

16.93 

!7-45 

17.97 

18.49 

19.01 

10 

10.5 

1  1 

11.5 

12 

12.5 

"3 

"3-5 

'4 

"4-5 

15 

12.03 

12.66 

13.28 

13.91 

14-53 

13-95 

14.67 

15.4 

16.13 

16.86 

17-59 

18.31 

19.04 

19.77 

20.5 

21.22 

15.83 

16.67 

17.5 

18.33 

19.17 

20 

20.84 

21.67 

22.5 

23-34 

24.17 

17.7 

18.63 

19.57 

20.51 

21-44 

22.38 

23.31 

24-25 

25.19 

26.12 

27.06 

19.53 

20.57 

21.61 

22.66 

23.7 

24.74 

25.78 

26.83 

27.87 

28.91 

29-95 

23-13 

24.38 

25.63 

26.88 

28.13 

29-37 

30.63 

31.88 

33-13 

34.38 

35.63 

12 

12.5 

13 

13.5 

•4 

"5 

16 

17 

18 

>9 

2O 

23-7 

24.74 

25.78 

26.83 

27.87 

29-95 

32.03 

34.12 

36.2 

38.28 

40.36 

28.13 

29.37 

30.63  31.88 

33-13 

35.63 

38.13 

40.63  41.13 

43.63 

46.13 

32.45 

33-91 

35.36  36.82 

38.28  41.19 

44.12 

47.02  49.95  52.87  55.78 

36.67 

38.33:40      141.67  43.33  46.67 

50 

53-33  56.67160       [63.33 

NOTE.— American  rolled  is  slightly  heavier. 


WEIGHT  OP  HOOP  IRON.  —  CAST  IRON.  —  METALS. 


Weight   of  HOOT>   Iron.    (D.  K.  Clark.) 

ONE   FOOT   IN   LENGTH. 

As  by  Wire-gauge  used  in    outh  Staffordshire  (England). 


THICKNESS. 

.625 

•75 

•875 

i 

.,  ID! 
1.     25 

•H  IN  INCHES. 
'•25      1-375 

'•5 

i.625|  1.75 

2 

No. 

Inch. 

Lb. 

Lb. 

Lb. 

Lb. 

Lb 

Lb. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

21 

.0344 

.0716 

.0861 

.1 

•"5 

.129 

.144       .158      .173-    .197 

.201 

.229 

2O 

•0375 

.0781 

.0938 

.109 

.125 

.141 

.156 

.172 

.188     .203 

.219 

•25 

19 

.0438 

.OQII    .109 

.128    .146 

.164 

.l82J      .2 

.219     .238 

•257 

.292 

18 

•OS 

.104 

.125 

.146    .167 

.188 

.208        .229 

•25 

.271 

.292 

•333 

*7 

•0563 

.117 

.141 

.164    .188 

.211 

•234 

.258 

.28l|    .305     .328 

•375 

16 

.0625    .13 

.156 

.182    .208 

•234 

.26 

.286 

•313    -339    -365    .417 

15 

•075 

.156 

.188 

.219    .25 

.281 

•313 

•344 

•375    -307    -438 

.5 

14 

.0875 

•I83 

.219 

.256    .293 

•329 

.366 

.402 

•438    -475 

.512 

•585 

13 

.1 

.208 

•25 

.292  !  .333 

•375 

.416 

.458     -5 

•543 

.584 

.667 

12 

.1125 

•234 

.281 

•328  .375 

422 

.469 

.516     .563    .609 

.656 

•75 

II 

.125 

.26 

•313 

•365  -417 

.469 

•  521 

•573 

.625    .677 

•729     -833 

10 

.1406 

•293 

•352 

.41 

.469 

•527 

.586 

.645 

.703;   .762 

.82 

.938 

9 

.1563    .326 

•391 

•456 

.522 

•587 

.652 

.717 

.783    .848 

•913 

1.04 

8 

•I7I9  -358 

•43 

.501 

•573 

.644 

.7l6 

.78.8 

•859    -931 

1-15 

7 

.1875  !  .391 

.469 

•547   -625 

•703 

.781 

•859       ^38  1-02 

.09 

1.25 

6 

.2031 

•423 

•508 

•593  -677 

.762 

.836 

.931    1.02 

i.i 

.19 

i-35 

5 

.2188   .456 

•547 

,638 

.729 

.82 

.912 

I 

1.09 

1.19 

.28 

1.46 

4   -2344 

.488 

.586 

.683 

.781 

.879 

•977 

1.07 

1.17 

1.27 

•37 

1.56 

CAST  IRON. 
To  Compute  "Weight  of  a  Cast  Iron  Bar  or  Hod, 

Ascertain  weight  of  a  wrought  iron  bar  or  rod  of  same  dimensions  hi 
preceding  tables,  or  by  computation,  and  from  weight  deduct  ^th  part. 

Or,  As  .1000  :  .9257  ::  weight  of  a  wrought  bar  or  rod  :  to  weight  re- 
quired. Thus,  what  is  weight  of  a  piece  of  cast  iron  4  x  3.75  X  12  inches? 

In  table,  page  128,  weight  of  a  piece  of  wrought  iron  of  these  dimensions 
is  50.692  Ibs.  Tnen?  I000  .  9 


Braziers'   and   Sheathing   Copper. 

BRAZIERS'  SHEETS,  2X4  feet  from  5  to  25  Ibs.,  2.5  x  5  feet  from  9  to  150  Ibs.,  and 
3X5  feet  and  4X6  feet,  from  16  to  300  Ibs.  per  sheet. 
SHEATHING  COPPER,  14  x  48  inches,  and  from  14  to  34  oz.  per  square  foot 
YELLOW  METAL,  14  x  48  inches,  and  from  1  6  to  34  oz.  per  square  foot. 


Weight    o±    Corrugated    Steel    Root    P»lates. 

Carnegie  Steel  Co. 


Dimensions, 

Thickness. 

Per  Sq.  Ft.  | 

Dimensions. 

Thickness. 

Per  Sq.  Ft. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

8.75X1.5 
8.75  X  1.5625 

•25 
•3125 

,ol 

12.1875X2.75 
12.1875  X  2.8125 

•375 
•4375 

17-75 
20.71 

8.75  X  1.625 

•375 

!2                1 

12.1875X2.875. 

•5 

23.67 

METALS. 

To   Com.pu.te  "Weight   of  Mietals  of  any  Dimen- 
sions  or  Form- 
By  rules  in  Mensuration  of  Solids  (page  360 ),  ascertain  volume  of  the 
piece,  multiply  it  by  weight  of  a  cube  inch,  and  product  will  give  weight 
in  pounds. 


132  WEIGHT    OF    CAST   IRON   PIPES. 

"Weight   of  Cast   Iron   3?ipes   or   Cylinders. 

From  i  to  70  Inches  in  Internal  Diameter. 

ONE   FOOT   IN   LENGTH. 


Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

IllS. 

Inch. 

Lba. 

Ins. 

Inch. 

Lba. 

lus. 

Inch. 

Lks. 

Z 

•25 

3-06 

4-75 

•375 

18.84 

II 

.875 

IOI.85 

-375 

5-05 

•5 

25.72 

"•5 

•5 

58.81 

1.25 

•25 

3-68 

•625 

32.93 

.625 

74.28 

•3125 

4-79 

•75 

40-43 

•75 

OX).o6 

•375 

5-97 

5 

•375 

19.76 

•875 

106.13 

i-5 

•375 

6.89 

•5 

26.95 

12 

•5 

61.26 

•4375 

8.31 

•625 

34-46 

.625 

77-34 

•5 

9.8 

•75 

42.27 

•75 

93-73 

x-75 

•375 

7.81 

5-5 

•375 

21.59 

•875 

110.42 

•4375 

9-38 

•5 

29-4 

12.5 

•5 

63.7I 

•5 

11.03 

•625 

37-52 

•625 

80.4 

2 

•375 

8-73 

•75 

45-95 

•75 

97-4 

•4375 

10.45 

6 

•375 

23-43 

•875 

114.71 

•5 

12.25 

•5 

31.86 

13 

•5 

66.16 

2.25 

•375 

965 

•625 

40-59 

.625 

8347 

•4375 

11.52 

•75 

49.62 

•75 

101.08 

•5 

13.48 

6-5 

•375 

25.27 

•875 

II9 

2-5 

•375 

10.57 

•5 

34-31 

13-5 

•5 

68.61 

•4375 

12.6 

.625 

4365 

.625 

86.53 

•5 

14.7 

•75 

53-3 

•75 

104.76 

2-75 

•375 

11.49 

7 

•5 

36-76 

•875 

123.29 

•4375 

14.67 

•5625 

41.7 

14 

•5 

71.06 

•5 

'5-93 

.625 

46.71 

.625 

89.6 

3 

•375 

12.4 

•75 

56-97 

•75 

108.43 

•5 

17-15 

7-5 

-5 

39.21 

•875 

127.58 

.625 

22.2 

•5625 

44-45 

14.5 

•5 

73-51 

•75 

27-57 

•625 

49-77 

•625 

92.66 

3-25 

•375 

I3-32 

•75 

60.65 

•75 

112.  II 

•5 

18.38 

8 

•5 

41.66 

•875 

131.87 

.625 

23-74 

•5625 

47-21 

15 

•5 

75-06 

•75 

29-4 

•625 

52.84 

•625 

95-72 

3-5 

•375 

14.24 

•75 

64.32 

•75 

115.78 

•5 

19.6 

9 

•5 

46.56 

•875 

136.16 

.625 

25.27 

-5625 

52.72 

15.5 

•5 

78.47 

•75 

31.24 

•625 

58.96 

•625 

98.78 

3-75 

•375 

I5.l6 

•75 

71.67 

•75 

119.46 

•5 

20.83 

9-5 

•5 

49.01 

•875 

140.44 

.625 

26.8 

•5625 

55.48 

16 

•625 

101.85 

•75 

33-08 

•625 

62.06 

•75 

123.14 

4 

•375 

16.08 

•75 

75-35 

•875 

144.73 

•5 

22.05 

10 

•5 

51.45 

i 

166.63 

.625 

28.33 

.625 

65.09 

16.5 

•625 

104.9 

•75 

34-92 

•75 

79-03 

•75 

126.75 

*25 

•375 

17     • 

•875 

93-2; 

•875 

149.02 

•5 

23.28 

10.5 

•5 

53.9i 

i 

I7L53 

.625 

29.86 

•625 

68.15 

!7 

•625 

107.97 

•75 

36-76 

•75 

82.7 

•75 

130.48 

4-5 

•375 

17.92 

•875 

97.56 

•875 

153-3 

•5 

23.88 

n 

•5 

56-36 

i 

176.43 

.625       31.4 

.625 

71.21 

17.5 

.625 

IJI.03 

•75       1  38-59 

•75 

86.38 

•75 

I34-I6 

WEIGHT   OP  CAST   IKON   PIPES. 


133 


tmeter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight.  I 

Diameter.  f 

Thickn. 

Weight. 

Ins. 

Inch. 

Lbs. 

Ins. 

Ina. 

Lbs. 

Ins. 

Ins. 

Lbs. 

J7-5 

.875 

157.59 

29 

218.7 

40 

•875 

350.56 

I 

I8L33 

.875 

256.23  ! 

I 

401.86 

*8 

.625 

II4.I 

I 

294.05 

I.I25 

453.46 

•75 

137-84 

30 

•75 

226.05  : 

1.25 

505.41 

.875 

I6I.88 

.875 

264.8  ; 

42 

.875 

367-69 

I 

186.23 

i 

303.86 

r 

421.45 

19 

.625 

120.23 

1.125 

343-22 

1.125 

472.52 

•75 

145.19 

31 

-75 

233-41 

1.25 

529.87 

.875 

1  70.46 

.875 

273-38 

44 

•875 

384.88 

i 

106.03 

i 

313.66 

i 

441.1 

20 

.625 

126.35 

1.125 

354.24 

1.125 

497-58 

•75 

152.54 

32 

•75 

240.75 

1.25 

554-42 

.875 

179.03 

•875 

281.95 

46 

•875 

402.OI 

i 

205.84 

i 

323-46  1 

i 

460.07 

21 

•625 

132.48 

1.125 

365.27 

1.125 

519.64 

•75 

159.89 

33 

•75 

248.11 

1-25 

578.88 

.875 

187.61 

•875 

290.53 

48 

•875 

419.17 

i 

215.64 

i 

333-26 

i 

480.29 

22 

•625 

138.61 

1-125 

376.29 

1.125 

541.69 

•75 

167.24 

34 

•75 

255-46 

1.25 

603.44 

.875 

196.19 

•875 

299.11 

50 

•875 

436.43 

I 

225.44 

i 

343.06 

i 

499.89 

23 

.625 

144-73 

1.125 

387.33 

1.125 

563.75 

•75 

174-59 

35 

•75 

262.81 

1.25 

627.93 

.875 

204.76 

•875 

307.68 

52 

•875 

453-49 

i 

235.24 

i 

352.87 

i 

5^-5 

24 

.625 

150.86 

1.125 

398.35  ! 

1.125 

585-81 

•75 

181.95 

36 

•75 

270.16  : 

1.25 

654.42 

•875 

213.34 

•875 

316.26  j 

55 

•875 

479-23 

i 

245-04 

i 

362.67  | 

i 

548.9 

25 

•625 

156.98 

1.125 

409.28; 

1.125 

618.91 

•75 

189.3 

fcfer-rttt- 

1.25 

456-37  { 

1.25 

689.21 

.875 

221.92 

37 

•75 

277-5I  1 

58 

•  i 

578.29 

i 

254.85 

•875 

324.84 

1.125 

651.96 

26 

.625     163.11 

i 

372.47  | 

1.25 

725.93 

•75    i  196-65 

1.125 

420.4 

1-375 

800.22 

•875 

230.5 

1.25 

468.65 

60 

i 

597-92 

i 

264.65 

38 

•75 

284.86 

1.125     674.01 

27 

.625 

169.23 

•875 

333-41 

1-25     i  750.45 

•75 

204 

i 

382.27 

1-375 

827.17 

•875 

239.07 

I.I25 

43I-4I 

65 

i 

646.93 

i 

274-45 

1.25 

480.89 

1.125 

729.18 

28 

.625 

175.36 

39 

•75 

292.21 

1.25 

811.73 

•75 

211-35 

•875 

341-97  i 

1-375 

894.6 

•875 

247.65 

i 

392.08  ! 

70 

i 

695.92 

i 

284.25 

1.125 

442.44 

1.25 

872.98 

29 

•625 

181.49 

1.25 

493-14  i 

1*5 

1051.25 

Equivalent  Length  of  Pipe  for  a  Socket. 
7  -J =  1.    d  representing  diameter  of  pipe  and  I  length  in  inchet. 

Additional  weight  of  two  flanges  for  any  diameter  is  computed  equal  to  a  line&J 
foot  of  the  pipe. 

NOTB.  — These  weights  do  not  include  any  allowance  for  spigot  and  socket  ends. 
2.— For  rule  to  compute  thicknesses  of  pipes,  flanges,  etc.,  see  page  560. 


134       WEIGHT    OF    STANDARD    ROLLED    STEEL    BEAMS. 


05  g 
0 

o  g 

S  * 

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lOiori-rororoPlNNiNwM  H  -e 

— — — — |  I 

'    ^  I 

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fe  9  IN 

8  S  s   | 

o5  J  ^i     o 

Q*H  10 


2J 


l  i 


illi^s^i  i  H  M  M  1 1   i 


^       loco  M  vo  oo   . 

S    M     M     10   «     ON  ^    | 


ld?d5  l  l  I  I  I  I  I  I  I  I  I  I  I  I 

£    C<     M     M 


1     1 


-£  iovO    t^OO   ^  O 


V  «0 

fQ  S 
V 

S  « 


0     8 

0 

4i        ^ 

?     i 

ft     ^ 

C          v. 

1 1 

0 

H 


WEIGHT  OF  ROLLED  STEEL,  SHEET  COPPER,  ETC.       135 

Weight    of  Round   Rolled.   Steel. 
From  .125  Inch  to  12  Inches  Diameter.    ONE  FOOT  IN  LENGTH. 


Diam. 

Weight. 

Diameter. 

Weight. 

Diameter.   Weight. 

Diam. 

Weight. 

Diam. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

.0417 

.875 

2.04 

1.625 

7-05 

2.875 

22 

5-75 

88.3 

•1875 

•0939 

•9375 

2-35 

1.6875 

7.6l 

3 

24 

.1 

6 

96.1 

•25 

.167 

i 

2.67 

1-75 

8.18 

3.25 

28.3 

6.5 

II3.2 

.26 

1.0625 

3 

I.8I25 

8.77 

3-5 

32 

7 

7       i  I3C-8 

•375 

•375 

1.125 

3.38 

1.875 

9.38 

3-75 

34-2 

7-5    1  136.8 

•4375 

•5" 

1.1875 

3.76 

2 

10.7 

4 

42 

•7 

8 

170.8 

•5 

.667 

1-25 

4.17 

2.125 

12 

4-25 

48.3 

8-5 

193.2 

•5625 

.845 

1.3125 

4.6 

2.25 

13.6 

4-5 

54.6 

9 

218.4 

.625 

1.04 

1.375 

5-05 

2-375 

I5.I 

4-75 

60.3 

9-5 

241.2 

•6875 

1.27 

1-4375 

2-5 

I6.7 

5 

66.8 

10 

267.2 

•75 

1 

6.01 

2.625 

18.4 

5.25 

73-6 

ii 

323 

.8125 

1^6 

1.5625 

6.52 

2-75 

20.  2 

5-5 

80.8 

12 

384-3 

Weiglit  of* 

Hexagonal,  Octagonal,  and    Oval    Steel. 

ONE   FOOT   IN   LENGTH. 

HEXAGONAL. 

OCTAGONAL. 

OVAL. 

Diam. 

Diam. 

Diam. 

Diam. 

over 

Sides. 

Weight. 

over 
Sides. 

Weight 

over 
Sides. 

Weight 

over 
Sides. 

We 

ght. 

Diam. 
over  Sides. 

Area. 

Weight. 

Inch. 

Lb». 

lns< 

Lbs. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Sq.In. 

Lbs. 

% 

.414 

I 

2-94 

% 

.396 

I 

2 

82 

%  X 

% 

.251 

•853 

.736 

*% 

3-73 

% 

.704 

1% 

356 

%  x 

5»| 

•344 

I.I7 

3 

I-I5 

1% 

4-6 

% 

I.I 

1% 

4.4 

!.; 

.446 

1.52 

% 

1.66 

I/8 

5-57 

% 

1.58 

1% 

5-32 

J/£  x 

% 

.697 

2-37 

%   2.25 

1% 

6.63 

% 

2.l6 

& 

6-34 

I^X 

% 

.884 

3 

Weight 

of  a    Square    Foot 

of*  Sheet   Copper. 

Wire  Gauge  of  Win.  Foster  $  Co.     (England.) 

Thickness. 

Weight. 

Thickness.              Weight. 

Thickness. 

Weight. 

w.o. 

Inch. 

Lbs. 

W.G. 

Inch. 

Lbs. 

W.G. 

Inch. 

Lbs. 

I 

.306 

I 

4 

II 

.123 

5.65 

21 

.034 

i-55 

a 

.284 

13 

12 

.109 

5 

22 

.029 

1-35 

3 

.262 

12 

13 

.098 

4-5 

23 

•025 

4 

.24 

II 

14 

.088 

4 

24 

.022 

i 

5 

.222 

10.15 

15 

.076 

3-5 

25 

.019 

.89 

6 

.203 

93 

16 

.065 

3 

26 

.017 

•79 

7 

,186 

8-5 

17 

•057 

2.6 

27 

.015 

•7 

8 

.168 

7-7 

18 

.049 

2.25 

28 

.013 

.62 

9 

.153 

7 

19 

.044 

2 

29 

.012 

.56 

10 

.138 

6-3 

20 

•038 

1-75 

30 

.Oil 

•5 

"Weight   of  Composition.   Sheathing   INTails. 


;  Number 

Number 

Number 

Number 

No. 

Length. 

in  a 

No. 

Length. 

in  a 

No. 

Length. 

in  a 

No. 

Length. 

in  a 

Pound. 

Pound. 

Pound. 

Pound. 

Inch. 

Ins. 

Ins. 

Ins. 

I 

•75 

290 

4 

I.I25 

201 

7 

I.I25 

I84 

10 

1.625 

101 

2 

.875 

260 

5 

1.25 

199 

8 

1.25 

168 

II 

1-75 

74 

3 

i 

212 

6 

100 

9 

no 

12 

2 

64 

136 


WEIGHT   OP   IRON,  STEEL,  COPPER,  ETC. 


Weight   of  Cast   and.    'Wrcmglit    Iron,   Steel,  Copper,  and 
Brass,  of  a   given    Sectional    Area. 

PER  LINEAL  FOOT. 


Sectional 
Area. 

Wrought 
Iron. 

Cast  Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun-metal 

Sq.  Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.1 

.336 

.313 

•339 

.385 

.492 

•357 

.38 

.2 

.67I 

.626 

.677 

.771 

.984 

•713 

•759 

•3 

I,OO7 

"939 

1.016 

1.156 

1.476 

1.07 

I.I39 

•4 

1-343 

I.25I 

J-355 

1.542 

1.967 

1.427 

L5I9 

•5 

1.678 

1.564 

1.694 

1.927 

2.461 

1.783 

1.894 

.6 

2.014 

1.877 

2.032 

2.312 

2-953 

2.14 

2.279 

.7 

2-35 

2.19 

2.371 

2.698 

3-445 

2.497 

2-658 

.8 

2.685 

2.503 

2.71 

3-083 

3-937 

2.853 

3.038 

•9 

3.021 

2.816 

3-049 

3-469 

4.429 

3.21 

3.418 

i 

3-357 

3.129 

3-387 

3-854 

4.922 

3-567 

3-798 

i.i 

3.692 

3.442 

3.726 

4.24 

5-4I4 

3-923 

4.177 

1.2 

4.028 

3-754 

4-065 

4.625 

5.906 

4.28 

4-557 

i-3 

4-364 

4.067 

4-404 

5-01 

6.398 

4-636 

4-937 

1.4 

4.699 

4-38 

4-742 

5.396 

6.89 

4-993        5-3J7 

J-5 

5-035 

4-693 

5.081 

5-781 

7.383 

5-35 

5-6o6 

1.6 

5-37i 

5.006 

5.42 

6.167 

7.875 

5-707 

6.076 

i-7 

5.706 

5-3I9 

5-759 

6.552 

8.367 

6.063 

6.456 

1.8 

6.042 

5-632 

6.097 

6-937 

8.859 

6.42 

6.836 

1.9 

6.378 

5-945 

6.436 

7-323 

9-351 

6-777 

7-215 

2 

6.714 

6.258 

6-775 

7.708 

9.843 

7.133 

7-595 

2.1 

7.049 

6-57 

7.114 

8.094 

10.33 

7-49 

7-97 

2.2 

7-385 

6.883 

7-452 

8.474 

10.83 

7.847 

8-35 

2-3 

7.721 

7.196 

7.791 

8.864 

11.32 

8.203 

8-73 

2.4 

8.056 

7-509 

8.13 

9-25 

n.8i 

8.56 

9.11 

25 

8.392 

7.822 

8.469 

9-635 

12.3 

8.917 

9-49 

26 

8.728 

8-135 

8.807 

10.02 

12.8 

9-273 

9.87 

2.7 

9.063 

8.448 

9.146 

10.41 

13.29 

963 

10.25 

2.8 

9-399 

8.76 

9-485 

10.79 

13.78 

9-98 

10.63 

2.9 

9-734 

9.073 

9.824 

ii.  18 

14.27 

10.34 

1  1.  01 

3 

10.07 

9-386 

10.  16 

11.56 

14.76 

10.7 

"•39 

3-i 

10.41 

9.699 

10.5 

".95 

15.26 

11.06 

11.77 

3-2 

10.74 

IO.OI 

10.84 

12.33 

!5.75 

11.41 

12.15 

3-3 

11.08 

10.32 

11.18 

12.72 

16.24 

11.77 

J2-53 

3-4 

11.41 

10.64 

11.52 

I3-I 

16.73 

12.13 

12.91 

3-5 

"•75 

10.95 

11.86 

13.49 

17.22 

12.48 

13.29 

3-6 

12.08 

11.26 

12.19 

13-87 

17.72 

12.84 

13-67 

3-7 

12.42 

11.58 

12.53 

14.26 

18.21 

13.2 

14.05 

3-8 

12.76 

11.89 

12.87 

14.64 

18.7 

13-55 

14-43 

3-9 

13.09 

12.2 

13.21 

15-03 

19.19 

J3-91 

14.81 

4 

13.43 

12.51 

13-55 

15.42 

19.69 

14.27 

15.19 

4.1 

I3-76 

12.83 

13.89 

15.8 

20.  1  8 

14.62 

15-57 

4.2 

14.1 

I3.J4 

14.23 

16.19 

20.67 

14-98 

15.95 

4-3 

1443 

13-45 

14.57 

16.57 

21.  l6 

15-34 

16.33 

4.4 

14.77 

13.77 

14.91 

16.96 

21.65 

15.69 

16.71 

4-5 

15-11 

14.08 

15.24 

!7«34 

22.15 

16.05 

17.09 

4.6 

15-44 

J4-39 

15-58 

!7-73 

22.64 

16.41 

17-47 

4-7 

15-78 

14.7 

15.92 

i8.ii 

23-13 

16.76 

17-85 

4.8 

16.11 

15.02 

16.26 

18.5 

23.62 

17.12 

18.23 

4.9 

16.45 

!5«33 

16.6 

18.88 

24.12 

17.48 

18.61 

5 

16.78 

1564 

16.94 

19.27 

24.61 

17-83 

18.99 

WEIGHT  OF  LEAD   AND  TIN  PIPE  AND  TIN  PLATES.        137 

"Weigh.!    of  Lead.   and.   Tin    Lined    IPipe    per    Foot. 

From  .375  Inch  to  5  Inches  in  Diameter.     (Tatham  fy  Bros.) 


Diam. 

WASTE 

Weight. 

-PIPE. 

Diam.  |  Weight. 

Diam. 

Weight.  ) 

BLOCK-l 
Diam. 

^IN   PIPE. 

Weight.    | 

Diam. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Inch. 

Lb. 

Inch. 

Lbs. 

Ins. 

Lbs. 

£ 

4 

8 

•375 

•3594 

-625 

•5 

1.25 

1.25 

2 

"  *3*   ; 

4-5 

6 

•375 

•375 

•625 

•625 

1.25 

1-5 

3 

3-5 

4-5 

8 

•375 

•5 

•75 

.625 

1-5 

2 

3 

5 

5 

8 

•5 

•375 

•75 

•75 

1'5 

2-5 

4 

5 

5 

IO 

•5 

.5 

i 

•9375 

2 

2-5 

4 

6 

5 

12 

•5 

.625 

i 

1.125 

2 

3 

WATER-PIPE. 

From  .375  Inch  to  5  Inches  in  Diameter. 


Diam. 

Thick- 
ness. 

Weight. 

Diam. 

Thick- 
ness. 

Weight. 

Diam. 

Thick- 
ness. 

Weight. 

Diam. 

Thick- 
ness. 

Weight. 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

•375 

.08 

.625 

•625 

•25 

3-5 

1.25 

.19 

4-75 

2-5 

•3125 

14 

•375 

.12 

I 

•75 

.1 

1.25 

1.25 

•25 

6 

2-5 

•375 

17 

•375 

.16 

1-25 

•75 

.12 

i-75 

i-5 

.12 

3 

3 

•1875 

9 

•375 

.19 

i-5 

•75 

.16 

2.25 

i-5 

.14 

3-5 

3 

•25 

12 

•375 

•34 

2-5 

•75 

.2 

3 

i-5 

•17 

4-25 

3 

•3125 

16 

•5 

.07 

•0545 

•75 

•23 

3-5 

i-5 

.19 

5 

3 

•375 

20 

•5 

.09 

•75 

•75 

•3 

4-75 

i-5 

•23 

6-5 

3-5 

•1875 

9-5 

•5 

.11 

i 

i;t 

j;i  ^ 

i-5 

i-5 

.27 

8 

3-5 

•25 

15 

•5 

•13 

1.25 

•2*1 

2 

i-75 

•13 

4 

3-5 

•3125 

18.5 

•5 

.16 

*-75 

.14 

2-5 

i-75 

•17 

5 

3-5 

•375 

22 

•5 

.19 

2 

•I? 

3-25 

i-75 

.21 

6-5 

•1875 

12.5 

•5 

•25 

3 

.21 

4 

i-75 

•27 

8-5 

4 

•25 

16 

.625 

.08 

.0727 

.24 

4-75 

2 

•15 

4-75 

4 

•3125 

21 

.625 

.09 

i 

•3 

6 

2 

.18 

6 

4 

•375 

25 

.625 

•13 

i-5 

•25 

.1 

2 

2 

.22 

7 

4-5 

•1875 

14 

•625 

.16 

2 

•25 

.12 

2-5 

2 

.27 

9 

4-5 

•25 

18 

•625 

.2 

2-5 

•25 

.14 

3 

2-5 

•1875 

8 

5 

•25 

20 

.625 

.22 

2-75 

•25 

.16 

3-75 

2.5 

•25 

ii 

5 

•375 

31 

JNlarks    and   Weight   of*  Tin-plates.    (English.) 


MARK 
OB  BRAND. 

Plates 
per  Box. 

Dimensions. 

Weight! 
per  Box. 

MARK 
OR  BRAND. 

Plates 
per  Box. 

Dimensions. 

Weight 
per  Box. 

No. 

1  88 
209 
230 
251 
272 

293 

112 
140 
112 
140 

168 
105 
126 

112 
126 

i  C  or  i  Com. 

2  C    

No. 
22.5 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 

100 
IOO 
IOO 
IOO 

Ins. 
13.75X10 
I3.25X  9-75 
i2.75X  9.5 
i3-75Xio 
i3-75Xio 
13.75X10 
I3-25X  9-75 
I2-75X  95 
i3-75Xio 
i3-75Xio 
i3-75Xio 
13.75X10 
13.75X10 
16.75X12.5 
16.75X12.5 
16.75X12.5 
16.7^X12.5 

No. 
112 

I02 
98 

II9 

157 
140 

133 
126 
161 

182 
203 
224 
245 

9l 
.126 

M7 
168 

DXXXX  
SDC  

No. 
100 
2OO 
200 
200 
2OO 
200 
200 
2OO 
112 
112 
225 
225 
200 
IOO 
IOO 
450 
4^0 

Ins. 
16.75X12.5 
15      Xi 
15      Xi 
15      Xi 
15      Xi 
15      Xi 
15      Xi 
15      Xi 

20        Xl 

20        Xl 

i3-75Xi 
i3-75Xi 
15      Xi 
16.75X1  .5 
16.75X1  .5 
13.75X10 
13.7^X10 

,  C 

SDX  .. 

HC'"::::' 

SDXX  

H  X  

SDXXX  
SDXXXX.  .  .  . 
SDXXXXX. 
SDXXXXXX. 
Leaded  1C... 
"      IX... 
ICW  

i  X 

2  X     

q  X  .. 

XX       .   . 

XXX  

xxxx.  .  .  . 
xxxxx  .  . 
xxxxxx. 

DC 

IXW  
CSDW  
CIIW  

DX  

XIIW  

DXX 

TT.  .  .  . 

DXXX.  . 

XTT.  .  .  , 

When  the  plates  are  14  by  20  inches,  there  are  112  in  a  box. 


138 


STEAM,    GAS,    AND    WATER    PIPES. 


Iron.   and.    Steel   Welded   Steam.,   Gras,  and   'Water 
Pipes.       STANDARD   DIMENSIONS.     National  Tube  Co. 


D 

Nomi- 
nal  In 
ternal. 

iamete 

Act'a 
Ex- 
t'nal 

r. 

Act'a 
In- 
t'nal 

|  Thickness. 

Circ 
ei 

Ex- 
t'nal 

umfer- 
ce. 

Inter 
nal. 

Tran 

Ex- 
ternal 

averse  A 

In- 
ternal 

reas. 
Metal 

Len{ 
Sq. 
Su 

Ex- 

tvi 

Cth  per 
?oot  o 
rface. 

t'nal. 

Pi 

.=  ..*• 

11 

Nominal 
Weight  per 
Foot. 

a  a  % 
ftl 

°  *"  2 

Ins. 
•  I21 

Ins. 

Ins. 

Ins 

•  °7 

Ins. 
1.27 

Ins. 
.85 

Sq.  Ins 
•*3 

Sq.  Ins 

06 

S.Ins 

Ft. 
9  43 

Ft. 

Ft. 
2513.1 

Lbs. 
.  24 

Int. 

•25 

•54 

•36 

.09 

'•7 

i.i- 

.2' 

.1 

.12 

7.07 

10.49 

1383-1 

.42 

•25 

-375 

•67 

-49 

.09 

2.12 

i-55 

•36 

.19 

.17 

5-68 

7-73 

751-2 

-56 

•375 

-5 

.84 

.62 

.  ii 

2.64 

1.96 

•55 

-3 

.25 

4-55 

6.13 

472-4 

.84 

•5 

•75 

1.05 

.82 

ii 

3-3 

2-59 

.87 

•53 

•33 

3-64 

4.63 

270 

1.  12 

•75 

i 

I*3I 

1.05 

13 

4-13 

3-29 

1-36 

.86 

•49 

2-9 

3-64 

166.9 

1.67 

i 

1-25 
'•5 

1.66 
1.9 

1.38 
1.61 

M 
M 

5-21 

5-97 

4-33 
5-o6 

2.16 
2-83 

2.04 

t 

2-3 
2.OI 

2-77 
2-37 

70.66 

2.24 

2.68 

1.25 

2 

2-37 

2.07 

7.46 

6.49 

4-43 

3-36 

1.07 

x.6i 

1.85 

42.91 

3-6i 

2 

2-5 

2.87 

2-47 

2 

9-°3 

7-75 

6-49 

4-78 

1.71 

i-33 

i-55 

30.1 

5-74 

2-5 

3 
3-5 

3-5 
4 

3-07 
3-55 

22 
23 

ii 

12.57 

11.15 

9.62 
'2-57 

7-39 
9.89 

2.23 
2.68 

1-09 
•95 

13 

19.49 
14.56 

7-54 
9 

3 

3-5 

4 

4-5 

4-03 

24 

14.14 

12.65 

iS-9 

12.73 

3-  17 

•95 

ii.  31 

10.66 

4 

4-5 

5 

25 

14-16 

19.63 

15.96 

3-67 

.76 

•85 

9.02 

12.49 

4-5 

5 

5-56 

5-04 

26 

17^8 

15-85 

24.31 

4-32 

.69 

-76 

7-2 

H-5 

5 

6 

6.62 

6.06 

28 

20.81 

19.05 

34-47 

28.85 

5-58 

.58 

-63 

4.98 

18.76 

6 

7 

7.62 

7.02 

3 

23-95 

22.06 

45-66 

38.74 

6-93 

•5 

•54 

3-72 

23-27 

7 

8 

8.62 

7-98 

32 

27.1 

25.08 

58-43 

50.04 

8-39 

•44 

.48 

2.88 

28  18 

8 

9 

9.62 

8-94 

34 

30.24 

28.08 

72.76 

62.73 

10.3 

•4 

-43 

2-3 

33-7 

9 

10 

IO-75 

O.O2 

37 

33-77 

31.48 

90.76 

78.84 

11.92 

•35 

.38 

1.83 

40 

0 

ii 

"•75 

I 

37 

36.91 

14.56 

108.43 

95-03 

13.4 

•32 

-35 

1-52 

45 

ii 

12 

I2-75 

2 

37 

0.05 

37-7 

127.68 

113.1 

14.58 

•3 

•32 

1.27 

49 

2 

3-25 

37 

43.98 

41-63 

153-94 

137.89 

16.05 

•27 

.29 

1.05 

53-89 

— 

'5 

14.25 

37 

47.12 

44-77 

176.71 

159.48 

17-23 

•25 

•27 

•9 

57-8i 

— 

16 

15-25 

— 

37 

50-27 

47.91 

201.06 

182.65 

18.41 

.24 

•25 

•79 

6i-77 

— 

18 

'7-25 

— 

37 

56.55 

54-19 

254-47 

233-78 

20.76 

.21 

.22 

.62 

69.66 

— 

20 

'9-25 

— 

37 

62.83 

60.48 

114.  16 

291.04 

23.12 

.19 

.2 

-49 

77-57 

— 

22 

21.25 

—  . 

37 

69.11 

66.76 

380.  13 

354-66 

25-4 

•  17 

.18 

•4 

35-47 

— 

24 

23.25 

— 

37 

75-4 

3-°4 

452.39 

424.  56 

27.83 

.16 

.16 

•34 

93-37 

— 

26 

25-25 

•  — 

37 

81.68 

9-32 

30-93 

00.74 

30.  19 

•  15 

.15 

.29 

02 

— 

28 

27.12 

— 

44 

7.96 

5-22 

77-87 

37-88 

.14 

.14 

•25 

27-34 

— 

30 

29 

— 

5 

34-25 

I.  II 

06^86 

60.52 

46.34 

•  13 

•13 

.22 

56 

— 

.Lap-welded  Steel,  Semi-Steel,  Special  Locomotive 
and  Franklinite  Boiler  Tubes. 

STANDARD  DIMENSIONS.    National  Tube  Co. 


Dia 

Ex- 
t'ual. 

meter. 

Inter- 
nal. 

|i 

£& 
*3 

Circum 

Exter- 
nal. 

ference. 

Inter- 
nal. 

Trai 

Exter- 
nal. 

isverse  Ar 

Inter- 
nal. 

eas. 
Metal. 

Length 
Foot  of 
Exter- 
nal. 

perSq. 
Surface 
Inter- 
nal. 

111 
fffi 

*n 

!l 

£Q 

l^sT 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.Ins 

Feet. 

Feet. 

Lbs. 

Ins. 

i 

-834 

.083 

4 

3»I42 

2.62 

.785 

.546 

•239 

3-82 

4-58 

.81 

i 

1.25 

1.084 

-083 

4 

3-927 

3-405 

1.227 

•923 

•304 

3-056 

3-524 

1.02 

1-25 

'•5 

I-3I 

•095 

3 

4.712 

4-iiS 

1.767 

1.348 

.419 

2-547 

2.916 

1.4 

i-5 

'•75 

1-532 

.109 

2 

5.498 

4.8i3 

2.405 

1.843 

.|62 

2.183 

2-493 

1.87 

i-75 

2 

1.782 

.109 

2 

6.283 

5-598 

3.142 

2-494 

.648 

1.91 

2.144 

2.17 

2 

2.25 

2.032 

.109 

2 

7.069 

6.384 

3-976 

3-243 

•733 

1.698 

1.88 

2-45 

2.25 

2-5 

2.26 

.12 

I 

7.854 

7.1 

4.909 

4.011 

•897 

1.528 

1.69 

3 

2-5 

2-75 

2.51 

.12 

I 

8.639 

7.885 

5«94 

49.48 

.992 

1-389 

1.522 

3-3i 

2-75 

3 

2.76 

.12 

I 

9-425 

8.671 

7.069 

5-983 

1.086 

1-273 

1.384 

3-63 

3 

3  25 

2.982 

-134 

0 

10.21 

9.368 

8.296 

6.984 

1-312 

I-I75 

1.281 

4-39 

3-25 

3-5 

3-232 

•134 

O 

10.996 

10.154 

9.621 

8.204 

1.417 

1.091 

1.182 

4-74 

3-5 

3-75 

3482 

•134 

o 

11.781 

10.939 

11.045 

9.522 

1.523 

1.019 

1.097 

5-09 

3-75 

4 

3-7°4 

.I48 

9 

12.566 

11.636 

12.566 

io-  775 

K.OXD 

•955 

1.031 

6 

4 

NOTE  i  —For  diameters  from  13  up  to  and  including  30  ins.  O.  D.,  details  are 

n  conformance  with 

the  circumstances,  as  there  is  not  a  standard,  the  thickness  varying.     NOTB  2.—  In  estimating  effective 

heating  or  evaporating  surface  of  tubes,  as  heating  liquids  by  steam,  superheating  steam,  or  trans- 
ferring heat  from  one  liquid  or  one  gas  to  another,  mean  surface  of  tubes  is  to  be  computed. 

IKON    BOILER   TUBES. 


Lap  -  welded. 


Charcoal     Iron. 
Tubes. 


andx    Steel 


STANDARD  DIMENSIONS. 
National  Tube  Co. 


Diarr 

Exter- 
nal. 

eter. 

Inter, 
nal. 

Thickness. 

y 

Circ 
en 

Exter 
nal. 

umfer- 
ce. 

Inter- 
nal. 

Tran 

Exter- 
nal. 

averse  A 

Inter. 

nal. 

reas. 
Metal. 

Leng 
Squar 
of  Su 
Exter- 
nal. 

thper 
9  Foot 
rface. 
Inter- 

i! 

External 
Diameter. 

Ins. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.Ins 

Ft. 

Ft. 

Lba. 

Ins. 

.86 

.072 

'5 

3-14 

2.69 

.78 

•57 

.21 

3-82 

4.463 

•71 

.125 

.98 

.072 

15 

3-53 

3.08 

•99 

•75 

.24 

3.396 

.8 

•125 

•25 

i.  ii 

.072 

15 

3-93 

3-47 

1.23 

.96 

-27 

3-056 

3-453 

.89 

•25 

1.15 

.083 

14 

4.12 

3.6 

1.35 

1.03 

-32 

.911 

3-333 

.08 

-312 

•375 

I.  21 

.083 

14 

4-32 

3-8 

1.48 

•34 

-778 

3.16 

.13 

-375 

•  5 

I*3' 

.083 

M 

4.71 

4.19 

1.77 

1.4 

•37 

•547 

2.863 

•24 

.5 

.625 

1.4; 

.095 

13 

2.07 

1.62 

.46 

•352 

2.662 

•53 

.625 

•75 

1.56 

•095 

13 

5-5 

4-9 

2-4 

1.91 

•49 

•183 

2-448 

.66 

•75 

•875 

1.68 

•095 

13 

5-89 

5-29 

2.76 

2.23 

•53 

•037 

2.267 

.78 

-875 

1.81 

•095 

'3 

6.28 

5-69 

3-M 

2.57 

•57 

.91 

2.  II 

.91 

.125 

1.9" 

.095 

6.68 

6.08 

3«55 

2-94 

.61 

•797 

1.974 

.04 

•125 

•25 

2.06 

•°95 

13 

7.07 

6-47 

3.98 

3-33 

.64 

.698 

1.854 

.16 

.25 

•375 

2.16 

.109 

12 

7-46 

6.78 

4-43 

3-65 

.78 

.608 

I.77I 

.61 

•375 

•  5 

2.28 

.109 

12 

7.85 

7.17 

4.91 

4.09 

.82 

.528 

1.674 

•75 

•  5 

2-53 

.109 

12 

8.64 

7-95 

5-94 

5-03 

•9 

•389 

1.508 

3.04 

•75 

•875 

2.66 

.  109 

12 

9-°3 

8-35 

6-49 

5-54 

•95 

•329 

1.438 

3.18 

•875 

3 

2.78 

.  109 

12 

9-42 

8.74 

7.07 

6.08 

•99 

•273 

1-373 

3-33 

3 

3-25 

3.01 

.12 

II 

10.21 

9.46 

8-3 

7.12 

1.18 

•'75 

1.269 

3.96 

3-25 

3-5 

3-26 

.12 

I 

IX 

10.24 

9.62 

8.35 

1.27 

.091 

I.I72 

4.28 

3-5 

3-75 

3-51 

.12 

I 

11.78 

11.03 

11.04 

9.68 

'•37 

.019 

1.  088 

4.6 

3-75 

4 

3-73 

-134 

0 

12.57 

11.72 

12.57 

10.94 

1.63 

•955 

1.024 

5-47 

4 

4-25 

3.98 

.'34 

O 

13-35 

12.51 

14.19 

12.45 

i-73 

•899 

'959 

5.82 

4-25 

4-5 

4.23 

•»34 

0 

14.14 

13-29 

'5-9 

14.07 

1.84 

.849 

-9°3 

6.17 

4-5 

4-75 

4.48 

•'34 

10 

14.92 

14.08 

17.72 

15.78 

1.94 

.804 

.852 

6-53 

4-75 

5 

4-7 

.148 

9 

15.71 

14.78 

19.63 

17.38 

2.26 

•764 

.812 

7.58 

5 

5-125 

4-95 

.148 

9 

16.49 

15-56 

21.65 

19.27 

2-37 

.728 

.771 

7-97 

5-25 

5.25 

5-2 

.148 

9 

17.28 

16.35 

23.76 

21.27 

2-49 

•694 

•734 

8.36 

5-5 

6 

5.67 

.165 

8 

18.85 

17.81 

28.27 

25-25 

3.02 

•637 

•674 

10.16 

6 

8 

6.67 
7-67 

.165 
.165 

8 
8 

21.99 
25.13 

20.95 
24.1 

38.48 
50.27 

34-94 
46  2 

3-J4 
4.06 

•546 
•477 

•573 
.498 

xx.o 

13.65 

7 
8 

9 

8.64 

.18 

7 

28.27 

27.14 

63.62 

58.63 

4-99 

.424 

.442 

16.76 

9 

o 

9-59 

203 

6 

31.42 

30.14 

78.54 

72.29 

6.25 

•  382 

20.99 

10 

i 

10.56 

22 

5 

34-56 

33-17 

95.03 

87.58 

7-45 

•347 

.362 

25.03 

iz 

2 

"•54 

229 

4-5 

37-7 

36.26 

113.1 

104.63 

8-47 

.318 

•33 

28.46 

12 

3 

12.52 

238 

4 

40.84 

39-34 

132.73 

123.19 

9-54 

•294 

•3°5 

32.06 

13 

4 

13.5 

248 

3-5 

43-98 

42-42 

153-94 

143.22 

10.71 

•273 

•283 

36 

14 

5 

14.48 

259 

3 

47.12 

45-5 

176.71 

164.72 

11.99 

•255 

.264 

40.3 

6 

15.46 

27I 

2-5 

50.27 

48.56 

201.06 

187.67 

X3'39 

•239 

•247 

45-2 

1  6 

8 

'7-43 

284 

2 

56.55 

54.76 

254.47 

238.66 

15.81 

.212 

.219 

52.87 

18 

20 

19.38 

3I2 

.31 

62.83 

60.87 

314.16 

294.86 

i9.3 

.191 

.197 

64.84 

20 

22 

21.31 

343 

•03 

69.11 

66.96 

380.13 

356.8 

23-34 

.174 

.179 

78.5 

22 

24 

23.25 

375 

•37 

75-4 

73-04 

452.39 

424.56 

27.83 

•'59 

.164 

93-37 

24 

26 

25.25 

375 

•37 

81.68 

r9.32 

530.93 

500.74 

30.19 

.147 

•'S1 

102 

26 

28 
30 

27.25 
29.25 

375 
375 

•37 
•37 

87.96 
94-25 

91.89 

706.'  86 

583.21 
671.96 

32.54 
34.9 

.136 
.127 

.14 

no 
118 

28 
|O 

NOTE  i. — For  diameters  from  13  up  to  and  including  30  ins.  O.  D.,  details 
are  in  conformance  with  the  circumstances,  as  there  is  not  a  standard,  the 
thickness  varying. 

NOTE  2.— In  estimating  effective  heating  or  evaporating  surface  of  tubes* 
as  heating  liquids  by  steam,  superheating  steam,  or  transferring  heat  from 
one  liquid  or  one  gas  to  another,  mean  surface  of  tubes  is  to  be  computed. 

M* 


140 


WEIGHT    OF    COPPER    TUBES. 


"Weiglit   of  Seamless   Drawn    Copper   Tiabes. 

American    Tu.be    "Works.     (Boston.) 

BY   EXTERNAL   DIAMETER.      ONE   FOOT  IN    LENGTH. 

Stubs1  W.  G.     From  .25  Inch  to  12  Ins.— f full,  I  light. 


No. 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

Ins. 

'/W 

3/64* 

3/64  / 

Vi6  / 

x/x6/ 

5/64l 

5/64  / 

3/32  / 

7/64 

i/8* 

Diamet'r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs.    |     Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbr. 

•25 

.09 

.1 

.12 

•13 

.14 

•15 

•17 

.18 

.19 

.19 

•375 

.14 

.16 

.19 

•23 

.24 

.26 

.29 

•32 

•35 

•37 

•5 

.2 

•23 

.27 

•31 

•34 

•37 

.42 

•47 

•52 

•56 

.625 

•25 

.29 

•34 

•4 

•44 

.48 

•55 

.61 

.69 

•74 

•75 

•3 

•36 

.42 

•49 

•54 

•59 

.67 

.76 

-85 

.92 

•875 

•36 

.42 

•49 

•58 

.64 

•7 

.8 

•9 

i.  02 

i.  ii 

i 

.41 

.48 

•57 

.67 

•74 

.81 

•93 

1.05 

1.18 

1.29 

1.125 

.46 

•55 

.64 

.76 

•83 

.92 

1.05 

1.19 

i-35 

1.47 

1.25 

•52 

.61 

•7i 

.84 

•93 

1.03 

1.18 

i-34 

1.52 

1-65 

1-375 

•57 

.68 

•79 

•93 

1.03 

1.14 

i-3i 

1.48 

1.68 

1.84 

i-5 

.62 

•74 

.86 

1.02 

J-i3 

1.25 

i-43 

1.63 

1.85 

2.02 

1.625 

.68 

.8 

•94 

I.  II 

-23 

1.36 

1.56 

1.77 

2.02 

2.2 

!-75 

•73 

.87 

1.  01 

1.2 

•33 

1.47 

1.69 

1.92 

2.18 

2-39 

1-875 

.78 

•93 

1.09 

1.29 

•43 

1.58 

1.81 

2.06 

2-35 

2-57 

2 

.84 

.16 

i-37 

•53 

1.69 

1.94 

2.21 

2.51 

2-75 

2.125 

.89 

.06 

.24 

.46 

•63 

1.8 

2.07 

2-35 

2.68 

2-93 

2.25 

•94 

•13 

•3i 

•55 

•73 

1.91 

2.19 

2-5 

2.85 

3.12 

2-375 

.19 

•39 

.64 

.82 

2.02 

2.32 

2.64 

3.01 

3-3 

2-5 

•05 

•25 

.46 

•73 

1.92 

2.13 

2-45 

2.79 

3.l8 

3.48 

2.625 

.1 

•32 

•54 

.82 

2.02 

2.23 

2-57 

2-93 

3-35 

3>67 

2-75 

.16 

•38 

.61 

•9 

2.12 

2-34 

2.7 

3.08 

3-5i 

3-85 

2.875 

.21 

•45 

1.68 

•99 

2.22 

2-45 

2.83 

3.22 

3.68 

4-03 

3 

.26 

•5i 

1.76 

2.08 

2.32 

2.56 

2-95 

3-37 

3-84 

4.22 

3-25 

•37 

.64 

1.91 

2.26 

2.52 

2.78 

3.21 

3.66 

4.18 

4.58 

3-5 

.48 

•77 

2.06 

2-43 

2.72 

3 

3'46 

3-95 

4-5i 

4-95 

3-75 

•58 

•9 

2.21 

2.61 

2.92 

3-22 

3-71 

4.24 

4.84 

5-31 

4 

.69 

2.02 

2.36 

2.79 

3-n 

3-44 

3-97 

4-53 

5-i7 

5-68 

4-25 

.8 

2.15 

2.51 

3-14 

3-3i 

3-66 

4.22 

4.82 

5-5i 

6.05 

4-5 

1.9 

2.28 

2.65 

6-32 

3-5i 

3-88 

4-47 

5-" 

5-84 

6.41 

4-75 

2.01 

2.4I 

2.8 

3-49 

3-7i 

4.1 

4-73 

5-4 

6.17 

6.78 

5 

2.12 

2-54 

2-95 

3-67 

3-9i 

4-32 

4.98 

5-69 

6-5 

7.14 

5-25 

2.23 

2.66 

3-i 

3.85 

4.11 

4-54 

5-23 

5-98 

6.84 

7-51 

5-5 

2-34 

2.79 

3-25 

3.85 

4-3 

4.76 

5-49 

6.27 

7.17 

7.87 

5-75 

2-44 

2.92 

3-4 

4.02 

4-5 

4.98 

5-74 

6.56 

7-5 

8.24 

6 

2-55 

3-05 

3-55 

4.2 

4-7 

5-2 

5-99 

6.85 

7-83 

8.61 

6.25 

2.66 

3-i8 

3-7 

4.38 

4.9 

5-4i 

6.25 

7.14 

8.17 

8-97 

6.5 

2.76 

3-3i 

3-85 

4.55 

5-i 

5-63 

6.5 

7-43 

8-5 

9-34 

6-75 

2.87 

3-44 

4 

4-73 

5-3 

5-85 

6-75 

7.72 

8.83 

9-7 

7 

2.98 

3.56 

4-15 

4.91 

5-49 

6.07 

7.01 

8.01 

9.16 

10.07 

7-25 

3-09 

3-69 

4-3 

5-09 

5-69 

6.29 

7.26 

8.30 

9-5    i  10.44 

7-5 

3-19 

3-82 

4-45 

5.26 

5-89 

6.51 

7-51 

8-59 

9.83     10.8 

8 

341 

4.08 

4-74 

5-62 

6.29 

6-95 

8.02 

9.17 

10.49     "-53 

8-5 

3-62 

4-33 

5-04  1  5-97 

6.68 

7-39 

8.52 

9-75 

ii.  16  ,  12.26 

9 

3-83 

4-59 

5-34    6.33 

7.08 

7-83 

9-03 

10-33 

11.82 

i3 

9-5 

4-05 

4-85 

5.64    6.68 

7.48 

8.26 

9-54 

10.91 

12.49 

*3-73 

10 

4.26 

5-11 

5-94    7-03 

7.87 

8-7 

10.05 

11.49 

13.15     14.46 

10.5 

4-47 

5-37 

6.24    7.39 

8.27 

9.14 

10.55 

12.07 

13.82     15.19 

ii 

4.69 

5-62 

6-54  !  7-74 

8.67      9.58 

11.06 

12.65 

14.48  !  15.92 

"•5 

4.9 

5-88 

6.84  -8.1 

9.06       10.02 

11.56 

13-23 

15.15     16.66 

12 

5-" 

6.13 

7-13  !  8-45 

9.46      IO.45   '    12.07 

13.81 

15.81     17.29 

WEIGHT   OF   COPPER  TUBES. 


141 


No. 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

Ins. 

9/641 

9/64  / 

»/64  / 

3/i6  / 

13/64 

7/32  / 

x5/64  / 

'/4/ 

9/32  / 

'9/64  / 

Dumet'r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

•4 

.41 

.42 

.44 

— 

— 

— 

— 

— 

— 

•5 

.61 

.64 

.67 

-71 

•73 

•75 

.76 

— 

— 

— 

.625 

.81 

.86 

.92 

•99 

1.04 

1.09 

1.  12 

I-I3 

1.18 

— 

•75 

I.OI 

1.09 

I.I7 

1.26 

i-35 

1.42 

1.49 

1-53 

1.61 

1.63 

.875 

1.22 

i-3i 

1.42 

i-53 

1.66 

1.76 

1.85 

1.92 

2.04 

2.09 

i 

1.42 

1-54 

1.67 

1.81 

i-97 

2.09 

2.21 

2.32 

2.48 

2-55 

1.125 

1.63 

1.78 

i-93 

2.08 

2.28 

2-43 

2.58 

2.71 

2.91 

3 

1.25 

1.83 

2 

2.18 

2.36 

2-59 

2.76     2.94 

3-" 

3-34 

3-46 

J-375 

2.03 

2.22 

2-43 

2.63 

2.9 

3-i 

3-3 

3-5 

3-77 

3-92 

i-5 

2.24 

2.44 

2.68 

2.91 

3.21 

3-43 

3-67 

3-9 

4.21 

4-38 

1.625 

2.44 

2.67 

2-93 

3-i8 

3.52 

3-77 

4-03 

4.29 

4.64 

4.83 

i-75 

2.65 

2.89 

3.18 

3-45 

3.83 

4.11 

4-39 

4.69 

5-07 

5-29 

1.875 

2.85 

3.12 

3-44 

3-73 

4.14 

4-44 

4.76 

5-o8 

5-51 

5-75 

2 

3-06 

3-34 

3-69 

4 

4-45 

4-78 

5-12 

5-48 

5-94 

6.21 

2.125 

3.26 

3-57 

3-94 

4.28 

4-75 

5-" 

5-48 

5-87 

6-37 

6.66 

2.25 

3.46 

3-8 

4.19 

4-55 

5.06 

5-45 

5-84 

6.27 

6.81 

7.12 

2-375 

3«67 

4.02 

4.44 

4.82 

5-37 

5-78 

6.21 

6.66 

7.24 

7-57 

2-5 

3.87       4-25!     4-69 

5-i 

5-68 

6.12 

6-57 

7.06 

7.67 

8.04 

2.625 

4.08 

4-47!    4-95 

5-37 

6 

6-45 

6-93 

7-45 

8.1 

8.49 

2-75 

4.28 

4-7       5-2 

5-65 

6.3 

6.79 

7.29 

7-85 

8-54 

8.95 

2.875 

4.48 

4.92 

5-45 

5-92 

6.61 

7.12 

7.66 

8.24 

8-97 

9.41 

3 

4.69 

5-i5 

5-7 

6.2 

6.92 

7.46 

8.02 

8.64 

9.4 

9.87 

3-25 

5-i 

5-6 

6.2 

6.74 

7-54 

8.13 

8.75 

9-43 

10.27 

10.78 

3-5 

5-51 

6.05 

6.71 

7.29 

8.16 

8.8 

9-47 

IO.22 

11.14 

11.7 

3-75 

5-9i 

6-5 

7.21 

7-84 

8.78 

9-47 

10.2 

II.OI 

12 

12.61 

4 

6.32 

6-95 

7.71 

8-39 

9-4 

10.14 

10.92 

II.8 

12.87 

I3«53 

4-25 

6-73 

7-4 

8.22 

8-94 

IO.O2 

10.81 

11.65 

12.59 

13.73 

14.44 

4-5 

7.14 

7-85 

8.72 

9-49 

10.64 

11.48 

12.37 

13.38 

14.6 

'5.36 

4-75 

7-55 

8-3 

9.22 

10.04 

11.26 

12.  16 

J3-i 

14.17 

15.46 

16.27 

5 

7-96 

8-75 

9-73 

10.58 

11.88 

12.83 

13-83 

14.96 

16.33 

17.19 

5-25 

8.36 

9-2! 

10.23 

11.13 

12.49 

13-5 

J4-55    !5-75 

17.2 

18.1 

5-5 

8.77 

9.66 

10.73 

11.68 

13.11 

14.17 

15.28    16.54 

18.06 

19.02 

5-75 

9.18 

IO.II 

11.24 

12.23   13.73 

14.84 

16 

17-33 

18.93 

T993 

6 

9-59 

10.56 

11.74 

12.78    14.35 

i5-5i 

16.73 

18.12    19.79 

20.85 

6.25 

10 

II.OI 

12.24 

!3-33    14-97 

16.18  i  17.46    18.91 

20.66 

21.76 

6-5 

10.41 

11.46 

12.75 

,3.88 

15.59    l6-85 

18.18    19.7 

21-53 

22.68 

6-75 

10.82 

11.91 

13-25 

14.42 

16.21 

I7-52 

18.91    20.49 

22.39 

23-59 

7 

11.22 

12.36 

13-75 

14.97 

16.83 

18.19 

19.63   21.28 

23.26 

24.51 

7-25 
7-5 

11.63 
12.04 

12.81 
13.26 

14.26 
14.76 

15-52 
16.07 

17.45    J8.86 
18.07    19-54 

20.36   22.07   24.13 
21.08   22.86   25 

25.42 
26.34 

7-75 

12.45 

13.71  15.26 

16.62 

1  8.68   20.21 

21.  81    23.65 

25.86 

27-25 

8 

12.86    14.17  i  15.77 

17.17 

19.3     20.88 

22.54  24.44 

26.72 

28.17 

8.25 

13.27    14.62    16-27 

17.71 

19.92   21.55 

23.26  25.23 

27-59 

29.08 

8-5 

13.67    15.07 

16.77 

18.26 

20-54     22.22 

23.99  26.02 

28.45 

30 

8-75 

14.08  :  15.52     17.28 

18.81 

21.  16   22.89   24.71    26.81 

29.32 

30.91 

9 

14.49    15-97  !  17-78 

19.36 

21.78     23.56     25.44     27.6 

30.18 

31-83 

9-25 
9-5 
9-75 

10 

10.5 

14.9 
i5-3i 
J5-72 
16.12 
16.94 

16.42   18.28 
16.87    l8-79 
17.32    19.29 
17.77    19-79 
18.68    20.8 

19.91 
20.46 

21.01 

2L55 
22.65 

22.4        24.23 
23.02     24.9 
23.64     25.57 
24.26     26.24 
25-5        27.59 

26.17 

26.89 

27.62 

28.34 
29.79 

28.39  31.05 
29.18  31.92 
29.97  !  32.78 
30.76  1  33.65 
32.34  35.38 

32'Z2 

33-66 
34-57 
35-49 
37-32 

ii 

17.76    19.58   21.81 

23.75)26.73     28.93 

31-25 

33-92 

37-11 

39.15 

"•S 

18.57  1  20.48     22.  8l 

24.84 

27.97     30.27 

32.7 

35.5 

38.84 

40.98 

12 

19.39     21.38     23.82 

25-94 

29.21     3I.6l 

34.15 

37.08 

40.58  !  42.81 

142       WEIGHT    OF    COPPER    AND    BRASS    TUBES,   ETC. 


By   Internal   Diameter. 

Add  following  Units  to  Weights  for  External  Diameter  in  preceding  tables. 

No. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

2.21 

1.97 

1.66 

1.38 

1.18 

1.  01 

.78 

.67 

•53 

•43 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

•35 

.29 

.22 

•I? 

•13 

.11 

.08 

.06 

•05 

•03 

ILLUSTRATION.— What  is  weight  of  a  copper  tube  6  ins.  in  internal  diameter, 
No.  3  gauge,  and  one  foot  in  length  ? 

By  preceding  table  6  ins.  external,  No.  3  gauge  =  18.12,  and  18.12  +  1.66  = 
19.78  Ibs.  

WEIGHT    OF   BRASS    TUBES. 
To    Compute   "Weight   of*  Brass   Tiobes. 

American    Tnt>e  \Vorlis.    (Boston.) 
RULE. — Deduct  5  per  cent,  from  weight  of  Copper  tubes. 
EXAMPLE.  —  What  is  weight  of  a  brass  tube  6  ins.  in  external  diameter,  No.  3 
gauge,  and  one  foot  in  length  ? 
By  preceding  table  6  ins.  =  18.12,  from  which  deduct  5  per  cent.  =  17.21  Ibs. 

By   Internal   Diameter. 

RULE. — Proceed  as  above  for  internal  diameter,  and  deduct  5  per  cent. 
EXAMPLE.  —  Weight  of  a  copper  tube  6  ins.  internal  diameter,  No.  3  gauge,  and 
i  foot  in  length  —  19.78  Ibs. 

Hence,  19.78  —  5  per  cent.  =  18.79  ^s- 

NOTE. —  Diameter  of  Tubes,  as  for  Boilers,  is  given  externally,  and  that  for  Pipes 
internally. 

Weights  of  English  are  essentially  alike  to  the  preceding.    (D.  K.  Clark.") 

Seamless    Brass    Pipe. 

American.    Txi"be    "Worlrs.     (Boston.) 

Made  to  correspond  with  Iron  Pipe  and  to  Jit  Iron  Pipe  fittings. 


Diameters. 

3s« 

Diameters. 

|j| 

Diameters. 

f  1.8 

Same 
as  Iron 
Pipe. 

Ex 

Inter- 
nal. 

act 
Exter- 
nal. 

i 

Same 
as  Iron 
Pipe. 

Ex 

Inter- 
nal. 

act 
Exter- 
nal. 

P 

Same 

Ex 

Inter- 
nal. 

act 
Exter- 
nal. 

if! 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins 

Lbs. 

Ins 

Ins. 

Ins. 

Lbs. 

X 

.281 

.405 

•25 

1.368 

1.66 

2-5 

4 

4 

4-5 

12.7 

3^ 

•375 

•54 

•43 

1% 

1.6 

1.9 

3 

4% 

45 

5 

% 

.484 
.625 

•675 
.84 

.62 
•9 

2 

2.062 

2-5 

2-375 
2.875 

4 

5-75 

5 
6 

5.062 
6.125 

6^625 

15-75 
18.31 

X 

.808 

1.05 

1.25 

3 

3.062 

3-5 

8-3 

— 

— 

— 

— 

i 

1.062 

I-3I5 

1  7 

3>£ 

3-5 

4 

10.9 

— 

— 

— 

— 

amless  Copper  Pipe  of  like  diameter  is  5  per  cent,  heavier 
"Weight   of  Sheet   Brass. 
ONE  SQUARE  FOOT.     (HoltzapJfeVs  Gauge.) 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

No. 

3 
4 

6 

1 

Inch. 
•259 
.238 
.22 
.203 
.18 
.165 

Lbs. 
10.9 
10 
9.26 
8-55 
7.58 
6-95 

No. 

9 
10 
ii 

12 
13 
H 

Inch. 
.148 

•134 
.12 
.109 
•095 
.083 

Lbs. 

6.23 
5-64 
5-05 
4-59 
4 
3-49 

No. 
15 

16 

i? 
18 

19 

20 

Inch. 
.072 
.065 
.058 
.049 
.042 
•035 

Lbs. 
3-03 
2.74 
2.44 
2.06 
1.77 
1.47 

No. 
21 
22 

23 
24 

25 

Inch. 
.032 
.028 
.025 
.022 
.02 

Lbs. 

i-35 
1.18 
1.05 
.926 
.842 

WEIGHT   OF   WEOUGHT   IKON   TUBES. 


143 


"Weight   of  "Wrought   Iron   Tiobes.    (English.) 

EXTERNAL   DIAMETER.      ONE    FOOT   IN    LENGTH. 

HoltzapJeVs  Wire-Gauge,    ffull,  I  light. 


No. 

~            —             4 

5 

6 

7 

8 

9 

Ins. 

.3125         .281          .238 

.22 

.203 

.18 

.165 

.148 

5/i6            9/32         15/64  / 

7/32 

I3/64 

3/i6  / 

"/64  / 

9/64  / 

Diam. 

Lbs.             Lbs.             Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

7 

21-9           19.8           16.9 

15-6 

14.5 

12  9 

II.8 

10.6 

7-5 

235        21.3        18.1 

16.8 

15-5 

13.8 

I2.7 

11.4 

8 

25.2        22.7        19.3 

17.9 

16.6 

14.7 

13-5 

12.2 

85 

26.8        24.2        20.6 

19.1 

17.6 

15-7 

14.4 

12.9 

9 

284        25.7        21.8 

20.2 

18.7 

166 

15-3 

13-7 

9-5 

30.1        27.1        23.1 

21.4 

19.8 

17.6 

16.1 

14-5 

10 

31.7        28.6        24.3 

22.5 

20.8 

18.5 

17 

15-3 

No.     !      7 

8 

9 

IO 

1 

1 

12 

•  3 

•4 

•5 

18 

.165 

.148 

134 

.  12 

.109 

•095 

.083 

.072 

3/161 

»/64  I 

9/64  / 

9/64  / 

'/8J 

7/64 

3/32  / 

5/64  / 

5/64  / 

Diam 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lba. 

Lbs 

Lbs. 

Lbs. 

Lbs. 

i-55 

1.44 

1.32 

1.22 

I. 

II 

.02 

•9 

•797 

•7 

I.I25 

1.78 

1.66 

1-39 

1.26 

.16 

.906        .794 

.25           2  O2 

1.88 

I.7I 

1.57 

I.42 

•3 

.  .15      i.oi 

.888 

•375         2.25 

2.09 

1.9 

1.74 

1.58 

•45 

.27        1.  12 

•983 

•5 

2.49 

2.31 

2.1 

1.92 

I. 

73 

•59 

•4 

1.23 

i.  08 

.625 

272 

2.52 

2.29 

2.09 

1.89 

•73 

•52 

1-34     i   1.  17 

•75 

2.96 

2.74 

2.48 

2.27 

2.05 

.87 

.65      1.45        1.27 

1-875 

319 

2.96 

2.68 

2.45 

2.21 

2.02 

.77  i  1.56       1.36 

2 

343 

3-17 

2.87 

2.62 

2.36 

2.16 

1.9 

1.67 

x-45 

2125 

367 

3-39 

3.06 

2.8 

2.52 

2-3 

2.O2        1.78 

J«55 

2  25 

39 

3-6 

3-26 

2.97 

2.68 

2.44 

2.14      1.88 

i!64 

2375 

4.14 

3.82 

3-45 

3-15 

2.83 

2-59 

2.27  *  1.99 

1.74 

2-5 

4-37 

4.04 

3-65 

3-32 

2.99 

2-73 

2.39  i   2.1 

1.83 

2625 

4.61 

4-25 

3-84 

3-5 

3- 

15 

2.87 

2.52        2.21 

1-93 

275 

4.84 

4-47 

4-03 

3-67 

3i 

3-02 

2.64 

2.32 

2.02 

2875 

5-08 

4.68 

4-23 

3.85 

+6 

3.16 

2.77 

2-43 

2.II 

3 

5-32 

4.9 

4.42 

4.02 

3-62 

3-3 

2.89 

2-54 

2.21 

3-25 

5-79 

5-33 

4.81 

4-37 

3-94 

3-59 

3-14 

2-75 

2.4 

3-5 

6.26 

5-76 

5-2 

4.72 

425 

387     3-39 

2.97 

2-59 

3-75 

6-73 

6.19 

5-58 

5-07 

4-57 

4.16     3.64 

3-19 

2.77 

4 

7-2 

6.63 

5-97 

5-43 

4-88 

4.44  i  3.89     3.4 

2.96 

4-25 

7.67 

7.06 

6.36 

5.78 

5-2 

4-73     4-i3     3-62 

3-15 

4-5 

8.14 

7-49 

6-45 

6.13 

5-5i 

5-oi 

408       3.84 

3-34 

4-75 

8.61 

7.91 

7-13 

6.48 

5-82 

5-3 

4.63     4.06 

3-53 

5 

9.08 

8-35 

7.52 

6.83 

6.13 

5.58  |  4.88     4.27 

3-72 

5-25 

9'56 

8.79 

7.91 

7.18 

6.44 

5-87  i  5  13     4-49 

3-9 

5-5 

10 

9.22 

8-3 

7-53 

6.76 

6.15  j  5-38     4-71 

4.09 

5-75 

10.5 

965 

8.68 

7-88 

7.07 

6-44     5-63     4-93 

4.28 

6 

6.25 

ii 
11.4 

IO.I 

10.5 

9.07 
9.46 

8.23 

8.58 

7-39  !  6.73  i  5-87     5.14 
7.7     i  7.01      6.12     5.316 

4-47 
4.66 

6-5 

11.9 

109 

9-85 

8-93 

8.02 

7-3 

6-37        5.58 

4-85 

6-75 

12.4 

11.4 

10.2 

9.28 

8.33 

7.58  ]  6.62     5.79 

5-03 

7 

12.9 

n.8 

10.6 

9-63 

8.64 

7.87     6.87 

6.01 

5-22 

725 

13-3 

12.2 

ii 

999 

8.96 

8.15      7.12 

6.23 

75 

138 

12.7 

11.4 

10.3 

9.27 

8.44      7.37  |  6.45 

5-61 

7-75 

14-3 

I3-1 

n.8 

10.7 

9-59 

8.72      7.62  !  6.66 

5-79 

14-7 

13-5 

12.2 

ii 

99 

9.01      7.86     6.88       5.98 

144 


WEIGHT  OF  COPPEK  TUBES. 


Weigh-t  of  Seamless  Drawn.  Copper  Tubes.    (English. 
For  Diameters  and  Thicknesses  not  given  in  preceding  Tables.  (D.  K.  Clark.) 

INTERNAL  DIAMETER.      ONE   FOOT   IN    LENGTH. 

HoltzapffeVs  Wire-Gauge,    ffull,  I  light. 
Specific  Weight  =  1.16.    Wrought  Iron  =  i. 


No. 

0000 

ooo 

oo 

0 

No. 

oooo 

ooo 

oo 

o 

Ins. 

•454 

29/64 

425 
27/64  / 

.38 

3/8  / 

•34 
11/32 

Ins. 

•454 
29/64 

•425 

27/64  / 

•38 

3/8  / 

•34 

,    "/32 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•75 

— 

— 

— 

4-5 

5-75 

34-2 

31-9 

28.3 

25.2 

•875 

— 



5-79 

5.02 

6 

35.6 

33-2 

29-5 

26.2 

8.02 

7.36 

6-37 

5-53 

6.5 

38.4 

35-8 

31.8 

28.3 

.125 

8.71 

8 

6-95 

6.05 

7 

41.1 

38.3 

34-i 

30.3 

•25 

9-4 

8.65 

7-52 

6-57 

7-5 

43-9 

40.9 

36.4 

32.4 

•375 

IO.I 

9-3 

8.1 

7.08 

8 

46.6 

43-5 

38-7 

34-5 

•5 

10.8 

994 

8.68 

7.6 

9 

52.1 

48.7 

43-3 

38.6 

.625 

"•5 

10.6 

9.26 

8.12 

10 

57-7 

53-8 

47-9 

42.7 

•75 

12.  1 

II.  2 

9-83 

8.63 

ii 

63.2 

59 

52-5 

46.8 

•875 

12.8 

11.9 

10.4 

9-15 

12 

68.7 

64.2 

57-2 

5i 

2 

13-5 

12.5 

ii 

9.66 

13 

74-2 

69-3 

61.8 

55-i 

2.125 

14.2 

13-3 

n.6 

10.2 

14 

79-7 

74-5 

66.4 

59-2 

2.25 

14.9 

13.8  - 

12.  1 

10-7 

15 

85-2 

79.6 

7i 

63-4 

2-375 

15-6 

14.5 

I2.7 

II.  2 

16 

90.7 

84.8 

75-6 

67.7 

2.5 

I6.3 

15-1 

13-3 

II.7 

17 

96.3 

90 

80.2 

71.8 

2.625 

J7 

15.8 

13-9 

12.2 

18 

101.8 

95-  1 

84.9 

76 

2-75 

17.7 

16.4 

14-5 

12.8 

19 

107.3 

100.3 

89-5 

80,1 

3 

19.1 

17.7 

15-6 

I3'8 

20 

II2.8 

105-5 

94.1 

84.2 

3-25 

20.4 

19 

16.8 

14.8 

21 

118.3 

110.7 

98-7 

88.3 

3-5 

21.8 

20.3 

17.9 

15-9 

22 

123.8 

115.8 

103-3 

92-5 

3-75 

23.2 

21.6 

19.1 

16.9 

23 

129.3 

120.9 

107.9 

96.6 

4 

24.6 

22.9 

2O.2 

17.9 

24 

134.8 

126.1 

II2.6 

100.6 

4-25 

25.9 

24.2 

21.4 

19 

26 

146 

136.4 

I2I.8 

108.8 

4-5 

27-3 

25-4 

22.5 

20 

28 

157-2 

146.7 

131 

117.1 

4-75 

28.7 

26.7 

23-7 

21 

30 

168.4 

i57-i 

140.2 

125.4 

5 

30.1 

28 

24.8 

22.1 

32 

179.6 

167.4 

149-5 

133-6 

5-25 

31.5 

29-3 

26 

23.1 

34 

190.7 

177.7 

158.7 

141.9 

5-5 

32.8 

30.6 

27.1 

24.1 

136 

201.9 

188 

167.9 

150.1 

13 
14 
15 
16 

*7 

18 

19 

20 
21 
22 

23 
24 


For  Diameters  from  13  to  24  Inches. 
5 


i 

2 

3 

4 

•3 

J9/64  / 

.284 
9/32  / 

•259 
x/4/ 

•238 
I5/64/ 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

48.5 

45-8 

41.7 

38.3 

52.1 

49-3 

44-9 

41.2 

55-8 

52.7 

48 

44.1 

59-4 

56.2 

51-2 

46.9 

63 

59-6 

54-3 

49-8 

66.7 

63.1 

57-4 

52.7 

70.3 

66.5 

60.6 

55-6 

74 

70 

63-7 

58.5 

77-6 

73-4 

66.9 

61.4 

81.3 

76.9 

70 

64-3 

84.9 

80.3 

73-2 

67.2 

88.6 

83.8 

76.3 

70.1 

7/32  / 


35-3 

38 

40.7 

43-4 

46 

48.7 

5i-4 

54 

56.7 

59-4 

62.1 

64.7 


6 

7 

8 

9 

10 

•203 
13/64 

.18 
3/i6  / 

.165 

«/64  I 

.148 
9/64  / 

•134 

9/64  / 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

32-6 

28.8 

26.4 

23.6 

21.4 

35-i 

31 

28.4 

25-4 

23 

37-6 

33-2 

30-4 

27.2 

24.6 

40 

35-4 

32-4 

29 

26.3 

42.5 

37-5 

34-4 

30.8 

27.9 

45 

39-7 

36-4 

32.6 

29-5 

47-4 

41.9 

38-4 

34-4 

31.2 

49.9 

44.1 

40.4 

36.2 

32.8 

52.4 

46-3 

42.4 

38 

34-4 

54-9 

48.5 

44-4 

39-8 

36 

57-3 

50-7 

46.4 

41.6 

37-7 

59-8 

52.9 

48.5 

43-4 

39-3 

WEIGHT  OF  COPPER  AND  WROUGHT  IKON  TUBES.       145 


For  Diameters  from  13  to  24  Inches. 

No. 

1  1 

12 

"3 

'4 

"5 

16 

17 

•8     |     19 

20 

Ins. 

.12 

.109 

•095 

.083 

.072 

.065 

•058 

.049    j    .042 

•035 

1/8  I 

7/64 

3/32  / 

5/64  / 

5/64  / 

Vi6/ 

Vi6  / 

3/64/1  3/64  / 

x/32/ 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs.         Lbs. 

Lbs. 

13 

19.1 

17.4 

I5.I 

13.2 

II.4 

10.3 

9.3 

7-77      6.65!    5.55 

14 

20.  6 

I8.7 

16.3 

14.2 

12.3 

II.  I 

9.9 

8.37      7.16      5  98 

15 

22.1 

20 

17.4 

15-2 

13.2 

II.9 

10.6 

8.96      7.67      6.4 

16 

23-5 

21-3 

18.6 

16.2 

I4.I 

12.7 

"•3 

9.56,    8.18      6.82 

17 

25 

22.7 

19.7 

17.2 

14.9 

13-5 

12.  1 

10.2 

8.69 

7.27 

18 

26.4 

24 

20.9 

18.2 

15.8 

-14-3 

I2.7 

10.7 

9.2 

7.69 

19 

27.9 

25-3 

22 

19.2 

I6.7 

I5-I 

13.4 

"•3 

9.71 

8.12 

20 

29-3 

26.6 

23.2 

20.2 

17.6 

15.9 

11.9 

10.2 

8.54 

21 

30.8 

27.9 

24-3 

21.3 

I8.4 

16.6 

14.8 

12.5 

10.7 

8.96 

22 

32.3 

290 

25-5 

22.3 

19.3 

17.4 

15-5 

13-1 

II.  7. 

9-39 

23 

33-7 

30.6 

26.7 

23-3 

20.2 

18.2 

16.2 

13-7 

u.8 

9.81 

34 

35-2 

31.9    i    27.8 

24-3 

21.  1 

19 

16.9 

14-3 

12.3 

IO.2 

"Weight    of  Wrought    Iron    Tn"bes.    (English.) 
For  Diameters  and  Thicknesses  not  given  in  preceding  Tables.  (D.  K.  Clark.) 

INTERNAL  DIAMETER.   ONE  FOOT  IN  LENGTH. 

HoltzapffeVs  Wire-Gauge,   ffull,  I  light. 


No. 

4 

5 

6 

7 

THICKNESS  IN  INCHES. 

.238 

.22 

.203 

.18 

Ins. 

5/8 

9/i6 

«/2 

7/i6 

3/8 

5/i6    j    «/4 

'5/64  / 

7/32  / 

«3/64     3/i6  / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs.    1   Lbs. 

Lbs. 

Lbs. 

Lb«. 

Lbs. 

19   |  128.5 

II5.2|  102.  1 

89.1 

76.1 

63.2    50.4 

48 

44-2  140.8 

36.2 

20      135 

121.  1 

107.3 

93.6      80 

66.5  53     i  50-4    46.5 

42.9     38 

21    '1141.5 

127 

II2.6 

98.2       83.9      69.7    55.6     52.9      48.8 

45-1 

39-9 

22    i  148.1 

132.9    117.8 

102.8 

87.9 

73      58.3    55-4     51-1 

47.2 

41.8 

23  i  154.6 

138.8    I23.I 

107.4 

91.8 

76.3  60.9    57.9 

53-4 

49-3 

43-7 

24   ;  l6l.2 

144.7     128.3     112 

95-7 

79.6  63.5    60.4 

55-7 

5i.5 

456 

26  :  174.3 

156.5     138.8 

121.  1     IO3.6 

86.1   68.7 

05-4 

60.3 

55-7 

49-3 

28  !  187.4 

168.3     149.2 

130.3     III.4 

92.7   74 

70.4 

64.9 

60 

53-i 

30  200.4 

180     ,  159.7 

139-5     1  19-3 

99.2   79.2 

75-4 

09-5 

64.2 

56-8 

32  213.5 

191.8   170.2 

148.6    I27.I 

105.7   84.4 

80.4 

74.1 

68.5 

60.6 

34    226.6 

203.6   180.6 

157-8 

135 

112.3  89.7 

85-4 

78.7 

72.8 

64.4 

36    239.7 

215.4!  191.1 

I67 

142.9 

118.8  94.9 

90.4 

83-4 

77 

68.1 

No. 

8 

9 

IO 

1  1 

12 

«3 

'4 

•  5 

16 

i? 

18 

Ins 

.165 

.148 

•134 

.12 

.109 

.095. 

.083 

.072 

.065 

.058 

.049 

»/64  / 

9/64  / 

9/64  / 

1/8  I 

7/64 

3/32  / 

5/64  / 

5/641 

x/i6/ 

Vi6  / 

3/64  / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbt. 

19      33-i 

29.7 

26.9 

24 

21.8 

19 

16.6 

14.4 

13 

u.6 

9.78 

20        34.8 

31.2 

28.3 

25-3 

22.9 

20 

17-5 

I5-I 

13.7 

12.2 

10.3 

21         36.6 

32-8 

29.7 

26.5 

24.1 

21 

18.3 

15-9 

14.3 

12.8 

10.8 

22 

38.3 

34-3 

31-1 

27.8 

25.2 

22 

19.2 

16.6 

15 

13-4 

"•3 

23 

40       1  35-9 

32.5 

29.1 

26.4 

23 

20.1 

17.4 

15-7 

14 

u.8 

24 

41.8 

37-4 

33.9 

30.3 

27.5 

24 

2O.9 

iS.i 

16.4 

14.6 

12.6 

26 

45-2 

40-5 

36.7 

32.8 

29.8 

26 

22.6 

19.7 

17.7 

15.8 

13-4 

28 

48.7 

43-6 

39-5 

35-3 

32.1 

28 

24.4 

21.2 

I9.I 

17 

14.4 

30 

52.1 

46.7 

42-3 

37-8 

34-4 

3° 

26.1 

22-7 

20.5 

I8.3 

i5-4 

32 

55-5 

49.8 

45-i 

40.4 

36.7 

32 

27.9 

24.2 

21.8 

19-5 

16.5 

34 

59 

52.9 

48 

42.9 

39 

34 

29.7 

25.8 

23.2 

20-7 

i7-5 

36 

62.4 

56 

50.8 

45-4 

41-3 

36 

3T-4 

27-3 

24.6 

21.9 

18.6 

146 


WEIGHT    OF    IRON,   STEEL,   COPPER,   ETC. 


"Weight    of  a   Square   IToot   of  "Wrought    and   Cast 
Iron,  Steel,  Copper,  Lead,  Brass,  and  Zinc  3?lates. 

From  .0625  to  i  Inch  in  Thickness. 


Thickness. 

Wrought 
Iron. 

Cast  Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun- 
metal. 

Zinc. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.0625 

2.5I7 

2.346 

2.541 

2.89 

3.691 

2.675 

2.848 

2-34 

.125 

5-035 

4-693 

5-o8l 

5.781 

7.382 

5-35 

5-696 

4.68 

•1875 

7.552 

7-039 

7.6s2 

8.672 

11.074 

8.025 

8-545 

7-02 

•25 

IO.07 

9-386 

10.163 

11.562 

14.765 

10.7 

".393 

9-36 

•3125 

12.588 

"•733 

12.703 

14-453 

18.456 

J3-375 

14.241 

II.7 

•375 

I5.I06 

14.079 

15.244 

17.344 

22.148 

16.05 

17.089 

14.04 

•4375 

17.623 

16.426 

17-785 

20.234 

25^39 

18.725 

19.938 

16.34 

•5 

20.141 

18.773 

20.326 

23.125 

29-53 

21.4 

22.786 

18.72 

•5625 

22.659 

21.119 

22.866 

26.0l6 

33-222 

24-075 

25-634 

21.06 

.625 

25.176 

23.466 

25.407 

28.906 

36.913 

26.75 

28.483 

23.4 

.6875 

27.694 

25.812 

27.948 

3L797 

40.604 

29-425 

3i.33i 

25.74 

•75 

30.211 

28.159 

30.488 

34.688 

44.296 

32.1 

34-179 

28.68 

.8125 

32.729 

30-505 

33.029 

37-578 

47.987 

34-775 

37.027 

30.42 

•875 

35-247 

32.852 

35-57 

40.469 

51.678 

36.656 

39-875 

32.76 

•9375 

37.764 

35-J99 

38.11 

43-359 

55-37 

39-331 

42.723 

35-i 

i 

40.282 

37-545 

40.651 

46.25 

59.061 

42.8 

45-572 

37-44 

From  One  Twentieth  Inch  to  Two  Inches  in  Thickness. 

Thickness. 

Wrought 
Iron. 

Cast  Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

metal. 

Zine. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.05 

2.014 

1.877 

2-033 

2.312 

2-593 

2.14 

2.279 

1.872 

.1 

4.028 

3-754 

4.065 

4.625 

5.906 

4.28 

4-557 

3-744 

•15 

6.042 

5.632 

6.098 

6.938 

8.859 

6.42 

6.836 

5-6i6 

.2 

8.056 

7-509 

8.13 

9-25 

II.8I2 

8.56 

9.114 

7.488 

•25 

10.071 

9.386 

10.163 

11.562 

14.765 

10.7 

"-393 

9-36 

•3 

12.085 

11.264 

12.195 

I3-875 

17.718 

12.84 

13.672 

11.232 

•35 

14.099 

13.141 

14.228 

16.187 

20.671 

14-98 

15-95 

13.104 

.4 

16.113 

I5.0I8 

16.26 

I8.5 

23.624 

17.12 

18.229 

14.976 

•45 

18.127 

16.895 

18.293 

20.812 

26.577 

19.26 

20.507 

16.848 

•5 

20.141 

18.773 

20.325 

23.125 

29-53 

21.4 

22.786 

18.72 

•55 

22.155 

20.65 

22.358 

25-437 

32.484 

23.54 

25.065 

20.592 

.6 

24.169 

22.527 

24.391 

27.75 

35-437 

25.68 

27-343 

22.464 

•65 

26.183 

24.409 

26.423 

30.063 

38.39 

27.82 

29.622 

24-336 

•7 

28.197 

26.281 

28.456 

32.375 

41-343 

29.96 

31-9 

26.208 

•75 

30.211 

28.154 

30.488 

34.687 

44.296 

32.1 

34.179 

28.08 

.8 

32.226 

30.035 

32.521 

37 

47-249 

34.24 

36.458 

29-95 

•85 

34-24 

31.912 

34-553 

39-312 

50.202 

36,38 

38.736 

31.824 

•9 

36.254 

33-79 

36.586 

41.625 

53-154 

3852 

41.015 

33.696 

•95 

38.268 

35-668 

38.628 

43-937 

56.108 

40.66 

43-293 

35.568 

i 

40.282 

37-545 

40.651 

46.25 

59.o6l 

42.8 

45.572 

37-44 

1.125 

45.317 

42.238 

45.732 

52.031 

66.443 

48.15 

51.268 

42.12 

1.25 

50.352 

46.931 

50.814 

57.8i3 

73.826 

53-5 

56.065 

46.8 

1-3125 

52.87 

49.278 

53-354 

60.703 

77.517 

56-17 

59-8i3 

49.14 

1-375 

55.387 

51.624 

55-895 

63.594 

81.209 

58.85 

62.661 

51.48 

1-4375 

57-905 

53.971 

58.436 

66.484 

84.9 

6i.53 

65-51 

53-82 

i-5 

60.422 

56-31  7 

60.976 

69.375 

88.591 

64.2 

68.358 

56-16 

1.5625 

62.94 

58-663 

63-517 

72.266 

92.283 

66.88 

71.206 

58.5 

1.625 

65.458 

61.011 

66.058 

75-156 

95.974!    69.55 

74-054 

60.84 

i-75 

70-493 

65.704 

7i-i39 

80.938 

!03-356     74>9 

79-751 

65-52 

1-875 

75.528 

70.397 

76.22 

86.719 

110.739    80.25 

85447 

70.2 

a 

80.564 

75-09 

81.3 

92.5 

118.122     85.6 

91.144 

74.88 

WEIGHT    OF    ROLLED   STEEL   T.  PIPES    AND   TUBES. 

"Weights,    etc.,    of  Rolled    Steel    T. 

Safe  Load  for  One  Foot  Uniformly  Distributed. 


Dimen- 
sions. 

Area. 

Weight 
per 
foot. 

Load. 
Tensile  Strength 
per  Sq.  Inch. 

Dimen- 
sions. 

Area. 

Weight 
per 
foot. 

Load. 
Tensile  Strength 
per  Sq.  Inch. 

12500 

16000 

12500 

16000 

Ins. 

Sq.  Ins. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

4-5X2.5 
4-5X2.5 

2-79 
2.4 

r3 

5220 
4520 

6950 
6030 

4X4 
4X4 

3.21 
4.02 

10.9 
J3-7 

13  loo 
16  170 

17470 
21550 

4-5X3 

3 

10 

754° 

10050 

4X4.5 

3.36 

11.4 

15840 

21  120 

4-5X3 

2-55 

8.5 

6490 

8650 

4X4.5 

4-29 

14.6 

20400 

27200 

4-5X3.5 

4-65 

15.8 

17020 

22690 

4X5 

3-54 

12 

19410 

25880 

5    X2.5 

3-24 

II 

6900 

9  200 

4X5 

4.56 

15.6 

24800 

33°7° 

5    X3 

3-99 

13.6 

9410 

12550 

— 

— 

— 

To   Compute   Weight   of  Metal   Pipes. 


D2  —  d2  C.    D  and  d  representing  external  and  internal  diameters  in  inches, 
and  C  coefficient. 
Cast  Iron  2.45.   Wrought  Iron  2.64.   Brass  2.82.   Copper  3.03.  Lead  3.86. 

To   Compute  "Weight   of  Metal  Tu.bes   and   Pipes 
per   .Lineal   iToot. 

From  .5  Inch  to  6  Inches  Internal  Diameter. 


Diam. 

Area  of  Plato. 

Diam. 

Area  of  Plate. 

Diam. 

Area  of  Plate. 

Diam. 

Area  of  Plat* 

Ins. 

Sq.  Foot. 

Ins. 

Sq.  Foot. 

Ins. 

Sq.  Feet. 

In*. 

Sq.  Feot. 

•5 

.1309 

L3I25 

.3436 

2-75 

.7199 

4-5 

1.1781 

.5625 

•1473 

1-375 

.36 

2-875 

.7526 

4.625 

1.  2108 

.625 

•1636 

i  4375 

•3764 

3 

.7854 

4-75 

1-2435 

.6875 

.18 

i-5 

•3927 

3-125 

.8l8l 

4875 

1.2763 

•75 

.1964 

1.625 

.4254 

3-25 

.8508 

5 

1.309 

.8125 

.2127 

i-75 

.4581 

3-375 

.8836 

5-125 

I'34I7 

.875 

.2291 

1-875 

.4909 

3-5 

.9163 

5-25 

1-3744 

•9375 

•2454 

2 

•5236 

3625 

•949 

5-375 

1.4072 

.2618 

2.125 

•5543 

3-75 

.9818 

5-5 

J  -4399 

.0625 

.2782 

2.25 

•587 

4 

1.0472 

5-625 

1.4726 

.125 

•2945 

2-375 

.6198 

4.125 

1.0799 

5-75 

I-5053 

•i875 

•3*05 

2-5 

•6545 

4-25 

1.1126 

5.875 

i-538i 

'25 

.3272 

2.625 

.6872 

4-375 

I-I454 

6 

1.5708 

Application,   of  Table. 

When  Thickness  of  Metal  is  given  in  Divisions  of  an  Inch. 
To  internal  diameter  of  tube  or  pipe  add  thickness  of  metal ;  take 
area  of  the  plate  in  square  feet,  from  table  for  a  diameter  equal  to 
sum  of  diameter  and  thickness  of  tube  or  pipe,  and  multiply  it  by 
weight  of  a  square  foot  of  metal  for  given  thickness  (see  table,  page 
146),  and  again  by  its  length  in  feet. 

ILLUSTRATION.— Required  weight  of  10  feet  of  copper  tube  i  inch  in  diameter  and 
125  of  an  inch  in  thickness. 

i  +  .125  =  1.125  X  3. 1416 -f- 12  =  .2945  square  feet  for  ifoot  of  length. 

Weight  of  i  square  foot  of  copper  .i2sth  of  an  inch  in  thickness,  per  table,  page 
135,  =5.781  Ibs.;  then,  .2945  (from  table  above)  x  5.781  x  10  =  17.025  Ibs. 

When  T/iickness  of  Metal  is  given  in  Numbers  of  a  Wire -Gauge. 
To  internal  diameter  of  tube  or  pipe  add  thickness  of  number  from 
table,  pp.  120  or  121 ;  multiply  sum  by  3.1416,  divide  product  by  12,  and 
quotient  will  give  area  of  plate  in  square  feet.    Then  proceed  as  before, 


148      WEIGHT  OF  IRON  AND  COPPER  PIPES,  BOLTS,  ETC. 

ILLUSTRATION.— Required  weight  of  10  feet  of  copper  pipe  2  inches  in  diameter 
and  No.  2  American  wire-gauge  in  thickness. 

2  +  -257 63X3. 1416 -f-  12  =  2. 257 63  X  3. 1416 -r- 12  =  .591  square  feet;  then,  .591 
X  11.6706  (weight  from  table,  page  118)  =6.897  Ibs. 

of  Riveted.   Iron   and.    Copper   IPipes, 

From  5  to  30  Inches  in  Diameter. 

ONE   FOOT   IN    LENGTH. 


Diameter. 

Thicknesi. 

Iron. 

Copper. 

Diameter. 

Thickness. 

Iron. 

Copper. 

Ins. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs, 

Lbs. 

5 

.125 

7.12 

8.14 

9 

•25 

25.01 

28.58 

•1875 

10.68 

12.21 

-25 

26.33 

30.09 

•25 

14.25 

16.28 

10 

•25 

27-75 

3I-7I 

5-5 

.125 

7.78 

8.89 

10.5 

•25 

29.19 

33-22 

•1875 

11.66 

13-33 

ii 

•25 

30-49 

34.85 

•25 

15-56 

I7.78 

12 

-25 

33-!3 

37-86 

6 

.125 

8.44 

9.64 

J3 

•25 

35-88 

4i 

•1875 

12.65 

14.46 

14 

•25 

38-52 

44.02 

•25 

16.88 

19.29 

15 

-25 

41.26 

47-J5 

6-5 

•  125 

9,1 

10.4 

•3125 

51-57 

58.94 

.1875 

13-65 

15-6 

16 

•25 

43-9 

50-17 

•25 

18.2 

20.8 

•3125 

54-87 

62.71 

7 

•  125 

9.78 

ii.  18 

17 

•25 

46.53 

53-iS 

•1875 

14.68 

16.78 

•3125 

58.17 

66.48 

•25 

19-57 

22.37 

18 

•25 

49.17 

56.2 

7-5 

•125 

10.49 

11.99 

•3125 

61.47 

70.25 

•1875 

15-73 

17.98 

20 

•3I25 

68.07 

77-79 

•25 

20.89 

23.87 

24 

•3125 

Si-33 

92.95 

8 

•1875 

16.7 

19.08 

25 

•3125 

84-57 

96.65 

•25 

22.26 

25-44 

28 

•3125 

94-56 

107.95 

8-5 

•25 

23-59 

26.96 

30 

•3I25 

101.14 

"5-59 

Above  weights  include  laps  of  sheets  for  riveting  and  calking. 

Weights  of  the  rivets  are  not  added,  as  number  per  lineal  foot  of  pipe  depends 
upon  the  distance  they  are  placed  apart,  and  their  diameter  and  length  depend 
upon  thickness  of  metal  of  the  pipe. 


"Weight   of  Copper   Rods   or  Bolts, 

From  .125  Inch  to  4  Inches  in  Diameter. 


ONE   FOOT   IN    LENGTH. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lb». 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

•047 

.8125 

1.998 

i-5 

6.8II 

2-75 

22.891 

•1875 

.106 

•875 

2.318 

•5625 

7-39 

.875 

25.019 

•25 

.189 

•9375 

2.66 

.625 

7-993 

3 

27.243 

•3125 

.206 

i 

3-03 

•75 

9.27 

.125 

29-559 

•375 

.426 

1.0625 

342 

.875 

10.642 

•25 

31.972 

•4375 

•579 

.125 

3-831 

2 

12.108 

•375 

34.481 

•5 

•757 

•1875 

4.269 

.125 

13.668 

•5 

37.081 

•5625 

.958 

•25 

4-723 

•25 

15-325 

.625 

39-777 

.625 

1.182 

•3I25 

5-2i 

•375 

17.075 

•75 

42.568 

.6875 

L431 

•375 

5.723 

•5 

18.916 

.875 

45-455 

•75 

1.703 

•4375   !    6.255 

.625 

20.856 

4 

48.433 

WEIGHT    OF   METALS. 

Weiglit    of  M!etals   of  a   Griven.    Sectional 

From  .1  Square  Inch  to  10  Square  Inches. 

PER   LINEAL   FOOT.      (Z).  K.  Clark) 


149 


&flCI. 
AREA. 

Wrought 
Iron. 

Cast 
Iron. 
•9375' 

Steel. 

I.O2. 

Brass. 

1.052. 

Gun- 

metal. 
1.092. 

SECT. 
ARKA. 

Wrought 
Iron. 

!  'l.'  '    ' 

Cast 
•9375- 

Steel.    B^l^a 

I.O2.        I.O52.      1.092. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

,1 

•33 

•31 

•34 

•35 

.36 

5-i 

17 

15-9 

17-3 

17.9 

18.6 

,2 

.67 

.62 

.68 

•7 

•73 

5-2 

17-3 

I6.3 

17.7 

18.2  j  18.9 

.3 

i 

.94 

i.  02 

1.05 

1.09 

5-3 

17.7 

16.6 

18 

18.6    19.3 

•4 
•5 

i 

1.25 
1.56 

1.36 
i-7 

1-43 
1-75 

1.46 
1.82 

5-4 
5-5 

18 

18.3 

16.9 
17.2 

18.4 
18.7 

18.9  !  19.7 

I9.3      20 

.6 

2 

1.88     2.04 

2.II 

2.18 

5-6 

18.7 

17-5 

19 

19.6      20.4 

•7 

2.33       2.19       2.38 

2.46 

2-55 

5-7 

19 

17.8 

19.4 

20       26.8 

.8 

2.67       2.5          2.72 

2.81 

2.91 

5-8 

19-3 

18.1 

19.7 

20-3      21.  1 

•9 

3       |    2.81     3.06 

3.16 

3-28 

5-9 

19.7    18.4 

20.1 

2O.7 

21.5 

i 

3-33     3-i5;   3-4 

3-51 

3.64 

6 

20        !  18.8 

2O.4 

21 

21.8 

i.i 

3-67     3-44     3-74 

3-86 

4 

6.1 

20.3    19.1 

20.7 

21.4 

22.2 

1.2 

4 

3-75     4-o8 

4.21 

4-37  i 

6.2 

20.7    19.4 

21.  1 

21.7 

22.6 

J-3 

4.33     4.06     4.42 

4-56 

4-73 

6-3 

21             19.7 

21.4 

22.1 

22-9 

1.4 

4.67     4.38     4-76 

4.91 

5-i   1 

6.4 

21-3    ^20 

21.8 

22-4 

23'3 

i-5 

5 

4-69     5-1 

5-26 

5-46; 

6.5 

21.7       20.3 

22.1 

22.8 

23-7 

1.6 

5-33 

5          5-44 

5-6i 

5-82; 

6.6 

22           2O.6 

22-4 

23.1 

24 

i-7 

5-67     5-31     5.78 

5-96 

6.19 

6.7 

22.3       20-9 

22.8 

23-5 

244 

1.8 

6 

5.63     6.12 

6.31 

6-55 

6.8 

22.7     i  21.3 

23.1 

23-9 

24.8 

1.9 

6-33     5-94     6-4o 

6.66    6.92 

6.9 

23        21.6 

23-5 

24.2      25.1 

2 

6.67     6.25     6.8 

7.01 

7.28, 

7 

23.3    21.9 

23.8 

24.6 

25-5 

2.1 

7 

6.56     7-14 

7-36 

7-64; 

7*i 

23.7       22.2 

24-1 

24.9 

25.8 

2.2 

7.33     6.88     7-48     7-73 

8.01, 

7.2 

24            22-5 

24-5 

25-3 

26.2 

2.3          7.67        7.19       7-82       8.67 

8.37J 

7-3 

24.3       22.8 

24.8 

25.6 

26.6 

2.4  i    8 

7-5 

8.16 

8.42 

8-74, 

7-4 

24.7       23.1 

25.2 

26 

26.9 

2-5          8.33 

7.81 

8.5 

8.77 

9.1   | 

7-5 

25 

23-4 

25-5 

26.3 

27-3 

2.6       8.67 

8.13 

8.84 

9.12 

9.46 

7.6 

25-3       23.8 

25-9 

26.7 

27.7 

2.7       9 

8.44 

9.18 

9-47 

9-83 

7-7 

25.7       24.1 

26.2 

27 

28 

2-8  i    9-33 

8-75     9-52 

9.82 

10.2 

7.8 

26 

24.4 

26.5 

27.4 

28.4 

2.9  !    9.67 

9.06     9.86 

10.2 

10.6 

7-9 

26.3 

24.7 

26.9 

27.7 

28.8 

3        10 

9.38     10.2 

10.5 

10.9 

8 

26.7 

25 

27.2 

28-1 

29.1 

3-i     10.3 

9.69    10.5 

10.9 

"•3 

8.1 

27 

25-3 

27-5 

28.4 

29-5 

3.2     10.7 

IO 

10.9 

II.2 

11.7 

8.2 

27-3 

25-6 

27.9 

28-8 

29.9 

3-3     « 

10.3 

II.  2 

n.6 

12 

8.3 

27.7 

25-9 

28.2 

29.1 

30.2 

3-4     "-3 

10.6 

n.6 

11.9 

I2.4 

8.4 

28 

26.3 

28.6 

29-5 

30.6 

3-5     "-7 

10.9 

11.9 

12.3 

I2.7 

8-5 

28.3 

26.6 

28.9 

29-8 

30-9 

3-6 

12 

"•3 

12.2 

12.6 

I3-I 

8.6 

28.7 

26.9 

29.2 

30.2 

31-3 

3-7     12.3 

n.6 

12.6 

13 

13-5 

8.7 

29 

27.2 

29.6 

30-5      31-7 

3-8     12.7 

11.9     12.9 

13-3 

13-8 

8.8 

29-3 

27-5 

29.9 

30-9     32 

3-9     J3 

12.2       13.3 

13.7 

14.2 

8.9 

29.7 

27.8 

30-3 

31-2     32.4 

4        13-3 

12-5       13.6 

14 

14.6 

9 

30 

28.1 

30.6 

31.6     32.8 

4.1      13.7      12.8     13.9 

14.4 

14.9 

9.1 

30-3 

28.4 

30-9 

3J-9 

33-i 

4.2 

14            13.1       14.3 

14.7 

15-3 

9.2 

30.7 

28.8 

31-3 

32-3    33-5 

4-3     J4-3      J3-4     !4-6 

15-1 

!5-7 

9-3 

31 

29.1 

31.6 

32.6    33.9 

4.4     14.7      13.8     15 

15-4 

16 

9-4 

31-3 

29.4 

32 

33 

34-2 

4-5      15 

I4-I        15-3 

15.8 

16.4 

9-5 

31-7 

29.7 

32.3 

33-3 

34-6 

4.6     15.3      14.4     15.6      16.1 

16.7 

9.6 

32 

30 

32.6 

33-7 

34-9 

4.7     15.7      14.7 

16         16.5 

17.1 

9-7 

32-3 

30-3 

33 

34 

35-3 

4-8 

16         15 

16.3     16.8 

17-5 

9.8 

32-7       30.6 

33-3  j  34-4 

35-7 

4.9 

16.3      15-3 

16.7     17.2 

17.8 

9.9 

33 

30-9 

33-7    34-7 

36 

5 

16.7      15.6     17 

17-5 

18.2 

IO 

33-3 

3i-3 

34 

35-1    3^4 

150 


LEAD   PIPES. — COPPEK   PIPES   AND   COCKS. 


Diam. 

Thick- 
ness. 

"TO 

Weight. 

reigl 

Diam. 

it   o 

ONE 

Thick- 
ness. 

f  Lej 

FOOT 

Weight. 

ad  i 

IN    LE 

Diam. 

Dipe 

NGTH 

Thick- 
ness 

.    (Eng 

Weight. 

lish.) 

Diam. 

Thick- 
ness. 

Weigkt 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

-5 

.097 

•93 

I 

.136 

2.4 

i-75 

.166 

5 

3 

•275 

H 

.112 

.07 

.156 

2.8 

.199 

6 

3-5 

.225 

13 

.124 

.2 

.2 

3-73 

.228 

7 

•273 

16 

.146 

•47 

.225 

4.27 

.256 

8 

4 

•257 

i7 

•625 

.089 

1.25 

•139 

3 

2 

.I78 

6 

•3125 

20.5 

.101 

•13 

.16 

3-5 

.204 

7 

•327 

22 

.121 

•4 

.18 

4 

.231 

8 

4-25 

•3125 

22.04 

.14 

2 

•193 

4-33 

.266 

9-33 

4-5 

.232 

17 

•75 

.112 

.6 

i-5 

.156 

4 

2-5 

.2 

8.4 

•295 

22 

.147 

1.87 

.179 

4.67 

.227 

9.6 

•3125 

23.25 

.l8l 

2.13 

.224 

6 

.261 

II.2 

4-75 

•3I25 

24-45 

.215 

2.4 

•257 

7 

3 

.218 

II.2 

5 

•3125 

25.66 

Dimensions    of  Copper   Pipes   and.   Composition 
Cocks. 

From  i  Inch  to  23  Inches  in  Diameter. 


•s  * 

« 

Flange  I 
Pipe. 

>iameter. 
Cock. 

Thick- 
ness. 

B 
No. 

olts. 
Diam. 

"8    -8 

a.S.3 

**1 

Flange 
Diam. 

Pipe. 

Thick- 
ness. 

B 

No. 

>lta. 
Diam. 

Ins. 

Ins. 

Ins. 

Inch. 

Inch. 

Ins. 

Ins. 

Inch. 

Inch. 

I 

3-375 

3-5 

-375 

3 

•5 

9 

12-75 

.625 

9 

.625 

1.25 

3-625 

3-75 

•375 

3 

-5 

9-25 

I3.I25 

.625 

10 

.625 

i-5 

3.875 

4-25 

•375 

3 

•5 

9-5 

13-375 

.6875 

IO 

.625 

i-75 

4.125 

4-375 

•4375 

4 

•5 

9-75 

I3-625 

.6875 

10 

.625 

2 

4-375 

4-75 

-4375 

4 

•5 

10 

13.875 

.6875     10 

.625 

2.25 

4.625 

5.25 

•4375 

5 

•5 

10.5 

14-5 

.6875 

10 

.625 

2-5 

4.875 

5-5 

•4375 

5 

-5 

ii 

15 

.6875 

IO 

.625 

2-75 

5.25 

5-75 

•4375 

5 

•5 

"•5 

15.625 

•75 

IO 

•75 

3 

6 

6.25 

•5 

5 

•625 

12 

16.125 

•75 

IO 

•75 

3-25 

6.125 

6.625 

-5 

6 

•625 

12.5 

16.625 

•75 

10 

•75 

35 

6-375 

6.875 

-5 

6 

•625 

13 

I7-25 

•75 

10 

•75 

3-75 

6.625 

7.25 

•5 

6 

•625 

13-5 

17.875 

•75 

10 

•75 

4 

6.875 

7-375 

•5 

6 

•625 

14 

18.375 

•75 

IO 

•75 

425 

7-125 

7.625 

•5 

6 

•625 

14-5 

18.875 

•75 

10 

•75 

45 

7-375 

8.25 

•5 

6 

•625 

15 

19-5 

-75 

10 

•75 

4-75 

7.625 

8-5 

•5 

6 

•625 

15-5 

20 

•75 

10 

•75 

5 

8 

9 

•5 

6 

•625 

16 

20.5 

•75 

10 

•75 

5-25 

8.25 

9-25 

•5 

6 

•625 

16.5 

21.125 

•75 

10 

•75 

5-5 

8-5 

9-5 

•5 

6 

•625 

17 

21.625 

•75 

II 

•75 

5-75 

9 

9-875 

«5 

6 

•625 

17-5 

22.125 

•75 

II 

•75 

6 

9-25 

.625 

8 

•625 

18 

22-75 

•75 

II 

•75 

6.25 

9-75 

-625 

8 

•625 

18.5 

23-25 

•75 

II 

•75 

6-5 

10 

.625 

8 

.625 

19 

23-75 

•75 

12 

•75 

6-75 

10 

.625 

8 

•625 

19-5 

24-375 

•75 

12       -75 

7 

10.5 

.625 

8 

•625 

20 

24.875 

•75 

12       .75 

7-25 

10.75 

-625 

8 

•625 

205 

25-375 

•75 

13     -75 

7-5 

11.125 

-625 

8 

•625 

21 

26 

•75 

13  1  -75 

7-75 

11  -375 

.625 

8 

•625 

21-5 

26.5 

•75 

13  !  -75 

8 

11.625 

-625 

9 

•625 

22 

27 

•75 

J3     -75 

8.25 

12 

-625 

9 

•625 

22.5 

27-625 

•75 

14     -75 

8-5 

12.25 

.625 

9 

•625 

23 

28.125 

•75 

14     -75 

8-75 

12.5 

.625 

9 

.625 

WEIGHT  OF  SHEET  LEAD,  LEAD  AND  TIN  PIPES,  ETC.       I  5  I 
Weight   of  Sheet   Lead. 

PER   SQUARE   FOOT. 


Thickness.      Weight.  ||   Thickness. 

Weight. 

Thickness. 

Weight.  ||   Thickness. 

Weight. 

Inch.              Lbs.              Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

.017                I                 .068 

4 

.118 

7 

.169 

10 

.034                 2                 .085 

5 

•135 

.186 

II 

•051            3     II      .101 

6 

.152 

9 

II        -203 

12 

Weight   of  Tin   !Pipe. 

ONE   FOOT   IN   LENGTH. 

Diam. 

THICKNESS. 

Diam 

THICKNESS. 

Diam. 

THICKN. 

Diam. 

THICKN. 

External. 

&  inch. 

%  inch. 

External. 

X  inch 

%  inch. 

External. 

%  inch. 

External 

.    %  inch. 

Inch. 

Lb. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

•25 

.148 

— 

1.25 

1.095 

1.417 

2.25 

5-04 

3.25 

7.56 

•5 

•384 

.472 

1.328 

1-732 

2-5 

5.67 

3«5 

8.19 

•75 

.62 

.787 

1.75 

1.564 

2.047 

2-75 

6-3 

3-75 

8.82 

i 

.856 

I.I03 

2 

1.  802 

2.362 

3 

6-93 

4 

9-45 

"Weight   of  Lead   Encased   Tin   3?ipes. 

Diameter. 

Light  Weights. 

ForS 
50  feet  and  under. 

upply  of  Water  Head 
51  to  250  feet. 

.* 

251  to  500  feet. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

I 

1-5 

2 

2.5    to    4 

3 

to   4.5 

3-5  to    5 

•5 

2 

2-5 

3 

3-5    "    5 

4 

"    6 

4-5  "     7 

.625 

3 

3-5 

4 

4-5    "    7 

5-25 

"    8 

6 

9 

•75 

3-5 

4 

4-5 

5.5    "    8 

6 

"    9 

7 

"    10 

i 

4-5 

5 

5-5 

7.25  "  10 

8 

"  ii 

9 

"   13 

1.25 

6-5 

7 

8 

9       "  12.5 

10 

"  14 

ia 

"  16 

8 

9 

10 

ii       "  16 

12.5 

"  18 

14 

"   21 

2 

ii 

13 

— 

16       "  23 

18.5 

"  26 

21 

"30 

*  The  extreme  weights  are  for  extra  heavy  pipe  with  less  proportion  of  tin. 

Dimensions  and  "Weight  of  Sheet  Zinc.    (Vielle-Montagne.) 

PER   SQUARE   FOOT. 


No. 

Thickness. 

sX.  5  metres; 
area,  i  square  metre. 

6.56X1.64  feet;  area, 
10.76  square  feet. 

2X  .65  metres  ; 
area,  1.3  sq.  metres. 

6.56X2.13  feet;  area, 
13.99  square  feet. 

2X.8  metres; 
area,  1.6  sq.  metre*. 

6  56X2.62  ft.  ;  area, 
17.22  square  feet. 

Weight. 

Millim. 

Inch. 

Kilom. 

Lbs. 

Kilom. 

Lbs. 

Kilom. 

Lbs. 

Lbs. 

9 

.41 

.Ol6l 

2.9 

6-39 

3-7 

8.16 

4.6 

10.14 

.589 

10 

•51 

.O2OI 

3-45 

7.6l 

4-45 

9.8l 

o-5 

12.12 

.704 

ii 

.6 

.0236 

4-05 

8-93 

5-3 

11.68 

6-5 

14-33 

.832 

12 

.69 

.0272 

4-65 

10.25 

6.1 

13-45 

7-5 

l6-53 

.96 

13 

.78 

.0307 

5-3 

11.68 

6.9 

15.21 

8.5 

18.74 

1.088 

14 

.87 

•0343 

5-95 

13.12 

7-7 

16.94 

9-5 

20.94 

1.216 

15 

.96 

.0378 

6-55 

14.44 

8-55 

18.85 

10.5 

23.I5 

1-344 

16 

i.i 

•0433 

7-5 

16.53 

9-75 

21.5 

12 

26.46 

I-536 

17 

1.23 

.0485 

8-45 

18.63 

10.95 

24.14 

13-5 

29.97 

1.74 

18 

1.36 

•0536 

9-35 

20.61 

12.2 

26.9 

15 

33-07 

1.92 

J9 

1.48 

.0583 

10.3 

22.71 

!3-4 

29-54 

I6.5 

36.38 

2.  112 

20 

1.66 

.0654 

11.25 

24.8 

14.6 

32.19 

18 

39.68 

2.304 

21 

1.85 

.0729 

12.5 

27.56 

16.25 

35.82 

20 

44-09 

2.56 

22 

2.  02 

•0795 

13-75 

30-31 

17.9 

39.46 

22 

48.5 

2.816 

23 

2.19 

.0862 

15 

33-07 

19-5 

42.99 

24 

52.91 

3-073 

24 

2-37 

•0933 

16.25 

35-82 

21.  1 

46.52 

26 

57-32 

3-329 

25 

2.52 

.0992 

17-5 

38.58 

22.75 

50.15 

28 

61.73 

3-585 

26 

2.66 

.1047 

18.8 

41.44 

24.4 

53-79 

31 

68.34 

3-060 

152        SHIP   AND    RAILROAD    SPIKES,   HORSESHOES. 

Railroad.    Spikes. 
(Dilworth,  Porter  &  Co.,  Pittsburg,  Pa.} 


Dimensions. 

In  keg 

Of  200 

Weight  of 
Rail  per 

Dimensions. 

In  keg 

Of  200 

Weight  of 
Rail  per 

Dimensions 

In  keg 

Of  200 

Weight  of 
Rail  per 

Lbs. 

Yard. 

Lbs. 

Yard. 

Lbs. 

Yard. 

Ins. 

No. 

Lbs. 

tns. 

No. 

Lbs. 

Ins. 

No. 

Lbs. 

2-5X-3I25 
3     X-3I25 

2-5X-375 

2230 
1880) 
1650) 

8  to  12 
12  to  16 

4-5X.375 

3-5X-4375 
4    X.4375 

780) 
89o| 
780) 

16  to  25 

4-5X-5 
5    X-S 

4-5X-5625 

5i8 
475} 
460) 

28  to  35 
35  to  40 

3    X-375 
3-5X.375 

1380) 
1250) 

16  to  20 

4-5X-4375 
3-5X-5 

690 
670) 

20  tO  30 

5    X-5625 
5-5X-5625 

405 
360 

40  to  56 
45  to  zoo 

4    X-375      1025 

16  to  25 

4 

X-5 

605 

24  to  35 

Street    Railway    Spikes. 

From  .25  to  .625  Inch.    Have  Countersunk  Heads. 

Square    Bolt    Spikes. 

.25  In. 

.25  In. 

.3i25ln. 

•375  In. 

•4375  In. 

•  Sin. 

.625  In. 

.752  In. 

Length. 

Length. 

Length. 

Length. 

Length. 

Length. 

Length. 

Length. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3  to  3-5      4  to  8 

4  to  8 

4  to  12 

6  to  12 

6  to  16 

8  to  16 

12  to  24 

Ship   and.   Railroad    Spikes. 

DIMENSIONS   AND   NUMBER   PER   POUND.      (P.  C.  Page,  MOSS.) 
Sliip    Spikes. 


X  In.  Sq. 

%,  In.  Sq. 

%  In.  Sq. 

tf  In.  Sq. 

%  In.  Sq. 

^  In.  Sq. 

Xln.Sq. 

1 

fifS 

1 

fl  T3 

43 
gi 

•s-g 

i 

e  ^ 

• 

°i 

ID 

S"g 

| 

C  "C 

J 

II 

1 

-0  § 
fcfc 

,2 

o  § 

fc<£ 

! 

<3  | 

^&H 

a 

0  § 

^p^ 

J_ 

il. 

J 

M 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3 

19 

3 

10 

4 

5-4 

5 

3-4 

6 

2.2 

8 

1.4 

10 

.8 

3-5 

15-8 

3-5 

9.6 

4-5 

5 

5-5 

3-i 

6-5 

2 

9 

1.2 

15 

.6 

4 

13.2 

4 

8 

5 

4.6 

6 

3 

7 

1.9 

10 

I.I 

— 

4-5 

12.2 

4-5 

6 

5-5 

4.2 

6-5 

2.8 

7-5 

1.8 

ii 

I 

— 

— 

5 

10.2 

5 

5-8 

6 

4 

7 

2.6 

8 

i«7 

— 



— 

— 

— 



6 

5-2 

6.5 

3-2 

7-5 

2.4 

8-5 

1.6 

— 



— 

— 

— 



— 

— 

— 

— 

8 

2.2 

9 

i-5 

— 



— 

—  > 

— 



— 

— 

— 

— 

— 



10 

1.4 

— 



— 

— 

Railroad  Spikes  5        inch  square  X  5 

r   ine    o        r>pr  IK 

"             "                 .  .$621     "          "       *  c.e    "     r.fi       " 

Spikes    and   Horseshoes. 
LENGTH  AND  NUMBER  PER  POUND.     (//.  Burden,  Troy,  N.  F.) 


d 

be 
1 

Boat  g 

pikes. 

1 

3 

d    . 

0-3 

i 
I 

Ship 

£ 

Spikes. 

I 

.S  • 

Hook  Hea 
Length. 

d. 

c   . 

Horse 

§ 

1 

shoes. 

.£  • 

3 
3-5 
4 
4-5 
5 
5-5 
6 

17-5 
14.68 

12-57 
9.2 

7-2 

6.3 

4-97 

Ins. 

6.5 

7 

7-5 
8 

8-5 
9 

10 

4.78 
3-62 
3-37 
2-95 
2.9 

2.1 
1.08 

Ins. 

4 

4.5 

5 
5-5 
6 

6-5 
7 

8 
6.5 
4-37 
4-3 
4.2 

3-77 
2-75 

Ins. 

7-5 
8 

8-5 
9 

10 

2-5 
1.74 
1.63 

Ins. 

4    X.375 
4-5  X.  4375 

5-5X-5 
5.5  X.  5625 
6    X  .5625 
6    X  .625 

5-55 
4.14 
2.52 
2.41 
1.87 
1.72 
1.38 

Ins. 
I 
2 

3 
4 
5 

.84 
•75 
•65 
•56 
•39 

CAST   IEON   AND   LEAD    BALLS. — NAILS. 


153 


\Veight  and  "Volume  of  Cast  Iron  and.  Lead  Sails. 

From  i  Inch  to  20  Inches  in  Diameter. 


Diameter. 

Volume. 

Cast  Iron. 

Lead. 

Diameter. 

Volume. 

Cast  Iron. 

Lead. 

lus. 

Cube  Ins. 

Lbs. 

Lbs. 

Ina. 

Cube  Ins. 

Lbs. 

Lbs. 

I 

.523 

.136 

.215 

9 

381.703 

99-51 

156.553 

i-5 

1.767 

.461 

.725 

9-5 

448.92 

117.034 

184.121 

a 

4.189 

1.092 

1.718 

JO 

523-599 

136.502   214.749 

2-5 

8.181 

2.133 

3-355 

10.5 

606.132 

158.043  j  248.587 

3 

14.137 

3-685 

5.798 

II 

696.91 

181.765 

285.832 

3-5 

22.449 

5-852 

9.207 

11.5 

796.33 

207.635 

326.591 

4 

33-5  i 

8.736 

13-744 

12 

904.778 

235.876 

371.096 

4-5 

47-7J3 

12.439 

19.569 

12.5 

1022.656 

266.647 

419.512 

5 

6545 

17.063 

26.843 

13 

1150.346 

299.623 

471.806 

5-5 

87.114 

22.721 

35-729 

14 

1436.754 

374-563 

589-273 

6 

113.097 

29.484 

46-385 

15 

1767.145 

460.696 

724.781 

6-5 

143-793  i  37-453 

58.976 

i  16 

2144.66 

559-  "4 

879.616 

7 

179-594 

46.82 

73-659!;  17 

2572.44 

670.717 

1055.066 

7-5 

220.893 

57.587 

90.598 

18 

3053.627 

796.082 

1252.422 

8 

268.082 

69.889 

109.952 

19 

359I-363 

936.271 

1472.97 

8-5 

321.555 

83-84 

131.883 

20 

4188.79 

1092.02 

1717-995 

NOTE.  —To  compute  weight  of  balls  of  other  metals,  multiply  weight  given  in 
table  by  following  multipliers: 


Steel                                      i  088 

Gun-metal...                             ..  i.i6c. 

Weight  and  Diameter  of  Cast   Iron  Balls. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

I 

1.94 

12 

4-45 

50 

7.l6 

224 

n.8 

1344 

21.44 

2 

2-45 

14 

4.68 

56 

7-43 

336 

13.51 

1568 

22.57 

3 

2.8 

16 

4.89 

DO 

7-6 

448 

14-87 

1792 

23-6 

4 

3-08 

18 

5-09 

70 

8.01 

560 

16.02 

20l6 

24-54 

5 

3-32 

20 

5-27 

80 

8-37 

672 

17.02 

2240 

25.42 

6 

3-53 

25 

5-68 

90 

8.71 

784 

17.91 

2800 

27.38 

7 

3-72 

28 

5-9 

IOO 

9.02 

896 

18.73 

3360 

29.1 

8 

3-89 

30 

6.04 

112 

9-37 

I008 

19.48 

3920 

30.64 

9 

4.04 

40 

6.64 

168 

10.72 

1  120 

20.17 

4480 

32.03 

No.  5  , 
"   6 


Length,   of  Horseshoe   Nails. 
By  Numbers. 

.  1.5  Ins.  I  No.  7 1.875  Ins.  I  No.  9 2.25  Ins. 

,  1.75  "     1    "    8 2  "     I    "10 2.5      " 


Lengths   of  Iron   Nails,   and   Number  in   a 


3d. 


L'gth. 


1.25 


420 

270 


L'gth. 


i-75 


L'gth, 


2-5 


Size. 

L'gth. 

,20 

Ins. 
3-25 

3-5 

40 


L'gth. 


4 
4.25 


1 54 


NAILS,  SPIKES,  TACKS,  ETC. 


Wrough-t    Iron.  Cut   Nails,  Tacks,  Spikes,  etc. 

(Cumberland  Nail  and  Iron  Co.) 
Lengths  and  Number  per  Lb. 


c 

Size. 

>rdinar 

Length. 

y- 

No.  per  Lb. 

I 

Size. 

~*inish 

Length. 

ing. 

No.  per  Lb. 

Size. 

Shingl 

Length. 

3. 

No.  per  Lb 

Ina. 

Ina. 

Ina. 

2d 

7l6 

4d 

1-375 

384 

5d 

i-75 

I78 

3  fine 

1.0625 

588 

5 

256 

8 

2-5 

74 

3 

1.0625 

448 

6 

2 

204 

9 

2-75 

00 

4 

1-375 

336 

8 

2-5 

102 

10 

3 

52 

5 

i-75 

216 

10 

3 

80 

Tacks. 

6 
7 

2 
2.25 

166 
118 

12 
2O 

3.625 
3.875 

65 

46 

I  OZ. 

.125 

•1875 

16000 
10666 

8 

2-5 

94 

Core. 

2 

•25 

8000 

10 
12 

2-75 

3-5 

72 

50 

6d 

8 

2 

143 

68 

2.5 

3 

•3125 
•375 

-  6400 
5333 

20 
30 
40 
50 
00 

4d 

3-75 
4-25 
4.75 
5 
5-5 
Light 
1-375 

32 

20 

14 
10 

373 

10 

12 
20 
30 

40 

WH 
WHL 

2333 
3-125 
3-75 
4.25 
4-75 
2-5 
2  25 

60 
42 
25 
18 

14 
69 

72 

6 
8 

10 
12 

16 

•4375 
-5625 
.625 
.6875 
•75 
.8125 

.875 

4000 
2666 

2000 
I  000 
1333 
H43 
I  000 

5 

*-75 

272 

18 

•9375 

888 

6 

2 

196 

Clinch. 

20 

i 

800 

Brads. 

6d 

2 

152 

Boat. 

6d 

2 

163 

2.25 

133 

Size. 

No.  per  Lb. 

8 

2-5 

96 

2-5 

92 

Ina. 

10 

2-75 

74 

10 

2-75 

72 

206 

12 

3-125 

Fence 

50 
'. 

— 

3 
3-25 

60 

43 

Spi 

3-5 

fees. 

6d 

2 

96 

Slate. 

4 

15 

7 

2.25 

66 

3d 

1.625 

288 

4-5 

13 

8 

2-5 

56 

4 

1-4375 

244 

5 

10 

10 

2-75 

50 

5 

i-75 

187 

5-5 

9 

— 

3 

40 

6 

2 

146 

6 

7 

Railroad.   Spikes. 

Number  in  a  Keg  of  150  Ibs. 


Length. 

No. 

Length. 

No. 

Length. 

No. 

Length. 

No. 

No. 

3  X  .375 
3-5  X  .375 
4  X  -375 

930 
890 
760 

5-5  X 

3-5  X  .4375 
4  X  .4375 
4-5  X  .4375 
.5625  standai 

675 
540 
510 

•d  for 

Ins. 

4X.5 
4-5  X  .5 
5X  .5 

a,  gauge  of  * 

450 
400 
340 

1-feet 

Ins. 

5  X  .5625 

5-5  x  .5625 
3.5  ins. 

300 
280 

Sliip   and.  Boat   Spikes. 

Number  in  a  Keg  of  150  Ibs. 


Length. 

No. 

Length. 

No. 

Length. 

No. 

455 
424 
390 
384 
300 

Length. 

No. 

Ina. 
4     X.25 
4-5  X.25 
5    X.25 
6    X.25 
7    X.25 

1650 
1464 
I380 
1292 

1161 

Ina. 

5  X.  3125 
6X-3I25 
7  X.  3125 
6X.375 
7X-375 

930 

868 
662 

570 
482 

Ins. 

8X.375 
9X-375 
10  X.  375 
8  X.  4375 
9X-4375 

Ina. 

10  X,  4375 
8X.5 
9X.5 
iox.5 
iiX-5 

270 
256 
240 
222 
203 

VAEIOUS   METALS. 


155 


"Weight   of  "Various   Mletals. 
Per  Cube  Inch  and  Foot. 


AfCTALS. 

Spec. 
Gravi- 
ty. 

W'ght 
in  an 
Inch. 

Ins. 
Lb! 

Weight 
in  a 
Foot. 

METALS. 

Specific 
Gravi- 
ty. 

W'ght 
in  an 
Inch. 

ini. 
in* 
Lb. 

Weight 
in  a 
Foot. 

Wrought-iron 
plates 
"    wire. 
Cast  iron  
Steel  plates.. 
"    wire... 
Copper,    (  .  .  . 
rolled  {... 
Gun  -metal,) 
cast  j 

Wrought  iron 
Cast  iron  .... 
Steel 

7734 
7774 
7209 

gj 
8750 

7.698 
7.217 
7.852 
8.805 
8.404 

Lb. 

•2797 
.2812 
.2607 
.2823 
.2838 
.3146 
.3212 

•3^5 

.278 
.26 
.283 
.318 
•304 

3-57 
3-55 
3-84 
3-54 
3-52 
3-i9 
3-" 
3.16 

En* 

31  ' 
3-84 

3-53 
3-i5 
3-29 

Lba. 
483-38 
485.87 
450-  54 
487.8 
490.45 
543-6 
555 

546-875 

ylish. 
48o 

450 
489.6 

549 
524 

Brass,  rolled. 
"     cast.  .  . 
Lead,  rolled  . 
Tin,  cast  
Zinc,  rolled.. 
Alumini-     ) 
um,  cast  j 
Silver 

8217 
8080 
11340 
7292 
7188 

2560 
10480 
8379 

7.409 
7.008 
11.418 
8.099 
8.548 

Lb. 
.2972 
.2922 
.4101 
•2673 
.26 

.0926 

•3791 
.3031 

.268 

•253 
.412 
.292 
.308 

3-37 
3-42 
2-44 
3-74 
3-85 
10.8 
2.64 
3.299 

3-74 
3-95 
2-43 
3-42 
3-24 

Lb. 

5i3-6 
505 
708.73 
462 
449.28 

1  60 

655 
523.69 

462 

437 
712 

5<>5 
533 

Tobin  Bronze. 
(D.K.  Clark.) 
Tin  

Zinc 

Lead.   .   .   . 

Copper  plates 
Gun-metal.  .  . 

Brass,  cast.  .  . 
u     wire.. 

WROUGHT  AND  CAST  IRON. 
To   Compute  "Weight  of  "Wrought  or  Cast   Iron. 

RULE.— Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .2816*  for 
wrought  iron  and  .2607*  for  cast,  and  product  will  give  weight  in  pounds. 

Or,  for  cast  iron  multiply  weight  of  pattern,  if  of  pine,  by  from  18  to  20,  accord- 
ing to  its  degree  of  dryness. 

EXAMPLE.— What  is  weight  of  a  cube  of  wrought  iron  10  inches  square  by  15 
inches  in  length  ? 

10 X  10 X  15  X- 2816  =  422.4  Its. 

COPPER. 

To    Compute   "Weight   of  Copper. 

RULE.— Ascertain  number  of  cube  inches  in  piece  ;  multiply  sum  by  .321 18,* 
and  product  will  give  weight  in  pounds. 

Sheathing   and    Braziers'    Sheets. 

For  dimensions  and  weights  see  Measures  and  Weights,  pages  118-121, 131, 142. 


LEAD. 

To    Compute   Weight   of   Lead. 

RULE. — Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .41015,* 
and  product  will  give  weight  in  pounds. 

EXAMPLE. — What  is  weight  of  a  leaden  pipe  12  feet  long,  3.75  inches  in  diameter, 
and  i  inch  thick? 

By  Rule  in  Mensuration  of  Surfaces,  to  ascertain  Area  of  Cylindrical  Rings. 
Area  of  (3.75  +  1  +  1)  =  25. 967 
"   3-75  —".044 

Difference,  14.923  (area  of  ring)  X  144  (12  feet)  =  2148.912 
X. 410 15  =  881.376  Ibs. 

BRASS. 
To   Compute  "Weight   of  Ordinary   Brass    Castings. 

RULE. — Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .2922,*  and 
product  will  give  weight  in  pounds. 

*  Weights  of  a  cube  inch  as  here  given  are  for  the  ordinary  metals ;  when,  however,  the  specific 
gravity  of  the  metal  under  consideration  is  accurately  known,  the  weight  of  a  cube  inch  of  it  should 
be  substituted  for  the  units  here  given. 


156      DIMENSIONS  AND  WEIGHTS  OF    BOLTS  AND  NUTS. 

Dimensions  and  "Weights  of  "Wrought  Iron  Bolts 
and.  IN"irts. 

SQUARE   AND   HEXAGONAL  HEADS   AND   NUTS. 

ftcmgli,  and  from  .25  Inch  to  4  Inches  in  Diameter. 
Square   Head   and   JSTut. 


Diameter 
of  Bolt. 

Wid 
Head. 

th. 

Nut. 

Diagc 
Head. 

nal. 
Nut. 

De 
Head. 

Pth. 

Nut. 

Weight. 
Head           Bolt 
and  Nut.  jper  Inch. 

Threads 
per  Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

No. 

•25 

.36 

•49 

•51 

.69 

•25 

•25 

.024 

.014 

20 

•3125 

•45 

.58 

.64 

.82 

•3 

•3125 

•043 

.022 

18 

•375 

-54 

.67 

.76 

•95 

-34 

•375 

.068 

.031 

16 

•4375 

•63 

.76 

.89 

1.07 

•4 

•4375 

.104 

.042 

14 

•5 

.72 

.84 

1.02 

1.19 

•44 

•5 

•145 

•055 

13 

•5625 

.82 

•94 

1.16 

i-33 

.48 

•5625 

.204 

.07 

12 

.625 

.91 

1.03 

1.29 

1.46 

•53 

.625 

•273 

.086 

II 

.6875 

i 

1.  12 

1.41 

1.58 

•58 

.6875 

•356 

.IO4 

II 

•75 

1.09 

1.  21 

1.54 

1.71 

•63 

•75 

•454 

.124 

10 

.8125 

1.18 

•3 

1.67 

1.84 

.67 

.8125 

•565 

•145 

10 

.875 

1.27 

•39 

1.8 

1.96 

.72 

.875 

.696 

.168 

9 

i 

J-45 

•57 

2.05 

2.22 

.81 

i 

1.013 

.22 

8 

1.125 

1.63 

•75 

2-3 

2.47 

•9 

1.125 

1.416 

.278 

7 

1.25 

1.81 

•94 

2.56 

2.74 

i 

1.25 

1.923 

•344 

7 

1-375 

1.99 

2.12 

2.81 

3 

i.i 

1-375 

2-543 

.416 

6 

i-5 

2.17 

2-3 

3-°7 

3-25 

1.18 

i-5 

3-234 

•495 

6 

1.625 

2.36 

2.48 

3-34 

3.5I 

1.28 

1.625 

4.105 

.581 

5-5 

1-75 

2-54 

2.66 

3-59 

3.76 

i-37 

1-75 

5.087 

.674 

5 

1.875 

2.72 

2.84 

3-85 

4.02 

1.46 

1.875 

6.182 

•773 

5 

2 

2.9 

3.02 

4.1 

4.27 

1.56 

2 

7.491 

.88 

4-5 

2.125 

3.08 

3.21 

4-35 

4-54 

1.65 

2.125 

8.936 

•993 

4-5 

2.25 

3-26 

3-39 

4.61 

4-79 

1.75     2.25 

iQ-543 

1.113 

4-5 

2-375 

3.44 

3-57 

4.86 

5-05 

1.84     2.375 

12-335 

1.24 

4-375 

2-5 

3-62 

3-75 

5.12 

5-3 

1.94 

2-5 

14-359 

1-375 

4-25 

2.625 

3-81 

3-93 

5-49 

5-56 

2.03 

2.625 

16.549 

i.5i5 

4 

2-75 

3-99 

4.11 

5-64 

5-8i 

2.12 

2-75 

18.897 

1.663 

4 

2.875 

4.17 

4.29 

5-9 

6.07 

2.22       2.875 

21-545 

1.818 

3-75 

3 

4-35 

4-47 

6.15 

6-32 

2-31     3 

24.464 

1.979 

3-5 

3-25 

4.71 

4.84 

6.66 

6.84 

2.5 

325 

30.922 

2.323 

3-5 

3-5 

5-07 

5-2 

7.17 

7-35 

2.68 

3-5 

38-391 

2.694 

3-25 

3-75 

5-44 

5.56 

7.69 

7.86 

2.87 

3-75 

47.168 

3-093 

3 

4 

5-8 

5-92 

8.2 

8-37 

3.06  I  4 

56.882 

3-518 

3 

FINISHED. — Deduct  .0625  from  diameters  of  bolts  and  depths  of  all  heads 
and  nuts.  For  Steel  Bolts,  add  i.  3  per  cent. 

Screws  with  square  threads  have  but  one  half  number  of  threads  of  those 
with  triangular  threads. 

NOTE.— The  loss  of  tensile  strength  of  a  bolt  by  cutting  of  thread  is,  for  one  of  i.  25 
ins.  diameter,  8  per  cent.  The  safe  stress  or  capacity  of  a  wrought  iron  bolt  and  nut 
may  be  taken  at  5000  Ibs.  per  square  inch. 

Preceding  width,  depth,  etc.,  are  for  work  to  exact  dimensions,  whether 
forged  or  finished. 

To    Compute    AVeiiglit    of  a,    Bolt    and    Nxit. 

Operation. — Ascertain  from  table  weight  of  head  and  nut  for  given  di 
ameter  of  bolt,  and  add  thereto  weight  of  bolt  per  inch  of  its  length,  multi- 
plied by  full  length  of  its  body  from  inside  of  its  head  to  end. 

NOTE.— Length  of  a  bolt  and  nut  for  measurement,  as  such,  is  taken  from  inside 
of  head  to  inside  of  nut,  or  its  greatest  capacity  when  in  position. 


DIMENSIONS  AND   WEIGHTS   OF  BOLTS   AND  NUTS.      157 

ILLUSTRATION. — A  wrought-iron  bolt  and  nut  with  a  square  head  and  nut  is  i  inch 
in  diameter  and  10  inches  in  length;  what  is  it's  weight? 

Weight  of  head  and  nut 1.013  \  7/10 

"       bolt  per  inch  of  length  .22  X  10  =  2.2     )  3'213 
For  Steel  Bolts,  add  1.3  per  cent. 

Hexagonal   Head.   and. 


Diameter 
of  Bolt. 

Wic 
Head. 

Ith. 
Nut. 

Diagonal. 
Head.  I   Nut. 

D 
Head. 

;pth. 
Nut. 

Weig 
Head 
and  Nut. 

ht. 

Bolt 
jer  Inch. 

Threads 
per  Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ibs. 

Lbs. 

No. 

•25 

•375 

•5 

•43 

•58 

•25 

•25 

.022 

.OI4 

20 

•3I25 

•4375 

.5625 

•5 

•65 

•3 

.3125 

•037 

.022 

18 

•375 

.5625 

•6875 

.65 

•79 

•34 

•375 

.002 

.031 

16 

•4375 

.625 

•75 

•72 

•87 

•4 

•4375 

.094 

.042 

14 

-5 

•75 

•875 

.87 

i 

•44 

•5 

•134 

•055 

13 

•5625 

.8125 

•9375 

•94 

i.  08 

.48 

•5625 

.18 

.07 

12 

•625 

•9375 

1.0625 

i.  08 

1.23 

•53 

.625 

.249 

.086 

II 

•6875 

i 

1.125 

1.16 

•58 

.6875 

•318 

.104 

II 

•75 

1.125 

1.25 

1.3 

1.44 

•63 

•75 

•413 

.124 

IO 

•8125 

1.25 

1-375 

1.44 

1-59 

.67 

.8125 

•522 

•145 

IO 

.875 

1-3125 

1-4375 

1.52 

1.66 

.72 

•875 

•639 

.168 

9 

1-5 

1.625 

i-73 

1.88 

.81 

•931 

.22 

8 

.125 

1.6875 

1.8125 

i-95 

2.09 

•9 

.125 

1.299 

.278 

7 

•25 

1.875 

2 

2.17 

2.31 

i 

•25 

1-759 

•344 

7 

•375 

2 

2.1875 

2.31 

2-53 

i.i 

•375 

2.263 

.416 

6 

•5 

2.25 

2-375 

2.6      2.74 

1.18 

.5 

2.958 

•495 

6 

1.625 

2-4375 

2.5625 

2.81     2.96 

1.28- 

.625 

3-741 

•581 

5-5 

1.75 

2.625 

2-75 

3-03  i  3.i8 

i-37  !     -75 

4-654 

.674 

5 

1.875 

2.8125 

2-9375 

3.25 

3-39  !  1-46 

1.875 

5.675 

•773 

5 

2 

3 

3-125 

3-46 

3.61 

1.56 

2 

6.854 

.88 

4-5 

2.125 

3-1875 

3-3I25 

3.68    3.83 

1.65    2.125 

8.163 

•993 

4-5 

2.25 

3-375 

3-5 

3-9 

4.04     i-75 

2.25 

9-658 

4-5 

2-375 

3-5625 

3.6875 

4.11     4.26     1.84 

2-375 

11.263 

.24 

4-375 

2-5 

3-75 

3*875 

4-33    4-47     1-94    2.5 

13-149 

•375 

4-25 

2.625 

3-9375 

4.0625 

4-55    4.69  ;  2.03  !  2.625 

4 

2-75 

4.125 

4-25 

4.77    4.91     2.12     2.75 

17-285 

^663 

4 

2.875 

4-3125 

4-4375 

4-99      5-12  ,  2.22      2.875 

19«75I 

.818 

3-75 

3 

4-5 

4.625 

5-2      5-34 

2-31     3 

22.378 

•979 

3-5 

3-25 

4.875 

5 

5-63     5-77 

2-5      3-25 

28.258 

2.323 

3-5 

3-5 

5-25 

5-375 

6.06    6.21     2.68    3.5 

35-o8i 

2.694 

3.25 

3-75 

5-625 

5-75 

6.5    6.64  2.87  ;  3.75 

43.178 

3.093 

3 

4 

6 

6.125 

6-93    7-07    3«o6    4 

51.942    3.518 

3 

FINISHED. — Deduct  .0625  from  diameters  of  bolts  and  depths  of  all  heads 
and  nuts. 

For  Wood  or  Carpentry. 

Head  and  Nut  (Square),  1.75  diameter  of  bolt.     Depth  of  Head,  .75,  and 
of  Xut,  .9. 

Washer. — Thickness,  .35  to  .4  of  diameter  of  bolt,  on  Pine  3.5  diameter, 
and  Oak  2.5. 

English. 

Moleswcn*th  gives  following  elements  of  Thread  of  Bolts : 

Angle  of  thread,  55°.     Depth  of  thread  =  Pitch  of  screw. 

Number  of  threads  per  Inch.  —  Square,  half  number  of  those  in  angular 
threads. 

Depth  of  thread. — .64  pitch  for  angular  and  .475  for  square  threads. 

0 


IJ8      DIMENSIONS  AND  WEIGHTS  OF  BOLTS  AND  NUTS. 
ITrencli   Standard.   Bolts   and  RTirts.    (Armengau&t.) 


HEXAGONAL,  HEADS  AND  NUTS. 


Equ 

Diainet 
of  Bolt. 

ilatei 

sr 

Si 

c( 

•alT 

•si 
ft 

'iangt 

Thicl 
Head. 

ilar  1 

ness. 
Nut. 

rhrea 

ll 

*! 

i. 

Safe 
Tensile 
Stress. 

& 

Diameter 
of  Bolt. 

^quare 

""S 
!£ 

Thn 

•s-g 

I* 

ei 

ad. 

2- 

c  a 

P 

Safe 

Tensile 
Stress. 

Mm. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Ins. 

Lbs. 

Mm. 

Ins. 

Ins. 

No. 

Ins. 

Lbs. 

5 

.2 

•J3 

18.1 

.24 

.2 

•55 

44 

20 

•79 

.072 

6-57 

1.82 

717 

7-5 

•3 

.22 

16 

•3 

•3 

.68 

99 

25 

.98 

.O8l 

5-97 

2.01 

I  142 

IO 

•39 

•31 

14.1 

•38 

•39 

.88 

178 

30 

1.18 

•093 

5-4 

2.22 

1635 

12.5 

•49 

•39 

12.7 

•44 

•49 

.04 

277 

35 

1.38 

.1 

4-93 

2.41 

22l8 

15 

•59 

.4811.5 

•52 

•59 

.2 

400 

40 

J-57 

.106 

4-53 

2.63 

2912 

17.5 

.69 

.58  10.6 

.58 

.69 

•4 

545 

45 

1.77 

.II4 

4.2 

2.85 

3074 

20 

•79 

.66 

9.8 

.66 

•79 

•5 

7i3 

50 

1.97 

.128 

3-9i 

3-07 

4547 

22.5 

.89 

.76   9.1 

.72 

.89 

.68 

902 

55 

2.17 

•13 

3-65 

3-3 

5288 

25 

.98 

.84   8.5 

.8 

.08 

.84 

I  120 

00 

2.36 

.14 

343 

3-5 

6540 

30 

1.18 

i.  02 

7-5 

94 

1.18 

2.16 

1035 

65 

2.56 

•15 

3-23 

3-7 

7660 

35 

1.38 

12 

6.7 

i.  08 

1.38 

2.48 

22l8 

70 

2.76 

.158 

3-o6 

3-92 

8893 

40 

1.58 

1.4 

6 

.22 

158 

2.8 

2912 

75 

2-95 

.166 

2.92 

4-13 

10214 

45 

1.77 

1-56,  5-5 

.36 

1.77 

3-2 

3674 

80 

3-15 

.174 

2.76 

4-36 

11603 

50 

1.97 

i-74   5-i 

•5 

1.97 

3-44 

4547 

85 

3-35 

.183 

2.63 

4.58 

13100 

55 

2.17 

1.92    4.7 

.64 

2.17 

3-76 

5288 

90 

3-54 

.192 

2.51 

4.78 

14794 

60 

2.36 

2.08,  4.4 

•74 

2.36 

4.08 

6540 

95 

3-74 

.2 

2.41 

5 

16352 

65 

2.56 

2.26   4.1 

.92 

2.56 

4.4 

7660 

100 

3-94 

.209 

2.31 

5.22 

18144 

70 

2.76 

2.44'  3-8 

2.06 

2.76 

4-7 

8893 

105 

4-i3 

.22 

2.22 

5-43 

20000 

75 

295 

2-6  |  35 

2.2 

2-95 

5 

10214 

no 

4-33 

.226 

2.13 

5-66 

21950 

80 

3-15 

2.78,  3-4 

2-34 

3-15 

5-35 

11468 

"5 

4-53 

•23 

2.O6 

5-87 

23990 

English.   Bolts   and  iNTuts.     (WUtworWs.) 
Hexagonal    Heads    arid.    N"nts,  and    Triangular   Threads. 


Diame 
Bolt. 

ter. 

•s-i 
II 

l"i 

ei 

D 
Head. 

sptb. 

Nut. 

Width 
of 
Head 
and 
Nut. 

Diana 
Bolt. 

eter. 

Base 
of 
Thread. 

Threads 
per  Inch. 

DeF 
Head. 

th. 
Nut. 

Width 
of 
Head 
and 
Nut. 

Ins. 

Inch. 

No. 

Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Ins. 

-I25 

•093 

40 

.109 

.125 

•338 

1.25 

1.067 

7 

1.094 

1.25 

2.048 

.1875 

•134 

24 

.164 

•1875 

.448 

1-375 

1.161 

6 

1.203 

1-375 

2.215 

.2187 

24 

i-5 

1.286 

6 

1.312 

1.5 

2.413 

•25 

.186 

20 

.219 

•25 

•525 

1.625 

1.369 

5 

1.422 

1.625 

2.576 

•3I25 

.241 

18 

•273 

•3125 

.001 

i-75 

1.494 

5 

L53I 

i-75 

2.758 

•375 

.295 

16 

.328 

•375 

.709 

1-875 

!-59 

4-5 

1.641 

1-875 

3.018 

•4375 

•346 

14 

•383 

•4375 

.82 

2 

I-7I5 

4-5 

i-75 

2 

3-T49 

•5 

•393 

12 

•437 

•5 

.919 

2.125 

1.84 

4-5 

1.859 

2.125 

3-337 

•5625 

•456 

12 

.492 

•5625 

.Oil 

2.25 

i-93 

4 

1.969 

2.25 

3-546 

.625 

.508 

II 

•547 

.625 

.101 

2-375 

2.055 

4 

2.078 

2-375 

3-75 

.6875 

•571 

II 

.60! 

.6875 

.201 

2-5 

2.18 

4 

2.187 

2-5 

3.894 

•75 

.622 

10 

.656 

•75 

.301 

2.625 

2.305 

4 

2.297 

2.625 

4.049 

.8125 

.684 

10 

.711 

.8125 

•39 

2-75 

2.384 

3-5 

2.406 

2-75 

4.181 

•8/5 

•733 

9 

.766 

•875 

•479 

2.875 

2.509 

3-5 

2.516 

2.875 

4.346 

•9375 

•795 

9 

.82 

•9375 

•574 

3 

2.634 

3-5 

2.625 

3 

4-531 

i 

.84 

8 

•875 

i 

.67 

3-25 

2.84 

3-25 

— 

— 

1,125 

•942 

7 

.984 

1.125 

.86 

3-5 

3-o6 

3-25 

— 

— 

— 

RETENTION   OF   SPIKES    AND    NAILS. 


Square    Heads    and   l^uts.     ( Whitwortk* s. ) 


159 


Dia 

Bolt. 

meter. 
Base  of 
Thread. 

Threads 
per  Inch. 

Dia 

Bolt. 

meter. 
Base  of 
Thread. 

Threads 
per  Inch. 

Dia 
Bolt. 

meter. 
Base  of 
Thread. 

Threads, 
per  Inch. 

Ins. 

3-75 
4 
4-25 

Ins. 
3-25 
3-5 

3-75 

No. 

3 

3 
2-875 

Ins. 

4-5 
4-75 
5 

Ins. 
3.875 
4.0625 

4-25 

No. 
2.875 

2-75 
2-75 

Ins. 
5-25 

5-5 
6 

Ins. 

4-4375 
4.625 

4.875 

No. 
2.625 
2.625 
2-5 

"Weight   of*  Heads   and    Nuts    in.    Lt>s.     (Molesworth.) 

Hexagonal,  1.07  D3.      Square,  1.35  D3.     D  representing  diameter  of  bolt 
in  inches. 

Retentiveness  of  'Wrought  Iron  Spikes  and.  Nails. 

Deduced  from  Experiments  of  Johnson  and  Sevan. 

SPIKES. 


SFIKI. 

WOOD. 

1 

| 

Depth  of 
Insertion. 

Force  re- 
auired  to 
raw  it. 

Ratio  of 
force  to 
weight. 

REMARKS. 

Ins. 

Ins. 

Ins. 

Lbs. 

Square  

Hemlockf 
Chestnut 

•39 
•37 

•38 

3-5 
3-5 

1297 
1873 

2.16 

Seasoned  in  part. 
Unseasoned. 

"     *  .... 

"     *  .... 

Yellow  pine 

•375 

•375 

3-375 

2052 

2-37 

Seasoned. 

i  r*f?  '  *  .... 

White  oak 

•375 

•375 

3-375 

3910 

4-52 

0 

u 

Locust 

•4 

•4 

3.5 

5967 

6.33 

ii   ef^' 

Flat  narrow.  . 

Chestnut 

•39 

•25 

3-5 

2223 

3-93 

Unseasoned. 

«         it 

White  oak 

•39 

•25 

3-5 

3990 

7-05 

Seasoned. 

(t                 U 

Locust 

•39 

•25 

3-5 

5673 

9-32 

" 

"    broad  .  . 

Chestnut 

•539 

.288 

3-5 

2394 

2.66 

Unseasoned. 

u           u 

White  oak 

•539 

.288 

3-5 

5330 

5.71 

Seasoned. 

U               U 

Locust 

•539 

.288 

3-5 

7040 

7.84 

u 

Square}  ^   • 

Hemlockf 

•4 

•39 

3-5 

1638 

Seasoned  in  part. 

"          >•  2,2 

Chestnutf 

-4 

•39 

3-5 

1790 

i.Si 

Unseasoned. 

"     J  Q  « 

Locust  f 

•4 

•39 

3-5 

3990 

4.17 

Seasoned  in  part. 

Round  and) 
grooved..) 

Ash 

Diam.  .5 

3-5 

2052 

2.21 

Seasoned. 

M 

M 

"      -5 

3-5 

245  * 

2.41 

u 

M 

White  oak 

«      .48 

3-5 

3876 

3-2 

u 

*  Burden's  patent.                                      t  Soaked  in  water  after  the  spikes  were  driven. 

NAIL. 

Length. 

Depth  of 
Insertion. 

Pine. 

NAILS 

Force  r& 
Hemlock. 

;. 

quired  to 
Elm. 

draw  it. 
Oak. 

Beech. 

Pressure  required 
to  force  them 
into  Pine. 

Sixpenny 
u 

Ins. 
2 
2 
2 

Ins. 

I 

1-5 

2 

Lba. 
I87 
327 
530 

Lbs. 
3I2 

539 

857 

Lbs. 
327 
571 
899 

Lbs. 
507 
675 
1394 

Lbs. 
667 
889 
1834 

Lbs. 
235 
4OO 

610 

General    Remarks. 

With  a  given  breadth  of  face,  a  decrease  of  depth  will  increase  retention. 

In  soft  woods,  a  blunt-pointed  spike  forces  the  fibres  downwards  and 
backwards  so  as  to  leave  the  fibres  longitudinally  in  contact  with  the  faces 
of  the  spike. 


l6<D     ANGLES  AND  DISTANCES.  —  DISTANCES  AND  ANGLES. 


To  obtain  greatest  effect,  fibres  of  the  wood  should  press  faces  of  the  spike 
in  direction  of  their  length  ;  thus,  a  round  blunt  bolt,  driven  into  a  hole  of 
a  less  diameter,  has  a  retention  equal  to  that  of  any  other  form,  when  wholly 
driven,  as  without  boring. 

The  retention  of  a  spike,  whether  square  or  flat,  in  unseasoned  chestnut, 
from  two  to  four  inches  in  length  of  insertion,  is  about  800  Ibs.  per  square 
inch  of  the  two  surfaces  which  laterally  compress  the  faces  of  the  spike. 

When  wood  was  soaked  in  water,  after  spikes  were  driven,  order  of  their 
retentive  power  was  Locust,  White  oak,  Chestnut,  Hemlock,  and  Yellow  Pine. 


Diameter  in  Inches. . 
Threads  per  Inch . . . 


G-as   IPipe   Threads. 

.125  I  .25  I  .375  I  .5     .75      i 
28  I    19  I     19  |  14      14 


1.25 
ii 


2 
II 


ANGLES    AND   DISTANCES. 


Angles    and    Distances    corresponding    to    Opening   of  a 

JRvile    of  Two    Feet. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle.  1  Distance. 

0 

Ins. 

o 

Ins. 

o 

Ins. 

O 

Ins. 

O 

Ins. 

I 

.2 

19 

396 

37 

7.6l 

55 

II.08 

73 

14.28 

a 

.42 

20 

4.17 

38 

7.8l 

56 

11.27 

74 

14.44 

3 

.63 

21 

4-37 

39 

8.01 

57 

"•45 

75 

14.61 

4 

.84 

22 

4.58 

40 

8.2 

58 

11.64 

76 

14.78 

5 

1.05 

23 

4.78 

41 

8.4 

59 

11.82 

77 

14.94 

6 

1.26 

24 

4-99 

42 

8.6 

60 

12 

78 

15.11 

7 

1.47 

25 

43 

8.8 

61 

12.  l8 

79 

I5-27 

8 

1.67 

26 

5-4 

44 

8-99 

62 

12.36 

80 

15.43 

9 

1.88 

27 

5-6 

45 

9.18 

63 

12.54 

81 

J5-59 

10 

2.09 

28 

5.81 

46 

9-38 

64 

12.72 

82 

ii 

2.3 

29 

6.01 

47 

9-57 

65 

12-9 

83 

15-9 

12 

2.51 

30 

6.21 

48 

9.76 

66 

1307 

84 

16.06 

13 

2.72 

31 

6.41 

49 

9-95 

67 

I3-25 

85 

16.21 

14 

2.92 

32 

6.62 

50 

10.14 

68 

13,42 

86 

16.37 

15 

33 

6.82 

51 

10.33 

69 

13-59 

87 

16.52 

16 

3-34 

34 

7.02 

52 

10.52 

70 

13-77 

88 

16.67 

17 

3-55 

35 

7.22 

53 

10.71 

71 

13.94 

89 

16.82 

18 

3-75 

36 

7.42 

54 

10.9 

72 

14.11 

90 

16.97 

Distances    and    A.ngles    corresponding    to    Opening    of    a 

Rnle   of  Two    Feet. 

Distance 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance 

Angle. 

Ins. 

o 

Ins. 

0 

Ins. 

0 

Ins. 

0 

Ins. 

o 

•25 

1.  12 

3 

14.22 

6-5 

31.26 

10 

49.14 

13-5 

68.28 

•375 

1.48 

3-25 

15-34 

6-75 

32-4 

10.25 

50-34 

13-75 

69-54 

•5 

2.24 

3-5 

16.46      7 

33-54 

10-5 

51-54 

14 

71.22 

.625 

2-59 

3-75 

I7-58      7-25 

35-09 

10-75 

53-14 

14.25 

•75 

3-35 

4 

19.11 

7-5 

36.24 

II 

54-34 

14-5 

74-2 

•875 

4.12 

4-25 

20.24 

7-75 

37-4 

11.25 

55-54 

I4-75 

75-5 

£ 

4-48 

4-5 

21.37 

8 

38-56 

"•5 

15 

77.22 

1.25 

5-58 

4-75 

22.5 

8.25 

40.12 

"•75 

5^38 

15-25 

78.54 

*-5 

7-1 

5 

24.4 

8-5 

41.28 

12 

60 

15.5 

80.28 

I-75 

8.<22 

5-25 

25.16 

8-75 

42.46 

12.25 

61.23 

82.2 

2 

9-34 

5-5 

26.3 

9 

44.2 

12.5 

62.46 

16 

83-36 

2.25 

10.46 

5-75 

27-44 

9-25 

45-2 

1275 

64.1 

16.25 

85-14 

2-5 

11-58 

6 

28.58 

95 

46.38 

65.36 

16.5     1  86.52 

2-75 

13.1 

6.25 

30-12 

9-75 

47-56 

'3-25 

67.02 

16.75  [88.32 

WIRE  ROPE.  1 6 1 


WIRE  ROPE. 

Wire  rope  will  run  over  sheavee  of  like  diameter  to  Hemp  rope  of  same 
strength ;  but  larger  sheaves  reduce  wear.  Adhesion  is  the  same  as  that  of 
hemp  rope.  Wear  increases  rapidly  with  speed.  Short  bends  should  be 
avoided.  In  substituting  wire  rope  for  hemp,  allow  same  weight  per  foot. 
Kinking  wire  rope  materially  damages  and  often  destroys  it. 

For  transmission  of  power,  wire  rope  can  be  used  up  to  distances  of  3 
miles.  For  distances  less  than  100  feet,  it  is  not  advised  for  long  trans- 
mission; sheaves  are  placed  at  intervals,  dividing  it  into  a  number  of 
shorter  ones  of  250  to  300  feet. 

Strength  per  square  inch  of  section  of  rope  is  about  50  per  cent,  of  an 
equal  section  of  solid  metal  of  §ame  strength  per  square  inch. 

Stationary  wire  ropes  should  be  kept  well  painted  or  tarred  to  prevent 
their  oxidation.  Running  ropes  should  always  be  well  lubricated  and  pro- 
tected from  grit  with  linseed-oil,  pine  tar,  graphite  grease,  or  any  similar 
non-acid  substances. 

Standard  wire  rope  is  made  of  6  strands  of  7,  12,  or  19  wires  each,  with 
hemp  or  wire  centre.  Wire  centre  adds  10  per  cent,  to  strength  and  weight 
of  rope,  but  reduces  its  flexibility  proportionally. 

Safe  working  load  for  standing  ropes  is  about  one  fourth  ultimate  strength, 
arid  for  running  ropes  it  is  from  one  fifth  to  one  seventh. 

Ropes  for  hoisting  are  composed  of  6  strands  of  19  wires  each  around  a 
hemp  centre. 

Ropes  for  transmission  of  power,  for  guys  and  rigging,  are  composed  of 
6  strands  of  7  or  12  wires  each. 

The  ultimate  strength  of  wires  of  which  wire  ropes  are  made  are  for : 
Iron  wire,  70000  to  90000  Ibs.  per  sq.  inch;  Bessemer  steel  wire,  100000  to 
iioooo  Ibs.;  Crucible  cast -steel  wire,  150000  to  180000  Ibs.,  and  Special 
plough-steel  wire,  210000  to  300000  Ibs. 

Special  ropes  can  be  made  of  4,  6,  8,  etc.,  strands  of  varied  construction. 
Wire  ropes  are  also  made  flat,  composed  of  several  strands  alternately 
twisted  to  right  and  left,  laid  alongside  each  other,  and  sewed  together 
with  soft  iron  wire. 

Wire  hawsers  of  steel  are  made  of  6  strands  of  12  wires  each  with  hemp 
centre,  around  a  common  hemp  centre,  and  are  as  flexible  as  hemp  hawsers 
of  equal  strength. 

Galvanized  wire  rope  replaces  hemp  for  rigging,  because  of  its  cheapness, 
durability,  and  resistance  to  stretch.  It  is  one  fifth  bulk  for  equal  strength 
of  hemp  "rope,  and  offers  less  surface  to  wind. 

Tiller  ropes  for  vessel-steering  gear  are  made  of  6  smaller  ropes  around 
a  hemp  centre,  each  small  rope  composed  of  6  strands  of  7  wires  each  with 
hemp  centre — 252  wires  in  all  in  the  rope,  giving  great  flexibility. 

Yacht  rigging  of  galvanized  cast-steel  rope  is  one  third  to  one  half  weight 
of  iron  wire  rope  of  equal  strength. 


162 


WIRE    ROPES. 


Elements  of  Hoisting   and   Haulage  TVire    Rope. 

John  A.  Roebling's  Sons  Co. ,  Trenton,  N.  J. 
HOISTING  ROPE.      19  Wires  in  a  Strand.     Hemp  Centre. 


SWEDISH  IRON. 

CAST-STKEL. 

Diameter. 

Approx- 
imate 
Circum- 
ference. 

Weight 
Foot. 

Tons  of  2000  Lbs. 
Breaking         Safe 
Strain.        Strain. 

Least 
Diameter 
of  Drum 
or  Sheave. 

Tons  of  2 
Breaking 
Strain. 

ooo  Lbs. 
Safe 
Strain. 

Least 
Diameter 
of  Drum 
or  Sheave. 

Ins. 

lus. 

Lbs. 

No. 

N 

0 

Feet. 

No. 

No. 

Feet. 

2.25 

7-125 

8 

78 

15 

.6 

i 

3 

156 

31.2 

8-5 

2 

6.25 

6-3 

[ 

62 

12 

•4 

i 

2 

24.8 

8 

1.75 

5-5 

4-85 

48 

9 

.6 

10 

96 

19.2 

7-25 

1.625 

5 

4-15 

42 

8 

•4 

8-5 

84 

16.8 

6.25 

1-5 

4-75 

3-5 

) 

36 

7 

.2 

7-5 

72 

14.4 

5-75 

1-375 

4-25 

3 

3' 

6 

.2 

7 

62 

12.4 

5-5 

1-25 

4 

2-4. 

) 

25 

5 

6-5 

50 

10 

5 

1.125 

3-5 

2 

21 

4 

.2 

6 

42 

8.4 

4-5 

i 

3 

1-5* 

5 

17 

3 

•4 

5-25 

34 

6.8 

4 

.875 

2-75 

1.2 

13 

2 

.6 

4-5 

26 

S-2 

3-5 

•75 

2.25 

•8c 

) 

9- 

7 

X 

•94 

4 

19.4 

3-88 

3 

.625 

2 

.62 

6. 

8 

I 

.36 

3-5 

13.6 

2.72 

2.25 

•5625 

1.75 

•5 

5- 

5 

I 

.  i 

2-75 

ii 

2.2 

•5 

1.5 

) 

4- 

4 

.88 

2.25 

8.8 

X.76 

J-5 

•4375 

1.25 

•  3' 

3- 

4 

.68 

2 

6.8 

1.36 

'•25 

•375 

1.  12 

5 

.25 

2. 

5 

•5 

1.5 

5 

I 

i 

•3125 

I 

.1; 

I. 

7 

•34 

I 

3-4 

.68 

.667 

•25 

•75 

.1 

I. 

2 

.24 

•75 

2.4 

.48 

•  5 

'.Transmission    and    Haulage    Rope. 

7  Wires  in  a  Sti 

'and.     h 

emp  Cer 

itre.    No 

rn.—  Add 

10  per  cent,  to  weight  for  WIRE  CENTRE. 

Swi 

DISH  IRON. 

CAST-STKEL. 

Diameter. 

Approx- 
imate 
Circum- 
ference. 

Weight 
per 
Foot. 

Tons  of  2 
Breaking 
Strain. 

ooo  Lbs. 
Safe 
Strain. 

Least 
Diameter 
of  Drum 
or  Sheave. 

Tons  of  2 
Breaking 

Strain. 

ooo  Lbs. 
Safe 
Strain. 

Least 
Diameter 
of  Drum 
or  Sheave. 

Ins. 

Ins. 

Lbs. 

No. 

No. 

Feet. 

No. 

No. 

Feet. 

•5 

4-75 

3-55 

34 

6.8 

13 

68 

13-6 

8.5 

•375 

4-25 

3 

29 

5- 

S 

12 

58 

ii.  6 

8 

•25 

4 

2-45 

24 

4.8 

10.75 

48 

9.6 

7-25 

.125 

3-5 

2 

20 

4 

9 

-5 

40 

8 

6.25 

3 

1.58 

16 

3- 

2 

8-5 

32 

6.4 

5-75 

'.875 

2-75 

1.2 

12 

2. 

4 

7 

•  5 

24 

4.8 

5 

•75 

2.25 

.85 

9- 

3 

I. 

86 

6 

•75 

18.6 

3-72 

4-5 

.6875 

2.125 

•75 

7- 

9 

I.58 

6 

15-8 

3.16 

4 

.625 

2 

.62 

6. 

6 

I  . 

32 

5 

•25 

13.2 

2.64 

3-5 

-5625 

1.75 

•5 

5- 

3 

I  .06 

4 

-5 

10.6 

2.12 

3 

•  5 

1.5 

•3S 

4- 

2 

. 

84 

4 

8.4 

1.68 

2-5 

•4375 

1.25 

•  3 

3' 

3 

.66 

3 

•25 

6.6 

1.32 

2.25 

•375 

I.I25 

.22 

2. 

4 

.48 

2-75 

4.8 

.96 

2 

•3125 

I 

.15 

I. 

7 

34 

2 

-5 

3-4 

.68 

!-75 

.2813 

.875 

.125 

I. 

4 

28 

2 

-25 

2.8 

•56 

Galvanized    Charcoal    Iron.    "Wire    Rope. 

Vessels' 

Rigging    a 

lid    .Derrick    GJ-viys. 

7  or  12  Wire 

5  in  a  Strand.     Hemp  Centre. 

Approx- 
imate 
Diam- 
eter. 

3ircum- 
erence. 

Weight 
per 
Foot. 

Breaking 
Strain 
in  Tons 
of  2000 
Lbs. 

of  Maniltt 
Rope  of 
Equal 
Strength 

Approx- 
imate 
Diam- 
eter. 

Circum-          **% 
ference.        jf^ 

Breaking 
Strain 
in  Tons 

Of  2OOO 

Lbs. 

Circum. 
>f  Manila 
Rope  of 
Equal 
Strength. 

Ins. 

Ins. 

Lbs. 

No. 

Ins. 

Ins. 

Ins.            Lbs. 

No. 

Ins. 

•75 

5-5 

4-85 

44 

ii 

•25 

4             2.55 

23 

8 

.6875 

5-25 

4-4 

40 

10.5 

•1875 

3-75        2.25 

20 

7-5 

.625 

5 

4 

36 

10 

.125 

3-5          i-95 

18 

6.5 

•5 

4-75 

3-6 

32 

9-5 

.0625 

3-^5        i-7 

15 

6 

•4375 

4-5 

3-25 

29 

9 

3              i-44 

13 

5-75 

•375 

4-25 

2.9 

26 

8-5 

'-875 

2.75        i.  21 

II 

5-25 

WIRE    ROPES,   HAWSERS,   AND   CABLES. 


i63 


Gralvanized    Charcoal    Iron    "Wire    Rope. 
Vessels'    Rigging    and    Derriclz    Griiys. 

John  A.  Roebling'ls  Sons  C"o.,  Trenton,  N.  J. 
7  Wires  in  a  Strand. 


Approx- 
imate 
Diam- 
eter. 

Circum- 
ference. 

Weight 
Fwt. 

Breaking 
Strain 
in  Tons 
of  POOO 
Lbs. 

Circum. 
of  Manila 
Rope  of 
Equal 
Strength. 

Approx- 
imate 
Diam- 
eter. 

Circum- 
ference. 

Weight 

Breaking 
Strain 
in  Tons 

Of  2080 

Lbs. 

Circum. 
ofManila 
Rope  of 
Equal 
Strength. 

Ins. 

Ins. 

Lbs. 

No. 

Ins. 

Ins. 

Ins. 

Lbs. 

No. 

Ins. 

.8125 

2.5 

i 

9 

5 

•375 

1.125 

.2 

1.8 

2.25 

•75 

2.25 

.81 

7«3 

4-75 

•3125 

i 

.16 

1.4 

2 

.625 

2 

.64 

5.8 

4-5 

.2812 

.875 

.123 

i.i 

I-75 

•5625 

1.75 

4.4 

3-75 

.25 

.81 

1-5 

.5 

1.5 

•36 

3-2 

3 

.2188 

.625 

.063 

•56 

1.25 

•4375 

1.25 

.25 

2-3 

2.5 

•1875 

•  5 

.04 

•36 

1.125 

Gralvanized.    Steel    Hawsers. 
For    Sea    and.    Lalze    T<ywing. 


Approx- 
imate 
Diam- 

Circum- 
ference. 

Weight 
Foot. 

breaking 
Strain 
in  Tons 
of  2000 

Circnm. 
of  Manila 
Rope  of 
Equal 

Approx- 
imate 

Diam- 

Circum- 
ference. 

Weight 
t>er 
Foot. 

Breaking 
Strain 
in  Tons 

Of  2000 

Circum. 
ofManila 
Rope  of 
Equal 

eter. 

Lbs. 

Strength. 

eter. 

Lbs. 

Strength. 

Ins. 

Ins. 

Lbs. 

No. 

Ins. 

Ins. 

Ins. 

Lbs 

No. 

Ins. 

1-75 

5-5 

3.25 

61 

13-5 

I-4375 

4-5 

2.1 

3 

42 

"•5 

1.6875 

5.25 

2-95 

57 

13 

1.375 

4.25 

1.94 

39 

ii 

1.625 

5 

2.7 

53 

12.5 

i 

•25 

4 

1-7 

2 

32 

10 

1-5 

4-75 

2.42 

45 

12 

1.1875 

3-75 

1.5 

29 

9.25 

GJ-alvanized  Steel  Cables  for  Suspension  Bridges. 

Diam- 

Weight 

Breaking 
Strain  in         Diam- 

Weight 

Breaking 
Stra  n  in         Diam- 

Weight 

Breaking 
Strain  in 

eter. 

Foot. 

Tons  of  2000        eter 
Lbs. 

BE 

Tons  of  2000       eter 
Lbs. 

Ct. 

Tons  of  2000 
Lbs. 

Ins. 

Lbs. 

No.               Ins. 

Lbs. 

No.               Ins. 

Lbs. 

No 

2-75 

12.7 

3»o              2.375 

9-5 

232                 2 

6-73 

164 

2.625 

n.  6 

283              2.25 

8.52 

208                  I 

.875 

5-9 

144 

2-5 

10.5 

256              2.125 

7-6 

185                 I 

•75 

5-' 

124 

Grange,  Weight,  and    Length    of 

Iron    "Wire. 

1 

Diam. 

Weight 
per  loo 

Feet. 

Weight 
of  one 
Mile. 

63  Ibs. 
Bundle. 

Area. 

1 
I 

' 
1 

> 

Diam. 

Weight 
per  100 
Feet 

Weight 
of  one 
Mile. 

63  Ibs. 
Bundle. 

Area. 

N 

Inch. 

Lbs. 

Lbs. 

Feet. 

Sq.  Inch. 

NoT 

Inch. 

Lbs 

Lbs. 

Feet. 

Sq.  Inch. 

6/0 

.46 

56.1 

2962 

112 

.16619 

16 

.063 

1.05 

55 

6000 

.  003  117 

5/o 

•43 

49.01 

2588 

129 

.14522 

I 

.054 

•77 

41 

8182 

.00229 

4/0 

•393 

40.94 

2162 

154 

.1 

21304 

I 

8 

.04 

7 

-58 

31 

10 

362 

.001734 

3/o 

.362 

34-73 

1834 

.102921 

I 

.041 

•45 

24 

14000 

.00132 

2/0 

•331 

29.04 

1533 

217 

.  086  049 

20 

•035 

•32 

17 

19687 

.000962 

I/O 

•3°7 

27.66 

1460 

228 

.0 

74023 

2 

i 

-03 

2 

.27 

14 

23 

333 

.000804 

i 

.283 

21.23 

II2I 

296 

.062901 

22 

.028 

.21 

ii 

30000 

.000615 

2 

.263 

18.34 

968 

343 

.0 

54325 

2 

3 

.02 

5 

•17 

5 

9.: 

'4 

36 

DOO 

.000491 

3 

.244 

I5-78 

833 

399 

.046759 

24 

.023 

.14 

7-39 

45000 

.000415 

4 

.225 

13-39 

707 

470 

.03976 

25 

.02 

5 

6.124 

54310 

.000314 

5 

.207 

"•35 

599 

555 

.0 

33653 

2 

5 

.OI 

8 

.09 

3 

)i 

67 

742 

.000254 

6 

.192 

9-73 

647 

.0 

28952 

2 

7 

.01 

7 

.08 

3 

4-: 

582 

75 

?°3 

.000227 

7 

.177 

8.03 

439 

759 

.024605 

2 

S 

.Ol6 

.074 

3-9°7 

85135 

.000201 

8 

.162 

6.96 

367 

9°5 

.0- 

20612 

2 

9 

.01 

5 

.06 

r 

3-' 

22 

103 

278 

.000  176 

9 

.148 

5.08 

306 

1086 

.017  203 

30 

.014 

•054 

2.851 

116666 

.000154 

o 

•135 

4-83 

255 

1304 

•014313 

31 

•0135 

2.64 

126000 

.000133 

i 

.12 

3.82 

202 

1649 

.on  309 

32 

.013 

.046 

2.428 

136956 

.000132 

2 

.105 

2.92 

154 

2158 

.0< 

38659 

3 

3 

.01 

i 

•03 

7 

I. 

?53 

170 

270 

.000095 

3 

.092 

2.24 

118 

2813 

.006647 

34 

.01 

•03 

1.584 

2IOOOO 

.000078 

4 

.08 

1.69 

89 

3728 

.01 

35026 

3 

5 

.oc 

Q5 

.02 

5 

I. 

32 

252 

000 

.000071 

5 

.072 

72 

4598 

.004071 

36      .009 

.02 

1.161 

286363 

.000064 

164 


IKON,  STEEL,   AND    HEMP    HOPE. 


"Weight  and.  Strength,  of  Single  Strand  and  Cable 
laid   Fence   TV^ire.    (F.  Morton  &  Co.} 


Strands. 

N, 

Single  Wire 
of  equal 
Diameter. 

Len 
per  10 
Of  a 

Strand. 

gth 
oo  Ibs. 
Of 
Rope. 

Strands. 

No. 

Single  Wire 
of  equal 
Diameter. 

Len 
per  io< 
Of  a 
Strand. 

gth 
DO  Ibs. 
Of 
Rope. 

No. 

No. 

Inch. 

Feet. 

Feet. 

No. 

No. 

Inch. 

Feet. 

Feet. 

3 

2A 

8 

•159 

20090 

15270 

7 

00 

4 

.229 

8300 

7366 

4 

2 

7 

.174 

14730 

12790 

7 

3/o 

3 

-25 

8036 

6228 

7 

I 

6 

.191 

I3I25 

10580 

7 

4/o 

2 

.274 

7500 

5156 

7 

0 

5 

.209 

10446 

8928 

7 

5/o 

I 

•3 

5090 

4286 

No.  and  diameter  of  wire  is  that  of  Ryland's  Bros.,  pp.  122-4. 

Hemp,  Iron,  and    Steel.     (R.  S.  Newall  &  Co.) 


HEMP. 
Circumference. 

Weight 
per 
Foot. 

IRON. 

Circumference. 

Weight 
per 
Foot. 

STEEL. 
Circumference. 

Weight 
Foot. 

Tensile  5 

Safe 
Load. 

>trength. 

Ultimate 
Strength. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

2.75 

•33 

I 

.16 

— 

— 

672 

4480 

i-5 

•25 

I 

.16 

1008 

6720 

3-75 

.66 

1.625 

•33 

— 

— 

J344 

8960 

1-75 

.42 

i-5 

•25 

1680 

II  20O 

4-5 

.83 

1-875 

•5 

2016 

13440 

2 

•58 

1.625 

•33 

2352 

15680 

5-5 

1.16 

2.125 

.66 

i-75 

.42 

2688 

17920 

2.25 

•75 

— 

— 

3024 

20  160 

6 

i-5 

2-375 

•83 

1-875 

•5 

336o 

22400 

2.5 

.92 

3696 

24640 

6-5 

1.66 

2.625 

2 

•58 

4032 

26680 

2-75 

.08 

2.125 

.66 

4368 

29  1  20 

7 

2 

2.875 

.16 

2.25 

•75 

4704 

31360 

3 

•25 

— 

— 

5040 

33600 

7-5 

2-33 

3-125 

•33 

2-375 

-83 

5376 

36840 

3-25 

.41 

— 

5672 

38080 

8 

2.66 

3-375 

-5 

2-5 

.92 

6048 

40320 

3-5 

1.66 

2.625 

i 

6720 

44800 

8.5 

3 

3-625 

1-83 

2-75 

i.  08 

7392 

49280 

3-75 

2 

— 

8064 

5376o 

9-5 

3-66 

3-875 

2.16 

3-25 

i-33 

8736 

58  240 

10 

4-33 

4 

2-33 

9408 

62  720 

4-25 

2-5 

3-375 

i-5 

10080 

67200 

ii 

5 

4-375 

2.66 

— 

— 

10752 

71680 

4-5 

3 

3-5 

1.66 

12096 

80640 

12 

5-66 

4-625 

3-33 

3-75 

2 

13440 

89600 

FLAT. 

Dimensions. 

Dimensions. 

Dimensions. 

4      X   .5 

3-33 

2.25   X.5 

1-85 

— 



4928 

44800 

5      Xi.25 

4 

2.5     X-5 

2.l6 

— 



5824 

51520 

5-5  Xi.  375 

4-33 

2.75   X.625 

2-5 

— 



6720 

60480 

5-75XI.5 

4.66 

3        X  .625 

2.66 

2       X.5 

1.66 

7168 

62720 

6      Xi.5 

5 

3.25   X.625 

3 

2.25  x-5 

1.83 

8064 

71680 

7       Xi.  875 

6 

3-5     X.625 

3-33 

2.25  x.  5 

2 

8960 

80640 

8.25X2.125 

6.66 

3-75   X.6875 

3-66 

2.5   X-5 

2.16 

9850 

89600 

8.5    X2.25 

7-5 

4        X.6875 

4.16 

2-75  X.  375 

2-5 

II  200 

100800 

9      X2.5 

8-33 

4-25   X-75 

4.66 

3      X.375 

2.66 

12544 

II2000 

9-5   X  2.375 

9.16 

4-5     X-75 

5-33 

3-25  X.  375 

3 

14336 

125  440 

10      X2.5 

10 

1  4.625  X.  75 

5-66 

3-5  X.375 

3-23 

15232 

134400 

HOPES    AND    CHAINS.                                     165 

Ultimate    Strength,    arid.    Safe    Loads    of    Hemp,    Iron, 

and.    Steel. 

Ultimate 

SAFB  LOAD         <   | 

Ultimate 

SAFE  LOAD 

Strength 
per  Lb.  Weight 
per  Foot. 

perLb. 
Weight 
per  Foot. 

per  Square 
in  InXT/l 

Strength 
perLb.Weigbt 
per  Foot. 

per  Lb. 
Weight 
per  Foot. 

per  Square 
ofCircum. 
in  Inches. 

Lba. 

Lbs. 

Lbs.      1 

Lbs. 

Lbs. 

Lbs. 

Hemp  . 
Iron  .  .  . 

15000 
22OOO 

4550 
5000 

100      ! 
600 

Steel. 

f  30000 
(45500 

f6ooo 

(5000 

f  1000 
(1300 

PLOUGH    STEEL    FLAT   MINING   ROPES. 


John  A  .  Roebling's  Sons  Co. 

,  New  York 

Width. 

Thickness. 

Weight 
per  Foot. 

Ultimate 
Strength. 

Width. 

Thickness. 

Weight 
per  Foot. 

Ultimate 
Strength. 

Ins. 

Ins. 

Lbs.                 Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

2 

•375 

I.I9             63000 

5-5 

•375 

3-9 

156000 

2-5 

•375 

1.86     i      74000 

5-5 

•5 

4.8 

193000 

3 

•375 

2.32 

93000 

6 

•375 

4-34 

I73OOO 

3-5 

•5 

2.97         118000 

6 

•4375 

4-5 

lOOOOO 

4 

•375 

2.86        114000 

6 

•5 

5-1 

210000 

4 

-5 

3.3           130000 

6-5 

•5 

5-5 

224000 

4-5 

•375 

3.12         125000 

7 

•5 

5-9 

238000 

4-5 

-1V5  1;I 

4              160000 

7-5 

•5 

6.25 

250000 

5 

•375 

3-4 

125000 

8 

•5 

6-75 

270000 

sJ$uifri 

•5 

4.27 

I7QOOO 

For  Cast-  Steel  Flat  Ropes  see  page  1029. 

- 

Ropes   and   Chains 

of  Equal   Strength. 

CIRCUMFERENCE. 

WEIGHT  PER  FOOT. 

Diameter 
of 
Iron  Chain. 

Hemp 
Rope. 

Crucible 
Steel 
Rope. 

Charcoal 
Iron 
Rope. 

Steel 
Rope. 

Iron 
Rope. 

Hemp 
Rope. 

Iron 
Chain. 

Safe 
Load. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Tous. 

•218  75 

2-75 

— 

I 

. 

14 

•34 

«5 

-3 

•25 

3 

— 

1.18 

, 

21 

.46 

•65 

•4 

.28125 

3-5 

I 

1.39 

•17 

28 

.67 

.81 

•5 

•3125 

4-25 

1.26 

1.57 

•25 

33 

•75 

.96 

.6 

•375 

4-5 

1.45 

1.77 

•3 

•45 

•83 

1.38 

.8 

•4375 

5 

1.57 

1.97 

•35 

57 

1.16 

I.76 

i 

.46875 

5-5 

1.77 

2.19 

•45 

7 

1.2 

2.2 

1.3 

•5 

5-75 

1.06 

2.36 

•59 

-83 

1.6 

2.63 

1.5 

-625 

6-75 

2.36 

2-75 

•85 

i.  08 

2 

4.21 

2-3 

.6875 

7-75 

2-75 

3-14 

I 

.1 

1-43 

2.65 

4.83 

•75 

8-75 

2-95 

3-53 

I 

.28 

1.8 

3-35 

5-75 

3'8 

•875 

9-75 

3.14 

3-93 

I 

•45 

2-3 

4.6 

7-5 

4.8 

•9375 

10.5 

3-53 

4-32 

1.83 

2-94 

4-92 

9-33 

5-9 

1.0625 

n-75 

3-93 

4.71 

2-33 

3-56 

5-83 

10.6 

7 

1.125 

12.75 

4-32 

5.1 

2.98 

4 

6.2 

11.9 

8.2 

1.25 

J4-75 

4.71 

5-5 

3-58 

4.8 

8-7 

14-5 

9-5 

1-375 

15-25 

4.81 

5-89 

3-65 

5-6 

9 

17.6 

ii 

15-75 

5.1 

6.28 

4.04 

6.3 

10.  1 

20 

12.5 

1.625 

17-75 

5-8 

7.07 

5-65 

7-95 

13-7 

22.3 

J-75 

19-5 

6-35 

7.85 

6-5 

9 

81 

16.4 

24-3 

19.6 

By  experiments  of  U.  S.  Navy,  hemp  rope  of  this  circumference  has 
weight  of  71  309  Z&s.,  and  a  wire  rope  of  5.34  ins.  has  equivalent  strength. 

a  breaking 

1 66         WEIGHT,  STRESS,  AND   TENSION   OF   EOPES. 


\Veiglit  of*  Hemp   and   Wire   Rope.     (Molesworth.) 
In  Lbs.  per  Fathom. 


Circum- 
ference. 

H» 

Common. 

HP. 

Good. 

Wi 

Iron. 

SE. 

Steel. 

Circum- 
ference. 

Hi 

Common. 

HP. 

Good. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

I 

.18 

.24 

.87 

.89 

5 

4-5 

6 

i-5 

.41 

•54 

1.96 

2 

5-5 

5-45 

7.26 

•55 

•74 

2.66 

2-73 

6 

6.48 

8.64 

2 

.72 

.96 

3.48 

3.56 

6-5 

7.61 

10.14 

2.25 

.91 

1.22 

4.4 

4-51 

7 

8.82 

11.76 

2-5 

1*13 

**5 

5-44 

5.56 

7-5 

10.13 

13*5 

2-75 

1.36 

1.82 

6.58 

6-73 

8 

11.52 

15-36 

3 

1.62 

2.16 

7.83 

8.01 

8-5 

I3-05 

17-34 

3-25 

1.9 

2-54 

9.19 

9-4 

9 

14.58 

19.44 

3-5 

2.21 

2.94 

10.66 

10.9 

10 

18 

24 

3-75 

2-53 

3.38 

12.23 

12.52 

12 

26 

34-56 

4 

2.88 

3-84 

13.92 

14.24 

15 

40.52 

54 

To  Compvite  Stress  upon  a  Rope  set  at  an  Inclination. 

RULE. — Multiply  sine  of  angle  of  elevation  by  strain  in  Ibs.,  add  an  allow- 
ance for  rolling  friction  and  weight  of  rope,  and  multiply  by  factor  of  safety. 

Factor  of  safety.— For  standing  rope  4,  for  running  5,  and  for  inclined 
planes  from  5  to  7. 

ILLUSTRATION.— Inclination  of  rope  92.5  feet  in  100,  velocity  1500  feet  per  minute, 
and  strain  2000  Ibs. ;  what  should  be  diam.  of  iron  rope,  7  wires  to  a  strand  ? 

Angle  of  92.5  feet  in  100  =  43°,  and  sine  of  43°  — .682.  .682  X  2000=  1364,  to 
which  is  to  be  added  rolling  friction  and  weight  of  rope,  assumed  to  be  u ;  hence, 
1364  +  11  =  1375. 

Factor  of  safety  assumed  at  6,  consequently  1375  X  6  =  8250  Ibs. ,  capacity  or  break- 
ing weight  or  stress  of  rope. 

By  table,  page  162,  8200  Ibs.  is  breaking  weight  of  a  wire  rope  of  7  strands,  .625 
inch  in  diam. 

To   Compvite   Tension   of  a   Rope. 

TD 

—  =  t.    v  representing  velocity  of  rope  in  feet  per  minute,  EP  horses1  power, 

and  t  tension  in  Ibs. 

ILLUSTRATION.— Assume  wheel  7  feet  in  diameter,  revolution  140  per  minute,  and 
IP  as  per  preceding  table,  29.6. 

_,  29. 6  X  33  ooo         976  800 

7X3-  Hi6  X  140  ,       3079 

To   Compvite    Operative   Deflection   of  a   Rope. 

=  d.    D  representing  distance  between  centres  of  wheels  or  drums  in 

feet,  w  weight  of  rope  in  feet  per  lb.,  t  tension,  or  power  required  to  produce 
required  power  or  tension  of  rope  when  at  rest,  and  d  deflection  in  feet. 

ILLUSTRATION.— Take  elements  of  preceding  case:  diam.  of  wire  rope  of  7  strands 
=  .5625  inch,  and  by  table,  page  162,  w  =  .41  lb.,  and  D  =  300  feet. 


Then 


10.7  X  317-2 


=  10.87  feet. 


Capacity.— At  the  Falls  of  the  river  Rhine  there  is  a  wire  rope  in  operation 
that  transmits  the  power  of  600  horses  for  a  distance  exceeding  one  mile. 


TRANSMISSION  OF  POWEK  AND  EQUIVALENT  BELT. 


Endless   Ropes. 

Wire  Ropes,  when  practicable  and  proper  for  application,  can  be  used  for 
transmission  of  power  at  a  less  cost  than  belting  or  shafting. 

Transmission,   of  IPower. 


Diameter 

of  Wheel. 

Jifc 
Is! 

tf^S 

So 

ll 

Diameter 
of  Wheel. 

*N 

III 

Diamater 
of  Rope. 

|| 

1  Diameter 
(  of  Wheel. 

Revolu- 
tions per 
Minute. 

Diameter 
of  Rope. 

II 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

4 

80 

•375 

3-3 

7 

IOO 

.5625 

21.  1 

II 

I4O 

•6875 

132.1 

4 

IOO 

•375 

4.1 

7 

140 

.5625 

29.6 

12 

80 

•75 

99-3 

4 

1  20 

•375 

5 

8 

80 

.625 

22 

12 

IOO 

•75 

124.1 

4 

140 

•375 

5-8 

8 

IOO 

.625 

27-5 

12 

140 

•75 

173-7 

5 

80 

•4375 

6.9 

8 

140 

.625          38.5 

13 

80 

•75 

122.6 

5 

IOO 

•4375 

8.6 

9 

80 

.625 

41-5 

13 

IOO 

•75 

153-2 

5 

120 

•4375 

10.3 

9 

IOO 

.625 

51-9 

13 

120 

•75 

183.9 

5 

I4O 

•4375 

I2.I 

9 

140 

.625 

72.6 

14 

80 

•875 

I48 

6 

80 

•5 

IO.7 

10 

80 

•6875 

58.4 

14 

IOO 

•875 

I76 

6 

IOO 

•5 

13-4 

10 

IOO 

.6875 

73 

14 

120 

•875 

222 

6 

120 

•5 

16.1 

10 

140 

.6875 

102.2 

15 

80 

•875 

2I7 

6 

I40 

•5 

18.7 

ii 

80 

.6875 

75-5 

15 

IOO 

.875 

259 

7 

80 

•5625 

16.9 

ii 

IOO 

•6875 

94-4 

15 

120 

•875 

300 

Wire   Rope   and.   Equivalent  Belt. 

In  substituting  wire  rope  for  an  ordinary  flat  belt,  the  diameter  is  deter- 
mined by  rule  in  practice  for  estimating  power  transmitted  by  a  belt— viz., 

One  horse  power  for  every  70  square  feet  of  running  belt  surface  per 
minute.  Thus,  a  belt  15  inches  wide  running  at  rate  of  1400  feet  per  min- 
ute, its  power  would  be  equal  to  (1400  x  15)  •—  (70  x  12)  =  25  horses'  power. 

The  same  result  is  obtained  by  the  use  of  a  wire  rope  .5625  inch  in  diam- 
eter, running  over  a  wheel  6  feet  in  diameter,  making  130  revolutions  per 
minute. 

Average  life  of  iron  wire  rope  with  good  care  is  from  3  to  5  yearst  and 
that  of  steel  rope  is  greater.  Wear  increases  rapidly  with  velocity. 

GJ-eiieral    Notes. — Hemp    and.   "Wire    Ropes. 

White  Rope,  2  inches  in  circumference,  of  different  manufactures,  parted  at 
a  stress  of  from  4413  to  6160  Ibs. 

Specimens  of  Italian,  Russian,  and  French  manufacture  parted  with  an 
average  stress  of  5128  Ibs.  =  1633  Ibs.  per  square  inch  of  rope. 

Bearing  capacity  of  a  hemp  rope  is  proportional  to  its  thickness,  number 
of  its  strands,  slackness  with  which  they  are  twisted,  and  quality  of  the 
hemp. 

Hemp  and  Wire  Ropes. —  Ultimate  Strength  is  2240  Ibs.  per  Ib.  per  fathom 
for  round  hemp,  3300  Ibs.  for  iron,  7000  Ibs.  for  cast-steel,  and  10000  Ibs.  for 
plough-steel. 

Working  Load  is  336  Ibs.  per  Ib.  weight  per  fathom  for  round  hemp,  660 
Ibs.  for  iron,  1400  Ibs.  for  cast-steel,  and  2000  Ibs.  for  plough-steel. 

Or,  .83  times  square  of  circumference  in  inches  for  round  hemp,  5  times 
square  of  circumference  for  iron,  and  9  times  square  of  circumference  for 
steel.  (D.  K.  Clark.} 

Steel  Ropes  may  be  one  half  less  in  weight  than  iron  or  hemp  for  like 
working  loads. 


1 68 


ROPES   AND   CHAINS. 


IRON  WIRE  AND  UNITED  STATES  NAVY  HEMP  ROPE. 
Wire  6  Strands-,  Hemp  Core.    Rope  4  Strands. 


Ci 
Actual. 

rcumferenc 
Nominal. 

WIRE, 
e. 
Core. 

Wires. 

Breaking 
Weight. 

Circun 

Actual. 

HEl 

ference. 
Nominal. 

[p. 

Yarns. 

Breaking 
Weight. 

Ins. 

Ins 

Ins. 

No. 

Lbs. 

Ins. 

Ins. 

No. 

Lbs. 

7 

7 

2-35 

108 

187400 

12 

I3-25 

1168 

75966 

6 

6 

2.25 

108 

104050 

II 

12.25 

1036 

77633 

4-937 

4-9 

i-57 

114 

65409 

10.5 

11.875 

928 

76933 

4-375 

4-5 

i-57 

114 

55316 

10 

n-375 

876 

7°533 

3-5 

3-36 

1.27 

114 

34480 

95 

10.5 

800 

58766 

3-187 

2.98 

1.17 

114 

28606 

9 

10.312 

712 

56466 

2-75 

2.68 

.78 

114 

21846 

8-5 

9-437 

640 

42866 

2-5 

2-45 

.78 

114 

15692 

8 

8.812 

560 

4OOOO 

2-375 

2.4 

.78 

42 

I57I8 

7-5 

8-437 

484 

35500 

2 

2.06 

•39 

114 

10925    !     7 

7.812 

436 

32  166 

Weight  and.  Strength,  of  Stud-link  Chain  Cable. 

(English.) 


D 

Diam. 
of  each 
Side. 

MKNSION 

Length 
of 
Link. 

3. 

Width 
of 
Link. 

Weight 
Fathom. 

Admiralty 
Proof-stress 
(adopted  by 
Lloyds'). 

D 

Diam. 
of  each 
Side. 

IMKNSION 

Length 
of 
Link. 

8. 

Width 
of 
Link. 

Weight 
per 
Fathom. 

Admiralty 
Proof-stress 
(adopted  by 
Lloyds'). 

Ina. 

Ins. 

Ins. 

Lbs. 

Tons. 

Ins. 

Ins. 

Ins. 

Lbs. 

Tons. 

•4375 

2.625 

1-575 

"•3 

3-5 

I«J 

9 

5-4 

121 

405 

•5 

3 

1.8 

134 

4-5 

1.625 

9-75 

585 

I42 

47-5 

•5625 

3-375 

2.025 

17.2 

5-5 

i-75 

o-5 

6-3 

164.6 

55-125 

.625 

3-75      2.25 

21 

7 

1-875 

1.25 

6-75 

189 

6325 

.6875 

4.125    2.475 

25.4 

8-5 

2 

2 

7-2 

215 

72 

•75 

4-5    ;  2.7 

30.2 

10.125 

2.125 

2?5 

7-65 

242.8 

81.25 

.875 

5-25      3-!5 

41.2 

13-75 

2-25    !    35 

8.1 

276.2 

91-125 

i 

6 

3-6 

53-8 

18 

2-375 

425 

8-55 

303.2       IOI.5 

1.125 

6-75      4-05 

69 

22.75 

2-5 

5 

336 

112  5 

1.25 

75        4-5 

84 

28.125 

2-75 

16.5 

9-9 

406.6  ;  136.125 

1-375    [8.25      4.95 

101.6 

34 

i 

NOTE  i.— Safe  Working-stress  is  taken  at  half  Proof-stress,  3.82  tons  per  sq.  inch 
of  section. 

•2.— Proof -stress  and  Safe  Working  -  stress  for  close-link  chains  are  respectively 
two-thirds  of  those  of  stud-link  chains. 

3. — Proof-stress  averages  72  per  cent,  ultimate  strength,  and  Ultimate  Strength 
averages  8  tons  per  square  inch  of  section  of  rod  or  one  side  of  a  link. 

Weight  of  close-link  chain  is  about  three  times  weight  of  bar  from  which 
it  is  made,  for  equal  lengths. 

Karl  von  Ott,  comparing  weight,  cost,  and  strength  of  the  three  materials, 
hemp,  iron  wire,  and  chain  iron,  concludes  that  the  proportion  between  cost 
of  hemp  rope,  wire  rope,  and  chain  is  as  2  :  i  :  3 ,  and  that,  therefore,  for 
equal  resistances,  wire  rope  is  only  half  the  cost  of  hemp  rope,  and  a  third 
of  cost  of  chains. 

Safe  "Working   Load   of  Chains.     (Molesworth). 


Diameter 
of  Iron. 

Load. 

Diameter 
of  Iron. 

Load. 

Diameter 
of  Iron. 

Load. 

Diameter 

Load. 

Ins. 

•375 

•5625 
.625 

Lbs. 
2240 
3800 
4900 
6270 

Ins. 
.6875 

•75 
.8125 

•875 

Lbs. 
7390 
8960 
10280 
12320 

Ins. 

•9375 
I 
1.0625 
1.125 

Lbs. 
13700 
15680 
17920 

20  160 

Ins. 
1.1875 
1.25 
I-3I25 

1-375 

Lbs. 
22  4OO 
24640 
26680 
30240 

ROPES   AND   CHAINS. 


169 


Breaking   Strain   and   IProof  of   Chain    Cables. 


Diam. 
of  Chain. 

Breaking 
Strain. 

Diam. 
.of  Chain. 

Breakinf 
Strain. 

Diam. 
of  Chain. 

Breaking 
Strain. 

Diam. 
of  Chain. 

Breaking 
Strain. 

Ins. 
I 
1.0625 
I.I25 

Lbs. 
67700 
75640 
84  100 

I.l875 
1.25 

1-375 

Lbs. 
92940 

102  160 

121  840 

Ins. 
1-5 
1.625 

i-75 

Lbs. 
I43IOO 
165  920 
2l6  120 

Ins. 
2 
2.125 
2.25 

Lbs. 
243  1  80 
272580 
303280 

Proof-stress  is  50  per  cent,  of  estimated  strength  of  weakest  link  and  46 
per  cent,  of  strongest. 

Comparison,    of"  "Wire    Ropes    and    Tarred.    Hemp    Rope, 
Hawsers,  and   Cables. 


COARSE  LAID. 

FINE   LAID. 

Ropes. 

Haws'rs. 

Cables. 

Ropes. 

Haws'rs. 

Cables. 

Diam- 

| 

Safe 

11 

4 

§a 

Ii 

if 

Diam- 

Safe 

rt 

H 

II 

eter. 

g 

Load. 

£g 

tajl 

£g 

£  « 

eter. 

Load. 

(2S 

££ 

•F  £ 

5 

H35 

02 

H55 

H02 

02 

H£ 

Ho5 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

•25 

.78 

425 

1.25 

— 

— 

— 

•5 

1875 

3-12 

2.87 

— 

•3125 

i 

690 

2-43 

2.25 

3-32 

— 

•5625 

2420 

3-56 

3-25 

4.87 

•375 

1.25 

825 

2.68 

2-375 

3-5 

— 

625 

2900 

3-93 

3-62 

5-25 

•5 

1600 

2.87 

2.62 

3-87 

— 

•75 

4320 

4.81 

4-37 

6-37 

-5625 

!-75 

2800 

3-8i 

3-5 

— 

•875 

5700 

5-5 

5 

7-25 

.6875 

2.125 

3800 

4-75 

4-25 

6^2 

— 

i 

8200 

7-25 

6.25 

8-75 

•75 

2-375 

4400 

5-25 

4.87 

7 

— 

1.125 

IO  IOO 

8.18 

7 

9-5 

•875 

2.625 

6  150 

6.12 

5-75 

8 

8 

1.25 

13600 

8.81 

8.06 

ii 

3 

8400 

6.62 

6.12 

8.62 

8.62 

«•! 

17500 

10 

9-75 

12.5 

•25 

3-75 

13400 

8.81 

8-5 

10.93 

10.93 

1.625 

21800 

n.x8 

10.93 

•375 

4-25 

16800 

9.87 

9-56 

12.25 

12.12 

i-75 

27000 

12.5 

12.12 

— 

.5 

4.625 

20  160 

10.75 

10.5 

13 

13.12 

1875 

32500 



— 

.625 

5 

24600 

-      11.87 

ii  56 

"•75 

2 

37000 

— 

— 

— 

In  above  table,  determination  of  circumference  of  rope,  etc.,  is  based  upon 
Breaking  Weight  or  Tensile  resistance  of  wire  being  reduced  by  one  fourth, 
and  ultimate  resistances  of  rope,  etc.,  are  reduced  one  third. 

Result  of  Experiments  upon  "Wire  Rope  at  TJ.  S.  Navy- 
Yard,  Washington.    -(J  A  Roebling's  Sons.) 


Circumfe 
Actual. 

rence. 

Nom- 
inal. 

ill 

1§ 

Weight 
Foot. 

Breaking 
Weight. 

Circumft 
Actual. 

rence. 

Nom- 
inal. 

tii 

rf  * 
*~Z 

jj!^ 

No. 
13 
14 

14 
17 
20 

18 

19 

III 

Breaking 
Weight. 

Ins. 

4-9375 
4-375 
3-9375 
3-5 
3-1875 
2-75 
2.6875 

2-5 

Ins. 

4-9 

4-5 

3*36 
2.98 
2.68 
2.56 
2-45 

No. 
19 
19 
19 
19 
19 
19 

7 
19 

No. 
II 

13 
14 
14 
15 
17 
13 

18 

Lbs. 
3-14 
2.15 

2.0875 

I-I525 
1.09 
1.0275 
1.0225 
.14 

Lbs. 
65409 

55  3l6 
44420 
34840 
28606 
21846 
18810 
15692 

Ins. 

2-375 

2.1875 

2 

1-9375 

1-4375 
1-3125 
1.125 

Ins. 
2.4 
2.12 
2.06 
1.9 
1.85 

I.  II 

7 
7 

7 
7 
19 
7 
7 

Lbs. 
.14 
.11 
.1 
.1 
.07 
.06 
•05 
-035 

Lbs. 
I57I8 
14478 
10925 

10  118 
7880 
5687 
4428 
3729 

To  Compute   Circumference  of  Wire  Rope  with  Hemp 
Core,  of  Corresponding  Strength  to  Hemp  Rope,  and 
of  Hemp    Rope    to    Circumference   of  "Wire    Rope. 
RULE  i. — Multiply  square  of  circumference  of  hemp  rope  by  .223  for  iron 
wire  and  .12  for  steel,  and  extract  square  root  of  product. 

2.— Multiply  square  of  circumference  of  hemp-core  wire  rope  by  4.5  for 
iron  wire  and  8.4  for  steel  wire. 

EXAMPLE.— What  are  the  circumferences  of  an  iron  and  steel  wire  rope  corre- 
sponding to  one  of  hemp-core,  having  a  circumference  of  8  ins.  ? 


/82X  .223  =  3. 78  in*,  trow,  and  i 


'82x.i2  =  2. 77  ins.  steel. 


ROPES,  HAWSEKS,  AND    CABLES. 


ROPES,  HAWSERS,  AND  CABLES. 

Ropes  of  hemp  fibres  are  laid  with  three  or  four  strands  of  twisted  fibres, 
and  are  made  up  to  a  circumference  of  12  ins.,  and  those  of  four  strands  up 
to  8  ins.  are  fully  i6per  cent,  stronger  than  those  of  three  strands. 

Hawsers  are  laid  with  three  or  four  strands  of  rope.  Cables  are  laid  with 
but  three  strands  of  rope.  Hawsers  and  Cables,  from  having  a  less  propor- 
tionate number  of  fibres,  and  from  the  irregularity  of  the  resistance  of  their 
fibres  in  consequence  of  the  twisting  of  them,  have  less  strength  than  ropes, 
difference  varying  from  35  to  45  per  cent.,  being  greatest  with  least  circum- 
ference, and  those  of  three  strands  up  to  12  ins.  are  fully  loper  cent,  strong- 
er than  those  having  four  strands. 

Tarred  ropes,  hawsers,  etc.,  have  25  per  cent,  less  strength  than  white 
ropes ;  this  is  in  consequence  of  the  injury  fibres  receive  from  the  high  tem- 
perature of  the  tar,  viz.  290°. 

Tarred  hemp  and  Manila  ropes  are  of  about  equal  strength,  and  have  from 
25  to  30  per  cent,  less  strength  than  white  ropes. 

White  ropes  are  more  durable  than  tarred. 

The  greater  degree  of  twisting  given  to  fibres  of  a  rope,  etc.,  less  its 
strength,  as  exterior,  alone  resists  greater  portion  of  strain. 

Ultimate  strength  of  ropes  varies  from  7000  to  12000  Ibs.  per  square  inch 
of  section,  according  as  they  are  wetted,  tarred,  or  dry.  One  sixth  of  ulti- 
mate strength  is  a  safe  working  load=  1166  to  2000  Ibs.  per  square  inch. 

Units  for  computing  Safe  Strain  th.at  may  "be  "borne  "by 
N~e\v   Ropes,  Hawsers,  and.    Catoles.     (t^.  S.  Navy.) 


DESCRIP- 
TION. 

Circumference. 

Wl 

3  strands. 

Ron 
ite. 
4  strands. 

ts. 
Tar 

3  str'ds. 

red. 
4  str'ds. 

HAW 

White. 
3  str'ds. 

SEES. 

Tarred. 
3  str'ds. 

CAI 

White. 
3  str'ds. 

LES. 

Tarred. 
3  str'ds. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

White 

2.5  to   6 

II4O 

1330 

— 

— 

600 

u 

6     "    8 

1090 

I20O 

— 

— 

570 

— 

510 

— 

u 

8       "    12 

1045 

880 

— 

— 

530 

— 

530 

— 

u 

12       "    18 

— 

— 

— 

— 

550 

— 

550 

— 

it 

18     "  26 

— 

— 

— 

— 

— 

500 



Tarred 

2-5  "    5 

— 

— 

855 

1005 

— 

460 

— 

44 

5      "    8 

— 

— 

825 

940 

— 

480 

— 

— 

u 

8       "    12 

— 

— 

780 

820 

— 

505 

— 

505 

u 

12       "    18 

— 

— 

— 

— 

— 

— 

525 

It 

18     "  26 









__ 





55° 

Manila 

2.5  "    6 

810 

950 

— 

— 

440 

— 

— 

tt 

6        "    12 

760 

835 

— 

— 

465 

— 

510 

:  — 

41 

12        U    l8 

— 

— 

— 

— 

— 

535 

— 

u 

18     "  26 

— 

— 

— 

— 

— 

— 

560 

— 

ILLUSTRATION.—  What  weight  can  be  borne  with  safety  by  a  Manila  rope  of  3 
strands,  having  a  circumference  of  6  inches  ?    (See  Rule,  page  167.  ) 
62 


=  27360^5. 

When  it  is  required  to  ascertain  weight  or  strain  that  can  be  borne  by 
ropes,  etc.,  in  general  use,  preceding  Units  should  be  reduced  from  one  third 
to  two  thirds,  in  order  to  meet  their  condition  or  reduction  of  their  strength 
by  chafing  and  exposure  to  weather.  Molesworth's  table  is  based  upon  a 
reduction  of  three  fourths. 

ILLUSTRATION.  —  What  weight  can  be  borne  by  a  tarred  hawser  of  3  strands.  10 
inches  in  circumference,  in  general  use? 

io2  X  (505  —  505  -r-  3)  =  zoo  X  336.67  =  33  667  #*• 


HOPES,  HAWSERS,  AND    CABLES. 


Destructive    Strength,   of*  Tarred.    Hemp    Ropes. 
(D.  K.  Clark.) 


Circum. 

Diam. 

Res 
Common 
Cold. 

ister. 
Russian 
Warm. 

Circum. 

Diam. 

Reg 
Common 
Cold. 

wter. 
Russian 
Warm. 

Ins. 
3 

3-5 
4 
4-5 
5 

Ins. 

•95 
i.  ii 
1.27 
1-43 
i-59 

Lbs. 
7390 
II  200 
I3IOO 
16330 
19580 

Lbs. 
8620 
II  760 
15340 
19440 
23990 

Ins. 

5-5 
6 

6-5 
7 
8 

1-75 
1.91 
2.07 
2.24 
2-54 

Lbs. 
24800 
28985 
34030 
40320 
52480 

Lbs. 
29120 
33150 
40550 
47041 
61420 

Specimens  furnished  by  National  Association  of  Rope  and  Twine,  Spinner*, 
As  tested  by  Mr.  Kirkaldy. 


Ron. 

Circum- 
ference. 

Weight 
perLb. 

Extreme 
Strength. 

Breaking 
Weight 
per  Ib.per 
Fathom. 

Extend 
atStre 

1000  Ibs. 

>n  In  50  in 
ss  per  Ib. 
r  Fathom 
2000  IDS. 

s.  Length 
W*h. 

3000  Ibs. 

Rus 

Ma< 
Hai 

Br 

a 

9 
£ 

5 

sian 
3hine 
id-sp 

eaki 

5 

rope  ...  48  thr 
yarn.  .  .  50     ' 
an  yarn,  51     * 

ing    Streng 

Old  Method. 
Common      Best 
Hemp.    Russian. 

ds. 

tn 
I 

Cc 

Ins. 
5-26 

5-37 
5-39 

of  a 

y  Regist 
Id.      W 

Lbs 
.92 
.85 
1.  00 

^arr 
sr. 
arm. 

6 

i 
6 

ec 
i 
! 

I 
I 
I 

1 

> 

Lbs. 
1088 

1514 
8278 

Hei 

§ 

5 

Lbs. 
1933 
2152 

3024 
np    R 

OldM 
Common 
Hemp. 

Ins. 
5-29 

4-53 
4.46 

opes. 

ethod. 
Best 
Russian. 

Ins. 

6.56 
5-91 

(Mr. 
By  Re 

Cold. 

Ins. 
6.63 

Plynn.) 

gister. 
Warm. 

Ins. 

3 

3-5 
4 
4-5 
5 

Ins. 

•95 
i.  ii 
1.27 

J-43 
i-59 

Lbs. 
5056 
7466 
8780 
10300 
13328 

Lbs. 
6248 

8668 
10460 
12432 
15859 

Lbs. 
7392 
II  2OO 
I3I04 
16330 
20496 

Lbs. 
8624 
II  760 
17810 

19443 
23990 

Ins. 

565 
6.5 

8 

Ins. 
1-75 
I.9I 
2.O7 
2.24 
2-54 

Lbs. 
15456 
18144 
20518 
22938 
26680 

Lbs. 
18414 
2l6lO 

23  610 
27462 
32032 

Lbs. 
24797 
28986 
34630 
40320 
52483 

Lbs. 
29120 
33150 
40544 
47040 
61420 

To  Compute    Strain    that   may  "be  borne  witn  safety  t>y 
new    Ropes,  Hawsers,  and.    Cables. 

Deduced  from  experiments  of  Russian  Government  upon  relative  strength 
of  different  Circumferences  of  Ropes,  Hawsers,  etc. 

U.  8.  Navy  test  is  4200  Ibs.  for  a  White  rope  of  three  strands  of  best  Riga 
hemp,  of  1.75  inches  in  circumference  (=  17000  Ibs.  per  square  inch  ofjibre), 
but  in  preceding  table  (page  166)  14000  Ibs.  is  taken  as  unit  of  strain  that 
may  be  borne  with  safety. 

RULE.— Square  circumference  of  rope,  hawser,  etc.,  and  multiply  it  by 
Units  in  table. 

To  Compute  Circumference  of  a  Rope,  Hawser,  or  Cat>le 
for    a   GKven    Strain. 

RULE. — Divide  strain  in  pounds  by  appropriate  units  in  preceding  table, 
and  square  root  of  product  will  give  circumference  of  rope,  etc.,  in  ins. 

EXAMPLE  i.— Stress  to  be  borne  in  safety  is  165  550  Ibs. ;  what  should  be  circum- 
ference of  a  tarred  cable  to  withstand  it? 

165  552  -r-  550  =  301,  and  -v/301  =  J7-  35  ins- 

2.— What  should  be  circumference  of  a  Manila  cable  to  withstand  a  strain,  in 
general  ttse,  of  149  336  Ibs.  ? 

Assuming  circumference  to  exceed  18  ins.,  unit  =  560. 

149  336  -r-  (560  —  560-7-3)  =  400,  and  V400  =  20  *'«*• 


172 


KOPES,   HAWSERS,   AND    CABLES. 


To   Compute    "Weignt   of  Ropes,  Hawsers,  and    Catoles. 

RULE. — Square  circumference,  and  multiply  it  by  appropriate  unit  in 
following  table,  and  product  will  give  weight  per  foot  in  Ibs. : 


HAWSERS. 
HOPES.  CABLES. 

3  strand  Hemp 032    .031    .031 

3-strand  tarred  Hemp,  .042    .041    .041 
3-strand  Manila 032    .031    .031 


4-strand  Hemp 033     —      — 

4-strand  tarred  Hemp,  .048     —       — 
4-strand  Manila 035    .034    .034 


Units  for  Thread  Ropes  is  same  as  that  for  Ropes  of  like  material. 
EXAMPLE.—  What  is  weight  of  a  coil  of  lo-inch  Manila  hawser  of  4  strands  of  120 
fathoms? 

io2  X  .034  =  3.  4,  and  120  X  6  X  3-  4  =  2448  Ibs. 

Weignt   and.    Strength,   of  Hemp   and   Wire   Ropes. 

(Molesworth.) 


C  representing  circumference  in  ins.,  W  weight  of  rope  in  Ibs.  per  fathom, 
L  working  load  in  tons,  and  S  destructive  stress  in  tons. 

VALUES  OF  y,  X,  AND  k. 


ROPES. 

y 

X 

k 

ROPES. 

y 

X 

k 

Hawser,  hemp  

'  '31 

Warm  register,  hemp 

.7 

.116 

Cable         " 

Manila  hawser  

•  *77 

.27 

•045 

Tarred  hawser  hemp 

'  '      cable 

IQ 

03  3 

'  '     cable          '  ' 

Iron  rope  

.87 

1.8 

,2Q 

Cold  register.        "     . 

.6 

.1 

Steel    "    . 

.80 

2.8 

.4^ 

To    Compute    Circumference   of  Hemp   or    "Wire    Rope 
for    Fore    or    ]Vtaiii    Standing    Rigging.     (V.  S.  Navy.) 

RULE.  —  To  length  of  mast  between  partners  and  deck,  add  half  extreme 
breadth  of  beam  of  vessel  and  divide  sum  by  half  extreme  breadth.     Mul- 
tiply quotient  by  half  square  root  of  tonnage  (OM)  and  extract  square  root 
of  product. 

For  Mizzen,  take  .74  of  Fore  and  Main. 

EXAMPLE.  —  Required  circumference  of  hemp  rope,  for  main  -  mast  of  a  vessel 
having  a  breadth  of  beam  of  45  feet  and  a  burden  of  3213  tons? 

Extreme  length  of  mast  ____  ..................  94.4  feet. 

Depth  of  hold,  or  total  bury  of  mast,  21.4  feet. 

Head  ..............................  15       "     36.4    " 

Breadth  of  beam,  45  feet. 


58  +  15--:-  1s-  =  3.  58,  and 


58       " 
6  =  io.n  ins. 


Then  if  circumference  for  a  wire  rope  is  required,  see  table,  page  164. 
Thus,  a  hemp  rope  10  ins.  in  circumference  has  equivalent  strength  of  an  iron 
wire  rope  of  4  ins.  and  a  steel  rope  of  3.25+  ins. 

Galvanized  Iron  Wire.—  Experiments  at  Navy  Yard,  Washington,  gave  for  flex- 
ibility a  mean  loss  of  30  per  cent,  and  for  tensile  strength  a  like  loss  of  13.5  per 
cent. 

Relative  Dimensions  of  Kemp  Rope  and  Iron  and  Steel 
\Vire    Rope.     (U.  S.  Navy.) 

Circumference  in  Inches. 

Hemp.  2.5  3.125  4  4.5  5.25  6.5  7.75  8.5  9.5 
Iron..  1.25  1.625  2  2.125  2.5  3  3-5  4  4-5 
Steel..  .875  1.125  i-5  1-625  I-875  2-I25  2.5  2.75  3.25 


11.75  13-  5       '6.5 
5        5-5      6  7 

3.5    4         4.375    5.25 


ANCHORS,  CABLES,  ETC. 


173 


ANCHORS,  CABLES,  ETC. 
Anchors,  Chains,  etc.,  for  a   Griven   Tonnage. 

(American  Shipmasters'  Association.) 
SAILS. 


til 

|fi 

Bow 
With- 
out 
Stock. 

j 
en. 
Admi- 
ralty 
Test. 

ANCHORS 

lucl 
Stream. 

ading  Stc 
KedKe. 

ck. 

3d 

Kedge. 

Diameter. 

CHJ 

M 

! 

JN  CABLI 

Admi- 
ral ty 
Test. 

t.—  STUD 
Weigh 

Stud. 

t  per  Fa 

Short 
Link. 

bom. 

Eng- 
lish.t 

Lbs. 

Tons. 

Lbs. 

Lbs. 

Lbs. 

IllS. 

Fnths. 

Tons. 

Lbs. 

Lbs. 

75 

616 

7 

168 

84 

— 

.8125 

90 

II 

40 

42 

35 

IOO 

728 

8 

106 

112 

— 

.875 

105 

13 

44 

48 

125 

840 

9 

224 

112 

— 

•9375 

105 

15 

51 

55 

48 

150 

952 

IO 

280 

I4O 

— 

i 

1  20 

17-5 

59 

63 

54 

i?5 

1036 

II 

336 

168 

— 

1.0625 

1  20 

20 

66 

70 

200 

1  120 

12 

392 

196 

— 

1.125 

1  20 

22-5 

75 

79 

68 

250 

1288 

13 

448 

224 

112 

1.1875 

135 

25 

82 

88 

— 

300 

1456 

14 

504 

252 

126 

1.25 

135 

28 

91 

98 

84 

350 

1624 

T5-5 

560 

280 

140 

I-3I25 

150 

31 

IOO 

106 

4OO 

1848 

17 

616 

308 

154 

i-3I25 

150 

31 

IOO 

106 

— 

450 

1904 

18.5 

672 

336 

168 

i-375 

165 

37 

H5 

118 

102 

500 

20l6 

20 

784 

392 

196 

1-4375 

165 

40 

120 

— 



600 

2352 

22 

896 

448 

224 

i-5 

180 

44 

132 

— 

122 

700 

2688 

24 

1008 

504 

252 

1-5625 

180 

47 

145 

— 

— 

800 

3024 

26 

II2O 

560 

280 

1.625 

180 

5i 

156 

— 

143 

900 

3248 

28 

1232 

616 

308 

1.6875 

180 

55 

162 

— 

1000 

3584 

29-5 

X344 

672 

336 

i-75 

180 

59 

175 

— 

166 

1200 

3808 

31 

I456 

738 

364 

1.875 

180 

63 

!80 

— 

191 

1400 

4032 

32.5 

1568 

784 

392 

*-9375 

180 

67 

205 

— 

1000 

4256 

34 

1680 

840 

420 

2 

180 

72 

219 

— 

— 

I800 

4480 

35-5 

1792 

896 

448 

2 

180 

72 

240 

— 

217 

2OOO 

4704 

37 

1904 

952 

504 

2.0025 

180 

81 

— 

— 

2500 

5040 

39 

2128 

II2O 

560 

2.125 

180 

86 

— 

— 

244 

3000  |  5376 

4i 

2353 

1232 

616 

2.1875 

180 

96 

— 

— 

t  Brown,  Lennox,  &  Co. 
Xo   CompxTte    Tonnage. 

Take  dimensions  as  follows :  Length.  —  From  after-side  of  stem  to  for- 
ward-side of  stern-post,  measured  on  spar  or  upper  deck  in  vessels  having 
two  decks  and  under,  and  on  main  deck  in  vessels  having  three  or  more 
decks.  Breadth. — Extreme  at  widest  point.  Depth. — At  forward  coaming 
of  main  hatch,  from  top  of  ceiling  at  side  of  keelson  to  under  side  of  deck. 

Then  multiply  these  dimensions  together,  divide  product  by  100,  and 
take  .75  of  quotient. 

All  vessels  to  have  2  bowers  and  i  each  stream  and  kedge  anchor,  and  for 
a  tonnage  exceeding  1400  a  third  bower  is  recommended. 

Hawsers  and  Warps  to  be  90  fathoms  in  length. 

Snroxids. 

SQUARE-RIGGED.  Hemp.— 5.75  ins.  in  diameter  for  a  tonnage  of  75,  ini 
creasing  progressively  up  to  12.75  ins.  for  3000  tons. 

FORE-AND-AFT  RIGGED.  From  .25  to  i  inch  hi  diameter  progressively 
greater  than  for  square-rigged. 

Wire. — One  half  diameter  of  hemp,  increasing  very  slightly  as  tonnage 
increases.  Thus,  for  3000  tons,  12.75  ins.  for  hemp  and  6.875  ins.  for  wire. 


'74 


ANCHOKS,   CABLES,  ETC. 

(American  Shipmasters'  Association.) 
STEAM. 


Tonnage 
computed 
as  per  Rule 
preceding. 

Bow 

ill 

4 

en. 

m 

NCHORS 

luclu 

ding  Stc 

! 

ck. 

'! 

Diam- 
eter. 

C 

i 

Admiral-  > 
tyTest.  * 
O 

ABLE.  —  ST 

Diam. 
Stream. 

UD. 

Weig 

1 

02 

it  per 

«»  . 
5*3 

£3 

Path. 

Ii 

Lbs. 

Tons. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Paths. 

Tons. 

Ins. 

Lbs. 

Lbs. 

100 

336 

4.9 

112 

— 

— 

.6875 

165 

8.1 

•5 

— 

— 

25 

ISO 

448 

6.4 

106 

— 

— 

.8125 

120 

11.9 

.5625 

40 

42 

35 

200 

616 

7.6 

224 

— 

— 

.875 

120 

13-8 

•5625 

44 

48 

250 

672 

8.2 

280 

— 

— 

•9375 

120 

15.8 

.625 

5i 

55 

48 

300 

812 

9-5 

308 

— 

— 

i 

1  2O 

18 

.625 

59 

63 

54 

350 

924 

10.4 

336 

— 

— 

1.0625 

120 

20.3 

.6875 

66 

70 

400 

1  120 

12 

532 

252 

— 

1.125 

135 

22.8 

.6875 

75 

79 

68 

450 

1344 

13-9 

500 

280 

— 

1.1875 

135 

25-4 

•75 

82 

88 

— 

500 

1512 

15-2 

672 

336 

— 

1.25 

150 

28.1 

•75 

91 

98 

84 

600 

i',o8 

I6.7 

738 

364 

— 

1-3125 

150 

31 

.8125 

100 

106 

700 

1876 

18 

784 

392 

— 

1-375 

I65 

34 

.8125 

"5 

118 

104 

800 

2026 

19 

896 

448  224 

1-4375 

I65 

37-2 

.875 

120 

— 

QOO 

2352 

21.6 

I008 

504  252 

i-5 

180 

40.5 

.875 

I32 

— 

122 

1000 

2632 

23-5 

1  120 

560  280 

1-5625 

1  80 

44 

-9375  J45 

— 

— 

1200 

2856 

25.2 

1176 

588  308 

1.625 

180 

47-5 

•9375 

J56 

— 

!43 

1400 

3108 

26.9 

1232 

616  308 

1.6875 

180  51.2 

162 

— 

I6OO 

3360 

28.6 

1344 

672  336 

1-75 

180  55-1 

175 

— 

166 

I800 
2000 

3584 
3808 

30.1 
31.6 

1456 
1512 

738364 
766  364 

1.8125 
1-875 

180 
180 

59-i 
633 

.0625 
.0625 

189 
205 

— 

191 

2300 

4088 

33-4 

1568 

784  392 

1-9375 

180 

67.6 

.125 

215 

— 

2000 

4256 

34.5  1624 

812  392 

2 

270 

72 

.125 

240 

— 

217 

3000 

4480 

35-7 

1680 

840  420 

2.0625 

270 

76.6 

•1875 

— 

— 

3500 

4592 

37 

1792 

896  476 

2.125 

270  81.3  i  .1875 

— 

— 

244 

4000 

4816 

38 

1000 

952  i  504 

2.l875 

270 

86.1   .25 

— 

—  . 

4500 

5040 

39-2 

2128 

1064  532 

2.25 

270 

91.1   .25 

— 

-^ 

— 

SQOO 

5264 

4i 

2352 

1  1  20  560 

2.3125 

270 

96    .3125 

— 

— 

—  — 

*  Brown,  Lennox,  &  Co. 

ANCHORS  AND  KEDGES. 

(U.  8.  Navy.) 
To   Compute   "Weight  of  a   Bovver   A.nchor   for   a  Vessel 

of*  a  given    Character    and    Rate. 

RULE. — Multiply  approximate  displacement  in  tons,  by  unit  in  following 
table,  and  product  will  give  weight  in  Ibs.,  inclusive  of  stock. 

"Units   to   determine    "Weights    and.    !N"u.m"ber   of  Anchors 
or    Kl  edges. 


Displacement 
of  Vessel  in 
Tons. 

£ 
"3 
p 

1 

1 

35 

! 

Displacement 
of  Vessel  in 
Tons. 

"a 
D 

! 

2 
2 
2 

*s 

s 

S. 

3 

Over  3700 
"    2400 
"    1900 

i-75 

2 
2.25 

2 

2 

2 

2 
2 
2 

i 
i 

i 

4 
3 
3 

Over  1500  .  . 
"      900  .. 
900  and  under 

2-5 
2-75 

3 

2 
I 
I 

3 
3 

2 

EXAMPLE.— Tonnage  of  a  bark- rigged  steamer  is  1500. 

1500  X  2. 5  =  3750  Ibs. ,  weight  of  anchor. 

Bower  and  Sheet  Anchors  should  be  alike  in  weight. 

Stream  A  nchors  and  Kedges  are  proportional  to  weight  of  bowers.  Thus, 
Stream  Anchor  .25  weight.  Kedges.  —  If  i,  .125  weight;  if  2,  .16  and  .1 
weight;  if  3,  .16,  .125,  and  .1  weight. 


ANCHORS,  CABLEis,   ETC. — TONNAGE. 


175 


To  Com.pu.te  Diameter  of  a  Chain  Ca"ble   corresponding 
to   a   Q-iveii    ^Weight    of  Anchor. 

(U.  S.  Navy.) 

RULE. — Cut  off  the  two  right-hand  figures  of  the  anchor's  weight  in  Ibs., 
multiply  square  root  of  remainder  by  4,  and  result  will  give  diameter  of 
chain  in  sixteenths  of  an  inch. 
EXAMPLE.— The  weight  of  an  anchor  is  2500  Ibs. 

-^25.00  x  4  =  20  sixteenths  =  1.25  ins. 

NOTE.— Diam.  of  a  messenger  should  be  .66  that  of  the  cable  to  which  it  is  applied. 

Lengths   of  Chain    Cables    for   each   Anchor. 

(U.  S.  Navy.) 


Weight  of  Anchor. 

Bower. 

Sheet. 

Stream. 

Weight  of  Anchor. 

Bower. 

Sheet. 

Stream. 

Lbs. 

Fathoms. 

Fathoms. 

Fathoms. 

Lba. 

Fathoms. 

Fathoms. 

Fathom* 

Under    800 

60 

60 

60 

Over  2000 

120 

120 

90 

Over      800 

90 

90 

60 

"    3000 

120 

120 

00 

"          1200 

90 

90 

75 

"    5000 

120 

120 

105 

"          I600 

105 

105 

75 

"     75oo 

135 

135 

105 

ANCHORS. 

From  Experiments  of  a  Joint  Committee  of  Representatives  of  Ship- 
owners and  Admiralty  of  Great  Britain. 

An  anchor  of  ordinary  or  Admiralty  pattern,  Trotman  or  Porter's  im- 
proved (pivot  fluke),  Honiball,  Porter's,  Aylin's,  Rodgers's,  Mitcheson's,  and 
Lennox's,  each  weighing,  inclusive  of  stock,  27  ooo  Ibs.,  withstood  without 
injury  a  proof  strain  of  45000  Ibs. 

Breaking  weights  between  a  Porter  and  Admiralty  anchor,  as  tested  at 
Woolwich  Dock-yard,  were  as  43  to  14. 

Comparative    Resistance   to   Dragging. 

Trotman 's  dragged  Aylin's,  Honiball's  Mitcheson's  and  Lennox's ;  Aylin's 
and  Mitcheson's  dragged  Rodgers's ;  and  Rodgers's  and  Lennox's  dragged 
Admiralty's.  

(TONNAGE   OF   VESSELS. 
To   Compute   Tonnage   of  Vessels. 

For  Laws  of  United  States  of  America,  with  amendments  of  1882  relative 
to  Steam- vessels,  see  Mechanics'  Tables,  with  rule  and  illustrated  diagrams, 
by  Chas.  H.  Haswell,  3d  edition,  Harper  &  Bros.,  New  York,  1878. 

English  Registered  Tonnage.  (New  Measurement.) 
Divide  length  of  upper  deck  between  after-part  of  stem  and  fore-part  of  stern- 
post  into  6  equal  parts,  and  note  foremost,  middle,  and  aftermost  points  of  division. 
Measure  depths  at  these  three  points  in  feet  and  tenths  of  a  foot;  also  depths  from 
under-side  of  upper  deck  to  ceiling  of  limber-strake;  or  in  case  of  a  break  in  the 
upper  deck,  from  a  line  stretched  in  continuation  of  the  deck.  For  breadths,  divide 
each  depth  into  5  equal  parts,  and  measure  the  inside  breadths  at  following  points, 
viz. : — At  .2  and  .8  from  upper  deck  of  foremost  and  aftermost  depths;  and  from 
.4  and  .8  from  upper  deck  of  amidship  depth.  Take  length  at  half  amidship  depth 
from  after-part  of  stem  to  fore-part  of  stern-post. 

,     Then,  to  twice  amidship  depth  add  foremost  and  aftermost  depths  for  sum  of 
depths,  and  add  together  foremost  upper  and  lower  breadths,  3  times  upper  breadth 
with  lower  breadth  at  amidship,  and  upper  and  twice  lower  breadth  at  after  division 
for  sum  of  breadths. 
Multiply  together  sum  of  depths,  sum  of  breadths,  and  length,  and  divide  product 

I  by  35°o,  which  will  give  number  of  tons. 

If  the  vessel  has  a  poop  or  half  deck,  or  a  break  in  upper  deck,  measure  inside 
mean  length,  breadth,  and  height  of  such  part  thereof  as  may  be  included  within 
the  bulkhead ;  multiply  these  three  measurements  together,  divide  product  by  92.4, 
and  quotient  will  give  number  of  tons  to  be  added  to  result  as  above  ascertained. 


176 


TONNAGE   OP   VESSELS. 


For  Open  Vessels.  —Depths  are  to  be  taken  from  upper  edge  of  upper  strake. 

For  Steam  Vessels. — Tonnage  due  to  engine-room  is  deducted  from  total  tonnage 
computed  by  above  rule.  To  determine  this,  measure  inside  of  the  engine-room 
from  foremost  to  aftermost  bulkhead;  then  multiply  this  length  by  amidship  depth 
of  vessel,  and  product  by  inside  amidship  breadth  at  .4  of  depth  from  deck,  and 
divide  final  product  by  92.4. 

The  volume  of  the  poop,  deck-houses,  and  other  permanently  enclosed  spaces, 
available  for  cargo  or  passengers,  is  to  be  measured  and  included  in  the  tonnage, 
but  following  deductions  are  allowed,  the  remainder  being  the  Register  tonnage. 

Deductions. — Houses  for  the  shelter  of  passengers  only;  space  allotted  to  crew 
(12  square  feet  in  surface  and  72  cube  feet  in  volume  for  each  person);  and  space 
occupied  by  propelling  power. 

Approximate    Rule. 

Gross  Register.—  Tonnage  of  a  vessel  expresses  her  entire  cubical  volume  in  tons 
of  loo  cube  feet  each,  and  is  ascertained  by  following  formula  : 

• —  =  Gross  tonnage,  and  —   —  c  =  Register  tonnage.    L  representing  length 
of  keel  between  perpendiculars,  B  breadth  of  vessel,  and  D  depth  of  hold,  all  in  feet. 

Builders'    ^Measurement. 
(L-.6B)XBX.5B=  e 

94 

Fore-perpendicular  is  taken  at  fore-part  of  stem  at  height  of  upper  deck. 

Aft-perpendicular  is  taken  at  back  of  stern-post  at  height  of  upper  deck. 

In  three-deckers,  middle  deck  is  taken  instead  of  upper  deck. 

Breadth  is  taken  as  extreme  breadth  at  height  of  the  wales,  subtracting  differ- 
ence between  thickness  of  wales  and  bottom  plank.  Deductions  to  be  made  for 
rake  of  stem  and  stern. 

1 8     /  Girth +  Breadth\2 
Iron  Vessels.      I — )  X  length  =  Gross  tonnage. 

I0000\  2  ) 

Length  measured  on  upper  deck,  between  outside  of  outer  plank  at  stem  and 
the  after-side  of  stern  post  and  rabbet  of  stern-post,  at  point  where  counter  plank 
crosses  it.  Girth  measured  by  a  chain  passed  under  bottom  from  upper  deck  at 
extreme  breadth,  on  one  side,  to  corresponding  point  on  the  other. 

Register  tonnage  = X  C.     C  representing  a  coefficient  for  vessels  as 

IOO 

follows : 


of  usual  form  .................  7 


Clippers  and  Steamers  { 


Yachts  above  60  tons 5 


5    SmanveSse,s{^a;-;::::::::;-45 

"Units   for   Measurement   and   Dead --weight    Cargoes. 

(C.  Mackrow,  M.  S.  N.  A.) 

To  Compute  Approximately  for  an  Average  Length  of  Voyage  the  Measure- 
ment Cargo,  at  40  feet  per  Ton,  which  a  Vessel  can  carry. 

RULE.— Multiply  number  of  register  tons  by  unit  1.875,  ai*d  product  will 
give  approximate  measurement  cargo. 

To  Compute  Approximately  Dead-weight  Cargo  in  Tons  ivhich  a  Vessel  can 
carry  on  an  Average  Length  of  Voyage. 

RULE.— -Multiply  number  of  register  tons  by  1.5,  and  product  will  give 
approximate  dead-weight  cargo  required. 

With  regard  to  cargoes  of  coasters  and  colliers,  as  ascertained  above,  about 
10  per  cent,  may  be  added  to  said  results,  while  about  10  per  cent,  may  be 
deducted  in  cases  of  larger  vessels  on  longer  voyages. 


TONNAGE    OF   VESSELS.  177 

In  case  of  measurement  cargoes  of  steam-vessels,  spaces  occupied  by  ma- 
chinery, fuel,  and  passenger  cabins  under  the  deck  must  be  deducted  from 
space  or  tonnage  under  deck  before  application  of  measurement  unit  thereto. 

In  case  of  dead-weight  cargoes,  weight  of  machinery,  water  in  boilers,  and 
fuel  must  be  deducted  from  whole  dead  weight,  as  ascertained  above  by 
application  of  dead-weight  unit. 

The  deductions  necessary  for  provisions,  stores,  etc.,  are  allowed  for  in 
selection  of  the  two  units. 

To  A  scertain  Weight  of  Cargo  for  an  A  verage  Length  of  Voyage.    (Moorsom. ) 

Deduct  tonnage  of  spaces  of  passenger  accommodations  from  net  register 
tonnage,  and  multiply  remainder  by  1.5. 

Average  space  for  each  ton  weight  of  cargo  on  such  a  voyage  67  cube  feet. 

Freight   Tonnage   or    Measurement    Cargo. 

Freight  Tonnage  or  Measurement  Cargo  is  40  cube  feet  of  space  for  cargo, 
and  it  is  about  1.875  times  net  register  tonnage  less  that  for  passenger  space. 

Royal   Thames   Yacht   Cluto. 

Measure  length  of  yacht  in  a  straight  line  at  deck  from  fore-part  of  stem  to  after- 
part  of  stern-post,  from  which  deduct  extreme  breadth  (measured  from  outside  of 
outside  planking),  both  in  feet;  remainder  is  length  for  tonnage.  Multiply  length 
for  tonnage  by  extreme  breadth,  that  product  by  half  extreme  breadth,  divide  re- 
sult by  94,  and  quotient  will  give  tonnage. 

If  any  part  of  stem  or  stern-post  projects  beyond  length  as  taken  above,  such 
projection  or  projections  shall,  for  purpose  of  computing  tonnage,  be  added  to  length 
taken  as  before  mentioned. 

All  fractional  parts  of  a  ton  are  to  be  considered  as  a  ton. 
Measurements  to  be  taken  either  above  or  below  main  wales. 

'• —  =  Tons.    L  representing  length  and  B  breadth,  in  feet. 

Corinthian   and.   New   Thames    Yacht   Civil). 

Measure  length  and  breadth  as  in  foregoing  rule,  and  depth  to  top  of  covering 
board;  multiply  length,  breadth,  and  depth  together,  divide  result  by  200,  and  quo- 
tient will  give  tonnage. 


Suez    Canal    Tonnage. 

Gross  Tonnage.— Spaces  under  tonnage  deck,  below  tonnage  and  uppermost  deck, 
all  covered  or  closed  -  in  spaces,  such  as  poop,  forecastle,  officers'  cabins,  galley, 
cook,  deck,  and  wheel  houses,  and  all  inclosed  or  covered-in  spaces  for  working  the 
vessel. 

From  which  are  to  be  deducted  berthing  accommodations  for  crew,  not  including 
spaces  for  stewards  and  passengers'  servants ;  berthing  accommodations  for  officers, 
except  captain;  galleys,  cook-houses,  etc.,  used  exclusively  for  crew,  and  inclosed 
spaces  above  uppermost  deck,  designed  for  working  the  vessel.  In  none  of  these 
spaces  can  passengers  be  berthed  or  cargo  carried,  and  total  deduction  under  all  of 
these  spaces  must  not  exceed  5  per  cent,  of  gross  tonnage. 

In  steamers  witb  standing  coal-bunkers,  English  rule  may  be  followed,  or  owner 
may  elect  to  have  tonnage  of  his  vessel  computed  by  "Danube  rule,"  which  is  an 
allowance  of  50  per  cent,  above  space  allowed  to  machinery  in  side- wheel  steamers 
and  75  in  screw  steamers. 

In  no  case,  however,  except  with  tow-boats,  must  deduction  for  propelling  power 
exceed  50  per  cent,  of  gross  tonnage. 


1^8  WORKS   OF   MAGNITUDE. 

WORKS    OF   MAGNITUDE. 

American. 
Aqueducts,  Roads,  and    Railroads. 

Croton  Aqueduct,  N.  Y.  — Has  a  section  of  53.34  square  feet  and  capacity  of 
loooooooo  to  118000000  gallons  per  day.  and  from  Dam  to  Receiving  Reservoir  is 
38. 134  miles  in  length. 

Aqueduct,  Washington.  —Cylinder  of  masonry  9  feet  in  diameter.  Stone  arch 
over  Cabin  John's  Creek,  220  feet  span,  57.25  feet  rise. 

National  Road.— Over  the  Alleghany  Mountains,  Cumberland  to  Illinois  Town. 
650.625  miles  in  length,  and  80  feet  in  width.  Macadamized  for  a  width  of  30  feet. 

Illinois  Central  Railroad.—  Chicago  to  Cairo,  length  365  miles,  Centralia  to  Dun- 
leith  344  miles,  total  709  miles. 


Bridges. 

Suspension  Bridge,  Niagara  River.— Wire,  Span  1042  feet  10  ins. 

Suspension  Bridge,  New  York  and  Brooklyn.  —  Length  of  river  span  1595  feet  6 
ins. ;  of  each  land  span  930  feet;  length  of  Brooklyn  approach  971  feet;  of  N.  Y. 
approach  1562  feet  6  ins. ;  total  length  of  bridge  5989  feet;  width  85  feet;  number 
of  cables  4;  diameter  of  each  cable  15.5  ins. ;  each  consisting  of  6300  parallel  steel 
wires  No.  7  gauge,  closely  laid  and  wrapped  to  a  solid  cylinder;  ultimate  strength 
of  each  cable  11200  tons;  depth  of  tower  foundation  below  high  water,  Brooklyn, 
45  feet — New  York  78  feet;  towers  at  high-water  line  140X59  feet;  towers  at  roof 
course  136x53  feet;  total  height  of  towers  above  high  water  277  feet;  clear  height 
of  bridge  in  centre  of  river  span  above  high  water,  at  50°,  135  feet;  height  of  floor 
at  towers  above  high  water  119  feet  3  ins. ;  grade  of  roadway  3  feet  in  100;  anchor- 
ages, at  base  i2gX  119  feet,  at  top  ii7X  104  feet;  weight  of  each  anchor-plate  23  tons. 

Iron  Pipe  Bridge  over  Rock  Creek. — 200  feet  span,  20  feet  rise.  Arch  of  2  lateral 
courses  of  cast-iron  pipe,  4  feet  internal  diameter,  and  i  inch  thick.  These  pipes 
conveying  the  water  not  only  sustain  themselves  over  the  great  span,  but  support 
a  street  road  and  railway. 

Iron  Bridge  over  Kentucky  River  near  Shakers'  Ferry,  Md.— 3  spans,  each  375 
feet,  and  275.5  feet  above  low  water. 

Bridge  on  line  of  New  York,  Erie,  and  Western  Railroad  across  the  Kinxua,.-— 
Of  iron;  length  2060  feet;  central  span  301  feet  in  height. 

Iron  Truss.—  Cincinnati  and  Southern  Railway,  over  Ohio  River,  519  feet, 

Foreign. 
Pyramids,  Statues,  etc. 

Pyramid  of  Cheops,  Egypt.— Length  of  side  at  base  762  feet;  height  to  present 
summit  453.3  feet;  to  original  summit  485.2  feet;  inclined  length  568.25  feet;  angle 
of  side  51°  51'  14";  area  of  each  face  =  square  of  height;  weight  5272600  tons; 
built  2 1 70  years  B.C. 

Peter  the  Great,  St.  Petersburg.  Russia.— Bronze;  height  of  horse  17  feet;  of  man 
ii  feet;  base  of  rock  42  feet  at  bottom,  36  at  top,  21  wide,  and  17  high,  weighing 
ti  oo  tons. 

Liberty,  New  York  Harbor. — Bronze;  no  feet  in  height  from  head  to  foot  and 
151.1  feet  to  flambeau ;  including  base,  305.6  feet.  Weight  of  statue  225  tons. 

Daibutsu,  of  stone,  Japan. —Sitting  posture;  height  44  feet^  circumference  87 
feet;  face  8.5  feet;  circumference  of  thumb  3  5  feet. 

Colossus  of  Rhodes.  —Height,  105  feet. 

Bridge. 

Britannia  Tubular  Bridge.  —  Of  iron,  with  a  double  line  of  Railway,  964  feet  in 
length,  with  two  approaches  of  230  feet  each.  Weight  3658  tons. 


WORKS   OF   MAGNITUDE. 
UVIonolitlis. 


179 


Obelisk  at  Karnak,  Egypt— Of  granite,  108  feet  10  ins. ;  pedestal  13  feet  2  ins.: 
weight  400  tons. 

Obelisk  in  Central  Park,  N.  Y. — Of  granite,  68  feet  n  ins.;  weight  168  tons. 

U.  S.  Treasury,  Washington.— Some  stones  of,  are  heavier  than  any  in  the  Pyra* 
mids  of  Egypt. 

Steam.    Hammers. 

At  workshops  of  Herr  Krupp,  at  Essen,  there  is  a  steam  hammer  weighing  50  tone 
having  a  fall  of  3  metres;  and  at  Creusot  there  is  a  hammer  weighing  between  75 
and  80  tons  having  a  fall  of  5  metres. 

Crane. 

At  Creusot  there  is  a  steam  crane  having  a  capacity  to  lift  150  tons. 

Ch.imn.eys. 

J.  Townsend's  chemical  works,  Glasgow,  diameter  at  foundation  50  feet;  at  top 
12  feet  8  ins. ;  height  from  foundation  488  feet;  from  ground  474  feet. 

Metropolitan  Traction  Company,  N.  Y.,  diameter  at  base  85  feet;  at  top  25  feet, 
and  height  353  feet. 

Pillar. 

At  a  gate  near  Delhi  is  a  wrought-iron  pillar  having  diameters  of  16.4  ins.  at  22 
feet  in  its  height  above  ground  and  12  ins.  at  its  top.  It  is  estimated  from  the  re- 
sult of  excavations  at  its  base  to  be  60  feet  in  length  or  height  and  to  weigh  17 
tons.  Its  period  of  structure  is  assigned  to  the  3d  or  4th  century  A.D. 

Roofs. 

Midland  Railway  Station,  London.  240(1.  I  Union  Railway  Station,  Glasgow.  195  ft 
Imperial  Riding-School,  Moscow.  235  "    |  Grand  Central  Station,  N.  Y 200  " 

JDiameters    of  Domes. 


DOMES. 

Feet. 

DOMES. 

Feet. 

DOHXS. 

Capitol,  Washington 
Glasgow  W.  Railw'y 

124-75 
198 

St.  Paul's,  London. 
St.  Peter's,  Rome.. 

112 

139 

MidrndRail'y,Lon. 
Great  North'n,  Eng. 

Lengths   of  Tnnnels. 


TtTNNILS. 

Feet. 

TUNNKLS. 

Feet. 

TUNNILS. 

Feet. 

Blaizy  

13  455 

Gunpowder,  Md..  . 

06  EJOO 

Nerthe 

Blue  Ridge  

4280 

Sutro  

20028 

Nochistongo 

I5IS3 

Hoosac  

25  031 

Semmering  .  .  , 

1610 

Riauivel.  .  . 

18627 

Thames  and  Medway ,  1 1 880  feet.        Weehawken,  4000  feet 

Mont  Cenis  7.5  miles  242  yards,  rises  i  in  45,  and  descends  i  in  2000. 

St.  Gothard  Tunnels  and  Roads  o  miles  477  yards  in  length ;  tunnels  116 156.5  feet, 
and  rises  i  in  233  in  whole  length;  26.5  feet  in  width;  19  feet  10  ins.  in  height 
Maximum  grade  2.7  feet  per  100.  Schemnitz,  10.27  miles  in  length,  9  feet  10  ins. 
in  height  by  5.25  feet  in  width. 

Miscellaneous. 

Fortress  Monroe,  Old  Point  Comfort,  Va.— Largest  fortress. 

Telegraph  Wire.—  Span  over  river  Kistnah  between  Bezorah  and  Sectanagran, 
6000  feet  in  length. 

Deer  Park,  Copenhagen.— 4200  acres. 

Oxford  College,  England.  — Largest  University;  said  to  have  been  founded  by 
Alfred. 

Cathedral  St.  Peter's,  Rome.— Width  of  front  216  feet;  of  the  cross  251  feet;  total 
height  469. 5  feet. 

Steamer  Great  Eastern.— Of  iron,  680  feet  in  length;  83  feet  width  of  beam-  60 
feet  depth  of  hold;  22927  tons;  built  at  Millwall,  England,  1857. 

Chinese  Wall.— 25  feet  at  base;  15  at  top;  height,  with  a  parapet  of  5  feet,  20  feett 
length  1250  miles. 

ArUsian  Wett,  Perth.— 3050  feet  in  depth;  temperature  of  wat«r  99°:  volume 
of  discharge  18000  gallons  per  day. 


ISO       BELLS,  CHURCHES,  COLUMNS,  TO  WEBS,  ETC. 


Weights    of  Bells. 


BULLS. 

Lbs. 

BELLS. 

Lbs. 

BELLS. 

Lbs. 

Pekin          

I2OOOO 

Oxford,      "Great 

St.  Peter's,  Rome. 

1  8  ooo 

Lewiston  Me  

Tom  "  Eng.  .... 

17  O24. 

Vienna  

40  200 

Montreal,  Can  

10233 
28560 

Olmutz.  Bohemia. 

•/  "-'•*/t 
40320 

Westm'ster,  "Big 

Moscow,  Russia.  .  . 

44S772 

Sac'd  Heart,  Paris 

55  o°° 

Ben,"  England. 

35620 

Erfurt,  Saxony.  .  .  . 
Notre  Dame,  Paris 

30800 
28670 

St.  Paul's,  Eng.  .  . 
St.  Ivan's,  Moscow 

42000 
127  830 

York 
State  House,  Phila. 

24080 
13000 

Rangoon,  Burmah,  201 600  Ibs. 

Capacity-   of  Principal   Ch/urohes   and.   Opera   Houses. 

Estimating  a  person  to  occupy  an  Area  0/19.7  Ins.  Square. 
Clinrones. 

St.  Peter's 54  ooo  I  St.  John,  Lateran 22  900 

Milan  Cathedral 37  ooo    Notre  Dame,  Paris 21  ooo 

St.  Paul's,  Rome 32000  I  Pisa  Cathedral 13000 

St.  Paul's,  London 25  600  I  St.  Stephen's,  Vienna 12  400 

St.  Petronio,  Bologna 24  400  |  St.  Dominic's,  Bologna 12  ooo 


Florence  Cathedral 24  300 

Antwerp  Cathedral 24  ooo 

St.  Sophia's,  Constantinople 23  ooo 


Tabernacle,  London 7  ooo 

"         Brooklyn 5500 

St.  Mark's,  Venice 7  ooo 


Opera   Houses   and.  Theatres. 


Carlo  Felice,  Genoa 2560 

Opera  House,  Munich 2370 

Alexander,  St.  Petersburg 2332 

San  Carlos,  Naples 2240 

Imperial,  St.  Petersburg 2160 

La  Scala,  Milan 2113 

Academy  of  Paris 2092 


Teatro  del  Liceo,  Barcelona 4000 

Covent  Garden,  London 2684 

Opera  House,  Berlin 1636 

New  York  Academy 2526 

Metropolitan  Opera,  N.  Y 5000 

Philadelphia  Academy 3124 

Chicago  "        3000 


Heights   of  Columns,  Towers,  I3oraes,  Spires,  etc. 


LOCATIONS. 

Feet. 

CHIMNEYS. 

Townsend's  Glasgow.  .  . 
St  Rollox      u 

474 
455-  5 

Musprat's                     Liverpool 

406 

Gas  Works  Edinburgh 

New  E  ngland  Glass  Co  .  Boston  
Steam  Heating  Co  New  York. 
Metropolitan  Tract.  Co.         '  * 

COLUMNS. 

Alexander  St.  Peters'g 

230 

220 

353 
*75 

Bunker  Hill  Mass  
City  London... 
July  Paris  

221 

202 

157 
132 

Nelson's  London  .  .  . 
Place  Vendome  Paris  

171 

Pompey's  Pillar  Egypt  
Trajan  Rome  

114 
145 

Washington           .   .  Wash'gton 

555 

York  '  London  .  .  . 

I38 

TOWERS  AND  DOMES. 

Babel  

680 

Balbec 

Capitol  Wash'gton 

287  5 

St.  Peters  Rome  
Cathedral  Cologne  .  .  . 
"       Cremona.. 
"       Escurial.  .. 

469-5 
524-9 
392 
200 

LOCATIONS. 

Feet. 

TOWERS  AND   DOMES. 

Cathedral  Florence  .  . 
"         .Maedeb'rer 

390-5 

4$  9 

363 

188 

200 

328 

355-1 

325 

465-9 
4048 
216 
410 

450 

210 

2OO 
404 
286 

216 

232.9 

344 
473 
443-8 
486 
464 
314-9 

.Milan  
.Petersburg 

Leaning 

Pisa 

Porcelain  
St.  Mark's  
3t.  Paul's  

SPIRES. 

Catbedral  

.China  
.Venice.  .  .. 
.  London  .  .  . 

.New  York. 

Strasburg 

Grace  Church 

.Antwerp  .  . 
New  York 

Freiburg  

Salisbury 

St  John's  

New  York. 

fSt  Paul's 

St.  Mary's  
Trinity  Church  

.Lubeck... 
.New  York. 

Balustrade    of    Notre 
Dame  Paris  

Towers  of  ditto 

u 

HAtel  des  Invalides.. 
St.  Nicholas  
St.  Stephen  
Strasburg.  
Utrecht  

.Hamburg.. 
.Vienna  

Votive  Church  

.  Vienna  — 

BRIDGES,   CANALS,   BREAKWATERS,   ETC. 


Areas   of  Lakes   in.   Europe,  Asia,  and   Africa. 


LAKES. 

Mi?es. 

LAKES. 

Mrfes. 

LAKES. 

Sq. 

Miles. 

400 
11600 

L 

Feet. 

Dembia,  Abyssinia. 
Loch  Lomond  

engths   of  B 

BRIDGES. 

13000 
27 

ridge 

Feet. 

Lough  Neagh,lrel'd 
Tonting,  China  

S. 
<X';        HilwiM  ;  •  «fU$ 

80 
1200 

Feet. 

Tchad,  Africa  

BRIDGES. 

Avignon  

Btulaioz 

1710 
1874 
2500 
995 
3483 
050 

Lion,  China  
M  enai 

6600 
1050 

5989 
3060 

Potomac       .  .  .  •  • 

5300 
2600 
9144 

339° 
860 
1223 

Riga  

Belfast  

N.  Y.  and  Brook-  ) 
lyn  spans  andj 
approaches....  ) 
Pont  St.  Esprit.  .  . 

St.  Lawrence  Riv'r 
Strasburg  

Blackfriars 

Vauxhall  

London.  .  . 

Westminster  

Lengths  of  Spans   of  Bridges. 


BRIDGES.       ,-: 

Feet.' 

BKIDGES. 

Feet. 

.,,-,.       BRIDGES. 

Feet. 

Britannia 

A6o 

Niag'a  at  the  Falls 

1268 

Schuylkill  

34° 

Conway 

u     at   Queens- 

South  wark  

240 

Menai  .  . 

400 
*8o 

town.  .  . 

1040 

Wheeling.  

1010 

Canals. 

Lengths.—  Lake  Erie  to  Albany  352  miles;  Chesapeake  and  Ohio  307;  Schuylkill 
108;  Delaware  and  Hudson  109;  Rideau  132;  London  to  Liverpool  265;  Caledonia 
25;  Liverpool  and  Leeds  127.5;  Rhone  to  Rhine  203. 

Capacity  of  Locks  of  Erie  240  tons,  and  of  Welland  1500. 

Welland  26.77  miles.  Lake  Erie  to  Montreal  via  Canal  70.5;  Lake  and  River 
375  miles. 

Montreal  to  Kingston.— Canal  120  miles;  River  126.25.    Suez,  sge  page  183. 

Breakwaters. 

Delaware.—  Average  depth  of  water  29.4  feet  below  low- water  level;  range  of  tide 
6.66  feet;  Outer  slope  45°;  Inner  slopes  1.5,  5,  3,  and  1.3  to  i ;  length  of  base  172.12 
feet. 

Plymouth.— Outer  slopes  1.75  to  i  from  bottom  to  7  feet  6  ins.  below  low- water 
line;  4  to  i  to  low- water  line;  16  to  i  to  4  feet  6  ins  above  low-water  line;  5  to  i 
to  high  water;  Inner  slope  1.5  to  i  above  low  water  line;  2  to  i  below  low- water  line. 

Depth  of  water  at  high  tide  46.5  feet;  at  low  tide  30  feet. 

Body  of  breakwater  cased  with  large  squared  stones  cramped  together. 

Portland.—  Depth  of  high  water  58  feet;  of  low  water  51  feet ,  Outer  slopes  i  to  i 
from  bottom  to  20  feet  below  low  water;  2  to  i  to  12  feet  below  low  water;  6  to  i 
to  low- water  line;  4  to  i  to  high- water  line;  Inner  slope  1.25  to  i. 

Body  of  breakwater,  rubble,  with  crest  wall  of  ashlar. 

Dover. — Depth  of  high-water  line  61  feet;  of  low-water  line  42  feet. 

Body  of  breakwater,  concrete  blocks  faced  with  granite;  batter  3  inches  to  the 
foot,  stepped  up  in  each  course. 

Marseilles.—  Depth  of  water  33  feet;  Outer  casing  of  beton  25. 5  tons  each ;  average 
thickness  of  casing  from  14  to  20  feet;  slope  i  to  i  from  bottom  to  water  line;  2.5 
to  i  above  water-line;  all  other  slopes  .33  to  i;  Inner  casing  of  first-class  rubble 
(of  stones  2  to  5  tons  weight),  about  12  feet  thick;  Hearting,  second-class  rubble 
(of  stones  .5  to  2  tons  weight),  about  6  feet  thick;  Nucleus,  of  quarry  rubbish. 

Algiers. — Depth  of  water  50  feet;  rubble  base  carried  up  to  33  feet  from  surface  of 
water ;  the  remainder  composed  of  large  beton  blocks  25. 5  tons  each ;  slopes  of  rubble 
base  i  to  i ;  Outer  slope  of  beton  blocks  1.25  to  i ;  Inner  slope  of  beton  blocks  i  to  i. 

Port  Said  (Suez  Canal). — Concrete  blocks,  10  cubTc  metres  each,  composed  of  i 
of  hydraulic  lime  to  13  of  sand,  mixed  with  sea  water;  4  days  in  the  mold  and  dried 
for  4  months  before  being  put  in  position.  In  some  instances  the  composition  of 
beton  blocks  is  .33  lime  or  cement  to  .66  sand  and  broken  stone,  about  the  size  of 
ballasting. 

Rubble  or  Block  Filling.—  Proportion  of  interstices  to  volume  of  breakwater  fin- 
ished: First-class  rubble  of  2  to  5  tons,  .25;  second  class  rubble  of  ,5  to  2  tons,  .2; 
third-class  rubble,  quarry  chips,  etc.,  .16;  beton  blocks,  15  to  25  tons,  .33. 

NOTE. — For  force  of  water,  see  Waves  of  the  Sea,  page  853. 

Q 


182 


LAKES,   OCEANS,   SEAS,   MOUNTAINS,   ETC. 


Areas,  Depths,  and   Heights    of*  GJ-reat   Northern 
of  United    States. 


LAKES. 

Length. 

Breadth. 

Mean  Depth. 

Height 
above  Sea. 

Area. 

Miles. 
250 

Miles. 
80 

Feet. 
200 

Feet. 
564 

Sq.  Miles. 

Huron  . 

200 

1  60 

1  20 

2o  8OO 

360 

IOQ 

QOO 

587 

Ontario 

1  80 

6c 

5OO 

Superior*... 

400 

160 

288 

6« 

•12000 

*  Greatest  depth  5400  feet. 

Elevation  Above  Tide-water  at  Albany.  —  Lake  Erie  570.6  feet;  Hudson  River 
2. 46  feet. 

Meaift   Depths   and   A.reas   of*  the   Oceans   and    Seas. 
(Herr  Krummel) 


Fathoms. 

Area 
Sq.  Miles. 

Fathoms. 

Area 
Sq.  Miles. 

201^ 

29  514275 

Gulf  of  Mexico  

IOOI 

I  765  910 

Archipelago      

487 
407 

3  046  600 

"    "  St.  Lawrence 

1  60 

101  075 

Azof  f 

8800 

Indian  

1829 

28  369  59$ 

Baltic  Sea 

,6 

Japan 

•383  205 

Black  Sea  

150000 

Mediterranean  

720 

1  109  230 

Behrinc's  Straits 

gg  4  eee 

North  Sea 

8 

Caspian  Sea  

I2OOOO 

North  Ice  Sea  

845 

2*°  §°5 
5  264  ooo 

China  (East)  Sea  .  . 

66 

4.72  2IO 

Persian  Gulf  

20 

90  100 

Dead  Sea 

Pacific  

1887 

60  343  690 

Enelish  Channel,  etc. 

,7 

78^16 

Red  Sea... 

24^1 

170820 

Mean  depth  of  Ocean  surrounding  land  1877  fathoms  =  2. 19  miles. 

In  his  subsequent  computations  he  estimates  ocean  area  at  143703000  square 
miles  and  determines  area  of  land  to  water  as  i  to  2.75,  and  that  mean  height  of 
land  =  1377  feet,  or  one  eighth  that  of  Ocean. 

Heights    of  Mountains,  Volcanoes,  and    IPasses 
above   Level   of  Sea. 


MOUNTAINS. 

|    Feet. 

MOUNTAINS. 

Feet. 

MOUNTAINS. 

Feet. 

Mount  Everest  (Him- 

Mount Pitt  

9  549 

EUROPE. 

Azores  Pico  

7  61^ 

alaya,  highest)  .  .  . 
Mount  Libanus.... 

29003 
9  523 

Mount  Washington. 
Nevado  de  Sorata.  . 

6426 
25  248 

Barthe'lemy  France 

f> 

Petcha  

15  ooo 

Orizaba    .   . 

18870 

Sinai  .... 

Potosi 

Ben  Nevis 

4  38° 

Sierra  Nevada  .  .  . 

Elbrus  Caucasus 

AFRICA. 

Tahiti 

T^  R 

Guadarama  Spain 

8  520 

Atlas  

10  400 

White  Mountains 

Hacla  

5147 

Compass,   Cape   of 

* 

Ida           

f> 

Good  Hope  

IOOOO 

VOLCANOES. 

Jungfrau  Switz'd.  . 

13  725 

Dianai  Peak,  St  He- 

Cotopaxi   

18887 

Mont  Blanc  

je  707 

lena  

2  700 

Etna 

IO8?A 

"    Cenis  

6780 

20  ooo 

Hecla    

5  ooo 

Mont  d'  Or,  France. 
Mulahassen,Gren'a. 
Neph  in,  Ireland  
Olympus  

6510 
II  663 
2634 

6  cio 

Ruivo,  Madeira.  ... 
Teneriffe  Peak  

AMERICA. 

5160 
12300 

Popocatapetl  
Sahama  
St.  Helen's,  Oregon. 
Vesuvius  

17784 
22350 
13320 

•a  Q'IO 

Parnassus 

6  ooo 

Aconcagua  (highest 

Plynlimmon,  Wales. 
The  Cylinder,  Pyr.  . 
Wetterhorn  

2463 
10930 

12  154 

in  America)  
Blue  Mount,  Jam  'a. 
Catskill  

23910 
8000 
3  804 

PASSES. 

Cordilleras  { 

13525 

Chimborazo 

Mont  Cenis     . 

6778 

ASIA. 

Ararat  

Correde,  Potosi  .... 
Crows'  Nest   Hi^h- 

21  441 
16036 

"     Cervis  
Pont  d'  Or 

11  100 

0843 

16  433 

lands  N.  Y  

I  37O 

St  Bernard,  Great.  . 

8  172 

Phawalagheri  

28077 
8  500 

Great    Peak,    New 
Mexico  

19  788 

"          Little.  . 
St  Gothard  

7192 
6808 

Mount  Lebanon... 

12000 

Mauna  Lou.  Hawaii 

liSot: 

Simnlon.  .  . 

6«3 

CANAL   LOCKS,  ELEVATIONS,  AND   RIVERS.  183 


Dimensions    of*  Canal    Locks.  —  (U.  S.) 


CANAL. 

ja 

! 

Breadth- 

ja 

& 

Length  of 
Canal. 

CANAL. 

5 

! 

1 

! 

UJ^ 
CanaL 

Albemarle  and  ) 

Feet. 

Ft. 

Ft. 

Miles. 

Cham  plain  

Feet, 
no 

Feet. 
18 

Feet. 

Miles. 
66  7< 

Chesapeake.  .  J 
Black  River        ] 

220 

40 

6 

'4 

Cayuga  and        | 
Seneca  ) 

no 

18 

7 

*4-75 

Crook'd  L'ke, 
Chenango 

ll 

Delaware  and     I 
Raritan  .  .   .  .  } 

220 

24 

7 

43 

Chemung, 

and  Gencsce 

90 

15 

4 

P 

Dismal  Swamp.  .  . 
Erie  .   . 

90 

'7-5 
18 

5-5 

44 

Valley  

I  "3-  75 

Falls  of  Ohio  Ky 

35° 

80 

7-6o 

Chesapeake  and  ) 

Oswego 

18 

18 

Delaware  ) 

220 

24 

9 

M 

Welland,  Canada.  . 

270 

45 

M 

28 

Length  of  vessel  that  can  be  transported  is  somewhat  less  than  lengths  of  locks. 

Suez   Canal.  —  Width  196  to  328  feet  at  surface,  72  at  bottom,  and  26  deep. 
Length  99  miles. 

Heights  of*  obtained.  Elevations,  and.  various  I?laces 
and   ^Points   above   the    Sea. 


LOCATIONS. 

Feet. 

LOCATIONS. 

Feet. 

LOCATIONS. 

Feet. 

Aconcagua  Chili 

2-3  QIO 

Geneva  city  

I  22O 

Mont  Rosa  Alps 

Antisana     highest 

Geneva  Lake  

I  Oo6 

Mount  Adams 

established  eleva- 

Gibraltar 

Mount  Katahdin 

C  -jfvj 

tion  (Farmhouse) 

1-1  A  "3  A 

Humboldt's  highest 

Mount  Pitt 

Balloon  (Gay  Lussac) 

22  900 

19  4OO 

Mount  Washington 

9549 
6420 

u      (Green,  1837) 

27  ooo 

Isthmus  of  Darien. 

645 

us 

"    (Glaisher  and 
Coxwell) 

Jungfrau,  Switz'd.  . 
La  Paz  Bolivia 

13725 

Pont  d'  Oro,  Pyr's.  . 
Posthouse  Ap  Peru 

9843 

Brazil,  Quito,  and  ( 
Mexico  plains.  .  \ 

6000 
8000 

Laguna,Teneriffe.  .. 
London,  city  

2OOO 
64 

Potosi,  Bolivia  
Quito  

13223 

Condor's  flight 

Madrid        .   . 

St  Bernard's  Mon'y 

8 

Eagle's        "    

16  500 

Mexico  city  of  

7  WJ 

Vegetation  ...  . 

' 

Everest,  Himalaya. 

29003 

Mont  Blanc,  Alps.  .  . 

15797 

White  Mountain  .  .  . 

17000 
6230 

Lengths   of  Rivers. 


RlVKRS. 

Miles. 

RIVERS. 

EUROPE. 

Danube  

1800 
400 

'780" 
442 

545 
420 
760 
510 

45o 
050 
5io 

220 
100 
630 
2400 

2500 
I786 

Ganges  

Hoang  Ho 

Indus  
Jordan  
Lena     .  . 

.... 

Dnieper.  
Douro          .      • 

Dwina  

Tigris  

Elbe  

Yenesei    and 
lenga  .... 

Se- 

Garonne  

Loire  

Yang-Tse  

Po         

AFRICA. 

Gambia  

Rhone  
Seine  

Shannon 

Nile    

NORTH  AMERICA. 

Thames                  . 

Tiber     

Vistula  

Colorado  
Columbia.  .  .  . 

Volga,  Russia  

ASIA. 

Connecticut.  . 
Delaware  .  .  . 

Hudson    and 
hawk  

Mo- 

Euphrates  

Miles. 

RIVERS. 

Mile*. 

*5*4 

Kansas  

I4OO 

3040 

La  Platte  

g 

1800 
176 

Mackenzie  
Mississippi  

2440 
3160 

Missouri 

1  1  60 

Ohio  and  Allegheny 
Potomac  

1480 
42O 

0 

Red               

37    A 

2300 

Rio  Grande       .... 

l8oo 

700 

St.  Lawrence  
Susquehanna  
Tennessee 

2173 
62O 

4000 

SOUTH  AMERICA. 

Amazon  

2O7O 

Essequibo  

5  2O 

1050 
I2OO 

Magdalena  
Orinoco  

T6°£ 

410 

Platte  

2300 

420 
32  5 

Rio  Madeira  
Rio  Negro  

2300 
1650 

IIOO 

1 84 


SEA   DEPTHS,  BUILDING   STONES,  ETC. 


Large    Trees   in    California. 

"Keystone  State. "— Calavera  Grove,  is  325  feet  in  height. 

"Father  of  the  Forest."—  Felled,  is  385  feet  in  length,  and  a  man  on  horseback 
can  ride  erect  90  feet  inside  of  its  trunk. 

"  Mother  of  the  Forest.  "—Is  315  feet  in  height,  84  feet  in  circumference  (26.75  feet 
in  diameter)  inside  of  its  bark,  and  is  computed  to  contain  537  ooo  feet  of  sound  i 
inch  lumber. 

Sea  Depths. 


Feet. 

Feet. 

Feet. 

Baltic  Sea   

120 

I30 

JOO 

3000 

Of  Atl 
Pac 
B  Cod, 

Uasc 

Feet. 

Coast  of  Spain  
West  of  St.  Helena. 
Tortugas  to  Cuba  .  . 
Gulf  of  Florida.... 
Off  Cape  Florida... 

6000 
27000 
4200 

3720 
1950 

Off  Cape  Canaveral. 
u  Charleston  
"  Cape  Hatteras.  . 
"  Cape  Henry  
"  Sandy  Hook  

2400 
4200 
3120 
4200 
2400 

t-U 
Feet. 

Adriatic 

English  Channel.  .  . 
Straits  of  Gibraltar. 
Eastward  of    " 
Estimated  depth 

250  miles  off  Cap 

< 

LOCATION. 

iflc 

no  bottom  at  7800  feet. 
>ades   and.  "Waterfalls. 

LOCATION.               Feet.                LOCATION. 

Arve  Savoy 

1600 
2400 
(30 
34 
(40 
362 
197 

1000 

1260 

Genesee,  N.  Y  
Lidford,  England  .  .  . 
Lulea  Sweden         . 

.      IOO 
.      IOO 

600 

Niagara  
Great  Fall 

164 
152 
74 
74 

800 
798 
125 

Cascade  Alps 

Cataracts  of  the  Nile. 
Chachia  Asia 

Passaic  

Mohawk  

.      68 

Missouri     

(So 
.    '80 

Ribbon,  Yosemite) 
Valley  J 

Foyers,  Scotland 
Garisha,  India  

.       ^00 

(94 
.    2^0 

Ruican,  Norway 
Staubbach,  Switz'd.  . 
Tendon,  France  

Gavarny,  Pyrenees  .  . 

Nant  d'Apresias.  .  .  . 

.    800 

Sandstone 000009  532 

Whitepine 000002  55 


Yosemite  Valley 2600  feet. 

Expansion,  and.  Contraction  of  Building  Stones  for  each 
Degree    of  Temperature.     (Lieut.  W.  H.  C  Barllett,  U.  S.  E.) 

For  One  Inch. 

Granite. 000004  825 

Marble 000005  6°8 

Resistance    of  Stones,  etc.,  to   the    Effects   of  Freezing. 

Various  experiments  show  that  the  power  of  stones,  etc.,  to  resist  effects  of  freez- 
ing is  a  fair  exponent  of  that  to  resist  compression. 

Magnetic   Bearings   of  New  York. 

The  Avenues  of  the  City  of  New  York  bear  28°  50'  30"  East  of  North. 
Filters  for  Waterworks. 

i  square  yard  of  filter  for  each  840  U.  S.  and  700  Imp'l  gallons  in  24 
hours ;  formed  of  2.5  feet  of  fine  sand  or  gravel  and  6  inches  of  common 
sand  or  shells. 

Led  off  by  perforated  pipes  laid  in  lowest  stratum. 

Distances   between    New   York,  Boston,  Philadelphia, 
Baltimore,  and    Western    Cities    of  U.  S. 

Assuming  Boston  as  standard,  New  York  averages  12  per  cent,  nearer  to  these 
cities,  Philadelphia  18  per  cent.,  and  Baltimore  22  per  cent. 

Between  New  York  and  Chicago  the  line  of  the  Pennsylvania  Railroad  is  47  miles 
shorter  than  that  by  the  Erie  and  its  connections,  50  miles  shorter  than  that  by  the 
N.  Y.  Central  and  Hudson  River  and  its  connections,  and  114  miles  shorter  than  that 
by  the  Baltimore  and  Ohio  and  its  connections. 

For  Distances  between  these  and  other  cities  of  the  U.  S.,  see  page  88. 


WEATHER-PLANTS,   ANTIDOTES,  .ETC.  1 8$ 

"Weather-foretelling   [Plants.     (Hanneman.) 

If  Rain  is  imminent. — Chickweed,*  Stellaria  media ;  its  flowers  droop 
and  do  not  open.  Crowfoot  anemone,  Anemone  ranunculoides ;  its  blossoms 
close.  Bladder  Ketmia,  Hibiscus  trionum ;  its  blossoms  do  not  open.  Thistle, 
Ccirduus  acaulis ;  its  flowers  close.  Clover,  Trifolium  pratense,  and  its  allied 
kinds,  and  Whitlow  grass,  Draba  verna;  all  droop  their  leaves.  Nipple- 
wort, Lampsana  communis ;  its  blossoms  will  not  close  for  the  night.  Yel- 
low Bedstraw,  Galium  verum ;  it  swells,  and  exhales  strongly ;  and  Birch, 
Betula  alba,  exhales  and  scents  the  air. 

Indications  of  Rain. — Marigold,  Calendula  pluvialis ;  when  its  flowers  do 
not  open  by  7  A.  M.  Hog  Thistle,  Sonchus  arvensis  and  oleraceus ;  when  its 
blossoms  open. 

Rain  of  shoi*t  duration. — Chickweed,  Stellaria  media ;  if  its  leaves  open 
but  partially. 

If  cloudy.  —  Wind-flower,  or  Wood  Anemone,  Anemone  memorasa;  its 
flowers  droop. 

Termination  of  Rain.  —  Clover,  Trifolium  pratense  ;  if  it  contracts  its 
leaves.  Birdweed  and  Pimpernel,  Convolvulus  and  Anagallis  arvensis;  if 
they  spread  their  leaves. 

Uniform  Weather. — Marigold,  Calendula  pluvialis ;  if  its  flowers  open  early 
in  the  A.  M.  and  remain  open  until  4  P.  M. 

Clear  Weather. — Wind-flower,  or  Wood  Anemone,  Anemone  memorasa; 
if  it  bears  its  flowers  erect.  Hog  Thistle,  Sonchus  arvensis  and  oleraceus ; 
if  the  heads  of  its  blossoms  close  at  and  remain  closed  during  the  night. 

Treatment   and.  Antidotes  to   Severe  Ordinary  I?oisons. 

Antidotes  in  very  small  doses. 

Chloroform  and  Ether.— Cold  affusions  on  head  and  neck,  and  ammonia 
to  nostrils.  Antidote.—  Camphor,  .petroleum,  sulphur. 

Toadstools. — (Inedible  mushroom).     Antidote.—  Same  as  for  chloroform. 

Arsenic  or  Fly  Powder. — Emetic ;  after  free  vomiting  give  calcined  mag- 
nesia freely.  If  poison  has  passed  out  of  stomach,  give  castor  oil. 

Antidote. — Camphor,  nux  vomica,  ipecacuanha. 

Acetate  of  Lead  (Sugar  of  lead).  —  Mustard  emetic,  followed  by  salts, 
Large  draughts  of  milk  with  white  of  eggs. 

Antidote.—  Alum,  sulphuric  acid  alike  to  lemonade,  belladonna,  strychnine. 

Corrosive  Sublimate  (Bug  poison).  —  White  of  eggs  in  i  quart  of  cold 
water,  give  cupful  every  two  minutes.  Induce  vomiting  without  aid  of 
emetics.  Soapsuds  and  wheat  flour  is  a  substitute  for  white  of  eggs. 

Antidote. — Nitric  acid,  camphor,  opium,  sulphate  of  zinc. 

Phosphorus  Matches. — Rat  Paste. — Two  teaspoonfuls  of  calcined  magne- 
sia, followed  by  mucilaginous  drinks.  Antidote.—  Camphor,  coffee,  mix  vomica, 

Carbonic  Acid  (Charcoal  fumes),  Chlorine,  Nitrous  Oxide,  or  Ordinary 
Gas. — Fresh  air,  artificial  respiration,  ammonia,  ether,  or  vapor  of  hot  water. 

Antidote.—  Camphor,  coffee,  nux  vomica. 

Belladonna  (Nightshade).  —  Emetic  and  stomach  pump,  morphine  and 
strong  coffee.  Antidote.—  Camphor. 

Opium. — Stomach  pump  or  emetic  of  sulphate  of  zinc,  20  or  30  grains,  or 
mustard  or  salt.  Keep  patient  in  motion.  Cold  water  to  head  and  chest. 

Antidote.—  Strong  coffee  freely  and  by  injection,  camphor,  ether,  and  nux  vomica. 

Strychnine  (Nux  vomica). — Stomach  pump  or  emetic,  chloroform,  cam- 
phor, animal  charcoal,  lard,  or  fat. 
Antidote. — Wine,  coffee,  camphor,  opium  freely,  and  alcohol  in  small  doses. 

Vegetable  Poisons. — As  a  rule,  an  emetic  of  mustard  and  drink  freely  of 
warm  water. 

*  Spreads  ita  leaves  about  9  A.  M.,  and  they  remain  open  until  noon. 

o* 


186  VETEBINABY. 

"Veterinary. 

.  Horses.—  Cathartic  Ball—  Cape  Aloes,  6  to  10  drs.  ;  Castile  Soap,  x  dr.  ;  Spirit 
of  Wine,  i  dr.  ;  Sirup  to  form  a  ball.  If  Calomel  is  required,  add  from  20  to  50 
grains.  During  its  operation,  feed  upon  mashes  and  give  plenty  of  water. 

Cattle.—  Cathartic.—  Cape  Aloes,  4  drs.  to  i  oz.  ;  Epsom  Salts,  4  to  6  oz.  ;  Gin- 
ger, 3  drs.  Mix,  and  give  in  a  quart  of  gruel.  For  Calves,  one  third  will  be  sufficient. 

Horses  and.  Cattle.—  Tonic.—  Sulphate  of  Copper,  i  oz.  to  12  drs.  ;  Sugar, 
.5  oz.  Mix,  and  divide  into  8  powders,  and  give  one  or  two  daily  in  food. 

Cordial.  —  Opium,  i  dr.;  Ginger,  2  drs.;  Allspice,  3  drs.,  and  Caraway  Seeds,  4 
drs.,  all  powdered.  Make  into  a  ball  with  sirup,  or  give  as  a  drench  in  gruel. 

Cordial  Astringent  Drench,  for  Diarrhoea,  Purging,  or  Scouring.  —  Tincture  of 
Opium,  .5  oz.  ;  Allspice,  2.5  drs.  ;  powdered  Caraway,  .5  oz.  ;  Catechu  Powder,  2  drs.  ; 
strong  Ale  or  Gruel,  i  pint  Give  every  morning  till  purging  ceases.  For  Sheep 
.25  this  quantity. 

Alterative.  —  Ethiop's  Mineral,  .5  oz.  ;  Cream  of  Tartar,  i  oz.  ;  Nitre,  2  drs.  Divide 
into  from  16  to  24  doses,  one  morning  and  evening  in  all  cutaneous  diseases. 

Diuretic  Ball.—  Hard  Soap  and  Turpentine,  each  4  drs.  ;  Oil  of  Juniper,  20  drops; 
and  powdered  Resin  to  form  a  ball. 

For  Dropsy,  Water  Farcy,  Broken  Wind,  or  Febrile  Diseases,  add  to  above,  All- 
spice and  Ginger,  each  2  drs.  Divide  into  4  balls,  and  give  one  morning  and  evening. 

Alterative  or  Condition  Powder.  —  Resin  and  Nitre,  each  2  oz.  ;  levigated  Anti- 
mony, i  oz.  Mix  for  8  or  10  doses,  and  give  one  morning  and  evening.  When  given 
to  Cattle,  add  Glauber  Salts,  i  Ib. 

Fever  Ball.  —  Cape  Aloes,  2  oz.  ;  Nitre,  4  oz.  ;  Sirup  to  form  a  mass.  Divide  into 
12  balls,  and  give  one  morning  and  evening  until  bowels  are  relaxed;  then  give  an 
Alterative  Powder  or  Worm  Ball. 

Hoof  Ointment—  Tar  and  Tallow,  each  i  Ib.  ;  Turpentine  .5  Ib.     Melt  and  mix. 

Dogs.  —  Cathartic.—  Cape  .Moos.  .5  dr.  to  i  oz.  ;  Calomel,  2  to  3  grs.  ;  Oil  of 
Caraway,  6  drops;  Sirup  to  form  a  ball.  Repeat  every  5  hours  till  it  operates. 

Emetic.  —  2  to  4  grs.  of  Tartar  Emetic  in  a  meat  ball,  or  a  teaspoonful  or  two  of 
common  salt.  Give  twice  a  week  if  required. 

Distemper  Powder.  —  Antimonial  Powder,  2,  3,  or  4  grs.  ;  Nitre,  5,  10,  or  15  grs.  ; 
powdered  Ipecacuanha,  2,  3,  or  4  grs.  Make  into  a  ball,  and  give  two  or  three  times 
a  day.  If  there  is  much  cough,  add  from  .5  gr.  to  i  gr.  of  Digitalis,  and  every  3  or 
4  days  give  an  Emetic. 

Mange  Ointment.—  Powdered  Aloes,  2  drs.;  White  Hellebore,  4  drs.;  Sulphur,  4 
oz.  ;  Lard,  6  oz.—  Red  Mange,  add  i  oz.  of  Mercurial  Ointment,  and  apply  a  muzzle. 

NOTE:  —  Physic,  except  in  urgent  cases,  should  be  given  in  morning,  and  upon  an 
empty  stomach;  and,  if  required  to  be  repeated,  there  should  be  an  interval  of  sev- 
eral days  between  each  dose. 


of  Horses. 

To    Ascertain,    a    Horse's    Age. 

A  foal  of  six  months  has  six  grinders  in  each  jaw,  three  in  each  side,  and  also  six 
nippers  or  front  teeth,  with  a  cavity  in  each. 

At  age  of  one  year,  cavities  in  front  teeth  begin  to  decrease,  and  he  has  four 
grinders  upon  each  side,  one  of  permanent  and  remainder  of  milk  set. 

At  age  of  two  years  he  loses  the  first  milk  grinders  above  and  below,  and  front 
teeth  have  their  cavities  filled  up  alike  to  teeth  of  horses  of  eight  years  of  age. 

At  age  of  three  years,  or  two  and  a  half,  he  casts  his  two  front  uppers,  and  in  a 
short  time  after  the  two  next. 

At  four,  grinders  are  six  upon  each  side;  and,  about  four  and  a  half,  his  nippers 
are  permanent  by  replacing  of  remaining  two  corner  teeth;  tushes  then  appear, 
and  he  is  no  longer  a  colt. 

At  five,  a  horse  has  his  tushes,  and  there  is  a  black-colored  cavity  in  centre  of  all 
his  lower  nippers. 

At  six,  this  black  cavity  is  obliterated  in  the  two  front  lower  nippers. 

At  seven,  cavities  of  next  two  are  filled  up,  and  tushes  blunted;  and  at  eight,  that 
of  the  two  corner  teeth.  Horse  may  now  be  said  to  be  aged.  Cavities  in  nippers 
of  upper  jaw  are  not  obliterated  till  horse  is  about  ten  years  old,  after  which  time 
tushes  become  round,  and  nippers  project  and  change  their  surface. 


DISTANCES,  POPULATION,  DROWNING,  ETC.          187 

Distances  "between   Principal  Cities   of*  East   and.  "West. 
In  Miles. 


CITIES. 

Boston. 

New 
York. 

Phila- 
delphia. 

Balti- 
more. 

CITIES. 

Boston. 

New 
York. 

Phila- 
delphia. 

Balti- 
more. 

Burlington,  la. 
Chicago  .... 

I2l6 
1009 

1106 
900 

1030 
823 

995 
802 

Louisville  
Memphis  

1161 
1438 

870 
1247 

794 
1171 

706 
1083 

Cincinnati..  .  . 
Cleveland 

743 

ego 

667 

CQ4. 

576 
48q 

Milwaukee  
Omaha  

£>3 

I5O3 

947 
1393 

908 

1^17 

887 
1294 

Columbus,  O.. 
Detroit 

807 

724 

500 
623 
673 

547 
682 

512 
661 

St.  Joseph  
St.  Louis  

I478 

1212 

1356 
1050 

1280 

973 

1223 
917 

Indianapolis  .  . 
Kansas  Citv.  .  . 

951 

1487 

810 

1324. 

735 
1248 

700 

1  1  02 

St.  Paul  
Toledo... 

1418 
784 

1308 
603 

Z232 
6l7 

I2II 

SQ6 

Population    of   some    Principal    Cities    of  tne    World. 


London  
New  York. 

1901 

4  536  063 

Warsaw  

St.  Louis. 

1897 

638  208 

Copenhagen. 
Dublin 

1901 

378  235 

Paris..,..!!! 

1896 

3  437  2O3 
2  536  834 

Naples  

1899 

575  23° 
544057 

Cologne  

1900 

373  Z79 
372  229 

Berlin  

1900 

i  888  326 

Boston  

1900 

560  892 

Belfast  

1901 

348965 

Chicago  
Vienna  

1900 
1900 

i  698  575 
i  662  269 

Manchester.. 
Amsterdam.. 

1901 
1899 

543969 
523  557 

SanFrancis'o 
Cincinnati..  . 

1900 
1900 

342  782 
325902 

Peking  

Est. 

i  650000 

Madrid  

1897 

512  150 

Pittsburgh  .  . 

1900 

21* 

321  020 

Tokio  
Philadelphia 

1898 

I  440  121 
I  293  697 

Baltimore.  .  . 
Munich  

1900 

508  957 

Edinburgh..  . 
Lisbon* 

1901 

1890 

316479 

St.  Petersb'g 

1807 

Milan  

1900 

499  959 
491  460 

Stockholm..  . 

*              x- 

Constant'ple 

toy  7 

Est. 

7      3 

Lyons  

1806 

466  028 

Frankfort.  .. 

IQOO 

288  480 

iSm 

^  >• 

Rome 

•*  y 

New  Orleans. 

iy«_iv 
" 

^<ju  ^uy 

PI          *  '  " 

ioy/ 

i  035  004 

Leipsic    .   .  . 

1901 

4Cc  080 

Antwerp.  .  .  . 

285  6OO 

Buda-Pesth.. 
Hamburg.  ... 
Liverpool... 
BuenosAyr's 

1900 
1900 
1901 
1895 

700  493 
713383 
705  728 
685  276 
663854 

Marseilles.  .  . 
Mexico  
Dresden  
Cleveland.  .  . 

i8y? 
1900 
1900 
1900 

^->D  >Juy 
442239 
402000 

395  349 
381  768 

Washington. 
Montreal  
Manilla  
Havana  

I9OO 
I9OI 
1901 
l899 

278718 
265  826 
244  732 
235981 

Treatment  of  3Dr  owning  Persons. 
Practice   adopted   toy   Board   of  Health,  New   York. 

Place  patient  face  downward,  with  one  of  his  wrists  under  his  forehead.  Cleanse 
his  mouth.  If  he  does  not  breathe,  turn  him  on  his  back  with  shoulders  raised  on 
a  support.  Grasp  tongue  gently  but  firmly  with  fingers  covered  with  end  of  a  hand- 
kerchief or  cloth,  draw  it  out  beyond  lips,  and  retain  it  in  this  position. 

To  Produce  and  Imitate  Movements  of  Breathing.—  Raise  patient's  extended  arms 
upward  to  sides  of  his  head,  pull  them  steadily,  firmly,  slowly,  outwards.  Turn 
down  elbows  by  patient's  sides,  and  bring  arms  closely  and  firmly  across  pit  of 
stomach,  and  press  them  and  sides  and  front  of  chest  gently  but  strongly  for  a  mo- 
ment, then  quickly  begin  to  repeat  first  movement. 

Let  these  two  movements  be  made  very  deliberately  and  without  ceasing  until 
patient  breathes,  and  let  the  two  movements  be  repeated  about  twelve  or  fifteen 
times  in  a  minute,  but  not  more  rapidly,  bearing  in  mind  that  to  thoroughly  fill  the 
lungs  with  air  is  the  object  of  first  or  upward  and  outward  movement,  and  to  expel 
as  much  air  as  practicable  is  object  of  second  or  downward  motion  and  pressure. 
This  artificial  respiration  should  be  maintained  for  forty  minutes  or  more,  when  the 
patient  appears  not  to  breathe;  and  after  natural  breathing  begins,  let  same  motion 
be  very  gently  continued,  and  give  proper  stimulants  in  intervals. 

What  Else  is  to  be  Done,  and  What  is  Not  to  be  Done,  while  the  Movements  are 
Being  Made.— If  help  and  blankets  are  at  hand,  have  body  stripped,  wrapped  in 
blankets,  but  not  allow  movements  to  be  stopped.  Briskly  rub  feet  and  legs,  press- 
ing them  firmly  and  rubbing  upward,  while  the  movements  of  the  arms  and  chest 
are  in  progress.  Apply  hartshorn,  or  like  stimulus,  or  a  feather  within  the  nostrils 
occasionally,  and  sprinkle  or  lightly  dash  cold  water  upon  face  and  neck.  The 
legs  and  feet  should  be  rubbed  and  wrapped  in  hot  blankets,  if  blue  or  cold,  or  if 
weather  is  cold. 

What  to  Do  when  Patient  Begins  to  Breathe.— Give  stimulants  by  teaspoonful  two 
or  three  times  a  minute,  until  beating  of  pulse  can  be  felt  at  wrist,  but  be  careful 
and  not  give  more  of  stimulant  than  is  necessary.  Warmth  should  be  kept  up  in 
feet  and  legs,  and  as  soon  as  patient  breathes  naturally,  let  him  be  carefully  removed 
to  an  enclosure,  and  placed  in  bed,  under  medical  care. 


i88 


MISCELLANEOUS  ELEMENTS. 


MISCELLANEOUS  ELEMENTS. 
Earth.. 

Polar  diameter  7899. 3  miles.  Mean  density  or  specific  gravity  of  mass  5.672.  Mass 
5  272  600  ooo  ooo  ooo  ooo  ooo  tons.  Apparent  diameter  as  seen  from  Sun  17  seconds. 

Sun. 

Heat  of  Sun  equal  to  322  794  thermal  units  per  minute  for  each  sq.  foot  of  pho- 
tosphere or  solar  surface. 

Diameter  of  Sun  882000  miles,  tangential  velocity  1.25  miles  per  second  or  4.41 
times  greater  than  that  of  the  Earth. 

Distance  from  Earth  91.5  to  92  millions  of  miles. 

]VEason    and.    Dixon's    Line. 

39°  43'  26.3"  N.  mean  latitude.     68.895  miles. 

Area  and   Population.     (Behm  and  Wagner. ) 


Divisions. 

Area. 

Population. 

Divisions. 

Area. 

Population. 

America     

Sq.  Miles. 
14  491  ooo 

95  495  5^o 

Oceanica  

Sq.  Miles. 
4  500000 

Europe  

3  760000 

315  929  ooo 

Greenland  ) 

Asia  

16313000 

834  707  ooo 

Iceland       j  *  *  ' 

82000 

10936000 

Total  

50000000 

1455923500 

38-oco 

China 434  626  ooo 

France 37000000 

(United  States. 


Countries. 
Germany 43  900000 


Great  Britain..  34  ooo  ooo 

f  Russia 66  ooo  ooo 

}  Territories  ...  22  ooo  ooo 
.50000000  I 


India,  British  .  .240298000 

Canada 3  839  ooo 

Mexico 9485000 

Brazil n  106000 

(Turkey 8866000 

\Indians 300000  |    (     "    in  Asia.  .16320000 

About  one  thirtieth  of  whole  population  are  born  every  year,  and  nearly  an  equal 
number  die  in  same  time ;  making  about  one  birth  and  one  death  per  second. 
Earlier  authority  estimated  population  at  i  288000000,  divided  as  follows 


Caucasians 360  ooo  ooo 

Mongolians  ....  552  ooo  ooo 

Ethiopians 190000000 

Asiatics 60000000 


Malays  and 
Indo-Amer's 
Protestants.. 
Israelites.  .  .  . 

\  177000000 

.  .  80  ooo  ooo 
..     5000000 

Mohammedans.  190000000 

Pagans 300  ooo  ooo 

Catholics         ) 

Rom.  &  Greek  I  250000000 


Descent   of  \Vestern    Rivers. 

Slope  of  rivers  flowing  into  Mississippi  from  East  is  about  3  inches  per  mile; 
and  from  West  6  inches. 

Mean  descent  of  Ohio  River  from  Pittsburgh  to  Mississippi,  975  miles,  is  about  5.2 
inches  per  mile;  and  that  of  Mississippi  to  Gulf  of  Mexico,  1180  miles,  about  2.8 
inches. 

Transmission,    of  Horse    3?ower. 

Largest,  and  perhaps  most  successful,  wire  rope  transmission  is  one  at  Schaff- 
hauseu,  at  Falls  of  the  Rhine.  Here,  power  of  a  number  of  turbines,  amounting 
to  over  600  IP,  is  conveyed  across  the  stream,  and  thence  a  mile  to  a  town,  where  it 
is  distributed  and  utilized. 

At  mines  of  Falun,  Sweden,  a  power  of  over  100  horses  is  transmitted  in  like 
manner  for  a  distance  of  three  miles. 

A.cids. 

Acetic  Acid  (Vinegar),  acid  of  Malt  beer,  etc.     Tartaric  Acid,  acid  of  Grape  wine. 
Lactic  Acid,  acid  of  Milk,  Millet  beer,  and  Cider. 

IVtan-ures. 

Relative  Fertilizing  Properties  of  Various  Manures. 

Peruvian  Guano x         I  Horse 048  I  Farm-yard 0298 

Human,  mixed 069  |  Swine 044  |  Cow 0259 

Or,  i  Ib.  guano  =  14.5  human,  21  horse,  22.5  swine,  33.5  farm-yard,  and  38.5  cow. 
Relative  Value,  Covered  and  Uncovered,  on  an  A  ere  of  Ground. 

Covered n  tons  1665  Ibs.  potatoes,  61    Ibs.  wheat,  215  Ibs.  straw< 

Uncovered 7     "     1307    "          "        61.5"        "      156   "       " 


MISCELLANEOUS   ELEMENTS. 


Yield   of  Oil   of  Several    Seeds. 

PerCent.  I  Per  Cent.  I  Per  Cent.  I  Per  Cent.  I  Per  Cent. 

Poppy. .  56  to  63  I  Castor . .  25  I  Sunflower.  15  I  Hemp.  14  to  25  I  Linseed,  n  to  22 
Thickness   of  \Valls   of  Buildings.     (English.)    (Molesworth.) 


OUTER  WALLS. 

Maximum 
Height 
of  Wall. 

Width 
of 
Footings. 

Ground 
Floor. 

Mi 
ist 
Floor. 

nimum  Width  c 
2d     I     3d 
Floor.  |  Floor. 

f  Walls 
4th 
Floor. 

5th 
Floor. 

6th 
Floor. 

ist  class  dwelling. 

Feet. 
85 

38.5 

21.5 

21.5 

17-5 

T7-5 

17.5 

13 

J3 

2(1        "                " 

70 

30-5 

17-5 

17-5 

17-5 

13 

13 

13 

3d     " 

52 

30-5 

17-5 

13 

13 

13 

13 



4th  "       v^;! 

38 

21.5 

13 

13 

8-5 

8-5 





PARTY  WALLS. 

ist  class  dwelling. 

85 

38.5 

21.5 

21-5 

17-5 

17-5 

17-5 

13 

13 

2d     "           " 

70 

30-5 

17-5 

17-5 

17-5     13 

13 

13 

3d     " 

52 

30.5 

17-5 

13 

I3 

13 

8.5 

— 

4th    "           " 

38 

iv~~   __   r 

21.5 

13 

n/vtV.       41 

8-5 

8.5      8.5 

— 

.!„.,_  J 

If  walls  are  more  than  70  feet  in  length,  those  of  lower  stories  must  be  widened 
by  half  a  brick. 

Minimum 
Width 
of  Wall. 


Minimum 
Width 


"Warehouses 

1st   Class.  of  Wall. 

For  a  height  of  36  feet  from   ins. 

topmost  ceiling 17.5 

For  a  height  of  40  feet  lower . .  21.5 
"        "         24  feet  lower . .  26 

For  footings 43.5 

3d   Class. 
For  a  height  of  28  feet  below 

topmost  ceiling 13 

For  a  height  of  16  feet  lower . .  17.5 
For  footings 30.5 

"Wooden 


"Warehouses 
Sd    Class. 
For  a  height  of  22  feet  below 

topmost  ceiling 13 

For  a  height  of  36  feet  lower . .  17.5 

8  feet  lower..  21.5 

For  footings 34.5 

4th    Class. 
For  a  height  of  9  feet  below 

topmost  ceiling 8.5 

For  a  height  of  13  feet  below . .  13 
For  footings 21.5 

Roofs .     (English. ) 


Span 
in  Feet. 

Principal 
Beam. 

Tie  Beam. 

King 
Posts. 

Queen 
Posts. 

Small 
Queens. 

Straining 
Beam. 

Struts. 

20 

4X4 

9X4 

4x4 

— 

— 

— 

3X3 

25 

5X4 

10  x  5 

5x5 

— 

— 

— 

5     X3 

30 

6x4 

ii  X  6 

6X6 

— 

— 



6     X3 

35 

5X4 

ii  X  4 

— 

4X4 

— 

7X4 

4X2 

45 

6X5 

13  X  6 

— 

6x6 

— 

7X6 

5X3 

50 

8x6 

13x8 

— 

8x8 

8X4 

9X6 

5      X3 

55 

8X7 

H  X  9 

— 

9X8 

9X4 

10  X  6 

5-5X3 

60     |     8x  8 

15  X  10 

— 

10  X  8 

10  X  4 

ii  X  6 

6     X  3 

Mineral   Constituents    absorbed    or   removed  from    an 
Acre   of  Soil    t>y   several    Crops.     (Johnson.) 


CROPS. 

1J 

n 

H 

.ii 
gj 

£g 

*i 

K  10 

CROPS. 

H 

14 

«  5 

&2 
E5 
a  o 
H  « 

£-1 

PC    10 

Potassa  
Soda  

Los. 
29.6 
3 

Lbs. 
17-5 
5-2 

Lbs. 

IS 

Lbs. 
38.2 
12 

Sulphuric  ) 
Acid...)  " 

Lbs. 
10.6 

Lbs. 
2.7 

Lbs. 
13-3 

Lbs. 
9.2 

Lime 

17 

20.  Q 

A  A.  g 

Chlorine  

2 

16 

3.6 

4.  1 

Magnesia 
Oxide  of  Iron. 

12.9 
10.6 

2.6 

9.2 

2.1 

19.7 

7-i 

7:l 

Silica  
Alumina  

118.1 

129.5 

2.4 

247.8 

78.2 

Acid  j 

20.  6 

25.8 

46.3 

15-1 

Total.... 

210 

213 

423 

209 

190 


MISCELLANEOUS   ELEMENTS. 


Average  Quantity  of*  Tannin,  in  Several  Su.batan.ce0. 

(Morfit.) 


Catechu.                    Per  Cent. 

Oak.                            Per  Cent. 

Young,  inner  b'k    15.2 
u    entire  b'k.      6 
"    spring-     ) 
cut  bark}    22 
"    root  bark.      8.9 
Chestnut. 
Amer.  rose,  bark      8 
Horse,    "    2 
Sassafras,  root  bark    <;8 

Sumac.                   p( 
Sicily  and  Malaga 
Virginia 

rCemt, 
16 

10 

5 

16 
16 
16 
13 

2A 

Bengal        44 

Carolina  

Nutgalls. 

Willow. 
Inner  bark  

Weeping  
Sycamore  bark  .... 
Tan  shrub    "    .... 
Cherry-tree  .  .  . 

Oak. 
Old,  inner  bark  {     I4'2 

Alder  bark 36  per  cent. 

To  Convert  Chemical    Formulae  into  a  ^Mathematical 
Expression. 

RULE.— Multiply  together  equivalent  and  exponent  of  each  substance,  and  product 
will  give  proportion  in  compound  by  weight.  Divide  1000  by  sum  of  their  products, 
and  multiply  this  quotient  by  each  of  these  products,  and  products  will  give  re- 
spective proportion  of  each  part  by  weight  in  1000. 

EXAMPLE. — Chemical  formula  for  alcohol  is 
tional  parts  by  weight  in  1000? 

C4  Carbon      =6.1X4  =  24.4)  (525-82) 

#6  Hydrogen  =    i  X  6=  6    '/  X  21.55  {129.3 
02  Oxygen     =    8X2  =  16    )  ( 344.8 

looo  -7-46.4=    =21.55    999-92 


Required  their  propor- 


by weight. 


Elementary    Bodies,  Avith    their    Symbols    and. 
Equivalents. 


BODY. 

Symb. 

Equiv. 

BODY. 

Symb. 

Equiv. 

BODY. 

Symb. 

Equir. 

Aluminium 

Al 

jo.  7 

Gold  

Au 

196  6 

Platinum  .... 

Pt 

08  8 

Antimony  
Arsenic  

Sb 
As 

64-6 

37  7 

Hydrogen  
Iodine 

H 
I 

I 
126  5 

Potassium  .  .  . 
Rhodium  .  .  . 

K 
R 

39-2 

Barium  
Bismuth  .... 

Ba 
Bi 

68.6 

71   e 

Iridium  
Iron 

Ir 
Fe 

98.5 
28 

Selenium  .... 
Silicon  

Se 

Si 

4° 

Boron  

B 

II 

Lead  

Pb 

103*  7 

Silver  

Ag 

108  3 

Bromine  
Cadmium.  ... 
Calcium  
Carbon  ...... 
Chlorine  ... 

Br 
Cd 
Ca 
C 
Cl 

78.4 

55-8 
20.5 
6.1 

oe  e 

Lithium  
Magnesium  .  . 
Manganese.  .  . 
Mercury  
Molybdenum 

L 

Mg 
Mn 
Hg 
Mo 

7 

II7 

200 

Sodium  
Strontium  
Sulphur  
Tellurium  
Tin 

Na 
Sr 
S 
Te 
Sn 

23-5 
43-8 
16.1 
64.2 

58  o 

Chromium.  .  . 

Cr 

35-5 
26.2 

Nickel 

Ni 

Titanium  .... 

Ti 

Cobalt 

Co 

Nitrogen 

N 

Tungsten 

W 

Columbium.  . 
Copper  
Fluorine  

Ta 
Cu 
F 

184.8 

31-7 
18.7 

Osmium  
Oxygen  
Palladium 

Os 
0 
Pd 

T 

Uranium  
Yttrium  
Zinc  

U 
Y 
Zn 

92 
60 
32 

Glucinum  — 

G 

'1 
6.9 

Phosphorus.  . 

P 

i5-9 

Zirconium.  .  . 

Zr 

34 

Analysis   of*  certain   Organic    Substances   by  \Veight 


BODY. 

Car- 
bon. 

Hydro- 
gen. 

Oxy- 
gen. 

Nitro- 
gen. 

BODY. 

Car- 
bon. 

Hydro- 
gen. 

Oxy- 
gen. 

Nitro 
gen. 

Albumen    

52-9 

7'  5 

23.9 

ic  7 

Morphi  ne 

6  A. 

16  1 

Alcohol 

52.7 

12  Q 

34  4 

Narcotine 

7^-3 
6c. 

Atmospheric  air 
Camphor  
Caoutchouc  
Casein  

73-4 
87.2 

CQ.8 

10.7 

12.8 

7-  4 

lit 

11.4 

23 

•3 
21.4 

Oil,  Castor.  
Linseed  
Spermaceti. 
Quinine  

2 

£• 

10.3 
"•3 
xi.8 

iS-7 
12.7 

IO.  2 

8  6 

8  i 

Fibrin 

eo  A 

IQ  7 

Starch 

f>7 

Gelatine  
Gum 

47-9 

7-9 

6  4. 

27.2 
5O  Q 

i7 

Strychnine  
Sugar 

76.4 

£7 
6.7 

6  6 

ii.  i 

5-8 

Hordein  

44.  2 

0.4 
6  4 

? 
47.6 

1.8 

Tannin  

42.2 

52  6 

•*.8 

13* 

Lifirnin.  .  . 

52.  5 

«;.7 

41.8 

Urea.  .  . 

18.0 

O.  7 

&, 

<i<;.2 

MISCELLANEOUS   ELEMENTS. 


191 


Dilution   JPer   Cent.  Necessary  to    Redu.ce    Spirituous 
Liquors. 

Water  to  be  added  to  100  volumes  of  spirit  when  of  following  strength: 


Strength 
Required. 

90 

85 

80 

75 

70 

65 

60 

55 

50 

Per  cent, 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent.  ]  Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

85 

5-9 

— 

— 

— 

— 

— 

— 

— 

— 

So 

12.5 

6-3 

— 

— 

— 

— 

— 

— 

— 

75 

20 

13-3 

6.7 

— 

— 

— 

— 

— 

— 

70 

28.6 

21.4 

14.3 

7-i 

— 

— 

— 

— 

— 

65 

38.5 

30.8 

23.1 

15-4 

7-7 

— 

— 

— 

— 

60 

50 

41.7 

33-3 

25 

16.7 

8-3 

— 

— 

— 

55 

63.6 

54-5 

45-5 

36.4 

27.4 

1  8.  2 

9.1 

— 

— 

50 

80 

70 

60 

So 

40 

30 

20 

10 

— 

40 

125 

112.5 

100 

87-5 

75 

62.5 

50 

37-5 

25 

30 

200 

183.3 

166.7 

150 

J33-3 

116.7 

100 

83.3 

66.7 

ILLUSTRATION. — 100  volumes  of  spirituous  liquor  having  90  per  cent,  of  spirit  con- 
tains: alcohol  90,  water  10,  =  100. 
To  reduce  it  to  30  per  cent,  there  is  required  200  volumes  of  water. 

,90       30      30  spirit, 

Hence  200  -4-  10  =  210,  and  —  =  —  =  J  or  30  per  cent. 

210      70      70  water, 


Alcohol. 

!>r 

In  100 

Specific 
Gravity. 

oportic 

Parts  oj 

Alcohol. 

>n   of  A.] 
'Spirit,  by 

Specific 
Gravity. 

Looliol 

Weight  c 

Alcohol. 

!»er    Cei 
r  Volume, 

Specific 
Gravity. 

It. 

at  60°. 

Alcohol. 

Specific 
Gravity. 

0 

5 

10 

I 
.991 
.984 

20 

30 
40 

•972 
.958 
•94 

50 
60 
70 

.918 

.896 
.872 

80 
90 

100 

.848 
.823 
•794 

In  TOO  Parts  of  Alcohol  and  Water,  by  Weight,  at  60°. 


Alcohol. 

Specific 
Gravity. 

Alcohol. 

Specific 
Gravity. 

Alcohol. 

Specific 
Gravity. 

Alcohol. 

Specific 
Gravity. 

0 

•53 

1.02 

f    -JjOi 

1.99 
3-02 
4.02 

.996 
•994 
•993 

S-oi 

6.02 

7.02 

.991 
.988 

7-99 
90S 
10.07 

.987 
.985 
.984 

Tides   of  Atlantic   and   Pacific   Oceans    at  Isthmus   of 

Panama.     (Totten.) 

Atlantic,  Navy  Bay.—  Highest  tide  i.$feet;  lowest  .63  feet. 
Pacific,  Panama  Bay.—  Highest  tide  17.72  to  21. 3  feet;  lowest  9.7  feet. 


STATE. 

A.res 

Sq.  Miles, 

is   of  TJ.  S.  C 

STATE. 

oal   IT 

Sq.  Miles. 

ields. 

STATE. 

Sq.  Miles. 

Illinois 

44000  I 

21000 
15437 
13500   1 

tuminous  a 

Ohio 

ii  900 
7700 
6000 
5000 

Tennessee 

4300 
3400 
55<> 
150 

Virginia  
Pennsylvania*  .  .  . 
Kentucky  

Indiana.  
Missourif  

Alabama 

Maryland     

Michiganf 

Georgia 

*B 

nd  Anthracite. 

f  Anthracite. 

Extremes    of  Heat   in    Various    Countries. 


England 96° 

France 106.5 

Holland   1     _0 
Belgium 


99-5° 


Egypt 
Africa 
Asia 


Denmark 

Sweden 

Norway 

Russia  ...'...  102° 

Germany 103°  |  Manilla 113-5°  |  N.  America 102° 

Extremes  of  temperature  upon  the  Earth  240°. 

Extremes    of  Cold    in    "Various    Countries. 


Greece 105° 

Italy 104° 

Spain 102° 

Tunis 112.5° 


116.1° 

133-4° 
120® 
Suez  .......  126.5° 


England...—  5° 
Holland  )  0 
Belgium  } 


Denmark 

Sweden 

Norway 


France —24° 

Russia —46° 

Germany.  .—32° 


Italy —10° 

Fort  Reliance,  N.  A.  .—70° 
Semipalatinsk,  "  ..—76° 


I92 


MISCELLANEOUS    ELEMENTS. 


Mean.    Temperatures   of  "Various   Localities. 

London 51°  I  Rome 60°  I  Poles —13°  I  Polar  Regions. .  36° 

Edinburgh 41°  |  Equator 82°  |  Torrid  Zone.      75°  |  Globe 50° 

Line    of*  Perpetual    Congelation,  or    Sno\v   Line. 


Latitude. 

Height. 

Latitude. 

Height. 

Latitude. 

Height. 

Latitude. 

Height. 

10 

15 
20 
25 

14764 
14760 
13478 
12557 

30 
35 
40 

45 

11484 
10287 
9000 
7670 

50 

g 

65 

6334 
5020 
3818 
2230 

70 
75 
80 

85 

1278 
1016 
45i 
327 

At  the  Equator  it  is  15260  feet;  at  the  Alps  8120  feet;  and  in  Iceland  3084  feet. 
At  Polar  Regions  ice  is  constant  at  surface  of  the  Earth. 

Limits   of  "Vegetation   in   Temperate    Zone. 

The  Vine  ceases  to  grow  at  about  2300  feet  above  level  of  the  sea,  Indian  Corn  at 
2800,  Oak  at  3350,  Walnut  at  3600,  Ash  at  4800,  Yellow  Pine  at  6200,  and  Fir  at  6700. 

Periods   of  Q-estation    and    Number  of  Young.- 

Weeks.    No.  Weeks.    No. 

Elephant.  100      i 


Horse . . 
Camel. . 


Weeks. 

No. 

We 

eks. 

No. 

Sheep  ...    21 

2 

Dog  

9 

6 

Goat  ....    22 

2 

Fox  

5 

Beaver..     17 

3 

Cat  

6 

Pig  17 

12 

Rat  

5 

8 

Wolf  10 

5 

Squirrel  .  . 

4 

6 

Guinea  Pig.  3       3 

Cow 41 

{43  Buffalo . .  40 

50  Stag 36 

45      i      Bear 30 

43      i      Deer 24 

Rabbit. . .  4 

Periods    of  Inculoation   of  Birds. 

Swan,  42  days;  Parrot,  40  days;  Goose  and  Pheasant,  35  days;  Duck,  Turkey,  and 
Peafowl,  28  days;  Hens  of  all  gallinaceous  birds,  21  days;  Pigeon  and  Canary,  14 
days.  Temperature  of  incubation  is  104°. 

-A.ges   of  Animals,  etc. 

Whale,  estimated  1000  years;  Elephant,  400;  Swan,  300;  Camel,  100;  Eagle,  100; 

Raven,  100;  Tortoise,  100  to  ;  Lion,  70;  Dolphin,  30;  Horse,  30;  Porpoise,  30; 

Bear,  20;  Cow,  20;  Deer,  20;  Rhinoceros,  20;  Swine,  20;  Wolf,  20;  Cat,  15;  Fox,  15; 
Dog,  15;  Sheep,  10;  Hare,  Rabbit,  and  Squirrel,  7. 

Relative    "Weights    of   Brain. 
Man,  154.33;  Mammifers,  29.88;  Birds,  26.22;  Reptiles,  4.2;  Fish,  i. 

Buoyancy   of  Casks. 

Buoyancy  of  a  submerged  cask  in  fresh  water  in  lbs.=62.425  times  the  volumn 
of  it  in  cube  feet,  7.48  times  the  volume  in  U.  S.  gallons,  and  6.2355  times  in 
Imperial  gallons,  less  the  weight  of  the  cask. 

Transportation    of  Horses    and    Cattle. 

Space  required  on  board  of  a  Marine  Transport  is:  for  Horses,  30  ins.  by  9  feet; 
Beeves,  32  ins.  by  9  feet.  Provender  required  per  diem  is:  for  Horses,  Hay,  15  Ibs. ; 
Oats,  6  quarts;  Water,  4  gallons.  Beeves,  Hay,  18  Ibs. ;  Water,  6  gallons. 

Rods    and    Eartn.    Excavation    and    Embankment. 

Number  of  Cube  Feet  of  various  Earths  in  a  Ton. 

Loose  Earth 24     I  Clay 18.6  I  Clay  with  Gravel 14.4 

Coarse  Sand 18.6  |  Earth  with  Gravel. . .  17.8  |  Common  Soil 15.6 

The  volume  of  Earth  and  Sand  in  embankment  exceeds  that  in  a  primary  ex- 
cavation in  following  proportions: 
Rock,  large 1.5    I  Rock,  ballast 1.2      I  Clay m 

"     medium 1.25  |  Sand 143  |  Gravel 09 

Clay  and  Earth  will  subside  about  .12. 


MISCELLANEOUS   ELEMENTS. 


193 


Hills   or   Plants   in   an   Area   of*  One   Acre. 

From  i  to  40  feet  apart  from  centres. 


Feet  apart. 

No. 

Feet  apart. 

No. 

Feet  apart. 

No. 

Feet  apart. 

No. 

X 

4356o 

5 

1742 

9 

538 

16 

171 

i-5 

19360 

5-5 

1440 

9-5 

482 

17 

IS' 

2 
2-5 

10890 

6 
6.5 

1210 

IO3I 

IO 

10.5 

III 

18 
20 

135 

108 

3 

4840 

7 

889 

12 

302 

25 

69 

3-5 
4 

3556 
2722 

?85 

ffi 

13 

H 

258 
223 

30 
35 

48 
35 

4-5 

2  151 

85 

692 

i5 

193 

40 

27 

Number  of  several  Seeds  in  a  Bushel,  and  Number  per 
Square   Foot    per  Acre. 


Timothy. 
Clover... 


41  823  360 
16400960 


Sq.  Foot. 


No. 

Sq.  Foot. 

Rye  

888390 

20  4 

Wheat... 

q  =16200 

12.8 

"Volumes. 

Permanent  gases,  as  air,  etc.,  are  diminished  in  their  volume  in  a  ratio  direct 
with  that  of  pressure  applied  to  them.  With  vapor,  as  steam,  etc.,  this  rule  is 
varied  in  consequence  of  presence  of  the  temperature  of  vaporization. 

Minerals. 
Relative    Hardness   of  some    Minerals. 


Talc i 

Gypsum 2 

Mica 2.5 

Carbonate  of  lime.  3 


Bary tes 3. 5 

Fluor-spar....  4 

Feldspar 6 

Lapis  Lazuli  .  6 


Opal ,    6 

Quartz 7 

Tourmalin  ....    7 
Garnet 7. 5 


Emerald 8 

Topaz 8 

Ruby 9 

Diamond. 10 


Weight   of  Diamonds. 


Regent  or  Pitt 

Carats. 

Carats. 

Dresden  765 

Star  of  the  SouthJ 

125 

Sancy  53  5 

Koh  i-Noort  
Piggott  
Napac  

.  106.06 
.    82.25 
.    78  62*; 

Eugenie,  brilliant  .  51 
Hope  (blue)  48.5 
Polar  Star  40  25 

t  Rough  254.5. 

t  Originally  793. 

Carats. 

Mattam .367 

Grand  Mogul* 279.9 

Orloff. 194. 25 

Florentine,  brilliant .  139.5 
Crown  of  Portugal. . .  138.5 
*  Rough  900. 

Heat  of  the   Sun. 

Sir  Isaac  Newton 3138740°  I  Waterston 16000000° 

Capt.  John  Ericsson 4909860°  |  Soret 10443323° 

Sundry  others  ranging  from  2520°  to  183600°. 

Moon.— Distance  of  Moon  from  Earth  237000  miles. 

Krigorinc    Mixture. 

Lowest  temperature  yet  procured.  Faraday  obtained  166°  by  evaporation  of  a 
mixture  of  solid  carbonic  acid  and  sulphuric  ether. 

Current   of  Rivers. 

A  fall  of .  i  of  an  inch  in  a  mile  will  produce  a  current  in  rivers. 

Sandstones. 

Structures  of  sandstone  erected  in  England  in  i2th  century  are  yet  in  good 
condition. 

Canal    Transportation. 

Erie  Canal  and  Hudson  River. — From  Buffalo  to  New  York,  495  miles,  cost  of 
transportation  2.46  mills  per  ton  (inclusive  of  tolls)  per  mile.  Transportation  of 
wheat  costs  when  it  reaches  New  York  4.72  cents  per  bushel,  and  .61  cents  per 
bushel  for  elevating  and  trimming. 

Towing. — Erie  Canal. — Four  mules  will  tow  230  tons  of  freight  down  and  100 
tons  back,  involving  a  period  of  30  days,  at  a  cost  of  8  cents  per  mile  for  a  course 
of  690  miles. 


1 94  MISCELLANEOUS   ELEMENTS. 

Matter. 

Unit  of  the  Physicist  is  a  molecule,  and  a  mass  of  matter  is  composed  of  them, 
having  same  physical  properties  as  parent  mass. 

It  exists  in  three  forms,  known  as  solid,  liquid,  and  gaseous.  Solids  have  indi- 
viduality of  form,  and  they  press  downward  alone.  Liquids  have  not  individuality 
of  form,  except  in  spherical  form  of  a  drop,  and  they  press  downward  and  sideward. 
Gases  are  wholly  deficient  in  form,  expanding  in  all  directions,  and  consequently 
they  press  upward,  downward,  and  sideward. 

Liquids  are  compressible  to  a  very  moderate  degree.  Water  has  been  forced 
through  pores  of  silver,  and  it  may  be  compressed  by  a  pressure  of  one  pound  per 
square  inch  to  the  3  soooooth  part  of  its  volume. 

Gases  may  be  liquefied  by  pressure  or  by  reduction  of  their  temperature. 

Combustible  matter  (as  coal)  may  be  burned,  a  structure  (as  a  house)  may  be 
destroyed  as  such,  and  the  fluid  (of  an  ink)  may  be  evaporated,  yet  the  matter  of 
which  coal  and  house  were  composed,  although  dissipated,  exists,  and  the  water 
and  coloring  matter  of  the  ink  are  yet  in  existence. 

Spaces  between  the  particles  of  a  body  are  termed  pores. 

All  matter  is  porous.  Polished  marble  will  absorb  moisture,  as  evidenced  in  its 
discoloration  by  presence  of  a  colored  fluid,  as  ink,  etc. 

Silica  is  the  base  of  the  mineral  world,  and  Carbon  of  the  organized. 

Miuiiteriess   of  Matter. 

A  piece  of  metal,  stone,  or  earth,  divided  to  a  powder,  a  particle  of  it,  however 
minute,  is  yet  a  piece  of  the  original  material  from  which  it  was  separated,  retain- 
ing its  identity,  and  is  termed  a  molecule. 

It  is  estimated  there  are  120000000  corpuscles  in  a  drop  of  blood  of  the  musk-deer. 

Thread  of  a  spider's  web  is  of  a  cable  form,  is  but  one  sixth  diameter  of  a  fibre  of 
silk,  and  4  miles  of  it  is  estimated  to  have  a  weight  of  but  i  grain. 

One  imperial  gallon  (277.24  cube  ins.)  of  water  will  be  colored  by  mixture  therein 
of  a  grain  of  carmine  or  indigo. 

A  grain  of  platinum  can  be  drawn  out  the  length  of  a  mile. 

Film  of  a  soap-and- water  bubble  is  estimated  to  be  but  the  aoooooth  part  of  an 
inch  in  thickness. 

It  is  computed  that  it  would  require  12000  of  the  insect  known  as  the  twilight 
monad  to  fill  up  a  line  one  inch  in  length. 

A  drop  of  water,  or  a  minute  volume  of  gas,  however  much  expanded— even  to 
the  volume  of  the  Earth — would  present  distinct  molecules. 

Gold  leaf  is  the  zSooooth  part  of  an  inch  in  thickness. 

A  thread  of  silk  is  25ooth  of  an  inch  in  diameter. 

A  cube  inch  of  chalk  in  some  places  in  vicinity  of  Paris  contains  100000  of  shells 
of  the  foraminifera. 

There  are  animalcules  so  small  that  it  requires  75  ooo  ooo  of  them  to  weigh  a  grain. 

Velocity,  ^Weight,  and   "Volume    of  Molecules. 

Velocity.—  Collisions  among  the  particles  of  Hydrogen  are  estimated  to  occur  at 
the  rate  of  17  million-million-million  per  second,  and  in  Oxygen  less  than  half  this 
number. 

Weight.—  A  million-million-million-million  molecules  of  Hydrogen  are  estimated 
to  weigh  but  60  grains. 

Volume.— 19  million-million-million  molecules  of  Hydrogen  have  a  volume  of  .061 
cube  ins.  Diameter.— Five  millions  in  a  line  would  measure  but .  i  inch. 

Charcoal,  Alcohol. 
Charcoal  as  yet  has  not  been  liquefied,  nor  has  Alcohol  been  solidified. 

Metals. 
Metals  have  five  degrees  of  lustre— splendent,  shining,  glistening,  glimmering,  and 

All  metals  can  be  vaporized,  or  exist  as  a  gas,  by  application  to  them  of  their  ap- 
propriate temperature  of  conversion. 

Repeated  hammering  of  a  metal  renders  it  brittle ;  reheating  it  restores  its  tenacity. 

Repeated  melting  of  iron  renders  it  harder,  and  up  to  twelfth  time  it  becomes 
stronger. 

Platinum  is  the  most  ductile  of  all  metals. 


MISCELLANEOUS   ELEMENTS.  19$ 

Impenetrability. 

Impenetrability  expresses  the  inability  of  two  or  more  bodies  to  occupy  same 
space  at  same  time. 

A  mixture  of  two  or  more  fluids  may  compose  a  less  volume  than  that  due  to  sum 
of  their  original  volume,  in  consequence  of  a  denser  or  closer  occupation  of  their 
molecules.  This  is  evident  in  the  mixture  of  alcohol  and  water  in  the  proportion 
of  16. 5  volumes  of  former  to  25  of  latter,  when  there  is  a  loss  of  one  volume. 

Elasticity. 

Elasticity  is  the  term  for  the  capacity  of  a  body  to  recover  its  former  volume, 
after  being  subjected  to  compression  by  percussion  or  deflection. 

Glass,  ivory,  and  steel  are  the  most  elastic  of  all  bodies,  and  clay  and  putty  are 
illustrations  of  bodies  almost  devoid  of  elasticity.  Caoutchouc  (India  rubber)  is  but 
moderately  elastic;  it  possesses  contractility,  however,  in  a  great  degree. 

Momentum. 

Momentum  is  quantity  of  motion,  and  is  product  of  mass  and  its  velocity.  Thus, 
the  momentum  of  a  cannon-ball  is  product  of  its  velocity  in  feet  per  second  and  its 
weight,  and  is  denominated  foot-pounds. 

A  foot-pound  is  the  power  that  will  raise  one  pound  one  foot. 

Sound. 

Velocity  of  sound  is  proportionate  to  its  volume;  thus,  report  of  a  blast  with  2000 
Ibs.  of  powder  passed  967  feet  in  one  second,  and  one  of  1200  Ibs.  1210  feet.  It  passes 
in  water  with  a  velocity  of  4708  feet  per  second.  Conversation  in  a  low  tone  has 
been  maintained  through  cast-iron  water  pipes  for  a  distance  of  3120  feet,  and  its 
velocity  is  from  4  to  16  times  greater  in  metals  and  wood  than  air. 

T.jignt. 

Sun's  rays  have  a  velocity  of  185000  miles  per  second,  equal  to  7.5  times  around 
the  Earth. 

Color    Blindness 

Is  absence  of  elementary  sensation  corresponding  to  red. 
JLmminous   JPoint. 

To  produce  a  visual  circle,  a  luminous  point  must  have  a  velocity  of  10  feet  in  a 
second,  the  diameter  not  exceeding  15  ins. 

All  solid  bodies  become  luminous  at  800  degrees  of  heat. 
IVIirage. 

When  air  near  to  surface  of  Earth  becomes  so  highly  heated,  as  upon  a  sandy 
plain,  that  its  density  within  a  defined  distance  from  it  increases  upwards,  a  line 
of  vision  directed  obliquely  downwards  will  be  rendered  by  refraction,  gradually 
increasing,  more  and  more  nearly  horizontal  as  it  advances,  until  its  direction  is  so 
great  as  to  produce  a  total  reflection,  and  the  reflected  ray  then,  by  successive  re- 
fractious,  is  gradually  elevated  until  it  meets  the  eye  of  the  observer. 

Looming  is  inverted  mirage,  frequently  seen  over  calm  water,  and  is  effect  of 
lower  or  surface  stratum  of  air  being  colder  than  that  above  it. 

Snow    Flakes. 
96  forms  of  snow  flakes  have  been  observed. 

^^JMQ         Melted    Snow 

Produces  from  .25  to  .125  of  its  bulk  in  water. 

Strength,   of  Ice. 

Two  inches  thick  will  support  men  in  single  file  on  planks  6  feet  apart;  4  inches 
will  support  cavalry,  light  guns,  and  carts;  and  6  inches  wagons  drawn  by  horses. 

Temperature. 

Sulphuric  acid  and  water  produce  a  much  greater  proportionate  contraction  than 
alcohol  and  water.  Both  of  these  mixtures,  however  low  their  temperature,  pro- 
duce heat  which  is  in  a  direct  proportion  to  their  diminution  in  volume. 

At  the  depth  of  45  feet,  the  temperature  of  the  Earth  is  uniform  throughout  the 
year. 

Temperature  of  Earth  increases  about  i°  for  every  50  to  60  feet  of  depth,  and  its 
crust  is  estimated  at  30  miles. 

A  body  at  Equator  weighs  two  hundred  and  eighty-nine  parts  less  than  at  the  Poles, 


196 


MISCELLANEOUS    ELEMENTS. 


Ages    of*  Animals,   Fishes,  etc. 

(Additional  to  page  192.) 

Tiger,  Leopard,  Jaguar,  and  Hyena  (in  confinement),  25  years;  Beaver,  50;  Stag, 
under  50;  Ox  and  Ass,  30;  Chamois,  25;  Llama,  Monkey,  and  Baboon,  15  to  18;  Par- 
rot, 200;  Tortoise,  100  to  200;  Crocodile,  100;  Carp,  70  to  150;  Goose,  80;  Pelican, 
45;  Hawk,  30  to  40;  Crane,  24;  Peacock,  Goldfinch,  Chaffinch,  from  10  to  25;  Do- 
mestic Fowls,  Pigeons,  Blackbird,  Nightingale,  and  Linnet,  10  to  16;  Thrush,  Robin, 
and  Starling,  8  to  12;  Wren,  2  to  3]  Salmon,  16;  Eel,  10;  Codfish,  4  to  17;  Pike,  30 
to  40;  Queen  Bee,  4;  Bee,  6  months,  and  Drones,  4  months.  (Houghtaling.) 

Birds   and.   Insects.—  (M.  De  Lacy.) 

Elements  of  Flight.—  Resistance  of  air  to  a  body  in  motion  is  in  ratio  of  surface 
of  body  and  as  square  of  its  velocity. 

Wing  Surface.— Extent  or  area  of  winged  surface  is  in  an  inverse  ratio  to  weight 
of  bird  or  insect. 

A  Stag-beetle  weighs  460  times  more  than  a  Gnat,  and  has  but  one  fourteenth  of 
its  wing  surface;  150  times  more  than  a  Lady  Bird  (bug),  and  has  but  one  fifth. 
An  Australian  Crane  weighs  339  times  more  than  a  sparrow,  and  has  but  one  sev- 
enth; 3000000  times  more  than  a  Gnat,  and  has  but  one  hundred  and  fortieth.  A 
Stork  weighs  eight  times  more  than  a  Pigeon,  and  has  but  one  half.  A  Pigeon 
weighs  ten  times  more  than  a  Sparrow,  and  has  but  one  half;  97  ooo  times  more  than 
a  Gnat,  and  has  but  one  fortieth. 

A  resisting  surface  of  30  sq.  yards  will  enable  a  man  of  ordinary  weight  to  descend 
safely  from  a  great  elevation. 

Strength  of  Insects.— Insects  are  relatively  strongest  of  alf  animals.  A  Cricket 
can  leap  80  times  its  length,  and  a  F'ea  200  times. 

Application   for   Stings   and   Bnrns. 

Sting  of  Insects. — Ammonia,  or  Soda  moistened  with  water,  and  applied  as  a  paste. 
Burns. — Hot  alcohol  or  turpentine,  and  afterwards  bathed  with  lime  water  and 
sweet  oil.    Cold  water  not  to  be  applied. 

To   .Preserve   Meat. 

Meat  of  any  kind  may  be  preserved  in  a  temperature  of  from  80°  to  100°,  for  a 
period  of  ten  days,  after  it  has  been  soaked  in  a  solution  of  i  pint  of  salt  dissolved 
in  4  gallons  of  cold  water  and  .5  gallon  of  a  solution  of  bisulphate  of  calcium. 

By  repeating  this  process,  preservation  may  be  extended  by  addition  of  a  solution 
•f  gelatin  or  white  of  an  egg  to  the  salt  and  water. 

To   Detect   Starch,   in    Milk. 

Add  a  few  drops  of  acetic  acid  to  a  small  quantity  of  milk ;  boil  it,  and 
after  it  has  cooled  filter  the  whey.  If  starch  is  present,  a  drop  of  iodine 
solution  will  produce  a  blue  tint. 

This  process  is  so  delicate  that  it  will  show  the  presence  of  a  milligram  of  starch 
in  a  cube  centimeter  of  whey  (i  grain  of  starch  in  2.16  fluid-ounces). 

Retaining  "Walls    of  Iron    Piles. 

Sheet  Piles.— 7  feet  from  centres,  18  ins.  in  width  and  2  ins.  in  thickness,  strength- 
ened  with  2  ribs  8  ins.  in  depth. 

Plates.— 7  feet  in  length  by  5  feet  in  width  and  i  inch  in  thickness,  with  on» 
diagonal  feather  i  by  6  ins. 
Tie-rods  2  ins.  in  diameter 

Stone   Sawing. 

Diamond  Stone  Sawing. — (Emerson.)  Alabama  marble  6  feet  X  2.5  feet  in  22  min- 
utes =  41  sq.  feet  per  hour. 

"Wood    Sa-wing. 

7722  feet  of  poplar,  board  measure,  from  g  round  logs  in  i  hour.  Engine  12  ins. 
diameter  by  24  ins.  stroke. 


MISCELLANEOUS   ELEMENTS. 


197 


Cost   oi*  Dredging. 

Actual  cost,  if  on  an  extended  work,  inclusive  of  Delivery,  if  dredging  into  or  on  a 

vessel  alongside  of  dredger. — (Trautwine.) 
Labor  at  $  i  per  day  and  Repairs  of  Plant  included. 


Depth. 

Cents. 

Depth. 

Cents. 

Depth. 

Cents. 

Depth. 

Cents. 

Feet. 
10 

Cube  Yards. 
6 

Feet. 

20 

Cube  Yards. 
8 

Feet. 
25 

Cube  Yards. 

10 

Feet. 
35 

Cube  Yard*. 
18 

15 

7 

22 

9 

30 

13 

40 

25 

Discharge  of  Scows  or  Camels.— Towing  .25  mile  4  cents  per  cube  yard,  .5  mile  6 
cents,  .75  mile  8  cents,  and  i  mile  10  cents. 

NOTE.  —  A  Scow  is  a  flat-bottomed  vessel  or  boat  A  Camel  is  a  shallow,  flat- 
bottomed  and  decked  vessel,  designed  for  the  transportation  of  heavy  freight  or  the 
sustaining  of  attached  bodies,  as  a  vessel,  by  its  buoyancy. 

Dredging. 

A  steam  dredge  will  raise  6  cube  yards,  or  8.5  tons,  per  hour  per  H*. 

^Eetal    Boring    and.   Turning. 

BORING.— Cost  iron.— Divide  25  by  the  diameter  of  the  cylinder  in  inches  for  the 
revolutions  per  minute. 

Wrought  iron.— The  speed  is  one  fourth  to  one  fifth  greater  than  for  cast  iron. 
Brass.—  The  speed  is  about  twice  that  for  cast  iron. 

TURNING  —Cast  iron.— The  speed  is  twice  that  of  boring. 

Wrought  iron.— Theispeed  is  one  fourth  to  one  fifth  greater  than  that  for  cast  iron. 

Brass.— The  speed  is  twice  that  of  boring. 

Vertical  boring.— The  speed  may  be  twice  that  of  horizontal  boring. 

The  feed  depends  upon  the  stability  of  the  machine  and  depth  of  the  cut. 

"Well    Boring. 

At  Coventry,  Eng.,  750000  galls,  of  water  per  day  are  obtained  by  two  borings  of 
6  and  8  ins.,  at  depths  of  200  and  300  feet. 

At  Liverpool,  Eng.,  3000000  galls,  of  water  per  day  are  obtained  by  a  bore  6  ins. 
in  diameter  and  161  feet  in  depth. 

This  large  yield  is  ascribed  to  the  existence  of  &  fault  near  to  it,  and  extending  to 
a  depth  of  484  feet. 

At  Kentish  Town,  Eng.,  a  well  is  bored  to  the  depth  of  1302  feet. 

At  Passy,  France,  a  well  with  a  bore  of  i  meter  in  diameter  is  sunk  to  a  depth  of 
1804  feet,  and  for  a  diameter  of  2  feet  4  ins.  it  is  further  sunk  to  a  depth  of  109  feet 
10  ins. ,  or  1903  feet  10  ins. .  froii  which  a  y  'eld  of  5  582  ooo  galls,  of  water  are  obtained 
per  day. 

Tempering    Boring    Instruments. 

Heat  the  tool  to  a  bloca-red  heat;  hammer  it  until  it  is  nearly  cold;  reheat  it  to 
a  blood-red  heat,  and  plunge  it  into  a  mixture  of  2  oz.  each  of  vitriol,  soda,  sal-am- 
moniac, and  spirits  of  nitre,  i  oz.  of  oil  of  vitriol,  .5  oz.  of  saltpetre,  and  3  galls,  of 
water,  retaining  it  there  until  it  is  cool. 

Circular   Sa-ws. 
Revolutions  per  Minute.— 8  ins.  4500, 10  ins.  3600,  and  36  ins.  xcxxx 

^Easonry. 

Concrete  or  Beton  should  be  thrown,  or  let  fall  from  a  height  of  at  least  10  fe«t, 
•r  well  beaten  down. 
The  average  weight  of  brickwork  in  mortar  is  about  102  Ibs.  per  cube  foot. 

Plastering. 

In  measuring  Plasterers'  work  all  openings,  as  doors,  windows,  etc.,  are  com- 
puted at  one  half  of  their  areas,  and  cornices  are  measured  upon  their  extreme 
edges,  including  that  cut  off  by  mitring. 

GJ-lazing. 

In  Glaziers'  work,  oval  and  round  windows  are  measured  as  squares. 
R* 


198 


MISCELLANEOUS   ELEMENTS. 


Coi'n    Measure. 

Two  cube  feet  of  corn  in  ear  will  make  a  bushel  of  corn  when  shelled. 

Tenacity   of*  Iron.    Bolts    in    "Woods. 

Diameter  1.125  ins-  and  12  ins.  in  length  required  for  Hemlock  8  tons,  and  for 
Pine  6  tons  to  withdraw  them. 

Length   of*  G-vin    Barrels.     (C.  T.  Coathupe.) 

The  length  of  the  barrel  of  a  gun,  to  shoot  well,  measured  from  vent-hole,  should 
not  be  less  than  44  times  diameter  of  its  bore,  nor  more  than  47. 

Hay    and.    Stra\v. 

Hay,  loose,  5  Ibs.  per  cube  foot.  Ordinarily  pressed,  as  in  a  stack  or  mow,  8  Ibs. 
Close  pressed,  as  in  a  bale,  12  to  14  Ibs. 

Ordinarily  pressed,  as  in  a  wagon  load,  450  to  500  cube  feet  will  weigh  a  ton. 
Straw  in  a  bale  10  to  12  Ibs.  per  cube  foot. 

Natural    Powers. 

Sun. — The  power  or  work  performed  by  the  Sun's  evaporation  is  estimated  at 
90000000000  IP. 

Niagara.— Volume  of  water  discharged  over  the  falls  is  estimated  at  33000000 
tons  per  hour,  and  the  entire  fall  from  Lake  Erie  at  Buffalo  to  Lake  Ontario  is  323.35 
feet. 

Velocity  of  Stars. 

According  to  computation  of  Mr.  Trautwine  a  Star  passes  a  range  in  3'  55.91"  less 
time  each  day. 

Service    Train    of  a    Quartermaster. 

Quartermaster's  train  of  an  army  averages  i  wagon  to  every  24  men;  and  a  well- 
equipped  army  in  the  field,  with  artillery,  cavalry,  and  trains,  requires  i  horse  or 
mule,  upon  the  average,  to  every  2  men. 

Tides. 

The  difference  in  time  between  high  water  averages  about  49  minutes  each  day. 
Atlantic  and  Pacific  Oceans.— Rise  and  fall  of  tide  in  Atlantic  at  Aspinwall  2  feet, 
in  Pacific  at  Panama  17.72  to  21.3  feet. 

Dimensions  of  Drawings  and  Paper  for  TJ.  S.  Patents. 
Drawings,  8  x  12  inches,  one  inch  margin.    Paper,  8  x  12.5  inches. 

Latitude. 

One  minute  of  latitude,  mean  level  of  Sea,  nearly  6076  feet  =  1. 1508  Statute  miles. 

Artesian   Well. 

White  Plains,  Nev.,  Depth  2500  feet. 

IToxindation    Piles. 

A  pile,  if  driven  to  a  fair  refusal  by  a  ram  of  i  ton,  falling  30  feet,  will  bear  i  ton 
weight  for  each  sq.  foot  of  its  external  or  frictional  surface,  or  a  safe  load  of  750  Ibs. 
per  sq.  foot  of  surface. 

Eartlu- 
Density  of  its  mass  5.67. 

Tripolith. 

A  new  building  material,  compounded  of  Coke,  Sulphate  of  Lime,  and  Oxide  of 
Iron.  It  has  increased  tensile  strength  after  exposure  to  the  air,  being  much  in 
excess  of  that  of  lime  and  cement. 

Gras    and    Electric    Light. 

Gas  light  of  16  candle  power  costs  5  cent  per  hour;  Electric,  4.15  cents. 

Niagara. 

Discovered,  1678.  Falls  have  receded  76  feet  in  175  years.  Height,  American 
Falls,  164  feet;  Horseshoe,  158  feet. 


BRIDGES. — U.  S.  ENSIGNS,  PENNANTS,  AND  FLAGS.     1 99 

U.   S.    ENSIGN,   PENNANTS,  AND    FLAGS, 
(From  April  20,  1896.) 

Ensign. — Head  (Depth,  or  Hoist}. — Ten  nineteenths  of  its  length. 

Field. — Thirteen  horizontal  stripes  of  equal  breadth,  alternately  red  and 
white,  beginning  with  red. 

Union. — A  blue  field  in  upper  quarter,  next  the  head,  .4  of  length  of  field, 
and  seven  stripes  in  depth,  with  white  stars  ranged  in  equidistant,  horizontal 
lines  and  set  staggered,  equal  in  number  to  number  of  States  of  the  Union. 

Pennants  (Narrow).  —  Head.  —  6.24  ins.  to  a  length  of  70  feet;  5.04  ins.  to  a 
length  of  40  feet;  4.2  ins.  to  a  length  of  35  feet.  Night,  3.6  ins.  to  a  length  of  20 
feet,  and  3  ins.  to  a  length  of  9  feet. — Boat,  2.52  ins.  to  a  length  of  6  feet. 

Union.— A  blue  field  at  head,  one  fourth  the  length,  with  13  white  stars  in  a  hori- 
zontal line.  Field.— A  red  and  white  stripe  uniformly  tapered  to  a  point,  red  up- 
permost. Night  and  Boat  Pennants. — Union  to  have  but  7  stars. 

Union   Jacli. — Alike  to  the  Union  of  an  Ensign  in  dimensions  and  stars. 

Flags. — President. — Rectangle,  with  arms  of  the  U.  S.  in  centre  of 
a  blue  field,  over  which  are  13  stars  in  an  arc. 

Secretary-  of  Navy. — Rectangle,  with  a  vertical  white  foul  anchor 
in  centre  of  a  blue  field,  with  four  white  stars  in  a  rectangle,  set  quadrilateral 
around  a  foul  anchor. 

Admiral. — Rectangle,  with  4  white  stars  in  centre  of  a  blue  field,  set  as 
a  lozenge. 

Vice- Admiral. — Same  as  Admiral's,  with  3  white  stars  set  as  an 
equilateral  triangle? 

Rear- Admiral. — Same  as  Admiral's,  with  2  white  stars  set  vertically. 

If  two  or  more  Rear-Admirals  in  command  afloat  should  meet,  their  seniority  is 
to  be  indicated  respectively  by  a  Blue  flag,  a  Red  with  White  stars,  and  a  White 
with  Blue  stars,  and  another  or  all  others,  a  White  flag  with  Blue  stars. 

Commodore.  (Broad  Pennant.) — Blue,  Red,  or  White,  according  to 
rank,  with  one  star  in  centre  of  field,  being  white  in  blue  and  red  pennants, 
and  blue  in  white. 

Swallow-tailed,  angle  at  tail,  bisected  by  a  line  drawn  at  a  right  angle  from  centre 
of  depth  or  hoist,  and  at  a  distance  from  head  of  three  fifths  of  length  of  pennant; 
the  lower  side  rectangular  with  head  or  hoist;  upper  side  tapered,  running  the  width 
of  pennant  at  the  tails  .  i  the  hoist.  Head.— .6  length.  Fly  1.66  hoist. 

Divisional  Marks.  —  Triangle,  ist  Blue,  2d  Red,  3d  White,  Blue 
vertical.  Reserve  Division. — Yellow,  Red  vertical.  Division  mark  is  worn 
by  Commander  of  a  division  of  a  squadron  at  mizzen,  when  not  authorized 
to  wear  Broad  Pennant  of  a  Commodore  or  Flag  of  an  Admiral.  Fly  .8  hoist. 

Signal  Numbers. — Fly  1.25  hoist.  Signal  Pennants,  Fly  4.6  hoist. 
Repeaters  1.89  hoist. 

Distinctive  Pennants.— Of  a  Senior  Officer  Present,  is  the  Dis- 
tinctive Mark  of  the  First  Division  of  a  fleet. 

Nignt  Signals. — Very's  System. 

International,  Signal  Number,  Square,  and  Signal  Pennants.    Fly,  3  hoist. 


Suspension   Bridges.    Length  of  Spans  in  Feet. 


You-Mau,  China 330 

Schuylkill  (Phila.) 342 

Hammersmith,  Eng. 422 

Pesth  (Danube) 660 


Niagara 822 

Lewistown  and  Queenstown 1040 

Cincinnati 1057 

Niagara  Falls 1280 


New  York  and  Brooklyn,  930,  1595.5,  and  930;  clear  height  of  Bridge  above  high 
water,  at  90°,  135  feet. 


2OO 


ANIMAL  FOOD. 


Alimentary   ^Principles. 

Primary  division  of  Food  is  into  Organic  and  Inorganic. 

Organic  is  subdivided  into  Nitrogenous  and  Non-Nitrogenous ;  Inorganic 
is  composed  of  water  and  various  saline  principles.  The  former  elements 
are  destined  for  growth  and  maintenance  of  the  body,  and  are  termed  "  plas- 
tic elements  of  nutrition."  The  latter  are  designed  for  undergoing  oxidation, 
and  thus  become  source  of  heat,  and  are  termed  "  elements  of  respiration,"  or 
"  Calorificient." 

Although  Fat  is  non -nitrogenous,  it  is  so  mixed  with  nitrogenous  matter  that  it 
becomes  a  nutrient  as  well  as  a  calorificient. 

Alimentary  Principles.  —  i.  Water;  2.  Sugar;  3.  Gum;  4.  Starch;  5.  Pectine; 
6.  Acetic  Acid;  7.  Alcohol;  8.  Oil  or  Fat.  Vegetable  and  Animal. — 9.  Albumen; 
10.  Fibrine;  n.  Caseine;  12.  Gluten;  13.  Gelatine;  14.  Chloride  of  Sodium. 

These  alimentary  principles,  by  their  mixture  or  union,  form  our  ordinary  foods, 
which,  by  way  of  distinction,  may  be  denominated  compound  aliments  ;  thus,  meat 
is  composed  of  fibrine,  albumen,  gelatine,  fat,  etc. ;  wheat  consists  of  starch,  gluten, 
sugar,  gum,  etc. 


Analy 

FOOD. 

sis   o 

Water. 

f  IVte 

Nitro- 
genous 
Matter. 

ats,  I 
Fat. 

"isli, 

Saline 
Matter. 

Vegeta 

Non-Nitro- 

kfenoua 
Matter. 

bles,  < 

Sugar. 

3  to. 

Cellu- 
lose. 

Ash,  etc. 

18 
15 
9-9 
54 
Si 
72 
49.1 
91 
13 
15 
91 
83 
36.8 

6~3 
25-5 
*7.t 

14.6 

19-3 
29.6 
.1 
13-1 

2 
1-3 

33-5 

2<4> 

2.8 

15-45 

29.8 
3-6 

.2 

sf 

•5 

.2 

2*'3 

2 

82 
•69.4 
55-7 

8~7 
64-5 

5~8 
7-4 

57~6 

1.26 
1.38 

58~4 
48.2 
9.6 
50.2 

16.8 

78" 
69-5 
95 

4-3 
61.1 

£4 

weight  of  fl 

4.9 

6.1 

•4 

2.8 

5-2 

5-4 
5-8 

2 

3-2 

•4 
3-7 

2.1 

4-2 

3-6 
our,  the  b 

2-9 

3-5 
5-9 

7.6 
3-i 

i 

3-5 
est  qualit 

3-2 

25 

i 
i.  a 

3-3 

9.1 

'*•/"" 

I 

.8 
'•7 

2 

3 
y  absorb 

Barley  Meal      .... 

Beans  White 

Beef  roast. 

2.Q5 

44 

5-1 

21.  1 
.2 

•4 

2 

•7 

i 

5-4 

fat  

lean       .  .  . 

salt  

Beer  and  Porter.  .  .  . 
Buckwheat  

Butter  and  Fats  
Cabbage  .... 

Carrots  

Cheese  

Corn  Meal  

66 
74 

52 
78 

76.6 
80.39 

It'* 
53 
15 

21 

82 
15 

39 

is 

75 
74 
i3 
i5 

68 
91 
63 
i5 
37 
'3 
our  var 
field  130 

2.7 

11 

18.1 
9.9 
19.17 
14.01 

20.55 
4.1 
12.4 

12.6 

14.4 
i.i 

1.8 

8.8 

2.1 

f 

13.2 
1.2 
l6.5 

10.8 
8.1 
18 
es  from  4c 
Ibs.  brear 

26.7 
10.5 
30-7 
2.9 
13-8 
I.I7 
1.52 
5.58 

3-9 

3X 

5-5 
•5 

2.1 
48.9 

73-3 

.2 

3-8 
•7 

2 
I&i 

15^8 
2 

1.6 
6 
>  to  60  pe 

1.8 
i-5 
i-3 
i 
1.3 

2-7 

"f 

3-5 
3 

i 
2-5 

2-3 

2.9 
•7 

1.2 

1.8 

2.4 
.6 
4-7 
i-7 

2-3 

r  cent,  ol 

Egg  

yolk  

Fish,  white  flesh... 
Eels   

Lobster,  flesh. 
Oysters  
Liver,  Calf's  
Milk  Cow's  . 

Mutton,  fat  

Oatmeal  .... 

Oats  

Parsnips  ... 

Peas  

Pork,  fat  

Bacon,  dry.  .  . 
Potatoes  

Poultry    

Rice  

Rye  Meal 

Tripe  

Turnips...  . 

Veal  

Wheat  Flour  

Bread*  
Bran  

*  Water  absorbed  by  1 
ing  most.    100  Ibs.  flour 

ANIMAL    FOOD. 


201 


Analysis    of  Different   Foods 

In  their  Natural  Condition. 


Nf- 

trates. 

Carbon- 
ates. 

Phos- 
phates. 

Water. 

Ni- 
trates. 

Carbon- 
ates. 

Phos- 
phates. 

Water, 

Apples 

84 

Milk  of  cow  .  . 

5 

8 

i 

86 

Barley 

60  < 

•a  e 

Mutton  

12  S 

4.0 

BeanS 

JA  8 

Oats  

66  , 

tie 

Beef  

Ttf 

•3Q 

e 

KO 

Parsnips  

9-2 

1 

i 

82  8 

86 

is 

Pork  

_o   c 

Cabbage  
Chicken  
Corn  North  'n 

4 
'9 

5 
3-5 

i 

4-5 

90 
73 

Potatoes  
"     sweet 
Rice  

2.4 

g 

22.5 
28.4 

J 

c 

3°-5 
£5 

«    South'n 

12 

Turnips 

35 

48 

3 

Veal  

j 

16  s 

6?b 

Lamb.  .  . 

ii 

35.  «; 

*.«; 

97 
so 

Wheat... 

is 

60.2 

in 

14.2 

Nitrates— Are  that  class  which  supplies  waste  of  muscle. 

Carbonates— Are  that  class  which  supplies  lungs  with  fuel,  and  thus  furnishes  heat 
to  the  system,  and  supplies  fat  or  adipose  substances. 

Phosphates— Are  that  class  which  supplies  bones,  brains,  and  nerves,  and  gives 
vital  power,  both  muscular  and  mental. 

From  above  it  appears,  that  Southern  corn  produces  most  muscle  and  least  fat, 
and  contains  enough  of  phosphates  to  give  vital  power  to  brain,  and  make  bones 
strong.  Mutton  is  the  meat  which  should  be  eaten  with  Southern  corn. 

The  nitrates  in  all  the  fine  bread  which  a  man  can  eat  will  not  sustain  life  beyond 
fifty  days;  but  others,  fed  on  unbolted  flour  bread,  would  continue  to  thrive  for  an 
indefinite  period.  It  is  immaterial  whether  the  general  quantity  of  food  be  reduced 
too  low,  or  whether  either  of  the  muscle-making  or  heat-producing  principles  be 
withdrawn  while  the  other  is  fully  supplied.  In  either  case  the  effect  will  be  the 
same.  A  man  will  become  weak,  dwindle  away  and  die,  sooner  or  later,  according  to 
the  deficiency ;  and  if  food  is  eaten  which  is  deficient  in  either  principle,  the  appe- 
tite will  demand  it  in  quantity  till  the  deficient  element  is  supplied.  All  food,  be- 
yond the  amount  necessary  to  supply  the  principle  that  is  not  deficient,  is  not  only 
wasted,  but  burdens  the  system  with  efforts  to  dispose  of  it. 

Analysis   of  Fruits. 


FRUIT. 

Water. 

Sugar. 

Acid. 

Albumi- 
nous sub- 
stances. 

Insoluble 
matter. 

Pectous 
sub- 
stances. 

Ash. 

85 

III 
IJ 

85*6 

85-4 
80 

85 
83-5 
80.8 

79-7 
88.7 

85-3 
81.3 
83-9 
87 

73-9 
er  in 
tn 

T  M 

7        Le 
6.4     Bi 

7.6 
1.8 
4.44 
13  i 
8.77 
10.7 

7 

If 

3-4 

2 
2.25 
6.73 

3-6 
Sugar, 

Varic 
e    Tat> 

classes  . 

i 
i.i 
1.19 

*| 

.56 
i-7 
i-35 

1.2 

I 

.07 
.96 
.87 
1.27 

^84 

2 

*-5 
Pectin, 

>ns  f? 

le.     (1 

.22 

•51 

% 

I 
.36 
•44 
.46 

3- 

i 

•4 
•4 
•43 
•83 

:I5 

Salt,  Acic 

rod  vie 

*er  Cent.) 
Water. 
.  .    01 

1.83 

5'.83 
5-91 
6.04 

3-74 
2.92 

l:ll 

5-49 
3-54 
3-98 

4-23 
4.01 
8-37 
5-5 
I,  etc.,  26 

ts  not 

Cabbage 
Ale  and 
Coffee  a 

3.88 
7-55 
1.72 

3-73 
2.07 

i-33 
2.4 
1.26 
2.4 
1.44 
6.4 
4.8 
10.48 
"•3 

•4 
Inoluc 

» 

•11 
.48 

S 

i7 

•43 
•37 
•47 
.46 
•34 
•34 
.42 

!66 

•4 

i 

Led  in 

Water. 

Apricot,  average  
Blackberry     ......... 

sour     

black  

Currant  red              ... 

Gooseberry  red  

yellow  .... 
Grape  white  

Peach,  Dutch  

Pear  red 

Plum,  yellow  gage.... 
large       "    
black  blue  
"     red  
Italian,  sweet  .  . 
Raspberry  wild 

Strawberry    "    

Banana    

Sugar  and  "Wat 

Su 
Sugar  crude                 c 

an  beef, 
ittermilk 

T<> 

Beer  .  .             91 

Buttermilk.  .  . 

:::::::::::  11 

ndTea  100 

2O2 


ANIMAL    FOOD 


Relative  "Val  vies  of  "Vegetable  K"oods  to  procure  an  Equal 
"Volume  of  Flesh,  in.  Beeves  or  Sneep. 

(Ewart.) 


ARTICLE. 

Beeves. 

Sheep. 

ARTICLE. 

Beeves. 

Sheep. 

ARTICLE. 

Beeves. 

Sheep. 

Peameal  
Bean  meal.  .  . 
Oatmeal  
Cornmeal.  .  . 
Barley  
Wheat  braD 

•°3 
.06 
.09 

1.87 

32C 

Meadow  hay. 
Oat  straw.  .  .  . 
Turnips  
Oats  
Bean  straw.  . 
Potatoes 

3.12 
3-98 
6.24 

8  7 

3-12 

12.48 
2.18 
6.24 
6  24 

Parsnips  
Beans  or  Peas 
Buckwheat.. 
Pea  straw.  .  . 
Cabbage  
Beets  .... 

18.72 

l8  72 

6.24 

'•7 
2.03 
6.24 
7-8 

Linseed  cake 

•50 

Me'dow  grass 

10.7 
12.5 

Carrots  

10.72 
19.67 

NOTE.— When  these  values  express  weight  in  Ibs.,  then  such  food  will  produce 
about  4  to  5  Ibs.  beef  or  mutton. 


Relative    Nutritive    "Value    of    1OO    parts    of   Human 
Food. 

Nutrient  Ratio — Is  determined  by  the  ratio  of  albuminoids  to  the  digestible 
carbo-hydrates  and  oil,  considered  as  starch.  Nutrient  Value — Is  the  percentage 
of  starch,  albuminoids,  oil,  and  sugar  converted  into  their  equivalents  of  starch. — 
(A.  H.  Church.) 


Ratio. 

Value 

Ratio 

Value. 

Ratio. 

Value. 

Almond,  Sweet 
\  pple               • 

5-3 

158 

II     "i 

Fig,  Dried  
Grape      

10 

20 

65 
16 

Milk,  Human.  . 
"     Skim.... 

9 
I  7S 

— 

Gooseberry 

Rice 

8? 

Barley  

13 

8s 

Ground-nut.  .  . 

5'  ^ 

Rye  Flour  

7 

85 

Beans         .  . 

2   ? 

fio 

Macaroni  I  fan 

88  •} 

Tomato  

e 

8  5 

Buckwheat 

86 

Maise    Corn 

8  < 

RT 

Turn  ip 

6 

Beet-  root  . 

Oatmeal  

r    8 

IO2 

Marrow  Veg'e 

£ 

Carrot  
Celery  
Cabbage 

14 

4-5 

7-5 
5 

Onion  
Parsnip  
Pea  

3-5 

12 

2  S 

65 

16 

7Q 

Moss,  Iceland. 
"     Irish... 
Walnut  

8 

" 

70 
64 

Cocoanut  
Cheese  Glos'r 

16 

90 

Pistachio-nut. 
Potato        .  . 

5-7 

M3 

Wheat,  Indian. 
"       Flour 

15 

7  T. 

84.6 
86  5 

Date  

Eee.  .  . 

10 

Q 

40 

"      Sweet.. 
Milk,  Cow.  .  . 

'3 

4 

22 

1  '      Bran  .  . 
Bread... 

4.8 

67 

S3 

The  Nutrient  ratio  generally  adopted  for  Standard  diet  is  i  to  4.75,  and  the 
proportion  of  fat  or  oil  to  starch  is  i  to  3. 5. 

The  Full  Daily  Diet  of  a  man  is  held  to  be  12  oz.  bread,  8  oz.  potatoes, 
6  oz.  meat,  4  oz.  boiled  rice  with  milk,  .375  pint  of  broth  or  pea  soup,  i  pint 
milk,  and  i  pint  of  beer. 

Nutritive   Values   and   Constituents   of  Miillr.— (Payen.) 


Nitrogenous 

Lactic 

Nitrogenous 

Lactic 

ANIMAL. 

Matter  and 
insoluble 

Butter. 

and 
soluble 

Water. 

ANIMAL. 

Matter  and 
insoluble 

Butter. 

and 
soluble 

Water. 

Salts. 

Salts. 

Salts. 

Salts. 

Goat  

4-5 

4.1 

~*8~~ 

85-6 

Ass  

1-7 

1.4 

6.4 

9°-5 

Cow  
Woman. 

4-55 
3-35 

3-7 
3-34 

5-35 
3-77 

86.4 
89-54 

Mare  .  .  . 
Ewe  

1.62 
4.68 

.2 

4.2 

8.75 
5-5 

89.43 
85.62 

\Veight   of  some    Different   Foods    required    to    furnish 
122O   GJ-rains   of  Nitrogenous    ^Matter. 


Pease 7 

Meat,  lean o 

Fish,  White...  i 


Lbs. 

Meat,  fat 1.3 

Oatmeal 1.5 

Corn  Meal 1.6 

Wheat  Flour..  1.7 


Bread 2.1 

Rye  Meal 2. 3 

Rice 2.8 


Turnips,  15.9  Ibs. ;  Beer  or  Porter,  158.6  Ibs. 


Barley  Meal. 

Milk 

Potatoes 

Parsnips 


Lb§. 
2.9 


8.3 
15-9 


ANIMAL  FOOD. 


203 


Proportion 

FRUIT. 

of  Sug 

Sugar. 

ar   and 

(Fresi 
Acid. 

Acid   in    "Vari< 

'm'Ms.  ) 
FKUIT. 

MIS   Fr 

Sugar. 

tiits. 

Acid. 

Apple 

Per  Cent. 
'       8.4 
1.8 

7.2 
14.9 
9.2 
1.6 

f  Oil   i 

Mustard  . 
Flax.  ... 

Per  Cent. 
.8 
i.i 
1.2 
2 

•7 

1-9 

•7 
n  Vari 

30 
34 

Plum  

Per  Cent. 
2.1 
6.3 
4 

5-7 
10.8 
5-8 

eeds.     < 
Orange.  .  . 

Per  Cent. 
•9 

.1 

13 

Berjot.) 

Apricot    .  . 

Prune  

Blackberry  .  .  . 
Currants 

Raspberry 

Red  Pear  

Gooseberry  .  .  . 

Sour  Cherry  

Grape 

Strawberry.  .  .  . 

Mulberry  

Sweet  Cherry  

Peach 

Whortleberry 

Proportion   c 
Beechnut  ....     ">* 

ou.s   Air-dry    S 
Almond  40 
Colza                 (4° 

Hemp  

?8 

(  40 

Watermelon  .  . 

•  36 

Peanut  38 

"  US 

Is® 

Analysis   of  different  Articles  of  Food,  with.    Reference 

only  to  tlieir  Properties  for  giving  Heat  and  Strength. 

(Payen.)    In  100  Parts. 


SUBSTANCES. 

C»r- 
b  n. 

Nitro- 
gen. 

SUBSTANCES. 

Car- 
bon. 

Nitro- 
gen. 

SUBSTANCES. 

Car- 
bon. 

Nitro- 
gen. 

Alcohol  
Barley 

52 
40 

I  Q 

Coflee  
Corn  

9 
44 

I.I 
1.7 

Oil,  Olive  
Oysters  

98     o 
7.18 

2  11 

Beans 

4e 

Eels    .   ... 

•JQ  O^ 

2 

Pease  

166 

Beef  meat 

•a 

Eggs  

13.  e 

I.Q 

Potatoes  

II 

Beer  strong 

oR 

Pitrg  dried 

Rice 

i  3 

Bread,  stale.  .  . 
Buckwheat 

28 
42.  5 

1.07 

2.2 

Herring,    salt- 
ed   

23 

3-  n 

Rye  Flour  
Salmon  

?6 

i-7S 

2  OQ 

Butter  

83 

.64 

Liver,  Calf's.. 

15.68 

3.93 

Sardines  

2O 

6^ 

5-5 

.31 

Lobster  

IO.OO 

2-93 

Tea  

2.  1 

.2 

Caviare    .... 

27.41 

4..  4.Q 

Mackerel  

IQ.  26 

3.  74 

Truffes  

Q.  Af 

I.  ^"J 

Cheese  Chest  Y 

Milk  Cow's 

I 

66 

Wheat 

Chocolate 

58 

!•  52 

Nuts  

10  65 

"     Flour 

38  c; 

?6, 

Codfigtusalt'd 

16 

t;.02 

Oatmeal.  .  . 

44 

l.QS 

Wine  .  .  , 

A. 

.01* 

NOTE.— Multiply  figures  representing  nitrogen  by  6.5,  and  equivalent  amount  of 
nitrogenous  matter  is  obtained. 

Hximaii   and   Animal    Sustenance. 

Least  Quantity  of  Food  required  to  Sustain  Life.     (E.  Smith,  M.D.) 

Carbon.  Hydrogen. 

Grs.  Grs. 

Adult  Man,       4300)  «  200)  M 

Adult  Woman,  3900}  Moan'  4IOC  180}  Moan'  I9°* 

An  adult  man,  for  his  daily  sustenance,  requires  about  1220  grs.  nitrog- 
enous matter  or  200  of  nitrogen,  and  bread  contains  8.1  per  cent,  of  it. 

Hence,  ^^°  =  15  062  grains  which  -f-  7000  in  alb.  =2  Ibs.  2.43  oz.  of  bread. 
.081 

These  quantities  and  proportions  are  also  contained  in  about  16  Ibs.  of 
turnips. 

Thus,  by  table  of  nutritive  values,  page  202,  turnips  have  263  grains  of  carbon  and 
13  of  nitrogen. 

Hence,  — ^—  and  —  =  16.35  Ibs.  for  the  necessary  carbon  and  15.4  Ibs.  for  the 
nitrogen. 

Relative   "Value   of  Foods   compared   -with.   1OO   l"bs.  of 
Q-ood    May. 


Lbe. 

Clover,  green . .  400 
Corn,  green  ...  275 
Wheat  straw  . .  374 


Lb8. 

Rye  straw 442 

Oat  straw 195 

Cornstalks ....  400 


Lb». 
Carrots  .......  276 

Barley  ........    54 


Oats 


57 


Corn 


59 


Linseed  cake  .  .    69 
Wheat  bran.  ...  105 


204 


ANIMAL   FOOD. 


"Weight  of*  Articles  of*  Food,  required,  to  "be  consumed  in 
the  human  system  to  develop  a  power  eq.ua!  to  rais- 
ing 14O  Ifos.  to  a  height  of*  1O  OOO  feet.  (Frankland.) 


SUBSTANCES. 

Weight. 

SUBSTANCES. 

Weight. 

SUBSTANCES. 

Weight. 

Cod-liver  oil  

Lbs. 
•553 

Rice  

Lbs. 
I-341 

Salt  Beef  

Lbs. 
3.65 

Beef  fat 

eee 

Isinglass       .   ... 

Veal  lean  .   .  . 

Bacon  

67" 

Sugar  lump  

i-  5°5 

Porter  

4  615 

Butter 

693 

Cream 

Potatoes 

5  068 

Cocoa  

.7Q7 

Egg  boiled  

2  2OQ 

Fish  

6  316 

Fat  of  Pork  

«O7 

Bread  ,  

2.  34^ 

7.8iq 

Cheese    ... 

ft 

Salt  Pork  

2  826 

Milk  

8  O2  1 

Oatmeal  

1   'Si 

Ham,  lean,  boiled.  . 

3.OOI 

Egg  white  of.  

8.745 

Arrowroot  
Wheat  flour.  .  . 

1.287 
I   ^11 

Mackerel  
Ale.  bottled... 

3.124 
3.461 

Carrots  

Cabbage  .  .  . 

9-685 
12.02 

Relative  Value  of  Varioiis  Foods  as  Productive  of  Force 
•when   Oxidized   in   the    Body. 


Cabbage      ....  i 

Porter  

^  6 

Egg  hard  boil'd  5  4 

Oatmeal  ....     93 

Carrots                x  2 

Veal  lean  .... 

»>  8 

Cream                 5  o 

Cheese             10  4 

Skimmed  Milk    i  2 

Salt  Beef  

•a.  q 

Egg  yolk  79 

Fat  of  Pork    12  4 

White  of  Egg..  1.4 

3.3 

Cocoa  16.3 

Milk                    i^ 

Lean  Beef  .... 

O    A 

Isinglass  .   .     .87 

Pemmican       16  o 

Apples  1.5 

Mackerel  

3« 

Rice  89 

Butter  173 

Ale  1.8 

Ham,  lean  

4 

Pea  Meal  9. 

Bacon  J7-94 

Fish  i.  Q 

Salt  Pork  

4.0 

Wheat  Flour   .  9  i 

Fat  of  Beef     21  6 

Potatoes  2.4 

Bread,  crumb.  . 

5-i 

Arrowroot  9.  3 

Cod-liver  Oil.  21.7 

Nutritious    Properties    of   different   "Vegetables   and    Oil- 
cake, compared   \vith    each    other   in    Quantities. 


Oil-cake i 

Pease  and  Beans  1.5 
Wheat,  flour. ..  2 

"      grain  ..2.5 
Oats 2.5 


Rye 2.5 

Bran,   wheat  j*'75 

Corn 3 

Barley 3 


Clover  hay 4 

Hay 5 

Potatoes 14 

old. ...  20 


Wheat  straw..  26 
Barley  "  26 
Oat  "  27.5 


Carrots 17. 5  |  Turnips 30 


ILLUSTRATION.— i  Ib.  of  oil-cake  is  equal  to  18  Ibs.  of  cabbage. 

Volume  of  Oxygen  required  to  Oxidize  100  parts  of  following  Foods  as  con- 
sumed in  the  Body. 

Grape  Sugar  . .  106  |  Starch 120  |  Albumen 150  |  Fat 293 

Hence,  assuming  capacity  for  oxidation  as  a  measure,  albumen  has  half  value  of 
fat  as  a  food-producing  element,  and  a  greater  value  than  either  starch  or  sugar. 

Proportion  of  A.lcohol  in  1OO  Parts  of  follo\ving  Liquors. 


Small  Beer. ..  i     and  i.  08 

Porter 3.5  and  5.26 

Cider 5.2  and  9.8 

Brown  Stout.  5.5  and  6.8 

Ale 6. 87  and  10 

Rhenish 7.58 

Moselle 8.7 

Johannisberger 8.71 

Elder  Wine 8.79 

Claret  ordinaire 8.99 

Tokay 9. 33 

Rudesheimer 10.72 

Marcobrunner n.6 

Gooseberry  Wine  ...  11.84 

Hockheimer. 12.03 

Via  de  Grave 12.08 


(Brande.  ) 
Herm  i  tage  red 

Lisbon  

1  8  QA. 

Champagne  

12.  6l 

Lachryma  

.  .  IQ.  7 

Amontillado  
Frontignac  

12.63 

12  89 

Teneriffe  
Currant  Wine  

••  *9-79 

Barsac 

TO  86 

Madeira 

OT 

Sauterne     

14  22 

Port           

.   .     23 

Champagne  Burg'dy 

Sherry  old 

23  86 

White  Port      

Marsala       

Bordeaux  

*5'  * 

Raisin  Wine  

Malmsey. 

16  4 

Madeira  Sercial  .  . 

Sherry  

17.  17 

Cape  Madeira  

.  .  20  m 

Malaga 

Gin         

^y-5A 
51  6 

Alba  Flora.  
Hermitage  white 

17.2 
17.26 

Brandy  
Rum     

••  53-39 
co  68 

Cape  Muscat  
Constantia,  red  

18.25 
18.92 

Irish  Whiskey  
Scotch  Whiskey  .  . 

••  53-9 
••  54-3« 

ANIMAL   FOOD. 


205 


Proportion    of    Food    Appropriated     and    Expended    "by 
following    Animals. 


. 

Proportion  appropriated  ..................      6.2 

"        in  manure  .....................     36.5 

"        respired  .......................    57.3 


Sheep. 

8 
31.9 

60.  i 


Swine. 

17.6 
16.9 
65.5 


Specific   Gravity  of  IVlillz  and  [Percentage  of  Cream,  etc. 


MILK. 

Specific 
Gravity. 

Volume 
of 
Cream. 

Volume 
Curd. 

Specific 
Gravity  when 
skimmed. 

Milk, 

pure* 

1030 
1027 
1024 
102  1 

12 

10.5 

I5 

6-3 
5-6 
4-9 

4.2 

1032 
1029 
1026 
1023 

10  per  ( 

20    " 

•3Q       " 

(C                 (f 

*  For  a  method  of  testing  the  purity  of  milk,  see  Pavy  on  Food  (Philadelphia,  1874),  page  196. 

NOTE.— The  average  proportion  of  cream  is  10,  or  ioper  cent. 


Proportion    Per    cent,  of  Starch,    in    sundry   Vegetables. 
Arrowroot 82     |  Wheat  flour. . .  66.3  I  Oatmeal 58.4  I  Potatoes 18.8 


Rice. 


.  79. i  I  Corn  meal ....  64.7  |  Pease. 55.4  j  Turnips 5.1 


Compos 

ition 

Fat. 

of  01 

Nitrogen  . 

iees< 
Salt. 

4-25 
7.09 
5-63 
6.21 

J     Of 

Water. 

Different 

Con 

Fat. 

n  tries 

Nitrogen. 

—  (Pc 
Salt. 

TTS" 

4-29 
5-93 
4-45 

tyen.) 

Water. 

3°-39 
32-05 
40.07 
26.53 

Neufchatel.  . 
Parmesan  .. 
Brie  
Holland  .... 

18.74 
21.68 
24.83 
25.06 

2.28 
5-48 
2-39 
4.1 

61.87 
30-3I 
53-99 
41.41 

Chester  
Gruyeres  .  .  . 
Marolles  .... 
Roquefort.  .  . 

25.41 
28.4 
28.73 
32.31 

5-56 
5-4 
3-73 
5-07 

Nutritive  Equivalents.    Compnted  from  Amount  of  Ni- 
trogen in  Sn"bstances  -when  Dried.     Unman  jVlillc  at  1. 


Rice  .   . 

81 

Bread  White 

i  42 

Cheese  

•a  QI 

Lamb  

8  M 

Potatoes  
Corn 

.84 

Milk,  Cows'  .  . 
Pease 

2-37 

Eel  
Mussel  .... 

4-34 

e  28 

Egg,  White:.. 
Lobster 

8-45 
8  so 

Rye  

1.06 

Lentils      .... 

076 

Liver  Ox  . 

5  7 

Veal  

873 

Wheat 

Egg  Yolk 

Pigeon 

7    f.6 

Beef 

!U 

Barley  

Oysters 

o  QC 

Mutton  

7  73 

Pork  

8  01 

Oats  .  .  . 

I.,* 

Beans.  .  . 

1.2 

Salmon  .  .  . 

7-73 
7.76 

Ham.  .  . 

O.I 

Herring,  9.14. 


Thermometric  3?ower  and  Mechanical  Energy  of  1O 
GJ-rains  of  "Various  Sntostances  in  their  Natural  Con- 
dition, when  Oxidized  in  the  A.nimal  Body  into  Car- 
bonic A.oid,  Water,  and  "Urea.— (Frankland.) 


SUBSTANCE. 

Water 

raised 

Lifted 
i  foot 
high. 

SUBSTANCE. 

Water 
raised 
i°. 

Lifted  1 
i  foot 
high. 

SUBSTANCE. 

Water 
raised 
I*. 

Lifted 
i  foot 
high. 

Ale,  Bass's  .  . 

Lbs. 
1.99 

i  j.8 

Lbs. 
1-54 

Cheese  
Cocoa-nibs 

Lbs. 
II.  2 

Lbs. 
8.65 
7  7 

Mackerel.  .  .  . 
Milk        .... 

Lbs. 
4.14 
i  64 

Lbi. 
3-2 

I.2S 

Arrowroot.  .  . 
Beef,  lean  .  .  . 
Bread  
Butter  
Cabbage  
Carrots  

10.06 
3.66 

1  8.  68 
1.08 
'•33 

7-77 
2.83 
4.26 
14.42 
•83 
1.03 

Cod-  liver  oil. 
Egg,  h'd  boil. 
"   yolk.... 
"  white... 
Flour,  wheat. 
Ham,  boiled  . 

II 

5.86 
8.5 

!.48 
9.87 

4-3 

18.12 

*fl 

£g 

3-32 

Oatmeal  
Pea  meal  — 
Potatoes  
Porter  
Rice,  ground. 
Sugar,  grape. 

IO.  I 

9-57 
2.56 
2-77 
9.52 
8.42 

7.8 

7-49 
1.99 
2.19 
7-45 
6.51 

206 


ANIMAL   FOOD. 


Digestion- 
Time  required,  for  Digestion  of  several  Articles  of  Food. 

(Beaumont,  M.D.) 


FOOD. 

Time. 

FOOD. 

Time. 

Apple,  sweet  and  mellow  .... 
sour  End  mellow 

A.   m. 
*    50 
2 
2    50 
2 
2    30 

3  45 
3 
3  30 
3 
2  45 
3  30 
5  30 
4 
4  15 
3  45 
3  15 
3  30 
3  30 
2  30 

2 

4 
4  30 
3  15 
4  15 
3  30 
2  45 
2  45 
4 
4  30 
3 

2 
I    30 

3  30 
3 
3  30 
3  30 

2 

4 
i  30 

4 
3 

2    70 

A.    m. 

4 

2    30 
2 
2   30 
2 
2    15 
3    15 

3 
2  55 
3  15 
3  30 
2  30 

2   30 

I 
5  15 
4  30 
4  15 
3  15 
3 
3  30 
3  20 

2   30 

I 
i  45 
3  20 
i  30 
4 
3 
3  30 
2  30 
5  30 
4  30 

2 

I 

2    18 

2  3° 

2    25 

3  30 

4 
4  50 
i  45 
i  .is 

Lamb  boiled  

sour  and  hard  

Liver,  Beefs,  boiled  
Meat  and  Vegetables,  hashed  . 

TV/Till?-    KrvJlrt  1  /-*»•  <Wv-V»                        f 

Barley  boiled  

Bean,  boiled  

Bean  and  Green  Corn,  boiled  . 
Beef  roasted  rare  

broiled  or  boiled  .... 
Oyster  

Steak,  broiled  

boiled  

roasted  

boiled,  with  mustard,  etc. 
Tendon,  boiled  

Parsnip,  boiled  

"        fried        .    .    . 

Pig  sucking  roasted  

old  salted  boiled  

Feet  soured  boiled  

Beet  boiled  

Pork,  fat  and  lean,  roasted  .  .  . 
recently  salted,  boiled  .  . 
"     *       "      fried... 
"      broiled  . 
"      raw.  .  .  . 
Potato,  boiled  

Wheat,  baked,  fresh  .  .  . 
Butter,  melted  

crude,  vm'r,  boiled  j 
Carrot  boiled  

baked  

roasted  

Sago,  boiled  

Cheese  old  and  strong 

Chicken  fricasseed.    .... 

Soup  Barley  

Custard  baked  

Beef  and  Vegetables  .  .  . 
Chicken  

Dumpling  Apple  boiled  . 

Mutton  or  Oyster  

Iv'ir.    , 

Suet,  Beef,  boiled  

oo*     V.        j  

whipped  

Mutton,  boiled  

boiled  hard 

Tapioca  boiled  

"        soft     

fried 

,  f  Wild    , 

Fish,  Cod  or  Flounder,  fried  .  . 
Cod  cured,  boiled  

Turkey,  roasted  {^8tv;; 
boiled  

Salmon,  salt'd  and  boil'd 
Trout,  boiled  or  fried.  .  . 
Fowl  boiled  or  roasted  

Veal  roasted  

fried  

Brain,  boiled  

Venison  Steak,  broiled  .  . 

Greneral   Notes. 

The  per-centage  of  loss  in  the  cooking  of  meats  is  as  follows:  Boiling  23;  Baking 
31 ;  Roasting  34. 

Potatoes  possess  anti-scorbutic  power  in  a  greater  degree  than  any  other  of  the 
succulent  vegetables. 

The  average  yearly  consumption  of  wheat  and  wheat  flour  in  Great  Britain  is  5.5 
bushels  per  capita  of  its  population. 

The  daily  ration  of  an  Esquimaux  is  20  Ibs.  of  flesh  and  blubber. — (Sir  John  Ross.) 


ANIMAL    FOOD. 


207 


An  adult  healthy  man,  according  to  Dr.  Edward  Smith,  requires  daily  of 

Phosphoric  acid  from  . .  32  to    79  grains.        Potash 27     to  107  grains. 

(Chlorine 5«  "  '75  Soda 80     "171       u 

( Or  of  common  salt. ...  85  "  291  Lime 2.3"      6.3  " 

and  of  Magnesia  2. 5  to  3  grains. 

A  common  fowl's  egg  contains  120  grains  of  Carbon  and  17.75  of  Nitrogen. 

An  ordinary  working-man  requires  for  his  daily  sustenance 

Oxygen 1.47      I    Starch 66 

Albuminous  matter 305        Salts 04 

Fat 22     |    Water , 4.535 

=  7.23  Ibs.  avoirdupois. 

Milk.—  If  the  milk  of  an  animal  is  taken  at  three  immediately  successive  periods, 
that  which  is  first  received  will  not  be  as  rich  in  milk-fat  as  the  last. 

In  a  Devon  cow,  milked  in  this  manner,  the  first  milk  gave  but  1.166  per  cent,  of 
fat,  and  the  last,  or  that  known  as  "  strippings, "  5.81  per  cent. 

Relative   Richness   of  M-ills   of  Several   ^niinals. 
Human  Milk  =  i. 


Milk-fat. 

Cow 1.66 

Mare 1.19 

Goat 2 


Casein. 
1.38 
•75 
1.04 


Sugar. 
.69 
.94 
.69 


Sheep., 
Camel. . 


Milk-fat. 
...     -5 
...  2.52 
...1.4 


Casein. 
.38 
2.1 


Sugar. 
-94 
.72 
.96 


The  condensation  of  milk  reduces  it  to  about  one  third  of  its  original  volume. 

A  Farm  of  second  rate  quality,  properly  cultivated,  will  sustain  100  head  of  cattle 
per  ioo  acres,  besides  laboring  stock  (employed  in  cultivation  of  farm),  and  swine. 
— (Ewart.) 

Thus,  calves  25 ;  do.  i  year  25 ;  do.  2  years  25 ;  cows  25. 

Cane  Sugar  (Saccharose)— Is  insoluble  in  absolute  alcohol,  and  in  diluted  alcohol 
it  is  soluble  only  in  proportion  to  its  weakness.  Loaf  sugar,  as  a  rule,  is  chemically 
pure. 

Beet  Root  Sugar— Contains  85  to  96  per  cent,  of  cane  sugar,  1.6  to  5.  i  of  organic 
matter,  and  2  to  4. 3  of  water. 

Honey— Contains  32  per  cent,  of  sugar  (levulose),  25.5  of  water,  27.9  of  dextrine, 
and  14.6  of  other  matter,  as  mannite,  wax,  pollen,  and  insoluble  matter. 

Molasses— Contains  47  per  cent,  of  cane  sugar,  20. 4  of  fruit  sugar,  2.6  of  salts,  2.7 
extractive  and  coloring  matter,  and  27.3  of  water. 

Flour.— Tests  of  flour,  see  A.  W.  Blyth,  London,  1882,  page  152. 

Bread.—  Wheat,  water  lost  by  drying  after  i  day  7.71  per  cent.,  3  days  8.86  and 
7  days  14.05  per  cent. 

Sago.—  2.5  Ibs.  per  day  will  support  a  healthy  man. 

Fig— Contains  nearly  as  much  gluten  as  wheat  bread  (as  6  to  7),  and  in  starch  and 
sugar  it  is  16  per  cent,  richer. 

Gooseberry  (dry)— Is  as  nutritious  as  wheat  bread. 

Watermelon,  Vegetable  marrow,  and  Cucumber— Contain  94,  95,  and  97  per  cent 
of  water  respectively. 

Onion  (dry)— Contains  25  to  30  per  cent,  of  gluten.     Potato  containing  but  5. 
otatcf r/e'oorMZ^°Wer>  Broccoli> and  Leaves  are  generally  rich  in  gluten,  while  the 

Ratio   of  Flesli-forxxiers   of  Tubers. 
Per  Cent. 


TUBERS. 

Flesh- 
formers. 

Starch, 
etc. 

Ratio  to 
Heat-giv'rs. 

TUBERS. 

Flesh- 
formers. 

Starch, 
etc. 

Ratio  to 
Heat-gw'rt. 

Beet  root  . 
Turnip  

•4 

.  5 

13-4 

1:30 
i'8 

Parsnip  

1.2 

ii 

i:  10 

Carrot  

.  c 

* 

Sweet  Potato 

4.8 

i:   3-5 

Potato  

I.  2 

g 

i*  16 

Yam 

z-5 

-f.  _ 

»M3 

2.2 

10.3 

i:  7.$ 

2O8      GRAVITY  OF  BODIES.— GRAVITY  AND  WEIGHT. 

GRAVITY   OF  BODIES. 

GRAVITY  acts  equally  on  all  bodies  at  equal  distances  from  Earth's 
centre ;  its  force  diminishes  as  distance  increases,  and  increases  as  dis- 
tance diminishes. 

Gravitating  forces  of  bodies  are  to  each  other, 

1.  Directly  as  their  masses. 

2.  Inversely  as  squares  of  their  distances. 

Gravity  of  a  body,  or  its  weight  above  Earth's  surface,  decreases  as 
square  of  its  distance  from  Earth's  centre  in  semi-diameters  of  Earth. 

ILLUSTRATION  i.— If  a  body  weighs  goo  Ibs.  at  surface  of  the  Earth,  what  will  ii 
weigh  2000  miles  above  surface  ?— Earth's  semi-diameter  is  3963  miles  (say  4000). 

Then  2000  +  4000 = 6000  =  i.  5  semi-diam's,  and  900  -r-  x.  s2  =  ^—  =  4°°  los- 

Inversely,  If  a  body  weighs  400  Ibs.  at  2000  miles  above  Earth's  surface,  what  will 
it  weigh  at  surface  ? 

400  X  i.52  =  9ooZ6s. 

2.  —  A  body  at  Earth's  surface  weighs  360  Ibs. ;  how  high  must  it  be  elevated  to 
weigh  40  Ibs.? 

^  =  9  semi-diameters,  if  gravity  acted  directly;  but  as  it  is  inversely  as  square 

4° 
of  the  distance,  then  -^/g  =  3  semi-diameters  =  3  X  4000=  12000  miles. 

3.— To  what  height  must  a  body  be  raised  to  lose  half  its  weight? 
As  ^/i  :  T/2  : :  4000  :  5656  =  as  square  root  of  one  semi-diameter  is  to  square  root 
of  two  semi-diameters,  so  is  one  semi-diameter  to  distance  required. 

Hence  5656  —  4000  =  1656  =  distance  from  Earth's  surface, 

Diameters  of  two  Globes  being  equal,  and  their  densities  different,  weight 
of  a  body  on  their  surfaces  will  be  as  their  densities. 

Their  densities  being  equal  and  their  diameters  different,  weight  of  them 
will  be  as  their  diameters. 

Diameters  and  densities  being  different,  iveight  will  be  as  their  product. 

ILLUSTRATION.— If  a  body  weighs  10  Ibs.  at  surface  of  Earth,  what  will  it  weigh  at 
surface  of  Sun,  densities  being  392  and  100,  and  diameters  8000  and  883000  miles? 

883  ooo  x  ioo  -r-  8000  x  392  =  28. 157  =  quotient  of  product  of  diameter  of  Sun  and 
its  density,  and  product  of  diameter  of  Earth  and  its  density. 

Then  28.157  X  10  =  281.57  Ibs. 
NOTE.— Gravity  of  a  body  is  .00346  less  at  Equator  than  at  Poles. 


SPECIFIC  GRAVITY  AND   WEIGHT. 

Specific  Gravity  or  Weight  of  a  body  is  the  proportion  it  bears  to  the 
weight  of  another  body  of  known  density  or  of  equal  volume,  and  which  is 
adopted  as  a  standard. 

If  a  body  float  on  a  fluid,  the  part  immersed  is  to  whole  body  as  specific 
gravity  of  body  is  to  specific  gravity  of  fluid. 

When  a  body  is  immersed  in  a  fluid,  it  loses  such  a  portion  of  its  own 
weight  as  is  equal  to  that  of  the  fluid  it  displaces. 

An  immersed  body,  ascending  or  descending  in  a  fluid,  has  a  force  equal 
to  difference  between  its  own  weight  and  weight  of  its  bulk  of  the  fluid,  less 
resistance  of  the  fluid  to  its  passage. 

Water  is  well  adapted  for  standard  of  gravity ;  and  as  a  cube  foot  of  it 
at  62°  F.  weighs  997.68  ounces  avoirdupois,  its  weight  is  taken  as  the  unit, 
or  approximately  1000. 


SPECIFIC    GRAVITY   AND   WEIGHT.  2OQ 

French  standard  temperature  for  comparison  of  density  of  solid  bodies 
and  determination  of  their  specific  gravities,  is  that  of  maximum  density  of 
water,  at  4°  C.  or  39.1°  F.,  and  for  gases  and  vapors  under  one  atmosphere  or 
.76  centimeters  of  mercury  is  32°  F.  or  o°  C.,  and  specific  gravity  of  a  body 
is  expressed  by  weight  in  kilogrammes  of  a  cube  decimeter  of  that  body. 

Densities  of  metals  vary  greatly. 

Potassium,  Sodium,  Barium,  and  Lithium  are  lighter  than  water.  Mercury 
is  heaviest  liquid,  and  Iridium  heaviest  metal.  Volcanic  scoriae  are  lighter 
than  water. 

Pomegranate  and  Lignum-vitas  are  heaviest  of  woods.  Pearl  is  heaviest 
of  animal  substances,  and  Flax  and  Cotton  are  heaviest  of  vegetable  sub- 
stances, former  weighing  nearly  twice  as  much  as  water. 

Zircon  is  heaviest  of  precious  stones,  being  4.5  times  heavier  than  water. 
Garnet  is  4  times  heavier,  Diamond  3.5  times,  and  Jet,  lightest  of  all,  is  but 
.3  heavier  than  water. 

To  Ascertain.  Specific  Grravity  of  a  Solid.  Body  heavier 

than.    "Water. 

RULE. — Weigh  it  both  in  and  out  of  water,  and  note  difference  ;  then,  as 
weight  lost  in  water  is  to  whole  weight,  so  is  1000  to  specific  gravity  of  body. 

Or,         _ =  G,  W  and  w  representing  weights  out  and  in  water,  and  G 

specific  gravity. 

EXAMPLE.  —  What  is  specific  gravity  of  a  stone  which  weighs  in  air  15  Ibs.,  in 
water  10  !bs.? 

15  — 10  =  5;  then  5  :  15  ::  1000  :  3000  Spec.  Grav. 

To   Ascertain    Specific    Grravity   of  a   Body  lighter   than 

\Vater. 

RULE. — Annex  to  lighter  body  one  that  is  heavier  than  water,  or  fluid 
used ;  weigh  piece  added  and  compound  mass  separately,  both  in  and  out  of 
water,  or  fluid ;  ascertain  how  much  each  loses,  by  subtracting  its  weight 
from  its  weight  in  air,  and  subtract  less  of  these  differences  from  greater. 

Then,  as  last  remainder  is  to  weight  of  light  body  in  air,  so  is  1000  to 
specific  gravity  of  body. 

EXAMPLE. — What  is  specific  gravity  of  a  piece  of  wood  that  weighs  20  Ibs.  in  air; 
annexed  to  it  is  a  piece  of  metal  that  weighs  24  Ibs.  in  air  and  21  Ibs.  in  water,  and 
the  two  pieces  in  water  weigh  8  Ibs.? 

20  -f-  24  —  8  =  44  —  8  =  36  =  loss  of  compound  mass  in  water; 
24  —  21  =  3  =  loss  of  heavy  body  in  water. 

33  :  20  ::  1000  :  606.06  Spec.  Grav. 

To   Ascertain   Specific    Grravity   of  a   Fluid. 
RULE. — Take  a  body  of  known  specific  gravity,  weigh  it  in  and  out  of 
the  fluid ;  then,  as  weight  of  body  is  to  loss  of  weight,  so  is  specific  gravity 
of  body  to  that  of  fluid. 

EXAMPLE.  — What  is  specific  gravity  of  a  fluid  in  which  a  piece  of  copper  (spec, 
grav.  =  9000)  weighs  70  Ibs.  in,  and  80  Ibs.  out  of  it  ? 

80  :  80  —  70  =  10  : :  9000  :  1125  Spec.  Grav. 

To   Ascertain   Specific    Grravity  of  a    Solid    Body  -which 

is    solnble    in    "Water. 

RULE. — Weigh  it  in  a  liquid  in  which  it  is  not  soluble,  divide  its  weight 
out  of  the  liquid  by  loss  of  its  weight  in  the  liquid,  and  multiply  quotient 
by  specific  gravity  of  liquid ;  the  product  is  specific  gravity. 

EXAMPLE.— What  is  specific  gravity  of  a  piece  of  clay,  which  weighs  15  Ibs.  in  air 
and  5  Ibs.  in  a  liquid  of  a  specific  gravity  of  1500,  in  which  it  is  insoluble  ? 
15  -f- 10  X  1500=  2250  Spec.  Grav 

s* 


2IO 


SPECIFIC    GRAVITY    AND    WEIGHT. 


SOLIDS. 


SCBSTANCIB. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Inch. 

SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cub* 
Inch. 

IMetals. 

2560 
2670 
7700 
6712 
5763 
470 
9823 
2000 

8450 
8300 
8200 
8380 
8lOO 
8214 

Lb. 

.0926 
.0906 
2785 
.2428 
.2084 
.017 

•3553 
.0723 

•3056 
•2997 
.2966 
.3026 
.2930 

.  2Q  7  2 

Metals. 
Mercury     60°  

13569 
13370 
8600 
8800 
8279 

10  000 

11350 
20337 

16000 
22069 

865 

8940 

10650 
I  520 

8600 
4500 
10474 
10511 
970 
7700 
7900 
7806 
7833 

7818 

7847 
7823 
7842 
7848 
7852 
7834 
2540 

6  no 
11850 

739° 
7291 
53oo 
17000 
18330 
7119 
6861 
7191 

800 
793 
845 
690 
400 
822 
852 
690 

567 
720 
898 
1031 
1328 
912 
928 
176 

Lb. 
.4908 
.4836 

•3183 
.2994 

-3613 
.4105 
•7356 
•5787 
.7982 

•0313 
•324 
•3852 
•055 
.3111 
.1627 
.3788 
.3802 
•0351 
.2785 
.2857 
.2823 
.2833 

.2828 
•  2838 
•283 
.2836 
.2839 
.284 
.2916 
.0918 

.221 

.4286 
.2673 
.2637 
.1917 

•*I49 
.6629 

•2575 
.2482 
.26 

Cube 
Foot. 

5° 
49.562 
52.812 
43-125 
25 
51-375 
53-25 
43-I25 
35-437 

56.125 
64-437 
83 

H 

2*.* 

"         wrought.... 
"         Bronze  
Antimony  
Arsenic                      •  •  • 

"              212°                  ... 

Nickel  
"     cast  

Barium        .  .     ........ 

Bismuth 

Palladium    

Platinum,  hammered  .  . 
"        native 

Brass. 
Sheet,  cop.  75,  zinc  25. 
Yellow  "    66,    "    34. 
Muntz  "    60,    "    40. 
plate       

"        rolled.  
Potassium  59° 

Red  lead  

Cast 

Rubidium    

Wire                      .   . 

Ruthenium  

Bromine 

3000 
8750 
8217 
8832 
8700 
8379 
8060 

739° 
8650 
1580 
5900 
8098 
8600 
6000 
8608 
8698 
8880 
8880 
19258 
19361 
17486 

15709 
18680 
23000 
7308 
6900 
75oo 
7207 
7217 
7065 
7218 
7788 

7774 
7704 
7698 
7540 
7808 
8140 
7744 
H352 
11388 
590 
1750 
8000 
15632 
n<;a8 

.1085 
•3l65 
.2972 

•3194 
.2929 
.3021 
.291 
.2668 
3129 

057 
.2134 
.2929 

•SI" 
.217 

•3«3 
.3146 
.3212 
.3212 
.6965 
.7003 
•6325 
.5682 
.6756 
.8319 
.264 
.2491 
.2707 
.2607 
.2609 

•2555 
.2611 
.2817 
2811 
.2787 
.2779 
.2722 
2819 
.2938 
.2801 
.4106 
.4119 
.0213 
•  0633 
.2894 
.5661 
.4018 

Selenium 

Bronze  gun  metal 

Silver  pure  cast       .   .  . 

'  ordinary  mean  . 
cop.  84,  tin  16  .  . 
"    81,  "    19  .. 
Tobin  

"        "     hammered. 
Sodium 

Steel  minimum  

maximum 

•je  tin  6? 

plates,  mean  
soft 

"        21  tin  74. 

Cadmium          

temper'd  and  hard- 
ened    

Calcium  

Chromium                    . 

wire       .  . 

Cinnabar   

blistered  

Cobalt 

crucible        . 

Columbium  

cast  

Copper  cast 

Bessemer 

u       plates     

ordinary  mean  

"       wire  and  bolts.. 
"       ordinary  mean. 
Gold  pure  cast  

Tellurium  

Thalium  

u     hammered 

Tin,  Cornish,  hammered. 
"         "       pure  

"     22  carats  fine  

U       20        "            4i     

Iridium       

Titanium 

Tungsten  

"      hammered  
Iron,  Cast,  gun  metal.  .  . 
"     minimum  

Wolfram  

"    rolled  

u     ordinary  mean  .... 
"     mean  Eng  . 

Woods  (Dry). 
Alder      

"     cast  hot  blast  
"       "    cold    "   
**     Wrought,  bars  
"                wire  

Ash                               1 

"               rolled  plates 
"               average  
Eng.  rails  .  . 
Lowmoor.  .  . 

Bamboo 

Bay  tree  

Beech  j 

"     ordinary  mean  — 

I 
Birch  | 

•'  '  rolled 

Blackwood,  India  
Boxwood  Brazil  

Lithium  

Magnesium 

"         France 

Manganese  

"         Holland  
Bullet-wood  

Butternut,  .  . 

Mercury  —40°  

U            +12°... 

SPECIFIC    GRAVITY   AND   WEIGHT. 


211 


SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

SUBSTANCES. 

Specific 
Gravity 

Weight 
of  a  Cube 
Foot. 

Woods   (Dry). 
Campeachy  

9J3 
56l 
1315 
441 
38o 
1573 
280 
1380 
7'5 

OIO 

726 
1040 
240 
644 
756 
i33i 
1209 

695 
570 
671 
800 
1014 
600 
512 
582 
970 

1055 
843 

IOOO 

592 
910 
860 
368 

792 

990 
770 

566 

1171 
720 

544 
560 

703 
650 

i|33 

£4 
604 

728 
913 

11° 
1063 

560 
852 
750 
576 
849 
56i 
897 
823 
872 
759 
*U. 

Lbs. 

57-062 
35-062 
82.157 
27.562 
23-75 
98-312 

S25 

44-687 
38.125 

45-375 
65 
'5 
40.25 

47-25 
83-187 
75562 
43-437 
35625 

4i  937 

63-375 
37-5 

32 

36-375 
60.625 
65-95 
52.687 
62.5 

37 
56.875 

53-75 
23 
49-5 
43-  125 
47-5 
61.875 
48.125 
35-375 
73-187 
45 
34 
35 
43-937 
40.625 
83-312 
50-25 
37-75 
45-5^ 
57062 

&«7 

35 
53-25 
46.875 

S6 
53.062 

56.062 
Si-437 
54-5 
47-437 
5.  Ordnanc 

^Voods   (Dry). 
Oak,  English  J 

858 
932 
1146 
1170 
1260 
1068 
860 
680 
7°5 
66x 
710 
785 
660 

59<> 
554 
461 
740 
1354 
58o 
383 
529 
705 
728 
482 
885 

g 

8 

500 

486 
585 

788 

807 

722 
624 
606 

441 

838 
720 

473 

54>' 
587 
6B7 
759 

2730 
2699 
1714 
1078 
866 

3073 
2250 
4000 
4865 
2305 

Lbs. 

53-625 
58-25 
71-625 
73-125 

78-75 
66.75 

53-75 
42'5 
44.062 
41.312 
44-375 
49.062 
41-25 
36-875 
34-625 
28.812 
46-25 
84.625 
3625 
23-937 
33.062 
44.062 
45-5 
30-125 
55-3" 
3i  25 

38.937 
23-937 
41  062 
61.25 
41937 
31-25 
30-375 
36.562 
4925 
50-437 

45-125 
39 
37-875 
27.562 
52-375 

29-562 
33-812 
36-687 
42.937 
42.437 

170.625 
168.687 
107.125 
67-375 

192.062 
140.625 
250      ' 
304.062 
144.06 

"    Indian      ........ 

"    heart,  60  years.... 
u    live  green  

"        fresh  burned.. 

"      "    seasoned  
"    white  

**        soft  wood  .... 
"        triturated  

Olive  

Chestnut,  sweet  
Citron  

Pear  

Cocoa 

Plum           ...        .  . 

Cork  

Pine  pitch  

Cypress,  Spanish  

"     red  

Dog-  wood 

"     white  

Ebony,  American  

"    yellow.  
"    Norway  

Elder  

Pomegranate  

Elm  { 

Poon  

'  '  rock  

Poplar  
"     white  

Quince  

Filbert        

Rosewood  

Fir,  Norway  Spruce.  .  .  . 

Satinwood  

Fustic  

Spruce       

Greenheart  or  Sipiri.  .  .  . 
Gum,  blue  .  .  ,  

'  '     water  

Teak  (African  oak)..  .  .  { 
Walnut 

Hackmatack  

Hazel  

"     black  

Willow 

Hickory,  pig-nut.  
shell-bark  
Holly  

Yew  Dutch  

Iron-wood.  

(Well  Seasoned.*) 
Ash            

Juniper                   .. 

Khair,  India  

Beech  

Larch                           I 

Cypress  

I 
Lemon  

Hickory,  red  
Mahogany,  St.  Domingo. 
Pine,  white  

Lignum-vitse       ..  ..  | 

Lime  

"     yellow  
Poplar 

White  Oak,  upland  
"        u    James  River 

Stones,  Earths, 
etc. 

Locust  

"         Honduras.  .  .  . 
"         Spanish  
Maple 

*'         yellow      . 

"     bird's-eye 

Mastic 

Mulberry  { 

Asbestos,  starry  

Oak  African 

Bary  tes,  sulphate  ....  J 
Beton,  N.  Y.  StCon'g  Co. 

e  Manual,  1841. 

'  '     Canadian  ......... 

u    Dantzic  

212 


SPECIFIC    GKAVITY    AND   WEIGHT. 


SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cub« 
Foot. 

Stones,  Earths, 
etc. 

Basalt          | 

2740 

Lbs. 
171.25 

Stones,  Earths, 
etc. 

Glass,  green  

2642 

Lbs. 
165.125 

Bitumen  red  

2864 

179 

72.  3 

"     optical  

3450 
2802 

215.625 
180.75 

"        brown 

830 

u     window 

2642 

165.  125 

Borax  

107   125 

u     soluble  

1250 

78.125 

Brick                            1 

1714 
1367 

85-437 

Gniess,  common  

2700 

167.4 

'  '    pressed  

1900 
2400 

118.75 
ISO 

Granite,  Egyptian  red.  . 
"        Patapsco  

2654 
2640 

165.875 
165 

"    fire 

"       Quincy    .   .   .. 

2652 

165.7^ 

*'    work  in  cement 

1800 

II2.5 

"       Scotch  

2625 

164.062 

it        u     41  mortar.  4 
Carbon       .               .   . 

1600 

2000 

IOO 

125 

218  ?«; 

"       Susquehanna  .  . 
gray 
Graphite 

2704 
2800 

2  2OO 

169 

JOT    C 

Cement  Portland  

I3OO 

81.25 

Gravel,  common  

J749 

109.  312 

"       Roman 

Grindstone 

J-3-J    Q07 

Chalk 

1520 

95 

Gypsum,  opaque  

2168 

135-5 

Clay     

2784 

Hone,  white,  razor  
Hornblende        

2876 
3^4O 

179-75 

221.25 

u  with  gravel  

2480 

155 

Iodine  

4Q4O 

1350 

84.375 

Lava,  Vesuvius  | 

1710 
0oTr. 

106.875 

1436 

89-75 

...                                 I 
Lias 

175.625 
146  875 

"    Borneo  

* 

80  625 

Lime  quick  

804 

50.  25 

"    Cannel                 { 

1238 

77-375 

"     hydraulic  

2745 

171.562 

u    Caking  

I3l8 

82.375 

7Q  8l2 

Limestone,  white  
'  '          green  

3156 
3180 

197-25 

198.75 

"    Cherry 

1277 

Magnesia  carbonate 

150 

«    Chili    

' 

80  625 

Magnetic  ore  

^OQ4. 

317.6 

"    Derbyshire 

I2Q 

80  7S 

Marble  Adelaide 

169  687 

"    Lancaster  

* 

'       African  

2708 

169  25 

'355 

84.687 

'       Biscayan,  black. 

260^ 

168.437 

"    Newcastle 

'       Carrara.   .  . 

2716 

169  75 

"    Riv&deGier  

I7OO 

8l.25 

'       common  

2686 

167.871; 

"    Scotch  | 

78.687 

'       Egyptian  

2668 
f>f\tc\ 

,i     '-> 
166.75 

»    Splint  

1302 

81   O7e 

'       Italian,  white.  .  . 

2049 
2708 

169.  25 

<l    Wales  mean  

jojr 

82  187 

'       Parian  .... 

2838 

*77-  375 

Coke  

IOOO 

62.5 

u       Vermont,  white. 

2650 

16^.  s? 

u    Nat'l  Va  _ 

746 

46  64. 

"       Silesian... 

27OO 

170  625 

Concrete,  in  cement.  .  .  . 

22OO 

*3'7-  5 

I75° 

109.  375 

"       mean   .   ... 

'  '     tough  

2Q4.O 

146  25 

Earth,*  common  soil,dry 
u       loose  

1216 

76 

Masonry,  rubble  
u        Granite 

2050 

2640 

128.125 

"       moist  sand.  .... 
"       mold,  fresh  
*       rammed 

2050 
2050 

128.125 
128.125 

"        Limestone  
"       Sandstone  
"        Brick 

2640 
2160 

165 
135 

'       rough  sand  
'       with  gravel 

1920 

120 

"          "rough  work 
Mica  .... 

1600 
2800 

IOO 

'       Potters'  

IQOO 

118  7=; 

Millstone  

2484 

4       light  vegetable.  . 

1400 

87.5 

4°4 
1  200 

78  75 

Emery  \  

4OOO 

250 

Mortar  j 

1384 

86.5 

Flint  black  

2e82 

161  vjCi 

Mud  

1750 

109.375 

2CQX 

162.  125 

"    wet  and  fluid  

1782 

112 

Fluorine  

82  5 

"     "      "    pressed 

Fuel,  Warlich's  

II5O 

71.875 

Nitre  

IQOO 

118  75 

"     Lignite  

81.25 

Oyster-shell    .... 

Glass,  bottle  

2732 

170.75 

2416 

u     Crown  .  . 

2487 

Peat  Irish  light 

2?8 

TT 

«     flint  | 

2933 

183.312 

"       "      dense.,  

270 
562 

35-125 

I 

196 

675 

43.187 

*  Specific  gravity  of  earth  is  estimated  at  from  1520  to  2200. 


SPECIFIC   GRAVITY   AND   WEIGHT. 


213 


SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cuba 
Foot. 

Stones,  Earths, 
etc. 

Peat  black  { 

1058 

Lbs. 
66.125 

Q-ranite. 
(Gtetfl  Gillmore,  U.  S.  A.) 
Duluth,  Minn.,  dark.... 

2780 

Lbi. 
173-7 

1329 
I77O 

83.062 
110.625 

Garrison's,  N.  Y.      "   .. 

2635 
2580 

164.7 
161.2 

Plaster  of  Paris        *.  \ 

1176 

73-5 

Jersey  City,  N.  J.,  soap.. 

3?30 

l89-3 

1  1        (t       "    drv 

3400 

212.5 

87  t 

Keene,  N.  H.,  bluish  gray 
Maine       

2656 
263^ 

166 
164.7 

2IOO 

131.  25 

Millstone  Pt.  ,  Conn  

^"35 
2700 

169  i 

2300 

143.75 

New  London,      "    

2660 

166.25 

Porphyry,  red  
Pumice-stone 

2765 

172.812 

C7   187 

Quincy,  Mass.,  light  
Richmond  Va  

2695 
2727 

168.5 
170.5 

2660 

l66.25 

"          "  gray  

£. 

2630 

164.4 

Red  lead        

SQJ.O 

558  75 

Staten  Island,  N  Y  

286l 

178.8 

1080 

68.062 

Westchester  Co.,  N.  Y.  . 

2655 

165.0 

Rock  crystal  

27-3C 

170.937 

Westerly,  R.  I.,  gray.  .  .  . 

2670 

x66.9 

Rotten-stone  

1081 

123  812 

2  no 

133.  125 

T  A  TYI  *»«  IOTI  ft  . 

'  '    rock           ........ 

T  -37   e 

/  rtan  '  /  /^•nimfVff    7T    V     A    \ 

2OQO 

IJ7-5 
130  625 

Sand  coarse       •       •     • 

1800 

Bardstown,  Ky.,  dark  .  . 

2670 

166.9 

-f.-n 

Caen,  France  

1900 

1  1  8.  8 

i°4-375 

Canajoharie  N.  Y  

2685 

167.8 

'     damp  and  loose.  .  . 
'     dried    "      "    ... 
'     dry        

1392 

1560 

87 
97-5 
88  7$ 

,  Cooper  Co.  ,  Mo.  ,  d'  k  drab 
'Erie  Co.,  N.Y,  blue.... 

2320 
2640 

141-3 
165 

1420 

I.*. 

Garrison's  N  Y.  

2635 

164..  7 

1659 

103.66 

Glens'  Falls   "  

2700 

168.? 

'     silicious    

1716 

107.25 
106  31 

Joliet,  111.,  white  

2540 

158-7 

22OO 

I  -37  e 

Kingston,  N.  Y  

2690 

168.1 

"         Sydney  
Schorl      

2237 

I39-8I 

108  12^ 

1  Lake  Champlain,  N.Y.  . 
:  Lime  Island,  Mich.,  drab 

2750 
2500 

171.9 
156-3 

8^0 

51  875 

Marblehead,  Ohio,  white 

2400 

150 

Sewer  pipe  mean 

Marquette,  Mich.,  drab  . 

2340 

146.25 

Shale  .     ..'  

600 

f.     - 

1  Sturgeon  Bay,  Wis.,  blu- 

ish drab      .......... 

2780 

173.7 

Slate  { 

2672 

167 

1  '   purple  .  .  . 

2900 
2784. 

181.25 

Marble. 

Smalt  

2.1,10 

jC2  C 

(6?en'Z  Gillmore,  U.  S.  A.  ) 

Soapstone  

2730 

*5-«-5 
170.625 

Dorset  Vt. 

•i 

,<.  - 

Spar  calcareous  

"     Feld,  blue  

2693 

168.312 

2875 

179-7 
1  168  i 

"        "     green 

160 

1W  ii  tiu,  com     ou  .  ...... 

2090 

"     Fluor  

2570 

.  I7I-9 

52C  i 

528  187 

X75 

Stalactite 

Stone,  Bath  Engl  

"      Blue  Hill 

26 

ige 

(Gen'l  Gittmore,  U.  S.  A. 

"      Bluestone  (basalt) 
14      Breakneck,  N.Y.. 
"      Bristol,  Engl  
"      Caen,  Normandy. 
11      common  

2625 
2704 
2510 
2076 
2C2O 

164.062 
169 
156.875 

129.75 
I  C7   e 

Albion,  N.Y.,  brown  
Belleville,  N.  J.,  gray  .  .  . 
Berea,  Ohio,  drab  
Cleveland,  "  olive  green 
Edinb'h,Sc'tl,  Craigleith 

2420 
2259 

2110 
2240 
226o 

151-25 
141.2 

131-9 
140 
141.25 

"      Craigleith,  Scotl.  . 
"      Kentish  rag.  u.  .  . 
"      Kip's  Bay,  N.Y.  . 
"      Norfolk     (Parlia- 
ment House).  .  . 
<{     Portland,  Engl.  .  . 
u      StatenIsl'd,N.Y. 
"      Sullivan  Co.,  " 
Sulphur,  native  
Terra  Cotta  

^5^0 
2316 
2651 
2759 

2304 
2368 
2976 
2688 
2033 
1  0^2 

165'.  687 
172 

I44 
I48 

1  86 
168 
127.062 

122 

Fond  du  Lac,  Wis.,  purple 
Fontenac,Minn.,l'g'tbuff 
Haverstraw,  N.  Y.,  red.  . 
Kasota,  Minn.,  pink  
Little  Falls,  N.  Y.,  brown 
Marquette,  Mich.,  purple 
Masillon,  O.,  yellow  drab 
Medina,  N.Y.,  pink  
,  Middletown,  Ct.,  brown. 
Seneca,  Ohio,  red    " 

222O 
2325 
2130 
2630 
2250 
2285 
2110 
2410 
2360 
23QO 

138-7 
145  3i 
133-  1 
I64-375 
140.6 

142-5 
131.87 
150.6 
147-5 
149.4 

Tile  
Trap.  .  . 

1815 
2720 

"3-437 

170 

Vermillion,  Ohio,  drab.  . 
i  Warrensburgh,  Mo  

2IOO 
2140 

135 
J  33-  75 

214 


SPECIFIC    GEAVITY    AND    WEIGHT. 


Spec.  Grav. 

Agate 2590 

Amethyst 3920 

Carnelian 2613 

Chrysolite 2782 

Diamond,  Oriental. . .  3521 

Brazilian..  3444 


IPrecions    Stones. 

Spec.  Grav. 

Emerald,  aqua    ma- 
rine   2730 

Garnet 4189 

"     black 3750 

Jasper 2600 


pure. 


3520 


Emerald 3950 


Jet. 


1300 


Lapis  lazuli 2960 

Malachite 4020 


Spec.  Grav. 

Onyx  .............  ..  2700 

Opal  ................  2090 

Pearl,  Oriental  ......  2650 

Ruby  ...............  3980 

Sapphire  ...........  3994 


Topaz 


3500 


Tourmaline  .........  3070 

Turquoise  ..........  2750 


SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

SUBSTANCES. 

Specific 
Gravity. 

Weight 
ol  a  Cube 
Foot. 

Miscellaneous. 
Amber                

1090 

Lbs. 
68.125 

Liquids. 
Acid  Acetic  

Lbs. 

66.  37  s 

Atmospheric  Air 

001292 

080728 

"     Benzoic 

667 

41  687 

Beeswax             

065 

60.  312 

"     Citric 

64.625 

1900 

118.75 

"     Sulphuric,  Con'd.  . 

95.062 

Butter 

942 

58  875 

"     Fluoric 

QO     7C 

Camphor     .       

088 

6l.7c» 

'     Muriatic  

75 

Caoutchouc  

93° 

58.125 

'     Nitric  

1217 

76.062 

Cotton 

95° 

59.  375 

1     Nitrous     

96  875 

Dynamite             

1650 

103.  125 

'     Phosphoric  

TCcg 

97-  375 

Egg            

1090 

'            "          solid.. 

28OO 

J7S 

Fat  of  Beef 

Q23 

57.687 

"     Sulphuric        .... 

l8AC) 

i*5  562 

"     Hogs               .   .. 

"    £ 

Q^O 

58.5 

Alcohol,  pure,  60°  

49.622 

"     Mutton  

923 

57.687 

95  per  cent  .  .  . 

816 

51 

Flax  

1790 

111.875 

80 

863 

53-937 

Gamboge            

1222 

50                 .  .  . 

58.375 

Glycerine,  60°  
Grain  Barley               . 

I26l 
CQO 

78.752 
36.875 

40 
25                 ... 

95i 

59-437 

DO  625 

"    '  Wheat         

75° 

46.875 

10                   .  .  . 

986 

61.625 

"      Oats  

5OO 

31.25 

5 

OQ2 

62 

Gum  Arabic 

1452 

9O-  75 

proof  spirit  *SQ 

) 

Gunpowder  loose  .... 

QOO 

56.25 

per  cent.  ,  60° 

}  934 

58.375 

1  '          shaken  .... 
"          solid  { 
Gutta-percha  

IOOO 
1550 
I800 
980 

62.5 
96.875 
112.5 
61.25 

"       proof  spirit,  50 
percent,  80° 
Ammonia,  27.9  percent. 
Aquafortis,  double  

j  875 
891 
1300 

54-687 

55.687 
8l.25 

Hay  old  compact 

128  8 

8.05 

11          single  

75 

1689 

Beer  

i°34 

64.625 

Human  body 

66  03^ 

Benzine 

850 

53.  125 

Ice  at  32°          .... 

Q22 

ey.  e 

848 

53 

Indigo  

IOOQ 

6^.062 

Blood  (human).  

i°54 

65.875 

Isinglass 

69  437 

Brandy    83  or  .5  of  spirit 

924 

57-75 

Ivory  
Lard 

1825 

114.062 

Bromine  
Cider        

2966 
1018 

185.375 
63  625 

Leather     .               . 

947 
060 

60 

Ether,  Acetic  

866 

54.  125 

Mastic  

1074 

67.  125 

845 

52.812 

Myrrh 

1360 

85 

"      Nitric  

IIIO 

69*  375 

Nitro-Glycerine     

1600 

IOO 

"      Sulphuric  

7*5 

44.687 

Opium 

83  * 

QO  62? 

Potash 

Milk 

f)i     C 

Resin             

1089 

68.062 

Oil,  Anise-seed  

J986 

61.625 

Snow 

083^ 

5.2 

Codfish  

Q2  -3 

57.687 

Whale  .... 

c7  68? 

Spermaceti  

943 

58.937 

Linseed  

923 

940 

58.75 

Starch 

95° 

59-  375 

850 

53.  125 

Sugar    

1606 

100.375 

Olive  

9I5 

57.187 

"      66                         ( 

972 

60.25 

Palm  

60.  562 

•50  \ 
Tallow 

1326 

82.875 
58.812 

Petroleum  
Rape  

914 

55 
57.  125 

Wax                               f 

941 
964 

60.25 

Sunflower  

j 
920 

57-875 

970 

60.625 

Turpentine  

870 

54-375 

*  Specific  gravity  of  proof  spirit  according  to  Ure's  Table  for  Sykes'a  Hydrometer,  920. 


SPECIFIC   GRAVITY   AND    WEIGHT. 


215 


SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

SUBSTANCES. 

Specific 
Gravity. 

Weight 
of  a  Cube 
Foot. 

Lic^viicLs. 

Spirit  '  rectified 

8-4 
.00061 
1015 
1080 
998.7 
998.8 
997-7 
956-4 
998 

Lbs. 

5I:!38* 

63-437 
67.5 
62.418 
62.425 
62.355 
59-64 
62.  379 

•  i  cube  inc 

Liquids. 
Water  Dead  Sea 

1240 

1029 

1016 

IOOO 

992 
997 
1038 

997 
2-5954  grt 

Lbs. 

77-5 
64.312 
64.312 
63-5 
62.5 
62 
62.312 
64-375 
62.312 
tins. 

Steam  at  212^         

"      Mediterranean.  .  . 
"     sea. 

Tar     ' 

Vinegar 

'      Black  Sea      .   . 

Water  at  32°      

'     rain  

"      "  10  i*-5 

Wine  Burgundy 

«  «I£4  

4       Champagne  
'       Madeira  

"         "    212°               

"     distilled,  at  39°.  . 

*  .038  18. 

'       Port 

h  at  standard  temperature  —  25 

Compression  of  following  fluids  under  a  pressure  of  15  Ibs.  per  square  inch: 
Alcohol..  .0000216  |  Mercury..  .00000265  |  Water..  .00004663  |  Ether..  .00006158 

Elastic   Fluids. 

i  Cube  Foot  of  Atmospheric  Air  at  32°  weighs  .080728  Ibs.  Avoirdupois  =  565.096 
grains,  and  at  62°  532.679  grains. 

Its  assumed  Gravity  ofiis  Unit  for  Elastic  Fluids. 


Spec.  Grav. 

Acetic  Ether             3  04 

Spec.  Grav. 
Nitric  acid               i  217 

Spec.  Grav. 
Vapor. 
Alcohol.   .  .              i  613 

Ammonia                     589 

"     oxide  i  094 

Atmos.  air,  at  32°..  i 
Azote    976 

Nitrogen  974 

Bisulphuret        of 
Carbon  2  64 

Nitrous  acid  2.638 
Nitrous  oxide  1.527 
Olefiant  gas.  .             9672 

Carbonic  acid  1.53 
"        oxide  972 
Carburet'dHydrog.     .559 
Chlorine                     2  421 

Bromine  .               5  4 

Chloric  Ether  3.44 
Chloroform  4.2 

Oxygen  1.  106 

Phosphurett'd  Hy- 
drogen .   .            i  77 

Ether                       2  586 

Chloro-carbonic  ...  3.  389 
Chloroform    5.3 

Hydrochlor.  Ether  2.255 
Iodine  8  716 

Sulphuretted  Hy- 
drogen    I«I7 

Cyanogen  1.815 

Nitric  acid  3-75 

Gas'coal  {  :$ 

Hydrochloric  acid  .   i.  278 
Hydrocyanic     "   .     .942 
Hydrogen      .   ....      0692 

Sulphurous  acid..  2.21 
Steam,$at  212°.  ..     .47295 
Smoke. 
Bitum.  Coal  102 
Coke     .         .         105 

Spirits  of  Turpen- 

Sulphuric  acid  ...  2.  7 
Ether..  2.586 
Sulphur.        .          2  214 

Muriatic  acid  1-247 
t  Weight  of  a  cube  foot  267.26  j 

Wood                      oo 

W;it,er    .  .                         &>>-> 

iraint,  and  compared  with  water  at  62°  specific  gravity  =  .000  612  3. 

Weight  of  a  Cube  Foot  of  Gases  at  32°  F.,  and  under  Pressure  of  one  Atmos- 
phere, or  2116.4  Ibs.  per  Square  Foot. 


Lbs. 

Air,  at  32° 080728 

"    "  62° 076097 

Alcohol 1302 

Carbonic  acid 12344 

Carburet.  Hydrog.  .  044  62 


Lbs. 

Chlorine "...  .197 

Chloroform 428 

Coal  gas 035  36 

Ether,  Sulphuric. .  .2093 
Gaseous  steam 050  22 


Lbs. 

Hydrogen 005  594 

Nitrogen 078  596 

Olefiant  gas 079  5 

Oxygen 089  256 

Steam 05022 


Sulphurous  acid 1814  Ibs. 

To    Compute    "Weight    of    a    Body    or     Sul3staii.ee    when 
Specific    Gravity    is    given. 

RULE. — Multiply  specific  gravity  by  unit  or  standard  of  body  or  sub- 
stance, and  product  is  the  weight. 

Or,  Divide  specific  gravity  of  body  or  substance  by  16,  and  quotient  will 
give  weight  of  a  cube  foot  of  it  in  Ibs. 

EXAMPLE.— Specific  gravity  is  2250;  what  is  weight  of  a  cube  foot  of  it? 
2250  x  62.5  =  140.625  Ibs. 


216 


WEIGHTS  OF  VARIOUS  SUBSTANCES. 


"Weights   and.   A^olumes   of  various   Substances   in 
Ordinary   Use. 


SUBSTANCES. 

Cube  Foot. 

Cube  Inch. 

SUBSTANCES. 

Cube  Foot. 

Cube  Feet 
in  a  Ton. 

Mietals. 

B-s.  .{SSTg} 
u           gun  metal. 
u            sheets  
"           wire  

Lbs. 
488.75 

543-75 
5I3-6 
524-16 
547-25 
543-62S 

450-437 
466.5 

479-5 
481.5 
486.75 
709-5 
7"-75 
848.7487 

487-75 
489.  562 
455-687 
428.812 
449-437 

52.812 
Si-375 
64-3 

35.062 
38-125 
49-5 
43-125 
83-312 
57-062 
35 
66-437 
54-5 

&25 
66-75 

53-75 
4^-937 
41.25 
36-875 
34-625 
29.  562 
33-8i2 

-Tobin  Br< 

Lbs. 
.2829 

•3147 
.297 

•3033 
•3179 
•3167 
.2607 
.27 

•2775 
.2787 
.2816 
,4106 
.4119 
.491174 
.2823 
•2833 
•2637 
.2482 
.2601 

Cube  Feet 
in  a  Ton. 
42.414 
43-6oi 
34.837 

•gin 

58.754 
45-252 
51.942 
26.886 
39.255 
64 
33.714 

41.101 

38.455 
33-558 
41.674 
52.169 
54-303 
60.745 
64.693 

75-773 
66.  348 

mze.  .  . 

Woods. 
Spruce  

Lbs. 

31-25 
31-25 
36-562 
30-375 

.075291 

137.562 
1  02 

89-75 
102.5 
80 
94.875 
84.687 
81.25 
62.5 
14-5 
20 

*s 

120.625 

137-  J25 
109.312 
1  2O 

93-75 
128.125 
128.125 
101.875 
126.25 

I65-75 
l69 

135-5 
12 

25 

57-5 

56.437 

197-25 
167.875 
97.98 

73-5 
62.5 
64.312 
62.355 
ibe  incn. 
)2i  IbS. 

71.68 
71  68 
61.265 
73-744 

12.8 

16.284 

21.961 

24.958 
21.854 
28 
23.609 
26.45! 

27-569 
35-84 
154.48 

114 

89.6 

18.569 

16.335 
20.49 
18.667 
23-893 
17.482 
17.482 
21.987 
17.742 
13-514 
13-254 
16.531 
186.66 
89.6 
38.95 
39-69 

"•355 
13-343 
22.862 
30.476 
35-84 
34-83 
35-955 

Walnut,  black,  dry... 
Willow  

u      drv 

Miscellaneous. 
Air.  

Copper  cast 

"      plates  

Iron  cast 

BaS'ilt   mean 

u     gun  metal  

Brick  fire  

"     heavy  forging.. 
*'     plates  

u      mean  

Coal,  anthracite  j 

u     bitumin.,  mean. 
u     Cannel  

"     wrought  bars.  .  . 
Lead  cast           

"      rolled  

Mercury  60®        .  .   . 

"     Cumberland  
u     Welsh,  mean.  .  . 
Coke 

Steel  plates  

"      soft        .     .  . 

Tin       

Cotton,  bale,  mean  .  .  . 
"        "    pressed  J 
Earth,  clay  

"     rolled         .  .. 

Woods. 
Ash  

*'     common  soil.  . 
"       gravel 
dry,  sand  
'     loose  

Bay  

Blue  Gum  

Cork  

'     moist,  sand.  .. 
'     mold 

Cedar             .   .. 

Chestnut  

1     mud  

Hickory,  pig-nut  
"       shell-bark.. 

"     with  gravel.  .. 
Granite,  Quiucy  
"       Susquehanna 
Gypsum  

Logwood  

Mahoga'y,Hondur's  { 
Oak,  Canadian  

u    hard  pressed  — 
Ice  at  32°  

"    English 

India  rubber 

"    live,  seasoned... 
"    white,  dry  

"         "  vulcanized 
Limestone  

"        "      upland... 
Pine,  pitch  

Marble  mean  

Mortar,  dry,  mean  .  .  . 
Plaster  of  Paris  
Water,  rain  

"     red  

"     white  

.  "     well  seasoned.. 
Pine,  yellow  

"      salt 

"      at  62°  

Metals.  - 

Cube  foot.            Ci 
.  .    S22.02  IbS.          .^c 

To  Compnte  Proportions  of  Two  Ingredients  in  a  Com- 
pound, or   to    Discover   Adxdteration   in    IMetals. 

RULE. — Take  differences  of  each  specific  gravity  of  ingredients  and  spe- 
cific gravity  of  compound,  then  multiply  gravity  of  one  by  difference  of 
other ;  and,  as  sum  of  products  is  to  respective  products,  so  is  specific 
gravity  of  body  to  proportions  of  the  ingredients. 

EXAMPLE.  —A  compound  of  gold  (spec.  grav.  =  18.888)  and  silver  (spec.  grav.  = 
10.535)  has  a  specific  gravity  of  14;  what  is  proportion  of  each  metal? 

18.888—14=4.888X10.535=51-495.     14—10.535=3-465X18.888=65.447. 
65-447+51-495: 65-447 -«4: 7-835  0old,    65.447+51.495: 5i.495"H: 6-165  suvtfft 


WEIGHTS   OP  VARIOUS   SUBSTANCES   IN  BULK. 


2I7 


"Weights  of  "Various  Substances  per  Cn/be  Foot  in  Sulk. 


Lba. 

Lead,  in  pigs 567 

Iron,      "      360 

Marble,  in  blocks) 
Limestone,  "  ) 
Trap 


172 


170 

Granite,  in  blocks  ....  164 
Sandstone 141 


Ash,  dry,  ioo  feet  BM 

"    white,  "        *' 

Cement,  struck  bushel  and 

packed* ioo 

Cement,  Portland,  bushel,  no 

Cherry,  dry,  ioo  BM 

Chestnut,  dry,  I0o  BM  . . .      . 

Coal,  anthracite,  i  cub.  yd. 

broken  and  loose  ...     i 
:c       "       "     .1  ton..  41. 

Coke,  ton  = 80  to  97 

Earth,  common  soil 137 


Potters'  clay 130 

Loam 126 

Gravel 109 

Sand 95 

Bricks,  common 93 

Ice,  at  32° 57.5 

Oak,  seasoned 52 


Lba. 

Coal,  caking 50 

Wheat 48 

Barley 38 

Fruit  and  vegetables . .  22 

Cotton  seeds 12 

Cotton 10 

Hay,  old 8 


175  ton. 

141    « 

Ibs. 
Ibs. 

156  ton. 
i53    " 

,75  yds 
5  cub.  feet 
cub.  feet. 
.  125  Ibs. 

*  One  packed  bushel  =  1.43  loose. 


Earth, loose 93.75105. 

Elm,  dry,  ioo  feet  BM 13  ton. 

Gypsum,  ground,  str.  bush.   70  Ibs. 

"    well  shaken  80  " 
Hemlock,  dry,  ioo  feet  BM.       .093  ton. 
Hickory,     "        "        "    .       .197    " 
Masonry, Granite, dressed..  165  Ibs. 

"       rough...  126 
Limestone,  dres'd  165 

Sandstone 135 

Brick,  pressed  . . .  140 
"  com'n,  rough,  ioo 


Comparative   ^ 
TIMBER. 

Weight 

Weight  of  a 
Green. 

of  G-i 

Cube  Foot. 
Seasoned. 

een    and.   Seaso 
TIMBER. 

ned.   T 

Weight  of  a 
Green. 

Lm"ber. 

Cube  Foot. 
Seasoned. 

American  Pine  

Lbs. 
44-75 
58.18 
60 

Lbs. 
30-7 
50 
53.17 

Cedar  

Lbs. 

& 

48.  7  «; 

Lbs. 
28.25 
43-5 

is.  5 

Ash  

English  Oak  

Beech  .  .  . 

Riga  Fir.  .  . 

-Application,   of  th.e  Ta~bles. 

When  Weight  of  a  Solid  or  Liquid  Substance  is  required.  RULE. — Ascer- 
tain volume  of  substance  in  cube  feet ;  multiply  it  by  unit  in  second  column 
of  tables  (its  specific  gravity),  and  divide  product  by  16;  quotient  will  give 
weight  in  Ibs. 

When  Volume  is  given  or  ascertained  in  Inches.    RULE. — Multiply  it  by 
unit  in  third  column  of  tables  (weight  of  a  cube  inch),  and  product  will  give 
weight  in  Ibs. 
EXAMPLE.— What  is  weight  ol  a  cube  of  Italian  marble,  sides  being  3  feet? 

33  x  2708  =  73 1 16  oz. ,  which  -7-16  =  4569. 75  Ibs. 
Or  of  a  sphere  of  cast  iron  2  inches  in  diameter  ? 

23  X  .5236  X  -2607  weight  of  a  cube  inch=i  1.092  Ibs. 

When  Weight  of  an  Elastic  Fluid  is  required.  RULE.— Multiply  specific 
gravity  of  fluid  by  532.679  (weight  of  a  cube  foot  of  air  at  62°  hi  grains), 
divide  product  by  7000  (grams  hi  a  Ib.  Avoirdupois),  and  quotient  will  give 
weight  of  a  cube  foot  hi  Ibs. 

EXAMPLE.— What  is  weight  of  a  cube  foot  of  hydrogen? 

Specific  gravity  of  hydrogen  .0692. 

532.679  x -0692 -7- 7000  =  .005  265  9  Ibs. 

To  Compute  "Weight  of  Cast  iMetal  "by  "Weight  of  Pattern. 

When  Pattern  is  of  White  Pine.  RULE.— Multiply  weight  of  pattern  in 
Ibs.  by  following  multipliers,  and  product  will  give  weight  of  casting : 

Iron,  14 ;  Brass,  15 ;  Lead,  22 ;  Tin,  14 ;  Zinc,  13.5. 

When  the  Cores  or  Prints  are  of  White  Pine.  Multiply  the  product  of  their 
area  and  length  in  inches  by  .0175  or  .02,  according  to  the  dryness  of  the  wood,  and 
proportionately  for  other  woods,  and  result  is  weight  of  core  or  print  to  be  deducted 
from  weight  of  pattern. 

V 


2l8          BALLOONS,  SHRINKAGE    OP   CASTINGS,  ETC. 

To  Compxite  Weights  of  Ingredients,  that  of  Compound 
being  given. 

RULE.— As  specific  gravity  of  compound  is  to  weight  of  compound,  so  are 
each  of  the  proportions  to  weight  of  its  material. 

EXAMPLE.— Weight,  as  p.  216,  being  28  Ibs.,  what  are  weights  of  the  ingredients? 

14:28::  {7-835:  15-67  gold, 
(0.165  •'  I2-33  silver. 

N°TI?-  —  Specific  gravity  of  alloys  does  not  usually  follow  ratio  of  their  compo- 
nents, it  being  sometimes  greater  and  sometimes  less  than  their  mean. 

To    Compnte    Capacity    of  a    Balloon. 

RULE.— From  specific  gravity  of  air  in  grains  per  cube  foot,  subtract  that 
of  the  gas  with  which  it  is  inflated  ;  multiply  remainder  by  volume  of  bal- 
loon in  cube  feet;  divide  product  by  7000,  and  from  quotient  subtract  weight 
ot  balloon  and  its  attachments. 

EXAMPLE.— Diameter  of  a  balloon  is  26.6  feet,  its  weight  is  100  Ibs  and  soecifir 
gravity  of  the  gas  with  which  it  is  inflated  is  .07  (air  being  assumed  at  if  what  is 
its  capacity,  specific  gravity  of  air  assumed  at  527.04  grain! 

527-04  —  (527-04  X  .07)  X  26.63  x  .5236 

I00  — .  500.04  Ibt. 

7000 

To    Compute   Diameter   of  a   Balloon. 

Weight  to  be  raised  being  given.— Ey  inversion  of  preceding  rule. 

~/W  -T-  7000-f-  *  -T-  *'  ,, 

^7  —     — —    —  =  a .    s  and  s  representing  weight  of  air  and  gas 

in  grains  per  cube  foot,  W  weight  to  be  raised  in  Ibs.,  and  d  diameter  of  bal- 
loon in  feet. 

ILLUSTRATION. — Given  elements  in  preceding  case. 


Then    3/590-04  +  100X7000^527.04-36.89^    3 /9854.69  =  266feet 

v  .5236  V  -5236 

IProof  of*  Spirituous   Liquors. 

A  cube  inch  of  Proof  Spirits  weighs  234  grains ;  then,  if  an  immersed 
cube  inch  of  any  heavy  body  weighs  234  grains  less  in  spirits  than  air,  it 
shows  that  the  spirit  in  which  it  was  weighed  is  Proof. 

If  it  lose  less  of  its  weight,  the  spirit  is  above  proof;  and  if  it  lose  more, 
it  is  below  proof. 

ILLUSTRATION. — A  cube  inch  of  glass  weighing  700  grains  weighs  500  grains  when 
weighed  in  a  certain  spirit;  what  is  the  proof  of  it? 

700  —  500  =  200  =  grains  =  weight  lost  in  spirit. 

Then  200  :  234  ::  i  :  i.ij=  ratio  of  proof  of  spirits  compared  to  proof  spirits,  or 
i  = .  17  above  proof. 

NOTE.— For  Hydrometers  and  Rules  for  ascertaining  Proof  of  Spirits,  see  page 
67 ;  and  for  a  very  full  treatise  on  Specific  Gravities  and  on  Floatation,  see  Jamie- 
son's  Mechanics  of  Fluids.  Lond.,  1837. 

Shrinkage    of  Castings. 

It  is  customary,  in  making  of  patterns  for  castings,  to  allow  for  shrinkage 
per  lineal  foot  of  pattern  as  follows : 
Iron,  small  cylinders  ...  =  ^  in.  per  ft.    Ditto  in  length. . . .  =  %  in  16  ins. 

"     Pipes =K       " 

'    Girders,  beams,  etc.  =  %  in  15  ins. 

"     Large  cylinders,") 


the  contraction  >  =  %&  P61" 
of  diam.at  top.J 
Ditto  at  bottom  .  .  =  TV      " 


Brass,  thin  ........  =  %  in  9  ins. 

"      thick  .......  =  %  in  10  ins. 

Zinc  .............  =  j£  in  a  foot. 


Lead 

Copper 

Bismuth 


GEOMETRY. 

GEOMETRY. 
Definitions. 

Point  has  position,  but  not  magnitude. 

Line  is  length  without  breadth,  and  is  either  Right,  Curved,  or  Mixed. 

Right  Line  is  shortest  distance  between  two  points. 

Curved  Line  is  one  that  continually  changes  its  direction. 

Mixed  Line  is  composed  of  a  right  and  a  curved  line. 

Superficies  has  length  and  breadth  only,  and  is  plane  or  curved. 

Solid  has  length,  breadth,  and  thickness,  or  depth. 

Angle  is  opening  of  two  lines  having  different  directions,  and  is  either 
Right,  A  cute,  or  Obtuse. 

Right  A  ngle  is  made  by  a  line  perpendicular  to  another  falling  upon  it. 

Acute  Angle  is  less  than  a  right  angle. 

Obtuse  Angle  is  greater  than  a  right  angle. 

Triangle  is  a  figure  of  three  sides. 

Equilateral  Triangle  has  all  its  sides  equal. 

Isosceles  Triangle  has  two  of  its  sides  equal. 

Scalene  Triangle  has  all  its  sides  unequal. 

Right-angled  Triangle  has  one  right  angle. 

Obtuse-angled  Triangle  has  one  obtuse  angle. 

Acute-angled  Triangle  has  all  its  angles  acute. 

Oblique-angled  Triangle  has  no  right  angle. 

Quadrangle  or  Quadrilateral  is  a  figure  of  four  sides,  and  has  following 
particular  designations — viz., 

Parallelogram,  having  its  opposite  sides  parallel. 

Square,  having  length  and  breadth  equal. 

Rectangle,  a  parallelogram  having  a  right  angle. 

Rhombus  or  Lozenge,  having  equal  sides,  but  its  angles  not  right  angles. 

Rhomboid,  a  parallelogram,  its  angles  not  being  rig-nt  angles. 

Trapezium,  having  unequal  sides. 

Trapezoid,  having  only  one  pair  of  opposite  sides  parallel. 

NOTE. — Triangle  is  sometimes  termed  a  Trigon,  and  a  Square  a  Tetragon. 

Gnomon  is  space  included  between  the  lines  forming  two  similar  parallelo- 
grams, of  which  smaller  is  inscribed  within  larger,  so  as  to  have  one  angle 
in  each  common  to  both. 

Polygons  are  plane  figures  having  more  than  four  sides,  and  are  either 
Regular  or  Irregular,  according  as  their  sides  and  angles  are  equal  or  un- 
equal, and  they  are  named  from  number  of  their  sides  or  angles.  Thus : 


Pentagon  has  five  sides. 
Hexagon    "   six     " 
Heptagon   "  seven" 
Octagon     "   eight  " 


Nonagon    has  nine  sides. 
Decagon       "    ten       " 
Undecagon  "    eleven  " 
Dodecagon   "    twelve" 


Circle  is  a  plane  figure  bounded  by  a  curved  line,  termed  Circumference 
or  Periphery. 

Diameter  is  a  right  line  passing  through  centre  of  a  circle  or  sphere,  and 
terminated  at  each  end  by  periphery  or  surface. 

Arc  is  any  part  of  circumference  of  a  circle. 

Chord  is  a  right  line  joining  extremities  of  an  arc. 

Segment  of  a  circle  is  any  part  bounded  by  an  arc  and  its  chord. 

Radius  of  a  circle  is  a  line  drawn  from  centre  to  circumference. 

Sector-  is  any  part  of  a  circle  bounded  by  an  arc  and  its  two  radii. 

Semicircle  is  half  a  circle. 

Quadrant  is  a  quarter  of  a  circle. 

Zone  is  a  part  of  a  circle  included  between  two  parallel  cords. 

Lune  is  space  between  the  intersecting  arcs  of  two  eccentric  circles. 


22O  GEOMETRY. 

Secant  is  line  running  from  centre  of  circle  to  extremity  of  tangent  of  arc. 

Cosecant  is  secant  of  complement  of  an  arc,  or  line  running  from  centre  of 
circle  to  extremity  of  cotangent  of  arc. 

Sine  of  an  arc  is  a  line  running  from  one  extremity  of  an  arc  perpendicu- 
lar to  a  diameter  passing  through  other  extremity,  and  sine  of  an  angle  is 
sine  of  arc  that  measures  that  angle. 

Versed  Sine  of  an  arc  or  angle  is  part  of  diameter  intercepted  between  sine 
and  arc. 

Cosine  of  an  arc  or  angle  is  part  of  diameter  intercepted  between  sine  and 
centre. 

Coversed  Sine  of  an  arc  or  angle  is  part  of  secondary  radius  intercepted 
between  cosine  and  circumference. 

Tangent  is  a  right  line  that  touches  a  circle  without  cutting  it. 

Cotangent  is  tangent  of  complement  of  arc. 

Circumference  of  every  circle  is  supposed  to  be  divided  into  360  equal 
parts,  termed  Degrees ;  each  degree  into  60  Minutes,  and  each  minute  into  60 
Seconds,  and  so  on. 

Complement  of  an  angle  is  what  remains  after  subtracting  angle  from  90 
degrees. 

Supplement  of  an  angle  is  what  remains  after  subtracting  angle  from  180 
degrees. 

To  exemplify  these  definitions,  let  Acb,  in  following  Figure,  be  an  assumed 
arc  of  a  circle  described  with  radius  B  A : 

Ag  A  c  6,  an  Arc  of  circle  AGED. 

A  6,  Chord  of  that  arc. 

B  A,  an  Initial  radius. 

B  C,  a  Secondary  radius. 

e  D  d,  a  Segment  of  the  circle. 

A  B  &,  a  Sector. 

A  D  E,  a  Semicircle. 

C  B  E,  a  Quadrant. 

AedK.ii  Zone. 

n  °  A,  a  Lime. 

B  g,  Secant  of  arc  A  c  &;  written  Stec. 

b  k,  Sine  of  arc  A  c  6 ;  written  Siu. 

A  k,  Versed  Sine  of  arc  A  c  6;  written  Versin. 

B  k  or  m  6,  Cosine  of  arc  Acb. 

A  ff,  Tangent  of  arc  A  c  b. 

C  B  6,  Complement,  and  b  B  E,  Supplement  of 
D  arc  A  c  6. 

C*,  Cotangent  of  arc,  written  Cot.     B  s,  Cosecant  of  arc;  written  Cosec. 
m  C,  Coversed  sine  of  arc,  or,  by  convention,  of  angle  A  B  6 ;  written  Coversin. 

Vertex  of  a  figure  is  its  top  or  upper  point.  In  Conic  Sections  it  is  point 
through  which  generating  line  of  the  conical  surface  always  passes. 

Altitude,  or  height  of  a  figure,  is  a  perpendicular  let  fall  from  its  vertex 
to  opposite  side,  termed  base. 

Measure  of  an  angle  is  an  arc  of  a  circle  contained  between  the  two  lines 
that  form  the  angle,  and  is  estimated  by  number  of  degrees  in  arc. 

Segment  is  a  part  cut  off  by  a  plane,  parallel  to  base. 

Frustum  is  the  part  remaining  after  segment  is  cut  off. 

Perimeter  of  a  figure  is  the  sum  of  all  its  sides. 

Problem  is  something  proposed  to  be  done. 

Postulate  is  something  required. 

Theorem  is  something  proposed  to  be  demonstrated. 

Lemma  is  something  premised,  to  render  what  follows  more  easy. 

Corollary  is  a  truth  consequent  upon  a  preceding  demonstration. 

Scholium  is  a  remark  upon  something  going  before  it. 

For  other  definitions  see  Mensuration  of  Surfaces  and  Solids,  and  Conic  Sectiona 


GEOMETRY. 


221 


Leng 

ths   of*  To 

Angle  45°. 

llo-wing 

Angle  60°. 

Elements,  3 

Radius  = 

Angle  45°. 

1. 

Angle  60°. 

Sine  .  . 

.707  107 
.707  107 
.292893 
.292893 
1.414214 

a    Line, 
of  ] 

»—-?-— 

.866025 
•5 
•5 
•133975 

2 

Sea 
as   A  B,  ^ 
Equal   3? 

rr 
t=j. 

Cosecant  

1.414214 

I 

'.765366 
.785398 

quired.    ] 

draw  two  p 
definite  lengt 
equired  nurn 
5,  4,  and  B  i, 

i  A  o,  join  o 
arallel  there 

I.I547 
1.73205 

•577349 

i 
1.0472 

S'um'ber 

irallel  lines, 
h,  and  upon 
ber  of  equal 
2,3,4;  Joifc 

B,  and  draw 
to. 

Cosine 
Versed 
Covers( 
Secant 

To    3D 
1. 

Sine.  . 
id  "  .  . 

Cotangent  .  .  . 
Chord  

ivide 
jL 

les. 
;vitli    any   re 
arts.—  Fig.  1 

>      From  A  and  E 
Ao,  Br,  to  an  in 
them  point  off  r 
parts,  as  A  i,  2,  - 
o  B,  4  i,  etc. 
Or,  point  off  01 
the  other  lines  p 

Apcc 
1 

i 

j          j  /-— 

x.-t.~ 

-i—  "3          2 

n   A 


To   Construct   a  33iagonal    Scale,  as  A  B.—  Fig.  2. 

Divide  a  line  into  as  many  di- 
visions  as  there  are  hundreds  of 
feet,  spaces  of  ten  feet,  feet,  or 
inches  required. 

Draw  perpendiculars  from  each 
division  to  a  parallel  line,  C  D. 
Divide  one  of  divisions,  A  E,  C  F, 
into  spaces  of  ten  if  for  feet  and 
hundredths,  and  twelve  if  for  feet 
and  inches;  draw  the  lines  Ai, 


A  to  E,  E  to  G,  etc. ,  will  measure  one 
foot;' A  to"a"c  to",7ttT^etc7^n  measure  i-ioth  of  a  foot  The  several  lines 
A  i,  a  2,  etc,  will  measure  upon  lines  fc,  Z,  etc,  i-iooth  of  a  foot;  and  op  will 
measure  upon  fc,  I,  etc,  divisions  of  i-ioth  of  a  foot. 

Lilies. 

To  33ra\v  a  Perpendicular  to  a  Right  Line, 
o  as  or,  Fig.  3,  c  A,  Fig.  4,  or 

*%Scr'  from    a   3?oint   external   to 

it,  as  A,  Fig.  S,  and  from 
any  two  3?oints,  as  c  d, 
Fig.  6. 

With  any  radius  as  r  A,  r  B,  cut  line 
at  A  and  B ;  then  with  a  longer  radius, 
as  A  o,  B  o,  describe  arcs  cutting  each 
_j   other  at  o,  and  connect  o  r.    (Fig.  3. ) 
B     Or,  from  A,  set  off  A  B  equal  to  3  B 
parts  by  scale ;  from  A  B,  with  radii    g$ 
of  4  and  5  parts,  describe  arcs  cut- 
ting at  c,  and  connect  c  A.  (Fig.  4. ) 
NOTE.  —  This  method  is  useful 
where  straight  edges  are  inappli- 
cable.    Any  multiples  of  numbers 
3,  4,  5  may  be  taken  with  same  ef- 
fect, as  6,  8, 10,  or  9, 12, 15. 

From  A,  with  a  sufficient  radius,  " 
cut  line  at  o  c,  and  from  them  de- 
scribe arcs  cutting  at  ?',  and  connect 
Ar.     (Fig.  5.) 

From  any  two  points,  as  c  d,  at  a  proper 
distance  apart,  describe  arcs  cutting  at  A  B, 
and  connect  them.     (Fig.  6.) 
T* 


222 


GEOMETRY. 


To  Bisect  a  Flight  Line  or  an  Arc  of* a 
Circle,  and.  to  Draw  a,  l^erpeivdicvi- 
lar  to  a,  Circular  or  Right  Line,  or  a, 
Radial  Are.— Fig.  7. 

From  A  B  as  centres  describe  arcs  cutting  each  other 
at  c  and  d,  connect  c  d,  and  line  and  arc  are  bisected 
at  e  and  o. 

Line  cd  is  also  perpendicular  to  a  right  line  as  A  B, 
and  radial  to  a  circular  arc  as  A  o  B. 


To  Draw    a  Line  Parallel    to   a  Given 

Right    Line,  as   c  d,  Fig.  8. 
From  A  B  describe  arcs  Ac,  Bd,and  draw  a  line  par- 
allel thereto,  touching  arcs  c  and  d. 


To    Describe   Angles    of  3O°    and,   6O°,  Fig.  D,  and.    45° 
Fig.  10. 

From  A,  with  ar  r  radius,  A  o,  de- 
scribe or,  and  from  o  with  a  like  ra- 
dius cut  it  at  r,  let  fall  perpendicular 
rs;  then  o  Ar  =  6o°,  and  Ars  =  3o°. 
(Fig.  9-) 

Set  off  any  distance,  as  A  B,  erect 
perpendicular  Ao  =  A  B,  and  connect 
o  B.  (Fig.  10.) 


To  Bisect  Inclination  of  Two  Lines. 
when  IPoint  of  Intersection  is  Inac- 
cessible.— Fig.  11. 

•m  Upon  given  lines,  A  B,  C  D,  at  any  points  draw  perpen- 
diculars e  o,  s  r,  of  equal  lengths,  and  from  o  and  s  draw 
parallels  to  their  respective  lines,  cutting  at  n;  bisect 
angle  ons,  connect  n  m,  and  line  will  bisect  lines  as  re- 
quired. 


To 
12. 


Rectilineal 
Describe    an    Octagon    upon    a   Line,  as    A  B.—  Fig.  IS. 

From  points  A  B  erect  indefinite  perpendiculars  A/,  Be; 
produce  A  B  to  m  and  w,  and  bisect  angles  m  A  o  and  n  Ep 
With  A  u  and  B  r. 

Make  A  u  and  B  r  equal  to  A  B,  and  draw  u  z,  r  v  parallel 
to  A/,  and  equal  to  A  B. 

From  z  and  v,  as  centres,  with  a  radius  equal  to  A  B,  de- 
scribe arcs  cutting  A/,  B  e,  in  /  and  e.  Connect  «/,/«, 
and  e  v. 

To  Inscribe  any  Reg-alar  Polygon  in  a 
Circle,  or  to  Divide  Circumference  into 
a  given  !N"umber  of  EcLnal  .Parts.—  Fig.  13. 


If  Circle  is  to  contain  a  Heptagon.  —  Draw  angle  A  o  B  at 
centre  o  for  360°  -=-7  =  51°  42'  51"+,  or  51^,  then  set  off  upon 
circumference  distance  A  B  or  remaining  angles  A  o  B. 


GEOMETRY. 


223 


Xo    Inscribe    a   Hexagon    in. 
a   Circle.— Fig.  14. 


H.     c^ 


Draw  a  diam- 
eter, AoB.  From 
A  and  B  as  cen- 
tres, with  A  o  and 
B  o,  cut  circle  at 
cm  and  ew,  and 
connect. 


To     Describe     a     Hexagon 
about  a  Circle.— Fig.  l£5. 

Draw  a  diam- 
eter as  a  o  b ;  and 
with  a  o  cut  circle 
at  c ;  join  a  c,  and 
bisect  it  with  ra- 
dius o  r,  through 
r  draw  e  r  paral- 
lel to  c  a,  cutting 
diameter  at  m ; 
then  with  radius 

o  m  describe  circle,  within  which  describe 

a  hexagon  as  above. 


To  Inscribe    a  Pentagon    in 
a  Circle.— Fig.  16. 

16.  A  Draw  diameters 

A  c  and  m  n,  at 
right  angles  to 
each  other;  bisect 
o  n  in  ?-,  and  with 
r  A  describe  A  s ; 
from  A  with  A  * 
describe  s  B. 

Connect  A  B,  and 
distance  is  equal  to 

one  side  of  a  pentagon. 

To     Describe     a     Pentagon 
upon   a   Line,  as   A  B.— Fig. 

ir. 

17          n  Draw  B  m  per- 

pendicular to  A  B, 
and  equal  to  one 
half  of  it ;  extend 
A  m  until  m  n  is 
equal  to  B  m. 

From  A  and  B, 
with  radius  Bn,  de- 
scribe arcs  cutting 
each  other  in  o; 

then  from  o,  with  radius  o  B,  describe 


circle  A  C  B,  and  line  A  B  is  equal  to  one 
side  of  a  pentagon  upon  circle  described. 

To  Describe  a  Regular  Polygon  of*  any  required  Number 

of  Sides.— Fig.  18. 

From  point  o,  with  distance   o  B,  describe   semicircle 
B  6  A,  which  divide  into  as  many  equal  parts,  A  a,  a  6,  b  c, 
etc. ,  as  the  polygon  is  to  have  sides. 
Thus,  let  a  Hexagon  be  required : 

From  o  to  second  point  6  of  six  divisions  draw  o  &,  and 
a/     \\  •  /^\  //      through  other  points,  c,  d,  and  e,  draw  o  C,  o  D,  etc. 
/        N\!/,-^     v£/  Apply  distance  o  B,  from  B  to  E,  from  E  to  D,  from  D  to 

Jf"~      o*-— — ''JJ         C,  etc.    Join  these  points,  as  b  C,  C  D,  etc. 


To  Construct  a  Square  or 
a  Rectangle  on  a  given 
3L.irie.-Fig.  19. 

19.  mx-  X,n  On  A  B  as  cen 

tres,  with  ABas 
radius,  describe 
__________  arcs  cutting  at 

9<f '" — |  ^~x"~  |  '~ST  c ;  on  c  describe 
arcs  cutting  at 
o  r;  and  on  o  r 
describe  others, 
cutting  at  m  n ; 
draw  Am  and 


To  Construct  a  Hexagon 
upon  a  given  .Line.— Fig. 
2O. 

From  ends  of  line, 
A  B,  describe  arcs 
cutting  each  other 
at  o,  and  from  o  as 
jf  acentre,  with  radius 
o  A,  describe  a  cir- 
cle, and  with  same 
radius  set  off  A  c, 
cd,  B/,/*,  and  con- 
nect them. 


B  n,  and  join  o  r. 

21  _       To    Inscribe    an   Octagon    in    a    Circle.— Fig.  SI.       22. 
Draw  diameters,  A  C,  B  D,  at  right 

angles,  bisect  arcs,  A  B,  B  C,  etc. ,  at  5,  r, 

o,  e,  and  join  Ao,  o  B,  etc.     (Fig.  21.) 

To    Describe    an    Octagon    . 
about  a  Circle.— Fig.  22.     A 
Describe  a  square  about  circle  A  B, 
draw  diagonals  c/,  e  d,  draw  o  t,  etc., 
perpendicular  to  diagonals  and  touch- 
ing circle.     (Fig.  22.) 


224 


GEOMETRY. 


To    Inscribe    a    Square    in    a    Circle.— Fig 

Draw  line  A  B  through  centre  of  circle ; 
take  any  radius,  as  A  e,  and  describe  the 
arcs  Aee,  Bee;  connect  ee,  continuing 
line  to  C  and  D ;  join  AC,  AD, etc.  (Fig. 23.) 

To  Describe  a  Sqxiare  abo\rt  A 
a  Circle.— Fig.  24:. 

Draw  line  A  B  through  centre  of  circle. 
Take  any  radius,  as  A  e;  describe  arcs 
Aee,  Bee;  connect  ee,  continuing  line 
to  CD. 

Describe  B  r  and  D  r ;  draw  and  extend  B  r  and  D  r,  and  sides  A  and  C  parallel  to 
them.     (Fig.  24.) 

To    Describe    an    Octagon    in   a    Square.— Fig. 

Let  A  B  C  D  be  given  square. 

Describe  Aorr,  B  o  r  r,  etc. ;  join  in- 
tersections rrrr,  etc.,  and  figure  formed 
is  octagon  required.  (Fig.  25.) 

To  Inscribe  an  Equilateral 
Triangle  in  a  Circle.  — 
Fig.  26. 

From  point  A,  with  A  o  equal  to  radius 
of  circle,  describe  o  o;  from  o  and  o  describe  o  r,  o  r ;  join  A  r,  r  r,  and  r  A.  (Fig.  26.) 

NOTE. All  figures  of  10  or  20  sides  are  readily  determined  from  side  of  a  pentagon, 

being  halved  or  quartered;  and  in  like  manner,  all  figures  of  6,  12,  or  24  sides  are 
readily  determined  from  radius  of  a  circle,  being  equal  to  the  side  of  a  hexagon. 

Circles. 

To  Describe  an  Arc  of  a  Circle, 
through  T-wo  given.  ^Points, 
-with  a  given  Radius.— Fig. 
27. 

On  A  B  as  centres,  with  given  radius,  de-  ' 
scribe  arcs  cutting  at  o,  and  from  o  with 
same  radius  describe  arc  A  B.     (Fig.  27.) 

To  Ascertain  Centre  of  a  Circle 
or  of  an  Arc  of  a  Circle.— Fig. 
2&. 

Draw  chord  A  B.  bisect  it  with  perpendicular  c  d,  then  bisect  c  d  for  centre  o. 
(Pig.  28.) 

To  Describe  a  Circular  Segment  that 
•will  both  fill  the  angle  bet-ween  tAvo 
diverging  lines  and  touch  them.— 
Fig.  2&. 

Bisect  inclined  lines,  A  B,  D  E,  by  line  «/  and  connect 
perpendicular  thereto,  B  D,  to  define  boundary  of  seg- 
ment to  be  described.  Bisect  angles  at  B  and  D  by  lines 
cutting  at  o,  and  from  o,  with  radius  o  e,  describe  arc 
men. 

Draw    a    Series    of   Circles    bet-ween    T-wo    Inclined 
Lines,  touching   them    and    each    other.— Fig.  3O. 

_  Bisect  given  lines  AB,  CD,  by  line  oc. 
From  a  point  r  in  this  line  erect  rs  perpen- 
dicular to  A  B,  and  on  r  describe  circle  s  m, 
cutting  centre  line  at  w;  from  u  erect  u  n 
perpendicular  to  centre  line,  cutting  A  B  at 
w,  and  from  n  describe  an  arc  n  u  v,  cutting 
A  B  at  w,  erect  x  v  parallel  to  r  s,  making  x 
i  centre  of  next  circle  to  be  described,  with 

radius  x  u,  and  so  on. 
Nora.— Largest  circle  may  be  described  first. 


GEOMETKY. 


225 


To  Describe   a  Circle  that  sliall  pass  through  any   tjiree 
given.    Points,  as    A  B  C.— figs.  31   and   32. 

Upon  points  A  and  B, 
with  any  opening  of  a 
dividers,  describe  arcs 
cutting  each  other  at  ee. 

On  points  B  C  describe 
two  more  cutting  each 
other  in  points  c  c. 

Draw  lines  ee  and  cc, 
and  intersection  of  these 
lines,  o,  is  centre  of  circle 
ABC.  (Fig.  31.) 

When  Centre  is  not  attainable.  —From  A  B  as  centres,  describe  arcs  A  g,  B  ft; 
through  C  draw  A  e.  B  c.     Divide  A  e  and  B  c  into  any  number  of  equal  parts,  also 
c  g  and  B  h  into  a  like  number.     Draw  A  i,  2,  3,  etc.,  and  B  i,  2,  etc.,  and  intersec- 
tion of  these  lines  as  at  o  are  points  in  the  circle  required.    (Fig.  32.) 
$3.  _  Or,  let  A  B  C  be  given  points,  connect 

f. L_ -c  AB,  AC,  C  B,  and  draw  e  c  parallel  to  A  B. 

Divide  C  A  into  a  number  of  equal  parts, 
as  at  i,  2,  and  3,  and  from  C  describe  arcs 
through  these  points  to  meet  right  lines 
from  C  to  points  i,  2,  and  3,  on  A  e,  and 
these  are  points  in  a  circle,  to  be  drawn  as 


before  directed.    (Fig.  33.) 

To  Draw  a  Tangent  to  a 
Circle  from  a  given  Point 
in  Circumference.  —  Fig. 
34.  e 

34. 


To  Dra-w  Tangents  to  a 
Circle  from  a  Point  with- 
out it.— Fig.  35. 

m  35. 


From  A  draw  A  o,  and  bisect  it  at  s; 
Through  point  A  draw  radial  line  A  o,    describe  arc  through  o,  cutting  circle  at 
and  erect  perpendicular  ef.     (Fig.  34. )       win ;  join  A  m  or  A  n. 

To   Draw   from    or  to    Circumference   of  a   Circle,  Lines 
leading   to   an   Inaccessible   Centre.— Fig.  36. 

Divide  whole  or  any  given  portion  of 
circumference  into  desired   number  of 
parts ;  then,  with  any  radius  less  than 
^      distance  of  two  divisions,  describe  arcs 
'l      cutting  each  other,  as  A  r,  6  r,  c  r,  d  r, 
etc. ;  draw  lines  b  r,  c  r,  etc.,  and  they 
will  lead  to  centre. 

To  draw  end  lines,  as  A  r,  F  r.     From  6  describe  arc  o,  and  with  radius  6  i,  from 
A  or  F  as  centres,  cut  arcs  A  r,  etc.,  and  lines  A  r,  F  r,  will  lead  to  centre. 

To   Describe  an  A.rc,  or   Segment  of  a  Circle,  of  a  large 

Radius.— Fig.  37. 

i. — _«__  ;j  /  Draw  chord  A  c  B ;  also  line  h  D  i 
^-^-"-V  parallel  with  chord,  and  at  a  distance 
equal  to  height  of  segment ;  bisect 
chord  in  c,  and  erect  perpendicular 
cD;  join  AD,  DB;  draw  Ah  and  Bt 
perpendicular  to  A  D,  B  D;  erect  also  perpendiculars  An,  B  n;  divide  A  B  and  h  i 
into  any  number  of  equal  parts;  draw  lines  i  i,  2  ?,  etc.,  and  divide  lines  A  n,  B  n, 
each  into  half  number  of  equal  parts  in  A  B;  draw  'ines  to  D  from  each  division  in 
lines  A  n,  B  n,  and  at  points  of  intersection  with  former  lines  describe  arc  or  segment 


226 


GEOMETRY, 


Ellipse. 

To    Descri"be    an    Ellipse    to    any    Length    and    Breadth 
given.— Fig.  38. 

88.  E  Let  longest  diameter  be  C  D,  and  shortest  E  F.    Take 

distance  C  o  or  o  D,  and  with  it,  from  points  E  and  F, 
describe  arcs  h  and /upon  diameter  C  D. 

Insert  pins  at  h  and  at  /,  and  loop  a  string  around 
them  of  such  a  length  that  when  a  pencil  is  introduced 
within  it  it  will  just  reach  to  E  or  F.  Bear  upon 
string,  sweep  it  around  centre  o,  and  it  will  describe 


NOTE. — It  is  a  property  of  Ellipse  that  sum  of  two  lines  drawn  from  foci  to  meet  in  any  point  in 
curve  ia  equal  to  transverse  diameter. 

Bisect  transverse  axis  A  B  at  o,  and  on  centre  o 
erect  perpendicular  C  D,  making  o  D  and  o  C  each 
equal  to  half  conjugate  axis.  From  C  or  D,  with 
radius  A  o,  cut  transverse  axis  at  ss  for  foci.  Divide 
B  A  o  into  any  number  of  equal  parts,  as  i,  2,  3,  etc. 
With  radii  A  i,  B  i,  on  s  and  s  as  centres,  describe 
arcs,  and  repeat  this  operation  for  all  other  divis- 
ions i,  2,  3,  etc.,  and  these  points  of  intersection  will 
give  line  of  curve. 

To  Ascertain  Centre  and  Two  Diameters  of  an  Ellipse. 

—Fig.  4:0. 

Let  A  B  c  u  be  diameters  of  an  Ellipse. 

Draw  at  pleasure  two  lines,  q  q,  o  m,  parallel  to 
each  other,  and  equidistant  from  A  and  B ;  bisect 
them  in  points  few,  and  draw  line  u  r;  bisect  it 
in  s,  and  upon  s,  as  a  centre,  describe  a  circle  at 
pleasure,  as/Z  v,  cutting  figure  in  points  fv. 

Draw  right  line/u;  bisect  it  in  i,  and  through 
points  i  s  draw  greatest  diameter  A  B,  and  through 
centre, s,  draw  least  diameter  CM,  parallel  tofv. 

To  Descrilbe  an  Ellipse  approximately  toy  Circular  Arcs. 
c  -Fig.  41. 

Set  off  differences  of  axes  from  centre  o  to  a  and 
c  on  o  A  and  o  C;  draw  a  c  and  bisect  it,  and  set  off' 
its  half  to  r ;  draw  r  s  parallel  to  a  e,  set  off'  o  n 
equal  to  o  ?•,  connect  n  s,  and  draw  parallels  r  TO, 
n  TO;  from  ?»,  with  radii  m  s  and  s  ?«,  describe  arcs 
through  C  and  D,  and  from  n  and  r  describe  arcs 
through  A  and  B. 

NOTE.— This  method  is  not  satisfactory  when  con- 
jugate axis  is  less  than  two  thirds  of  transverse  axis. 

Semi-  Elliptic     Arc     -with    Three 
Centres.— Fig.   4:2. 

Draw  A  M,  B  M,  parallel  respectively  to  B  C,  A  C, 
meeting  in  M.  Draw  M  Oi  perpendicular  to  A  B, 
cutting  B  BI  in  0*,  and  A  A*  in  O3.  Find  a  mean 
proportional  (B  D)  between  C  A  and  C  B.  (This 
may  be  done  by  marking  B  c  on  M  B  produced,  equaj 
to  B  C  and  describing  a  semicircle  on  M  c  cutting 
B  C  in  D).  Make  A  E  equal  to  B  D.  With  centres 
O1,  03,  and  radii  O1  D.  03  E,  describe  arcs  intersect- 
ing in  O2.  Then  O«,  O2,  03  are  points  which  can  be 
used  as  centres  for  successive  arcs  of  the  required 
curve  —A.  L.  Lucas,  Ass't  Eng'r  U.  S.  Dep't. 


GEOMETRY. 


227 


43. 


To  Con.stru.et   an.  Ellipse    from   Two 

Circles.— Fig.  43. 

Describe  two  semicircles,  as  A  B,  C  D,  diameters  of 
which  are  respectively  lengths  of  major  and  minor 
axes.     The  intersection  of  the  horizontal  and  vertical 
_    lines  drawn  from  any  radial  line  will  give  a  point  in 
D  the  curve  C  D. 

To  Construct  an  Ellipse,  -when  T\vo 
Diameters  are  Griven.—  Fig.  44. 

Make  c  o  and  A  v  equal  to  each  other,  but  less 
than  half  breadth.  Draw  vo,  and  from  its  centre  t 
draw  and  extend  perpendicular  at  i  to  d,  draw  dvm, 
make  B  u  •=.  A  v,  draw  d  u  r,  from  u  and  v  describe 
B  r  and  A  m,  from  d  describe  m  c  r,  extend  c  z  to  *, 
and  it  will  be  centre  for  other  half  of  figure. 

To    Construct    an    Ellipse   t>y    Ordinates.— Fig.  45. 

Divide  semi  transverse  axis,  as  A  6,  into  8  or  10 
divisions,  as  may  be  convenient,  and  erect  ordi- 
nates,  the  lengths  of  which  are  equal  to  semi-con- 
jugate, multiplied  by  the  units  for  each  division  as 
follows: 


Eighths. 

1  — .48412   5  —  .92703 

2  — .  661  44   6  —  .  968  24 

3  — .78063    7—  .99216 

4  — .86603    8  —  i 


Divisions. 
i —  -435385 

2  — .6 

3  — .71414 
4-.8 


Tenths. 

5  — .86602 

6  — .91651 

7  —  -95394 

8  —  .97979 


9—  .99499 
10 —  i 


C 


To  Construct  an  Ellipse  -when  Diameters  do  not  Inter- 
sect at  Right  Angles.— Fig.  46. 

Let  A  B  and  C  D  be  given  diameters. 

Draw  boundary  lines  parallel  to  diameters, 
divide  longest  diameter  into  any  number  of 
equal  parts,  and  divide  shortest  boundary  lines 
into  same  number  of  equal  parts. 

From  one  end  of  shortest  diameter,  D,  draw 
radial  lines  through  divisions  of  longest  diame- 
ter, and  from  opposite  end,  C,  draw  radial  lines 
to  divisions  on  shortest  boundary  lines  ;  the 
intersection  of  these  lines  will  give  points  in  the 
curve. 


To   Describe   a   Grothic  Arc.— Fig.  47'. 

Take  line  A  B.    At  points  A  and  B  draw  arcs  B  a  and  A  c, 
and  it  will  describe  arc  required. 

[B 

To  Describe  an  Elliptic  Arc,  Chord  and  Height  being 
given.— Fig.  48. 

Bisect  A  B  at  c  ;  erect  perpendicular  A  g,  and 
draw  line  q  D  equal  and  parallel  to  A  c. 

Bisect  A  c  and  A  q  in  r  and  n;  make  c  I  equal  to 
c  D,  and  draw  line  I  r  q ;  draw  also  line  n  s  D ;  bisect 
s  D  with  a  line  at  right  angles,  and  cutting  line 
c  D  at  o;  draw  line  o  q;  make  cp  equal  to  c  A:,  and 
draw  line  o  p  i. 

Then,  from  o  as  a  centre,  with  radius  o  D,  describe 
arc  s  D  i;  and  from  k  and  p  as  centres,  with  radius 
A  k,  describe  arcs  A  s  and  B  i. 


228 


GEOMETRY. 


To   Describe    a   Grothie   Arc.— Figs.  4:9   and   SO. 

Divide  line  A  B  into  three  equal  parts,  e  c  ;  from  points 
A  and  B  let  fall  perpendiculars  A  o  and  B  r,  equal  in  length 
to  two  of  divisions  of  line  A  B; 
draw  lines  o  h  and  r  g  from  points 
e,  c;  with  length  of  cB,  describe  arcs 
Ag  and  B/i,  and  from  points  o  and  r 
describe  arcs  g  i  and  i  h.  (Fig.  49. )  ^1 

Or,  divide  line  A  B  into  three 
equal  parts  at  a  and  b,  and  on  points 
A,  a,  b,  and  B,  with  distance  of  two 
divisions,  make  four  arcs  intersect- 
ing at  c  and  o. 

Through  points  c,  o,  and  divisions  a,  &,  draw  lines  c/and  o  e,  on  points  a  and  fc 
describe  arcs  A  e  and  B/,  and  on  points  c  o  arcs/ 5  and  e  s.     (Fig.  50.) 


Cycloid,   and.   Epicycloid. 
To   Describe    a    Cycloid.— Fig.  si. 

When  a  circle,  as  a  wheel,  rolls  over  a 
straight  right  line,  beginning  as  at  A  and 
ending  at  B,  it  completes  one  revolution, 
and  measures  a  straight  line,  A  B,  exactly 
equal  to  circumference  of  circle  cer,  which 
is  termed  the  generating  circle,  and  a  point 
or  pencil  fixed  at  point  r  in  circumference 
traces  out  a  curvilinear  path,  A  r  B,  termed  a 
cycloid.  A  B  is  its  base  ar?d  cr  its  axis. 

Place  generating  circle  in  middle  of  Cy- 
cloid,  as  in  figure;  draw  a  line,  m  n,  paral- 
lel to  base,  cutting  circle  at  e;  and  tangent 
n  i  to  curve  at  point  n.  The  following  are  some  of  properties  of  Cycloid : 


Horizontal  line  e  n  =  arc  of  circle  e  r. 
Half-base  A  c— half-circumference  cer. 
Arc  of  Cycloid  r  n  —  twice  chord  r  e. 
Half-arc  of  cycloid  Ar=twice  diameter 
of  circle  r  c. 


Or,  whole  arc  of  Cycloid  A  r  B  =  four 
times  axiscr. 

Area  of  Cycloid  A  r  B  A  =  three  times 
area  of  generating  circle  re. 

Tangent  n  i  is  parallel  to  chord  e  r. 


To   Describe    Cnrve   of  a   Cycloid.— Fig.  £53. 

On  an  indefinite  line,  AB,  set  off  co- 
circumference  of  generating  circle,  di- 
vide this  line  into  any  number  of  equal 
parts  (8  in  figure),  and  at  points  of  divis- 
ion erect  perpendiculars  thereto.  Upon 
eacl1  of  these  lines  describe  a  circle  = 
generating  circle.  On  c  i  take  i  x  — 
.25  C  i,  and  with  «  as  a  centre,  with  radius  a?  c  =  .75  c  x,  describe  an  arc  cutting  circle 
at  x';  from  2  on  next  circle,  with  two  distances  of  i  i',  measured  as  chords,  cut 
circle  at  2';  from  3  on  next  circle,  with  three  distances  of  i  i',  cut  circle  at  3',  and 
proceed  in  like  manner  from  each  side  until  figure  is  complete. 


To   Describe    an   Interior   Epicycloid    or    Hypocycloid.— 
Fig.  S3. 

If  generating  circle  is  rolled  on  inside  of  fundamental 
circle,* as  in  Fig.  53,  it  forms  an  interior  epicycloid,  or 
hypocycloid,  A  c  B,  which  becomes  in  this  case  nearly  a 
straight  line.  Other  points  of  reference  in  figure  cor- 
respond to  those  in  Fig.  51.  When  diameter  of  generat- 
ing circle  is  equal  to  half  that  of  fundamental  circle, 
epicycloid  becomes  a  straight  line,  being  diameter  or 
the  larger  circle. 

S«e  explanation,  Fig.  54. 


GEOMETRY. 


229 


To  Describe  an  Exterior  Epicycloid.— 
Fig.  54. 

An  Epicycloid  differs  from  a  Cycloid  in  this,  that  it  is 
generated  by  a  point,  o'",  in  one  circle,  o  r,  rolling  upon 
circumference  of  another,  A  r  «,  instead  of  upon  a  right 
line  or  horizontal  surface,  former  being  generating  circle 
and  latter  fundamental  circle. 

Generating  circle  is  shown  in  four  positions,  in  which 
generating  point  is  indicated  by  oo'o^'o'".  A.O'"  * 
is  an  Epicycloid. 

Involute. 
To    Describe   an   Involute.— Fig.  55. 

Assume  A  as  centre  of  a  circle,  b  c  o ;  a  cord  laid  partly 
upon  its  circumference,  as  be;  then  the  curve  eimn, 
described  by  a  tracer  at  end  of  cord,  when  unwound  from 
a  circle,  is  an  involute. 

This  curve  can  also  be  defined  by  a  batten,  x,  rolling  on 
a  circle,  as  s  u. 

3?ara"bola. 

To    Construct    a    3Para"bola    "by    Ordinates    or 
-Abscissa. — Figs.  56    and    57". 

By  Ordinates. 

Divide  ordinate  a  b  into  10  equal  parts,  and  erect  perpendicu- 
lars, length  of  which  will  be  determined  by  multiplying  abscissa 
c  by  respective  units  for  each  perpendicular,  as  follows: 
Divisions. 


3  —  -Si      5  — -75 
4-64      6-. 84 


7  — .91 

8  — .96 


9—  -99 

10— I 


By  Abscissa. 

Divide  abcissa  a  c  into  8  or  10  equal  parts,  as  may  be  convenient, 
and  draw  ordinates  thereto,  the  lengths  of  which  will  be  deter- 
mined by  multiplying  half  ordinate  a  &  by  respective  units  for 


each  ordinate,  as  follows: 
Eighths. 


Divisions. 


5—  -79°  57 


7— 

8  —  i 


-93541 


6—  .7746 

7—  .83666 

8—  .89443 

9—  .94868 


Tenths. 

1  —  31623 

2  —  .44721 

3  —  -54772 

4  —  .63245 

5  —  .70711  10  —  i 

"With    a    Sq.-u.are    and   Cord.—  Fig.  58. 

Place  a  straight  edge  to  directrix  A  B,  and  apply  to  it  a 
square,  c  o. 

Attach  to  end  o  end  of  a  cord  equal  to  o  A,  and  attach  other 
end  to  focus  e;  slide  square  along  straight  edge,  maintaining 
cord  taut  against  edge  of  square,  by  a  point  or  pencil,  and  curve 
will  be  traced.  (Fig.  58.) 

When      Height      and  mr._r_^  ___  r.^ 
Base      are     given.  —      j     '  -L-~y 
Fig.  59.  •"]-"/ 

Assume  A  B  axis  and  c  d  a  double  ordinate  or  base.      '    ' 
Through  A  draw  m  n  parallel  to  c  d,  and  through  c 
and  d  draw  cm,dn,  parallel  to  axis  A  B.    Divide  c  m, 
d  n  into  any  number  of  equal  parts,  as  at  a  c  e  o,  also 
cB,Bd,  into  a  like  number  of  parts.   Through  points  c  X 
i,  2,  3,  and  4  draw  lines  parallel  to  axis,  and  through 
a  c  e  o  draw  lines  to  vertex  A,  cutting  these  perpendiculars,  and  through  these  pointe 
curve  may  be  traced.     (Fig.  59.) 


c  ' 


GEOMETRY. 


To  Describe  Curve  of  a  Parabola,  Base  and 

Height   being   given.— Fig.  6O. 
Draw  an  isosceles  triangle,  as  a  6  d,  base  of  which  shall  be  equal 
to,  and  its  height,  c  6,  twice  that  of  proposed  parabola.     Divide 
each  side,  a  &,  d  6,  into  any  number  of  equal  parts ;  then  draw  lines, 
i  i,  2  2,  3  3,  etc.,  and  their  intersection  will  define  curve.  (Fig.  60.) 

To  Describe  a  Parabola,  any  Ordinate  to  Axis 
and  its  Abscissa  being  given.— Fig.  61. 

Bisect  ordinate,  as  A  o  in  r;  join     m c ji 

B  r,  and  draw  r  s  perpendicular  to  it, 
meeting  axis  continued  to  s.    Set  off 
a    B  c,  B  e,  each  equal  to  o  s;  draw  m 
c  u  perpendicular  to  B  s,  then  m  w  is  directrix  and 
B  e  focus;  through  e  and  any  number  of  points,  i,  i, 
i,  etc.,  in  axis,  draw  double  ordinates  v  i  v,  and  on 
centre  e,  with  radii  e  c,  i  c,  etc.,  cut  respective  or- 
dinates at  v  v,  etc.,  and  trace  curve  through  these 
points. 
NOTE. — Line  vev  passing  through  focus  is  parameter. 

Spiral. 
To   Draw  a  Spiral   about   a   given   Point.— 

Fig.  62. 

Assume  c  the  centre.  Draw  A  h,  divide  it  into  twice  number 
of  parts  that  there  are  to  be  revolutions  of  line.  Upon  c  de- 
scribe re, os,  Ah,  and  upon  e  describe  rs,os,  etc. 

Hyperbola. 

To    Describe    a    Hyperbola,  Transverse    and    Conjugate 
Diameters    being   given.— Fig.  63. 

Let  A  B  represent  transverse  diameter,  and  C  D 
conjugate. 

Draw  C  e  parallel  to  A  B,  and  e  r  parallel  to  C  D ; 
draw  o  e,  and  with  radius  o  e,  with  o  as  a  centre, 
describe  circle  F  e  r,  cutting  transverse  axis  pro- 
duced in  F  and  /;  then  will  F  and  /be  foci  of  fig- 
ure. 

In  o  B  produced  take  any  number  of  points,  n,  w, 
etc.,  and  from  F  and /as  centres,  with  A  n  and  B  n 
as  radii,  describe  arcs  cutting  each  other  in  s,  s, 
etc.    Through  s,  s,  etc. ,  draw  curve  ssssBssss. 
NOTE. — If  straight  lines,  as  o  ey  and  o  r  y,  are  drawn  from  centre  o  through  ex- 
tremities e  r,  they  will  be  asymptotes  of  hyperbola,  property  of  which  is  to  ap- 
proach continually  to  curve,  and  yet  never  to  touch  it. 

When  Foci  and  Conjugate  Axis  are  given. — Let  F  and /be  foci,  and  C  D  conjugate 
axis,  as  in  preceding  figure. 

Through  C  draw  g  C  e  parallel  to  F  and  /;  then,  with  o  as  a  centre  and  o  F  as  a 
radius,  describe  an  arc  cutting  g  C  e  at  g  and  e;  from  these  points  let  fall  perpen- 
diculars upon  line  connecting  F  and/  and  part  intercepted  between  them,  as  A  B, 
will  be  transverse  axis. 

Catenary. 

To  Delineate   a   Catenary,  Span  and  Versed  Sine  being 
given.  —  Fig.  64.     ( W.  Hildenbrand. ) 

Divide  half  span,  as  A  B,  into  any  required 
number  of  equal  parts,  as  i,  2,  3,  and  let  fall  B  C 
and  A  o,  each  equal  to  versed  sine  of  curve ;  divide 
Ao  into  like  number  of  parts,  i',  2',  3',  as  A  B. 
Connect  C  i',  C  2',  and  C  3',  and  points  of  intersec- 
tion of  perpendiculars  let  fall  from  A  B  will  give 
points  through  which  curve  is  to  be  drawn. 

Or,  suspend  a  finely  linked  chain  against  a  ver- 
tical plane,  trace  curve  from  it  on  the  plane  in  accordance  with  conditions  of  given 
length  and  height,  or  of  given  width  or  length  of  arc. 
NOTE.— For  other  methods  see  D,  B,  Clark's  Manual,  pp.  18, 19. 


AREAS   OF   CIRCLES. 


231 


DlAM. 

. 

.OOOI92 

3 

7.0686 

7 

38-4846  ; 

14 

& 

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$ 

7.3662  ; 
7.6699 

% 

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41.2826  , 

^ 

X* 

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7-9798  i 

% 

42.7184  ; 

% 

% 

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% 

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8.618 
8.9462 

\ 

44.1787 
45-6636 
47.1731 

\ 

% 

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%» 

9.2807 

% 

48.7071 

% 

% 

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% 

9.6211 
9.968 

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50.2656 
51.8487 

15 

M 

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% 

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% 

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Y 

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% 

10.679 

% 

55.0884 

/to 

% 

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K 

56.7451 

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% 

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% 

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58.4264 

% 

%t 

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% 

n-7933 

% 

60.1322  i 

% 

% 

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12.177 
12.5664 

K 

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63.6174 

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% 

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% 

% 

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% 

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69.0293 

% 

% 

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% 

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% 

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% 

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14.606 

% 

72.7599 

% 

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I5-033 
15-465 

% 

74.6621 
76.5888 

% 

! 

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% 

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10 

78.54 

17 

Ae 

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%& 

16.349 

% 

80.5158 

% 

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% 

16.8002 

% 

82.5161 

% 

/J6 

1.1075 

% 

17-257 

% 

84.5409 

% 

3^ 

1.2272 

% 

17.7206 

% 

86.5903 

1A 

%& 

1-353 

% 

18.19 

% 

88.6643 

%• 

% 

1.4849 

% 

18.6655 

% 

90.7628 

% 

%& 

1.6229 

% 

19.147 

% 

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% 

K 

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5 

19-635 

ii 

95-0334 

18 

/M 

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x& 

20.129 

<^J 

97-2055 

^3 

% 

2.0739 

% 

20.629 

% 

99.4022 

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% 

2.2365 

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21.135 

% 

101.6234 

% 

M 

2.4053 

% 

21.6476 

% 

103.8691 

% 

% 

2.58 

%& 

22.166 

% 

106.1394 

% 

% 

2.761  2 

% 

22.6907 

% 

108.4343 

% 

M^6 

2.9483 

%B 

23.221 

110.7537 

% 

2 

3.1416 

% 

23-7583 

12 

113.098 

19 

xie 

3.338 

%> 

24.301 

% 

115.466 

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% 

24.8505 

% 

117.859 

% 

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% 

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% 

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% 

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% 

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% 

B* 

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K 

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28.2744 

13 

132-733 

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% 

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K 

% 

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H 

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K 

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% 

34-4717 

145.802 

% 

6.491  8 

% 

35-7848 

% 

148.49 

% 

% 

6.7772 

% 

1  37-1224 

% 

151.202 

% 

232 


AREAS   OP   CIRCLES. 


DlAM. 

AREA. 

DlAM. 

AREA. 

DlAM. 

AKIA. 

DlAM. 

AREA. 

21 

346.361 

28 

6I5-754 

35 

962.115 

42 

1385.45 

% 

350.497 

% 

621.264 

H 

069 

% 

1393-7 

& 

354.657 

y± 

626.798 

% 

975-909 

% 

1401.99 

i 

358.842 

% 

632.357 

% 

982.842 

% 

1410.3 

363-05I 

X 

637.941 

% 

989.8 

y* 

1418.63 

% 

367.285 

% 

643-549 

% 

996.783 

% 

1426.99 

% 

371-543 

% 

649.182 

% 

1003.79 

% 

1435-37 

% 

375.826 

% 

654.84 

% 

1010.822 

% 

1443-77 

22 

380.134 

29 

660.521 

36 

1017.878 

43 

1452.2 

/^ 

384.466 

% 

666.228 

% 

1024.96 

% 

1460.66 

% 

388.822 

x 

671.959 

H 

1032  065 

g 

1469.14 

% 

393-203 

% 

677.714 

% 

1039.195 

% 

1477.64 

% 

397.609 

y* 

683.494 

% 

1046.349 

% 

1486.17 

% 

402.038 

§ 

689.299 

% 

1053.528 

% 

1494-73 

% 

406.494 

% 

695.128 

% 

1060.732 

% 

I503-3 

% 

410.973 

% 

700.982 

% 

1067.96 

% 

I5II-9I 

23 

415477 

30 

706.86 

37 

1075.213 

44 

1520.53 

420.004 

% 

712.763 

% 

1082.49 

% 

1529.19 

/£ 

424.558 

y± 

718.69 

% 

1089.792 

y± 

1537-86 

% 

429'135 

% 

724  642 

% 

1097.118 

% 

1546.56 

% 

433-737 

% 

730.618 

i 

1104.469 

% 

1555.29 

% 

438-364 

$ 

736.619 

1111.844 

% 

1564.04 

% 

443.015 

742.645 

1119.244 

H 

1572.81 

% 

447.69 

% 

748.695 

I 

1126.669 

% 

1581.61 

24 

452-39 

31 

754.769 

38 

1134.118 

45  1 

I590-43 

457-115 

760.869 

Ys 

1141.591 

1599.28 

% 

461.864 

% 

766.992 

% 

1149.089 

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1608.  16 

% 

466.638 

% 

773-14 

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1156.612 

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471.436 

% 

779-3!3 

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1164.159 

•& 

1625.97 

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476.259 

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785-51 

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1171.731 

% 

1634.92 

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481.107 

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791.732 

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% 

1643.89 

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485-979 

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797.979 

% 

1186.948 

% 

1652.89 

25 

490.875 

32i 

804.25 

39 

1194.593 

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1661.91 

495.796 

810.545 

1202.263 

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1670.95 

% 

500.742 

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1209.958 

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1225.42 

% 

1698.23 

% 

515.726 

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1233.188 

% 

1707.37 

% 

520.769 

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842.391 

% 

1240.981 

K 

1716.54 

% 

525.838 

% 

848.833 

% 

1248.798 

% 

I725-73 

26 

530-93 

33 

855.301 

40 

1256.64 

47 

1734-95 

X^j 

536.048 

% 

861.792 

H 

1264.506 

% 

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% 

54I-I9 

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1790.76 

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567-267 

% 

901.259 

1312.219 

% 

1800.15 

27 

572.557 

34 

907.922 

41 

1320.257 

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1809.56 

K 

577.87 

914.611 

H 

1328.32 

% 

1819 

M 

583.209 

% 

921.323 

% 

1336.407 

y± 

1828.46 

X 

588.571 

% 

928.061 

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1344.519 

% 

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1352.655 

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599-371 

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1360.816 

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604.807 

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1369.001 

% 

1866.55 

K 

610.268 

% 

955-255 

% 

1377.211 

% 

1876.14 

AREAS    OF   CIRCLES. 


233 


DlAM. 

ARIA. 

DIAM.     AREA. 

DIAM. 

ARIA. 

DIAM. 

ARIA. 

49 

1885.75 

56 

2463.01 

63 

3"  7-25 

70 

3848.46 

% 

1895.38 

% 

2474.02 

% 

3129.64 

% 

3862.22 

U 

1905.04 

« 

2485.05 

& 

3142.04 

% 

3876 

1914.72 

% 

2496.  1  1 

% 

3I54-47 

M 

3889.8 

% 

1924.43 

X 

2507.19 

x 

3166.93 

K 

3903-63 

% 

1934.16 

% 

2518.3 

% 

3I79-4I 

% 

39I7.49 

% 

1943.91 

% 

2529-43 

% 

3191.91 

% 

393I-37 

% 

1953-69 

% 

2540.58 

% 

3204.44 

% 

3945-27 

So 

1963-5 

57 

2551-76 

64 

3217 

7* 

3959-2 

X 

1973-33 

% 

2562.97 

X 

3229.58 

% 

3973-15 

% 

1983.18 

& 

2574.2 

M 

3242  18 

% 

3987.13 

% 

1993.06 

% 

2585-45 

H 

3254.81 

% 

4001.13 

X 

2002.97 

X 

2596-73 

X 

3267.46 

B 

4015.16 

2012.89 

% 

2608.03 

/N* 

3280.14 

78 

4029.21 

a| 

2022.85 

% 

2619.36 

% 

3292.84 

74 

4043.29 

H 

2032.82 

% 

2630.71 

% 

3305-56 

% 

4057-39 

51 

2042.83 

58 

2642.09 

65 

33I8.3I 

72 

4071.51 

% 

2052.85 

2653.49 

333L09 

X 

4085.66 

¥ 

2062.9 

% 

2664.91 

% 

3343-89 

% 

4099.84 

2072.98 

% 

2676.36 

% 

3356.71 

% 

4114.04 

% 

2083.08 

X 

2687.84 

% 

3369-56 

X 

4128.26 

% 

2093.2 

2699.33 

% 

3382.44 

% 

4142.51 

% 

2103.35 

M 

2710.86 

% 

3395-33 

% 

4156.78 

% 

2113.52 

% 

2722.41 

% 

3408.26 

% 

4171.08 

S2 

2123.72 

59 

2733-98 

66 

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2780.51 

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3473-24 

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4242.93 

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2175.08 

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2792.21 

% 

3486.3 

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4257-37 

% 

2185.42 

% 

2803.93 

% 

3499-4 

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4271.84 

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% 

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% 

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% 

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% 

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% 

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54 

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61 

2922.47 

68 

3631.69 

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2300.84 

% 

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4432.61 

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2311.48 

% 

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3658.44 

% 

4447-38 

2322.15 

% 

2958.52 

% 

3671.86 

% 

4462.16 

% 

2332-83 

% 

2970.58 

H 

3685.29 

y* 

4476.98 

% 

2343-55 

2982.67 

% 

3698-76 

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4491.81 

% 

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% 

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% 

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% 

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% 

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% 

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% 

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% 

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3092.56 

% 

3821.02 

% 

4626  45 

% 

2452.03 

\  % 

3104.89 

% 

3834-73 

% 

464L53 

234 


AREAS    OF    CIRCLES. 


77 

4656.64 

84i 

554J-78 

9i 

6503.9 

98 

7542-c 

H 

4671.77 

5558.29 

H 

6521.78 

% 

7562.: 

X 

4686.92 

% 

5574-82 

y± 

653968 

% 

758i.. 

% 

4702.1 

% 

559I-37 

M 

6557.61 

% 

7600.? 

X 

47I7-31 

X 

5607.95 

M 

6575.56 

X 

7620.  i 

4732-54 

% 

5624.56 

% 

6593-54 

% 

7639-. 

% 

4747-79 

M 

5641.18 

% 

6611.55 

% 

7658.* 

% 

4763.07 

H 

5657-84 

% 

6629.57 

% 

7678.2 

78 

4778.37 

85 

5674-51 

92 

6647.63 

99 

7697.- 

4793-7 

5691.22 

y* 

6665.7 

K 

7717.1 

3€ 

4809.05 

3^ 

5707.94 

% 

66838 

H 

7736.< 

H 

4824.43 

% 

5724.69 

% 

6701.93 

% 

7756.] 

% 

4839.83 

% 

574I-47 

X 

6720  08 

% 

7775( 

4855.26 

% 

5758.27 

6738.25 

% 

7795- 

% 

4870.71 

% 

5775-1 

% 

675645 

M 

7814.' 

% 

4886.18 

% 

579L94 

% 

6774.68 

% 

7834< 

79 

4901.68 

86 

5808.82 

93 

6792.92 

100 

7854 

X 

4917.21 

X 

5825.72 

% 

68u.2 

H 

7893-: 

¥ 

4932.75 

X 

5842.64 

% 

6829.49 

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7932.' 

4948.33 

% 

5859.59 

% 

6847.82 

% 

7972- 

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6866.16 

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4979-55 

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6884.53 

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4995-19 

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8131-: 

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5026.56 

87 

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% 

5042.28 

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% 

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5996.05 

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8291.  c 

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103 

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6030.41 

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5121.25 

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6047.63 

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5232.84 

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7181.81 

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% 

6186.25 

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7200.6 

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% 

5264.94 

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6203.69 

% 

7219.41 

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8783.1 

82 

5281.03 

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& 

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% 

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7294.91 

% 

8950.C 

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6291.25 

% 

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107 

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% 

5361.84 

6308.84 

% 

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% 

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% 

6326.45 

% 

735L79 

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% 

5394.34 

% 

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% 

7370.79 

% 

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83 

5410.62 

90 

6361.74 

97 

7389-83 

108 

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% 

9203-: 

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5443-26 

% 

6397.13 

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6414  86 

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K 

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% 

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% 

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6468.21 

% 

7504.55 

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% 

5525.3 

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6486.04 

% 

7523-75 

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9460.] 

AREAS    OF    CIRCLES. 


235 


no 

9  503-34 

1  20 

11309.76 

130 

13273.26 

140 

1539384 

y± 

9546.59 

y± 

"356.93 

y± 

13324.36 

# 

15448.87 

x 

9589-93 

% 

n  404.2 

H 

13375-56 

H 

15503-99 

% 

9633-37 

% 

II45I-57 

% 

13426.85 

% 

15559-22 

in 

9676.91 

121 

11499.04 

131 

13478.25 

141 

15614-54 

y± 

9  72o-55 

X 

1  1  546.61 

y± 

13529-74 

¥ 

15669.96 

K 

9  764-29 

K 

11594.27 

y* 

13581.33 

15  725-48 

H 

9808.12 

H 

11642.03 

% 

13633.02 

% 

15781.09 

112 

9852.06 

122 

11689.89 

132 

13684.81 

142 

15836.81 

M 

9896.09 

u 

1I737-85 

X 

13736.69 

u 

15892.62 

2 

9940.22 

X 

11785.91 

m 

13788.08 

X 

15948.53 

% 

9984.45 

X 

11834.06 

% 

13840.76 

% 

16004.54 

1J3 

10028.77 

123 

11882.32 

133 

13892.94 

143 

16060.64 

y± 

10073.2 

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11930.67 

X 

13945.22 

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16116.85 

y* 

10117.72 

X 

11979.12 

y* 

13997.6 

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16173.15 

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10162.34 

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12027.66 

% 

14050.07 

% 

16229.55 

114 

10207.06 

124 

12076.31 

134 

14  102.64 

144 

16  286.05 

y± 

10251.88 

% 

12  125.05 

X 

14  155-31 

Sf 

16342.65 

8 

10296.79 

X 

I2I73.9 

K 

14208.08 

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i639935 

% 

10341.8 

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12222.84 

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14253.09 

% 

16456.14 

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13019.23 

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15120.18 

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17378.2 

119 

ii  122.05 

129 

13069.84 

139 

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149 

17436.67 

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n  168.83 

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3^ 

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15284.08 

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11262.69 

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17671.5 

To  Compvite  Area  of  a  Circle  greater  than  any  in  Tat>le. 

RULE. — Divide  dimension  by  two,  three,  four,  etc.,  if  practicable  to  do  so, 
until  it  is  reduced  to  a  diameter  to  be  found  in  table. 

Take  tabular  area  for  this  diameter,  multiply  it  by  square  of  divisor,  and 
product  will  give  area  required. 

EXAMPLE.— What  is  area  for  a  diameter  of  1050? 

1050-4-7  =  150;  tab.  area,  150  =  17671.5,  which  x  72  =  865 903.5,  area. 

To  Compute  Area  of*  a  Circle  in  Feet   and  Inches,  etc., 
"by   preceding    Table. 

RULE.— Reduce  dimension  to  inches  or  eighths,  as  the  case  may  be,  and 
take  area  in  that  term  from  table  for  that  number. 


AREAS   OF   CIRCLES. 


Divide  this  number  by  64  (square  of  8)  if  it  is  in  eighths,  and  quotient  will 
give  area  in  inches,  and  divide  again  by  144  (square  of  12)  if  it  is  in  inches, 
and  quotient  will  give  area  in  feet. 

EXAMPLE. — What  is  area  of  i  foot  6.375  ms<? 

i  foot  6. 375  ins.  =  18. 375  ins.  =  147  eighths.  Area  of  147  =  16  971.71,  which  -f-  64 
=  265. 181 25  ins.;  and  by  144  =  1.84 125  feet. 

To  Compute   Area  of  a  Circle  Composed,   of  an    Integer 
and.    a   Fraction. 

RULE. — Double,  treble,  or  quadruple  dimension  given,  until  fraction  is  in- 
creased to  a  whole  number,  or  to  one  of  those  in  the  table,  as  >g,  %,  etc., 
provided  it  is  practicable  to  do  so. 

Take  area  for  this  diameter ;  and  if  it  is  double  of  that  f  01  which  area  is 
required,  take  one  fourth  of  it ;  if  treble,  take  one  sixteenth  of  it,  etc. 

EXAMPLE.— Required  area  for  a  circle  of  2.1875  ins. 

2.1875  x  2  =  4.375,  area  for  which  =  15.0331,  which -1-4  =  3. 758  *»*. 

When  Diameter  is  composed  of  Integers  and  Fractions  contained  in  Table. 

RULE. — Point  off  a  decimal  to  a  diameter  from  table,  and  add  twice  as 
many  figures  or  ciphers  to  the  right  of  the  area  as  there  are  figures  cut  off 
from  the  diameter. 

EXAMPLE  i.— What  is  area  of  9675  feet  diameter? 

Area  of  96. 75  =  7351.79 ;  hence,  area  =  73  517  goo  feet. 
2.— What  is  area  of  24  375  feet  diameter? 

Area  of  2. 4375  —  4. 6664 ;  hence,  area  =  466  640  ooo  feet. 

To   Ascertain.   Area    of  a   Circle    as   3OO,  3OOO,  etc.,  not 
contained   in   Table. 

RULE.— Take  area  of  3  or  30,  and  add  twice  the  excess  of  ciphers  to  the 
result. 

EXAMPLE.— What  is  area  of  a  circle  3000  feet  in  diameter? 

Area  of  30  =  706. 86,  hence  area  of  3000  =  7  068  600  feet. 

To   Compute   Area  of  a   Circle   toy   Logarithms. 
RULE.— To  twice  log.  of  diameter  add  7.895091  (log.  of  .7854),  and  sum 
is  log.  of  area,  for  which  take  number. 

EXAMPLE.— What  is  area  of  a  circle  1200  feet  in  diameter? 

Log.  1200  x  2  -j-J 895  091  =  6.158  362  +  ^895091  =  6.053453,  and  number  for 
which  =  1 130  976  feet. 


Diam. 

Aret 

Area. 

is  of 

Diam. 

Birming 

Area. 

lam   ^ 

Diam. 

Wire   Q-a 

Area. 

uge. 

Diam. 

Area. 

No. 

Sq.  Inch. 

No. 

Sq.  Inch. 

No. 

Sq.  Inch. 

No. 

Sq.  Inch. 

I 

.070686 

IO 

.014  103 

19 

.001  385 

28 

.000154 

2 

•063347 

II 

.011309 

20 

.000962 

29 

.000  133 

3 

.052  685 

12 

.009331 

21 

.000804 

3° 

.OOO  113 

4 

.044  488 

13 

.007  088 

22 

.000616 

3i 

.000  078 

5 

.038013 

14 

.005411 

23 

.000491 

32 

.000064 

6 

•032365 

15 

.004071 

24 

.00038 

33 

.00005 

7 

.025  447 

16 

.003318 

25 

.000314 

34 

.000038 

8 

.021  382 

17 

.002  642 

26 

.OOO  254 

35 

.00002 

9 

.017203 

18 

.001  886 

27 

•O0020I 

36 

.000013 

CIRCUMFERENCES   OF   CIRCLES. 


237 


Ci 

DlAM. 

rcnmfe 

ClRCUM. 

rence 

DlAM. 

s   of  Circles, 

ClRCUM.       I  i        DlAM. 

from  ^ 

ClRCUM. 

^  to  1 

DlAM. 

50. 

ClRCUM. 

^ 

^.04909 

3 

9.4248 

8 

25.1328 

15 

47.124 

jy 

9.62II 

X 

25.5255 

47.5167 

/te 

.0961 

X 

9-8I75    i 

% 

25.9182 

% 

47.9094 

& 

.19635 

/M 

10.014 

% 

26.3IO9    i                 % 

48.3021 

K 

7Q2  7 

% 

10.2102   i 

% 

26.7036 

% 

48.6948 

/o 

%& 

10.406      I 

% 

27.0963 

49.0875 

_ 

-589 

% 

10.6029   ' 

%     27.489 

49.4802 

% 

.7854 

%& 

10.799  ! 

%    27.8817 

49.8729 

& 

-981  75 

M 

10.9956   | 
II.I9I 

9          28.2744 
%      28.6671 

16 

50.2656 
50.6583 

% 

1.1781 

11.3883   j 

%      29.0598 

/€ 

5L05I 

•y 

1.37445 

11.584 

%      29.4525 

M 

5L4437 

18 

29.8452 

X 

51.8364 

A 

1.5708 

% 

11.977 

ff 

30.2379 

% 

52.2291 

%L 

1.767  15 

% 

12.1737 

M 

30.6306 

% 

52.6218 

6/ 

% 

12.369 

% 

3I-0233 

% 

53-0145 

/B 

•9°35 

4 

12.5664 

10 

3I.4I6 

17 

53.4072 

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2.15985 

12.762 

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31.8087 

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53-7999 

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12.9591 

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32.2014 

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54.978 

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2.7489 

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13-547 

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33-3795 

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55.3707 

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2-945  25 

r 

13-7445 
13-94 

% 

33-7722 
34.1649 

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55.7634 
56.1561 

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3.1416 

% 

14.1372 

ii 

34-5576 

18  8 

56.5488 

3-3379 

%& 

H-333 

X 

349503 

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56.0415 

3-5343 

% 

14.5299 

% 

35-343 

3^ 

57.3342 

3.7306 

% 

I4'725 

% 

35-7357 

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57.7269 

/? 

% 

14.9226 

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36.1284 

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58.1196 

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4-1233 

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15.119 

36-5211 

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58.5123 

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JB 

I5-3I53 

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36.9138 

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58-905 

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4.516 

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15-5" 

% 

37-3065 

% 

59.2977 

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4.7124 

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15-708 

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37.6992 

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59.6904 

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5-1051 

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16.4934 

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54978 

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61.2612 

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5.6941 

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17.6715 

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39.6627 

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61.6539 

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5.8905 

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18.0642 

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40-0554 

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62.0466 

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6.0868 

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18.4569 

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40.4481 

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62.4393 

2 

6.283  2 

6 

18.8496 

40.8408 

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62.832 

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19.2423 

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8.2467 

22.7766 

% 

44.7678 

% 

66.759 

% 

8-443 

% 

23.1693 

% 

45.1605 

% 

67.1517 

% 

8.6394 

23.562 

% 

45.5532 

H 

67.5444 

% 

8-835  7 

% 

23.9547 

% 

45-9459 

% 

67.9371 

K 

9.032  i 

M 

24-3474 

M 

46.3386 

M 

68.3298 

% 

9.228  4 

K 

24.7401 

% 

46.7313 

% 

68.7225 

238 


CIRCUMFERENCES    OF   CIRCLES. 


DlAM. 

ClECUM. 

DlAM. 

ClHCUM. 

DlAM. 

ClRCUM. 

DlAM. 

CrRcuu. 

22 

69.1152 

29 

9I.IO64 

36 

113.098 

43 

135.089 

% 

69.5079 

% 

91.4991 

% 

H3.49 

^ 

I35.48I 

X 

69.9006 

% 

9I.89I8 

% 

113.883 

y± 

135.874 

% 

70.2933 

% 

92.2845 

% 

114.276 

% 

136.267 

y, 

70.686 

¥ 

92.6772 

X 

114.668 

% 

136.66 

71.0787 

93.0699 

% 

II5.o6l 

% 

137.052 

% 

71.4714 

% 

93.4626 

% 

115-454 

% 

137-445 

% 

71.8641 

% 

93.8553 

% 

115.846 

% 

137.838 

23 

72.2568 

30 

94.248 

37 

Il6.239 

44 

138.23 

72.6495 

94.6407 

% 

1  16.632 

% 

138.623 

/^ 

73.0422 

% 

95-0334 

% 

II7.O25 

% 

139.016 

% 

73.4349 

% 

954261 

% 

II7.4I7 

% 

139.408 

% 

73.8276 

% 

95.8188 

y* 

II7.8I 

!§ 

139.801 

% 

74.2203 

% 

96.2115 

% 

II8.203 

% 

140.194 

H 

74.6I3 

% 

96.6042 

% 

H8.595 

% 

140.587 

% 

75-0057 

% 

96.9969 

% 

118.988 

% 

140-979 

24 

75-3984 

31 

97.3896 

38 

II9.38I 

45 

141.372 

% 

75-  7911 

% 

97.7823 

/& 

119.773 

& 

141.765 

U 

76.1838 

X 

98.175 

M 

I20.I66 

% 

142.157 

%  76.5765 

% 

98.5677 

H 

120.559 

% 

142.55 

76.9692 

% 

98.9004 

% 

120.952 

% 

142.943 

% 

77.3619 

993531 

% 

121.344 

% 

143-335 

M 

77-7546 

% 

99'7458 

% 

121.737 

% 

143.728 

%  \  78.1473 

% 

100.1385 

% 

122.13 

% 

I44.I2I 

25  i  78.54 

32 

100.5312 

3V 

122.522 

46 

144.514 

%  78.9327 

100.9239 

122.915 

% 

144.906 

M  79-3254 

/^ 

101.3166 

% 

123.308 

% 

145.299 

% 

79.7181 

H 

101.7093 

% 

123.7 

% 

145.692 

% 

80.1108 

y* 

IO2.IO2 

y* 

124.093 

X  146.084 

%  80.5035 

% 

102.4947 

% 

124486 

%  146.477 

%  i  80.8962 

% 

102.8874 

M 

124.879 

%  i  146.87 

%  81.2889 

% 

103.2801 

K 

125.271 

%  I  147.262 

26    81.6816 

33 

103.673 

4°1 

125.664 

47    147.655 

>£   82.0743 

104.065 

126.057 

147.048 

3£   82.467 

% 

104.458 

% 

126.449 

% 

148.441 

82.8597 

% 

104.851 

% 

126.842 

% 

I48.833 

M 

83.2524 

* 

105.244 

y> 

127.235 

k 

149.226 

% 

83.6451 

105.636 

% 

127.627 

149.619 

M 

84.0378 

% 

106.029 

M 

128.02 

M 

I5O.OII 

ji 

84-4305 

% 

106.422 

% 

128.413 

% 

150.404 

27 

84.8232 

34 

106.814 

41 

128.806 

48 

150.797 

85.2159 

107.207 

% 

129.198 

X 

151.189 

/^ 

85.6086 

/£ 

107.6 

% 

129.591 

% 

151.582 

M 

86.0013 

% 

107.992 

% 

129.984 

% 

I5I-975 

/^ 

86.394 

% 

108.385 

y* 

130.376 

y* 

152.368 

% 

86.7867 

108.778 

% 

130.769 

% 

152.76 

K 

87.1794 

% 

109.171 

% 

I3I.I62 

% 

I53-I53 

X 

87.5721 

% 

109.563 

% 

I3L554 

% 

153.546 

28 

87.9648 

35 

109.956 

42 

I3L947 

49  \, 

153.938 

/^ 

88.3575 

H 

110.349 

% 

132.34 

H 

154.331 

M 

88.7502 

¥ 

IIO.74I 

% 

132.733 

% 

154.724 

% 

89.1429 

III.I34 

% 

I33-I25 

% 

I55-II6 

K 

89.5356 

K 

III.527 

% 

I33-5I8 

K 

155.509 

% 

89.9283 

% 

III.9I9 

% 

133-9" 

If 

155.902 

% 

90.321 

% 

II2.3I2 

/^ 

134303 

% 

156.295 

X 

90.7137 

% 

112.705 

% 

134.696 

% 

156.687 

CIRCUMFERENCES   OF   CIRCLES. 


239 


ClRCUM. 

DIAM.  (  Cracim.  [1  DIAM. 

ClRCtTM. 

DIAM.  1  CiKcrv. 

157.08 

571 

179.071 

64 

2OI  .062 

71 

223.054 

J57-473 

179.464 

y& 

201.455 

/^J 

.  223.446 

157-865 

% 

179.857 

% 

201.848 

/^ 

223.839 

158.258 

% 

180.249 

%> 

202.24 

% 

224.232 

158.651 

% 

180.642 

y* 

202.633 

224.624 

159.043 

% 

181.035 

% 

203.026 

% 

225.017 

159.436 

% 

181.427 

% 

203.419 

% 

225.41 

159.829 

H 

181.82 

% 

203.8II 

% 

225.802 

160.222 

58 

182.213 

65i 

204.204 

72 

226.195 

160.614 

182.605 

204.597 

X 

226.588 

161.007 

% 

182.998 

x 

204989 

% 

226.981 

161.4 

% 

183.39! 

% 

205.382 

% 

227.373 

161.792 

X 

183.784 

X 

205.775 

% 

227.766 

162.185 

184.176 

206.167 

% 

228.159 

162.578 

% 

184.569 

% 

206.56 

% 

228.551 

162.97 

% 

184.962 

% 

206.953 

% 

228.944 

163.363 

59 

185.354 

66 

207.346 

73 

229.337 

163.756 

185.747 

X 

207.738 

% 

229.729 

164.149 

3^ 

186.14 

X 

208.131 

% 

230.122 

164.541 

% 

186.532 

208.524 

% 

230.515 

164.934 

/& 

186.925 

% 

208.916 

X 

230.908 

165.327 

187.318 

% 

209.309 

% 

231-3 

165.719 

% 

187.711 

% 

209.702 

% 

231.693 

166.112 

% 

188.103 

% 

210.094 

% 

232.086 

166.505 

60 

188.496 

67 

210.487 

74 

232.478 

166.897 

% 

188.889 

X 

210.88 

232.871 

167.29 

% 

189.281 

211.273 

% 

233-264 

167.683 

% 

189.674 

% 

211.665 

% 

233-656 

168.076  ii   3^ 

190.067 

X 

212.058 

% 

234-049 

168.468    % 

190.459 

% 

212.451 

% 

234.442 

168.861  j 

% 

190.852 

% 

212.843 

% 

234.835 

169.254 

% 

191.245 

% 

213-236 

% 

235.227 

169.646 

61 

191.638 

68 

213.629 

75 

235.62 

170.039 

X 

192.03 

X 

214.021 

236.013 

170.432 

192.423 

214.414 

% 

236.405 

170.824 

% 

192  816 

% 

214.807 

% 

236.798 

171.217 

X 

193.208 

% 

215.2 

X 

237.191 

171.61 

193.601 

% 

215-592 

237.583 

172.003 

% 

193-994 

% 

215-985 

% 

237.976 

172.395 

% 

194.386 

% 

216.378 

% 

238.369 

172.788 

62 

194.779 

69 

216.77 

76 

238.762 

173.181 

/>£ 

195.172 

217.163 

% 

239.154 

1  73-573 

% 

195.565 

% 

217-556 

X 

239.547 

173.966 

% 

J95-957 

% 

217.948 

% 

239.94 

174-359 

X 

X 

218.341 

X 

240.332 

174.751 

196.743 

218.734 

240.725 

I75.I44 

% 

i97-I35 

% 

219.127 

% 

241.118 

175.537 

% 

197.528 

% 

219.519 

% 

241.51 

175-93 

63 

197.921 

70 

219.912 

77 

241.903 

176.322 

198.313 

% 

220.305 

X 

242.296 

176.715 

/^ 

198.706 

% 

220.697 

242.689 

177.108 

% 

199.099 

%$ 

221.09 

% 

243.081 

177-5 

X 

199.492 

sv 

221.483 

% 

243.474 

177.893 

% 

199.884 

% 

221.875 

% 

243.867 

178.286 

% 

200.277 

% 

222.268 

% 

244.259 

178.678 

% 

200.67 

y* 

222.661 

% 

244.652 

240 


CIRCUMFERENCES    OF    CIRCLES. 


DlAM. 

ClRCUM.     DlAM. 

ClRCUM. 

DlAM. 

ClRCUM. 

DlAM. 

ClRCUM. 

78 

245.045 

85 

267.036 

92 

289.027 

99 

3II.OI8 

H 

245-437 

« 

267.429 

H 

289.42 

% 

3".4II 

X 

245.83   , 

X 

267.821 

% 

289.813 

% 

311.804 

H 

246.223 

268.214 

% 

290.205 

% 

3I2.I06 

X 

246.6l6 

% 

268.607 

% 

200.598 

% 

312.589 

% 

247.008 

% 

268.999 

% 

290.991 

78 

312.082 

H 

247.401 

% 

209.392 

% 

291.383 

% 

313.375 

% 

247.794 

% 

269.785 

% 

291.776 

% 

313.767 

79 

248.186 

86 

270.178 

93 

292.169 

IOO 

314.16 

y* 

248.579 

% 

270.57 

292.562 

& 

3T4.945 

X 

248.972 

% 

270.963 

/^ 

292.954 

H 

315.731 

% 

249.364 

% 

271.356 

% 

293-347 

H 

3l6.5l6 

% 

249.757 

x 

271.748 

x 

293.74 

IOI 

317.302 

% 

250.15 

% 

272.141 

% 

294.132 

H 

318.087 

% 

250.543 

% 

272.534 

% 

294'525 

X 

318.872 

% 

250.935 

% 

272.926 

% 

294.918 

% 

3I9.658 

so 

251.328 

87 

273.3I9 

94 

295-31 

102 

320.443 

y* 

251.721 

H 

273.712 

H 

295.703 

% 

321.229 

% 

252.113 

X 

274.105 

% 

296.096 

% 

322.014 

% 

252.506 

% 

274.497 

% 

296.488 

H 

322.799 

X 

252.899 

% 

274.89 

y& 

296.881 

IQ3 

323.585 

% 

253.291 

% 

275.283 

% 

297.274 

% 

324.37 

% 

253.684 

K 

275-675 

% 

297.667 

% 

325.156 

% 

254.077 

% 

276.068 

% 

298.059 

H 

325.941 

81 

254-47 

88 

276.461 

95 

298.452 

104 

326.726 

/^ 

254.862 

H 

276.853 

X 

298.845 

H 

327.512 

/£ 

255.255 

H 

277.246 

X 

299.237 

H 

328.297 

% 

255.648 

% 

277.629 

% 

299.63 

H 

329.083 

% 

256.04 

¥ 

278.032 

¥ 

300.023 

105 

329.868 

% 

256.433 

278.424 

300.415 

H 

330.653 

% 

256.826 

% 

278.817 

% 

300.808 

X 

33  1  -439 

K 

257.218 

% 

279.21 

% 

301.201 

% 

332.224 

82 

257.6II 

89 

279.0O2 

96 

3OI«594 

106 

333-01 

H 

258.004 

279-995 

H 

301.986 

% 

333-795 

H 

258.397 

% 

280.388 

X 

302.379 

x  334-58 

% 

258.789 

% 

280.78 

% 

302.772 

%  335-306 

% 

259.182 

% 

281.173 

303.164 

107  336.151 

% 

259.575 

281.566 

6/ 

303.557 

fc 

336.937 

% 

259.967 

% 

281.959 

% 

303.95 

X 

337.722 

K 

260.36 

% 

282.351 

% 

304.342 

H  338.507 

83i 

200.753 

90 

282.744 

97 

304.735 

108    339.293 

26l.I45 

283.137 

H 

305.128 

%   340.078 

% 

261.538 

/^ 

283.529 

% 

305.521 

X  340.864 

% 

26I.93I 

% 

283.922 

%  \  305-913 

%  341.649 

% 

262.324 

% 

284.315 

Jf  306.306 

109 

342.434 

% 

262.716 

% 

284.707 

306.699 

y± 

343-22 

% 

263.109 

% 

285.1 

!% 

307.091 

X 

344.005 

% 

263.502 

% 

285493 

K 

307.484 

% 

344-  79  1 

84 

263.894 

91   i  285.886 

98i 

307.877 

no 

345.576 

H 

264.287 

y%  286.278 

308.27 

% 

346.361 

¥ 

264.68 

%  286.671 

% 

308.662 

X 

347-147 

265.072 

%  287.064 

% 

309055 

% 

347.932 

% 

265.465 

K  287.456 

X 

309.448 

in 

348.718 

% 

265.858 

287.849 

309.84 

H 

349-503 

% 

266.251 

% 

288.242 

% 

3J0.233 

X 

350.288 

n/. 

266.643 

%  \  288.634 

% 

310.626 

% 

35L074 

CIBCUMFEKENCES   OF   CIRCLES. 


241 


DlAM. 

ClECUM. 

DlAM. 

ClRCUM. 

DlAM. 

ClECUM. 

DlAM. 

ClBCTM. 

112 

35L859 

121 

380.134 

130 

408.408 

T39 

436.682 

/£ 

352.645 

/€ 

380.919 

409.192 

/^ 

437.467 

X 

353-43 

x5 

381.704 

ff 

409.979 

/^ 

438.253 

354-215 

K  382.49 

M 

410.763 

M 

439.037 

"3 

355-001 

122     383.275 

131 

4"-55 

140 

439.824 

¥ 

355.786 

%  \  384.061 

u 

412.334 

/€ 

440.608 

356.572 

K  !  384-846 

% 

413.12 

/^ 

441-395 

% 

357-357 

%  I  385.63I 

% 

413.905 

2i 

442.179 

114 

358.142 

123 

386.417 

i32i 

414.691 

141 

442.966 

358.928 

387.202 

415.476 

/^ 

443-75 

% 

359-  7  J3 

i 

387.988 

M 

416.262 

i^ 

444-536 

% 

360.499 

388.773 

M 

417.046 

K 

445-321 

"5 

361.284 

124 

389.558 

133 

417-833 

142 

446.107 

362.069 

390-344 

M 

446.891 

% 

362.855 

K 

391.129 

% 

419.404 

447.678 

H 

363-64 

39I-9I5 

H 

420.188 

M 

448.462 

116 

364.426 

125 

392.7 

134 

420.974 

143 

449-249 

% 

365.211 

* 

393.484 

X 

421.759 

450.033 

% 

365.996 

394.271 

B 

422.545 

K 

450.82 

% 

366.782 

% 

395-055 

M 

423-33 

% 

451-604 

117 

367-567 

126 

395.842 

135 

424.II6 

144 

452.39 

368.353 

3^ 

396.626 

M 

424.9 

/^ 

453-175 

M 

369.138 

K 

397.412 

x^ 

425.687 

K 

453.961 

« 

369.923 

398.197 

M 

426.471 

454-745 

118 

370.709 

127 

398-983 

136 

427^58 

J45 

455.532 

X 

371-494 

* 

399.768 

i 

428.042 

456.316 

% 

372.28 

400.554 

428.828 

% 

457.103 

% 

373.065 

9! 

401.338 

% 

429.613 

146 

458.674 

119 

373.85 

128 

402.125 

137 

430.399 

/^ 

460.244 

374.636 

3^ 

402.909 

431.183 

I47i 

461.815 

x^ 

375-421 

i 

403.696 

/? 

43L97 

463.386 

% 

376.207 

M 

404.48 

^ 

432-754 

148 

464.957 

1  20 

376.992 

129 

405.266 

138 

433-541 

% 

466.528 

xi 

377-777 

406.051 

434-325 

149 

468.098 

x£ 

378.563 

i 

406.837 

)2 

435-  "2 

K 

469.669 

X 

379-348 

407.622 

K 

435.8o6 

150 

471,24 

To  Compute  Circnmference  of*  a  Diameter  greater  tlian 
any   in    preceding    Ta"ble. 

RULE. — Divide  dimension  by  two,  three,  four,  etc.,  if  practicable  to  do  so, 
until  it  is  reduced  to  a  diameter  in  table. 

Take  tabular  circumference  for  this  dimension,  multiply  it  by  divisor, 
according  as  it  was  divided,  and  product  will  give  circumference  required. 

EXAMPLE.— What  is  circumference  for  a  diameter  of  1050? 

I050-j_  7  =  150;  tab.  circum.,  150  =  471.24,  which  X  7  =  3298.68,  circumference. 

To  Compute   Circumference  of*  a   Diameter  in  Feet  and. 
Indies,  etc.,  toy   preceding    Tatole. 

RULE.— Reduce  dimension  to  inches  or  eighths,  as  the  case  may  be,  and 
take  circumference  in  that  term  from  table  for  that  number. 

Divide  this  number  by  8  if  it  is  in  eighths,  and  by  12  if  hi  inches,  and 
quotient  will  give  circumference  hi  feet. 


242 


CIRCUMFERENCES    OF    CIRCLES. 


EXAMPLE. — Required  circumference  of  a  circle  of  i  foot  6.375  ins. 

i  foot  6.375  ins.  =  18.375  ins.  =  147  eighths.  Circum.  of  147  =  461.815,  which  -f-  8 
=  57.727  ins.;  and  by  12  =  4.8106  feet. 

To  Compute  Circumference  for  a  Diameter  composed  of 
an    Integer    and.   a   Fraction. 

RULE. — Double,  treble,  or  quadruple  dimension  given,  until  fraction  is  h> 
creased  to  a  whole  number  or  to  one  of  those  in  the  table,  as  %,  %,  etc.,  pro- 
vided it  is  practicable  to  do  so. 

Take  circumference  for  this  diameter ;  and  if  it  is  double  of  that  for  which 
circumference  is  required,  take  one  half  of  it ;  if  treble,  take  one  third  of  it  ; 
and  if  quadruple,  one  fourth  of  it. 

EXAMPLE.— Required  circumference  of  2.21875  ins. 

2.21875  X  2  =  4.4375,  which  X  2  =  8. 875;  circum.  for  which  =  27. 8817,  which -7-4 
=  6.9704  ins. 

When  Diameter  consists  of  Meyers  and  Fractions  contained  in  Table. 

RULE.— Point  a  decimal  to  a  diameter  in  table,  take  circumference  from 
table,  and  add  as  many  figures  to  the  right  as  there  are  figures  cut  off. 

EXAMPLB.— What  is  circumfe?snce  of  a  circle  9675  feet  in  diameter? 
Circumference  of  96. 75  =  303.95 ;  hence,  circumference  of  9675  =  30  395  feet. 

To    Ascertain    Circumference    for    a    Diameter,  as    5OO, 
COOO,  etc.,  not   contained    in    Table. 

Rule.— Take  circumference  of  5  or  50  from  table,  and  add  the  excess  of 
ciphers  to  the  result. 

EXAMPLE.— What  is  circumference  of  a  circle  8000  feet  in  diameter? 

Circumference  of  80  =  251. 38 ;  hence,  circumference  of  8000  =  25 138  feet. 


To  Compute  Circumference  of*  a  Circle  "by  Logarithms. 

RULE.— To  log.  of  diameter  add  .497  15  (log.  of  3.1416),  and  sum  is  log. 
of  circumference,  from  which  take  number. 

EXAMPLE.— What  is  circumference  of  a  circle  1200  feet  in  diameter? 

Log.  1200  =  3.079 18  +  .497 15  =  3.  57633,  and  number  for  which  =  3769.92 /«««. 

Circumferences   of  Birmingham    \Vire   Grauge. 


Diam. 

Circum. 

Diam. 

Circum. 

Diam. 

Circum. 

Diam. 

Circum. 

No. 

Ins. 

No. 

Ins. 

No. 

IDS. 

No. 

Ins. 

I 
2 

.94248 
.89221 

10 
II 

.42097 
.37699 

19 
20 

.13195 
.10995 

28 
29 

.04398 
.040  84 

3 

.81367 

12 

.34243 

21 

.10053 

30 

•0377 

4 

•7477 

I3             .29845 

22 

.08796 

31 

.031  41 

5 

.691  15 

14         .260  75 

23 

.078  54 

32 

.028  27 

6 

•637  74 

15     j     .226  19 

24 

.06911 

33 

•025  13 

7 

.565  49 

16 

.2042 

25 

.06283 

34 

.021  99 

8 

.51836 

I7 

.18221 

26 

•056  55 

35 

.015  71 

9 

.46495 

18 

•15394 

27 

.050  26 

36 

.01257 

AREAS    AND   CIRCUMFERENCES    OF   CIRCLES.         24J 
.A^reas   and   Circumferences.    (Advancing  by  Tenths.) 


DlAM. 

AREA. 

ClRCUM. 

DlAM. 

AREA. 

ClRCUM. 

.1 

.007854 

.31416 

.6 

24.6301 

17-593 

.2 

.031  416 

.62832 

•7 

25.5176 

17.0071 

.3 

.070686 

.94248 

.8 

26.4209 

I8.22I3 

•4 

.125664 

1.2566 

•9 

27.3398 

18.5354 

•5 

.19635 

1.5708 

6 

28.2744 

18.8496 

.6 

.282744 

1.885 

.1 

29.2247 

I9.I638 

•7 

.384846 

2.I99I 

.2 

30.1908 

19.4779 

.8 

.502656 

2.5133 

•3 

3I«I725 

19.7921 

•9 

.636174 

2.8274 

•4 

32.17 

20.1062 

i 

.7854 

3.I4I6 

20.4204 

.1 

•9503 

3-455  8 

.6 

34.212 

20.7346 

.2 

I.I3I 

3.7699 

•7 

35.2566 

2I.O487 

•3 

L3273 

4.0841 

.8 

36.3169 

21.3629 

•4 

1-5394 

4.3982 

•9 

37.3929 

21.677 

I 

1.7671 
2.0IO6 
2.2698 

5.0266 
5-3407 

7 
.1 

.2 

38.4846 
39.592 
40.7151 

21.9912 
22.3054 
22.6195 

.8 
•9 

2 
.1 
.2 
•3 

2-5447 
2.8353 
3.1416 
3-4636 
3-80I3 
4.1548 

5-969 
6.2832 

6-5974 
6.9115 
7-225  7 

•3 

•4 

i 

.8 

41.854 
43.0085 
44.1787 

45-3647 
46.5664 

47-7837 

22.9337 
23.2478 
23-502 
23.8762 
24.1903 
24-5045 

•4 

4-5239 
4.9087 
5.3093 

7*854 
8.1682 

•9 
8 

49.0168 

50.2656 

24.8l86 
25   1328 

•9 
3 
.1 

.2 

•3 
•4 

i 

6.1575 
6.6052 

7.0686 

75477 
8.0425 

8-553 
9.0792 
9.621  1 
10.1788 

8.4823 

8.7965 
9.1106 
9.4248 

9-739 
10.053  * 
10.3673 
10.681  4 
10.9956 
11.3098 

.1 

.2 

•3 
-4 

.6 

'.8 
•9 
9 

51.5301 
52.8103 
54.1062 
55-4*7* 
56.7451 
58.0882 

59-4469 
60.8214 
62.2115 
63.6174 

25-447 
25.76II 
260753 
26.3894 
26  7036 
27.0178 

27.3319 
276461 
27  9602 
28.2744 

.7 

10.752  i 

11.6239 

.1 

65-039 

28.5886 

.8 

11.3412 

11.9381 

.2 

664763 

28.OXD27 

«9 

11.9459 

12.2522 

•3 

67.9292 

29.2169 

4 

12.5664 

12.5664 

«4 

69-3979 

29531 

.1 

13.2026 

12.8806 

•5 

70.8823 

298452 

.2 

J3-8545 

.6 

72-3825 

3°-  1  594 

•3 

14.522 

I3-5°89 

•7 

73-8983 

30-4735 

•4 

15-2053 

13.823 

.8 

75.4298 

30-7877 

•5 

I5-9043 

14.1372 

«9 

76.9771 

31.1018 

.6 

16.619  1 

14.4514 

10 

78.54 

31.416 

•7 

1  7-349  5 

14-765  5 

.1 

80.1187 

31.7302 

.8 

18.0956 

15.0797 

.2 

81.713 

32-0443 

•9 

18.8575 

I5-3938 

•3 

83.3231 

32.3585 

5 

19-635 

15-708 

-4 

84.9489 

32.6726 

.1 

20.4283 

l6.022  2 

•5 

86.5903 

32.9868 

.2 

21.2372 

16.3363 

.6 

88.2475 

33-301 

•3 

22.0619 

16.650  5 

•7 

89.9204 

336151 

•4 

22.9023 

16.9646 

.8 

91.6091 

33-9293 

•5 

23-7583 

17.2788 

•9 

93-3134 

34-2434 

244        AREAS   AND    CIRCUMFERENCES    OF    CIRCLES. 


DlAM. 

AREA. 

CIRCUM. 

DlAM. 

AREA. 

CIRCUM. 

II 

95-0334 

34.5576 

•5 

213.8251 

51.8364 

.1 

96.7691 

34.8718 

.6 

216.4248 

52.1505 

.2 

98.5206 

35-I859 

•7 

219.0402 

52.4647 

.3 

100.2877 

35.5001 

.8 

221.6713 

52.7789 

•4 

102.0700 

35.8142 

•9 

224.3181 

53-093 

•5 
.6 

103.8691 
105.6834 

36.1284 
36.4426 

17 
.1 

226.9806 
229.6588 

53-4072 
53-72I4 

•7 

107.5134 

36.7567 

.2 

232.3527 

54-0355 

.8 

109.3591 

37-0709 

•3 

235.0624 

54-3497 

•9 

III  .22O5 

37-385 

-4 

237.7877 

54.6638 

12 

113.0976 

37.6992 

•5 

240.5287 

54.978 

.1 

114.9904 

38.0134 

.6 

243.2855 

55.2922 

.2 

116.8989 

38.3275 

•7 

246.058 

55.6063 

•3 

118.8232 

38.6417 

.8 

248.8461 

55-9205 

•4 

120.7631 

38.9558 

«9 

251-65 

56.2346 

•5 

122.7187 

39-27 

18 

254.4696 

56.5488 

.6 

.'s 

•9 

124.6901 
126.6772 
128.6799 
130.6984 

39.8983 
40.2125 
40.5266 

.1 

.2 

•3 

«4 

257.3049 
260.1559 
263.0226 
265.905 

56.863 
57-I77I 
57"49r3 
57.8054 

13 

132.7326 

40.8408 

•5 

268.8031 

58.1196 

.1 

134.7825 

41.155 

.6 

271.717 

58.4338 

.2 

136.8481 

41.4691 

•7 

274.6465 

58.7479 

-3 

138.9294 

4L7833 

.8 

277.5918 

59.0621 

•4 

141.0264 

42.0974 

-9 

280.5527 

59.3762 

i 

I43-I39I 
145.2676 

42.7258 

19 
.1 

283.5294 
286.5218 

59.6904 
60.0046 

•7 

I47.4II7 

43-0399 

.2 

289.5299 

60.3187 

.8 

149.5716 

43-3541 

•3 

292.5536 

606329 

•9 

I5I.747I 

43.6682 

•4 

295-593I 

60.947 

14 

153-9384 

43.9824 

•5 

298.6483 

61.2612 

.1 

156.1454 

44.2966 

.6 

301.7193 

6i.5754 

.2 

158.3681 

446107 

-7 

304.806 

61.8895 

•3 

160.6064 

44.9249 

.8 

307.9082 

62.2037 

•4 

162.8605 

45239 

•9 

311.0263 

62.5178 

•5 

165.1303 

45-5532 

20 

314.16 

62.832 

.6 

167.4159 

45.8674 

.1 

317.3094 

63.1462 

•7 

169.7171 

46.1815 

.2 

320  4746 

63.4603 

.8 

172034 

46.4957 

.3 

63.7745 

•9 

174.3667 

46.8098 

•4 

326.8521 

64.0886 

15 

176.715 

47.124 

•5 

330.0643 

64.4028 

.1 

179.0791 

47.4382 

.6 

333.2923 

64.717 

.2 

181.4588 

47.7523 

•7 

336.536 

65.0311 

•3 

183.8543 

48.0665 

.8 

339-7955 

65.3453 

•4 

186.2655 

48.3806 

•9 

343.0706 

65-6594 

•5 

188.6924 

48.6948 

21 

346.3614 

65.9736 

.6 

I9I.I349 

49.009 

.1 

349.6679 

66.2878 

is 

193.5932 
196.0673 

49.3231 
49.6373 

.2 
•3 

352.9902 
356.3281 

66.6019 
66.9161 

•9 

108.557 

49-95  14 

-4 

359.6818 

67.2302 

16 

201.0624 

50.2656 

•5 

3630511 

67.5444 

.1 

203.5835 

50.5797 

.6 

366.4362 

67.8586 

.2 

2O6.I2O4 

50.8939 

•7 

369.837 

68.1727 

•3 

208.6729 

51.2081 

.8 

373-2535 

68.4869 

-4 

211.2412 

5L5222 

•9 

376.6857 

68.801 

AREAS    AND    CIRCUMFERENCES    OF    CIRCLES.         245 


DlAM. 

ARIA. 

ClRCTJM. 

DlAM. 

ASIA. 

ClRCUM. 

22 

380.1336 

69.1152 

•5 

593-9587 

86.394 

.1 

69.4294 

.6 

598.2863 

86.7082 

.2 

387.0765 

69.7435 

•7 

602.6296 

87.0223 

•3 

390.5716 

70.0577 

.8 

606.9885 

87.3365 

•4 

394.0823 

70.3718 

•9 

611.3632 

87.6506 

•5 

397.6087 

70.686 

28 

6I5-7536 

87.9648 

.6 

401.1509 

71.0002 

.1 

620.1597 

88.279 

•7 

404.7088 

7L3I43 

.2 

624.5815 

88.593I 

.8 

408.2823 

71.6285 

•3 

629.019 

•9 

411.8716 

71.9426 

•4 

633.4722 

89.2214 

23 

415.4766 

72.2568 

•5 

637.9411 

89.5356 

.1 

419.0973 

72.571 

.6 

642.4258 

89.8498 

.2 

422.7337 

72.8851 

•7 

646.9261 

90.1639 

.3 

426.3858 

73-1993 

.8 

651.4422 

90.4781 

•4 

430.0536 

73-5I34 

•9 

655-9739 

9O.7922 

•5 

433-7371 

73.8276 

29 

660.5214 

9I.IO64 

.6 

437.4364 

74.I4I8 

.1 

665.0846 

91.4206 

•7 

441.1513 

74-4559 

.2 

669.6635 

9J-7347 

.8 
•9 

444.882 
448.6283 

74.7701 
75.0842 

•3 
•4 

674.258 
678.8683 

92.0489 
92.363 

24 

452.3904 

75-3984 

•5 

683.4943 

92.6772 

.1 

456.1682 

75.7126 

.6 

688.1361 

92.9914 

.2 

459.9617 

76.0267 

•7 

692.7935 

93-3055 

•3 

463.7708 

76.3409 

.8 

697.4666 

93.6197 

•4 

467-5957 

76.655 

•9 

702.1555 

93-9338 

47I-4363 

76.9692 

30 

706.86 

94.248 

.6 

475.2927 

77-2834 

.1 

711.5803 

94.5622 

•7 

479.1647 

77-5975 

.2 

716.3162 

94.8763 

.8 

483.0524 

77.9117 

•3 

721.0679 

95-I905 

-9 

486.9559 

78.2258 

•4 

725-8353 

95-5046 

25 

490.875 

^8.54 

•5 

730.6183 

95.8188 

.1 

494.8099 

78-8542 

.6 

735.4171 

96.133 

.2 

498.7604 

79.1683 

•7 

740.2316 

96.4471 

•3 

502.7267 

79-4825 

.8 

745.0619 

96.7613 

•4 

506.7087 

79.7966 

•9 

749.9078 

97-0754 

510.7063 

80.1108 

754.7694 

97.3896 

.6 

514.7196 

80.425 

.1 

759.6467 

97.7038 

•7 

518.7488 

80.7391 

.2 

764.5398 

98.0179 

.8 

522.7937 

81.0533 

•3 

769.4485 

98.3321 

-9 

526.8542 

81.3674 

•4 

774-373 

98.6462 

26 

530.9304 

81.6816 

•5 

779-3I3I 

98.9604 

.1 

535.0223 

81.9958 

.6 

784.269 

99.2746 

.2 

539-  J3 

82.3099 

•7 

789.2406 

99-5887 

•3 

543-2533 

82.6241 

.8 

794.2279 

99.9029 

•4 

547.3924 

82.9382 

•9 

799.2309 

100.217 

:1 

•7 

555-7176 
559.9038 

83.2524 
83.5666 
83.8807 

32 
.1 

.2 

804.2496 
809.284 
814.3341 

100.5312 
100.8454 
101.1595 

.8 

564.1057 

84.1949 

.3 

819.4 

101.4737 

'9 

568.3233 

84.509 

«4 

824.4815 

101.7878 

27 

572.5566 

84.8232 

•5 

829.5787 

102.102 

.1 

576.8056 

85.1374 

.6 

834.6917 

102.4162 

.2 

581.0703 

85-4515 

•7 

839.8204 

102.7303 

-3 

585.3508 

85-7657 

.8 

844.9647  :  103.0445 

•4 

589.6469 

86.0798 

•9 

850.1248  !  103.3586 

246        AREAS   AND    CIRCUMFERENCES   OF   CIRCLES. 


DlAM. 

AREA. 

CincuM. 

DlAM. 

AREA. 

Cinctrat. 

33 

855.3006 

103.6728 

•5 

1164.1591 

120.9516 

.1 

860.4921 

103.987 

.6 

1170.2146 

121.2658 

.2 

865.6993 

104.3011 

•7 

1176.2857 

121.5799 

.3 

870.9222 

104.6153 

.8 

1182.3726 

121.8941 

•4 

876.1608 

104.9294 

•9 

1188.4751 

122.2082 

•5 

881.4151 

105.2436 

39 

1194.5934 

122.5224 

.6 

886.6852 

105.5578 

.1 

1200.7274 

122.8366 

•7 

891.9709 

105.8719 

.2 

1206.8771 

123.1507 

.8 

897.2724 

106.1861 

•3 

1213.0424 

123.4649 

-9 

902.5895 

106.5002 

•4 

1219.2235 

123.779 

34 

907.9224 

106.8144 

•5 

1225.4203 

124.0932 

.1 

913.271 

107.1286 

.6 

1231.6329 

124.4074 

.2 

918.6353 

107.4427 

•7 

1237.8611 

124.7215 

•3 

924.0152 

107.7569 

.8 

1244.105 

125-0357 

•4 

929.4109 

108.071 

•9 

1250.3647 

125.3498 

•5 

934.8223 

108.3852 

40 

1256.64 

125.664 

.6 

940.2495 

108.6994 

.1 

1262.9311 

125.9782 

•7 

945.6923 

109.0135 

.2 

1269.2378 

126.2923 

.8 

951.1508 

109.3277 

•3 

1275.5603 

126.6065 

•9 

956.6251 

109.6418 

•4 

1281.8985 

126.9206 

35 

962.115 

109.956 

•5 

1288.2523 

127.2348 

.1 

967.6207 

IIO.27O2 

.6 

1294.6219 

127.549 

.2 

973.142 

110.5843 

•7 

1301.0072 

127.8631 

.3 

978.6791 

110.8985 

.8 

1307.4083 

128.1773 

•4 

984.2319 

III.2I26 

•9 

I3I3825 

128.4914 

•5 

989.8003 

III.5268 

4i 

1320.2574 

128.8056 

.6 

995.3845 

111.841 

.1 

1326.7055 

129.1198 

!s 

1000.9844 
1006.6001 

112.1551 
112.4693 

.2 

•3 

I333-  1694 
1339.6489 

129.4339 
129.7481 

•9 

1012.2314 

112.7834 

•4 

1346.1442 

130.0622 

36 

1017.8784 

113.0976 

•5 

1352-6551 

130.3764 

.1 

1023.5411 

113.4118 

.6 

1359.1818 

130.6906 

.2 

1029.2196 

113.7259 

•7 

1365.7242 

131.0047 

•3 

1034.9137 

1  14.0401 

.8 

1372.2823 

*3*'3*&9 

•4 

1040.6236 

"4.3542 

•9 

1378.8561 

131-633 

1046.3491 

114.6684 

42 

1385-4456 

131.9472 

'.8 

1052.0904 
1057.8474 
1063.6201 

114.9826 
115.2967 
115.6109 

.1 

.2 

•3 

1392.0508 
1398.6717 
1405.3084 

132.2614 

132.5755 
132.8897 

•9 

1069.4085 

II5.925 

•4 

1411.9607 

133.2038 

37 

1075.2126 

116.2392 

•5 

1418.6287 

i33-5i8 

.1 

1081.0324 

"6.5534 

.6 

1425.3125 

133.8322 

.2 

1086.8679 

116.8675 

•7 

1432.012 

134.1463 

•3 

1092.7192 

117.1817 

.8 

1438.7271 

vi  34.4605 

•4 

1098.5861 

117.4958 

•9 

1445.458 

134.7746 

•5 

1104.4687 

117.81 

43 

1452.2046 

135.0888 

.6 
•7 

1110.3671 
IIl6.28l2 

118.1242 
118.4383 

.1 

.2 

1458.9669 
1465.7449 

135.403 
I35-7I7I 

.8 
•9 

II22.2IO9 
1128.1564 

118.7525 
119.0666 

•3 
•4 

1472.5386 
1479.348 

136.0313 
136.3454 

38 

1134.1176 

119.3808 

•5 

1486.1731 

136.6596 

.1 

1140.0945 

119.695 

.6 

1493.014 

136.9738 

.2 

1146.0871 

120.0091 

•7 

1499.8705 

137.2879 

•3 

1152.0954 

120.3233 

.8 

1506.7428 

137.6021 

•4 

1158.1194 

120.6374 

•9 

1513-6307 

137.9162 

AREAS   AND    CIRCUMFERENCES    OF   CIRCLES.        247 


DlAM. 

AREA. 

ClRCCM. 

DlAM. 

ARIA. 

ClRCTM. 

44 

1520.5344 

138.2304 

•5 

1924.4263 

155.5092 

.1 

I527-4538 

138.5446 

.6 

1932.2097 

155.8234 

.2 

1534.3889 

138.8587 

-7 

1940.0087 

156.1375 

•3 

1  54  1  -3396 

139.1729 

.8 

1947.8234 

156.4517 

•4 

1548.3061 

139.487 

•9 

1955.6539 

156.7658 

•5 

1555-2883 

139.8012 

50 

1963.5 

I57-08 

.6 

1562.2863 

I40.II54 

.1 

1971.3619 

157.3942 

•7 

1569.2999 

140.4295 

.2 

1979.2394 

157-7083 

.8 

1576.3292 

140.7437 

•3 

1987.1327 

I58.O225 

•9 

I583.3743 

141.0578 

•4 

1995.0417 

158.3366 

45 

I590-435 

141.372 

-5 

2002.9663 

158.6509 

.1 

I597.5H5 

141.6862 

.6 

2010.9067 

158.965 

.2 

1604.6036 

142.0003 

•7 

2018.8628 

159.2791 

•3 

1611.7115 

142.3145 

.8 

2026.8347 

1  59-5933 

•4 

1618.8351 

142.6286 

•9 

2034.8222 

159.9074 

•5 

1625.9743 

142.9428 

51 

2042.8254 

160.2216 

.6 

1633.1293 

I43.257 

.1 

2050.8443 

160.5358 

•7 

1640.3 

I43.57H 

.2 

2058.879 

160.8499 

.8 

1647.4865 

I43.8853 

•3 

2066.9293 

161.1641 

•9 

1654.6886 

144.1994 

•4 

2074.9954 

161.4782 

46 

1661.9064 

144.5136 

•5 

2083.0771 

161.7924 

.1 

1669.1399 

144.8278 

.6 

2091.1746 

162.1066 

.2 

1676.3892 

I45.I4I9 

•7 

2099.2878 

162.4207 

•3 

1683.6541 

145.4501 

.8 

2107.4167 

162.7349 

•4 

1690.9348 

145.7702 

•9 

2H5-56I3 

163.049 

.5 

1698.2311 

146.0844 

52 

2123.7216 

163.3632 

.6 

I705-5432 

146.3986 

.1 

2131.8976 

163.6774 

•7 

1712.871 

146.7127 

.2 

2140.0893 

1639915 

.8 

1720.2145 

147.0269 

-3 

2148.2968 

164.3057 

•9 

I727-5737 

I47'34I 

•4 

2156.5199 

164.6198 

47 

1734.9486 

I47.6552 

•5 

2164.7587 

164.934 

.1 

1742.3392 

147.9694 

.6 

2173.0133 

165.2482 

.2 

1749-7455 

148.2835 

•7 

2181.2836 

165.5623 

•3 

1757.1676 

148.5977 

.8 

2189.5695 

165.8765 

•4 

1764.6053 

I48.9II8 

•9 

2197.8712 

166.1906 

•5 

1772.0587 

149.226 

53 

2206.1886 

166.5048 

.6 

I779.5279 

149.5402 

.1 

2214.5217 

166.819 

•7 

1787.0128 

149.8543 

.2 

2222.8705 

167.1331 

.8 

1794-5133 

I50.I685 

.3 

2231.235 

167.4473 

•9 

1802.0296 

150.4826 

•4 

2239.6152 

167.7614 

48 

1809.5616 

150.7968 

•5 

2248.0111 

168.0756 

.1 

1817.1093 

I5I.III 

.6 

2256.4228 

168.3898 

.2 

1824.6727 

I5L425I 

•7 

2264  8501 

168.7039 

•3 

1832.2518 

I5L7393 

.8 

2273.2932 

169.0181 

•4 

1839.8466 

152.0534 

•9 

2281.7519 

169.3322 

•5 

1847.4571 

152.3676 

54 

2290.2264 

169.6464 

.6 

1855.0834 

I52.68l8 

.1 

2298.7166 

169.9606 

•7 

1862.7253 

152.9959 

.2 

2307.2225 

170.2747 

.8 

1870.383 

I53.3IOI 

•3 

23*5.744 

170-5889 

•9 

1878.0563 

153.6242 

•4 

2324.2813 

170.903 

49 

1885.7454 

I53-9384 

•5 

2332-8343 

171.2172 

.1 

1893.4502 

154.2526 

.6 

2341.4031 

171-5314 

.2 

1901.1707 

154.5667 

•7 

2349.9875 

171.8455 

•3 

1908.9068 

154.8809 

.8 

2358-5876 

172.1597 

•4 

1916.6587 

155.195 

•9 

2367-2035 

372.4738 

248        AREAS  AND   CIRCUMFERENCES   OF   CIRCLES. 


DlAM. 

ABBA. 

ClRCUM. 

DlAM. 

ABBA. 

ClRCUM. 

55 

2375-835 

172.788 

•5 

2874.7603 

190.066 

.1 

2384.4823 

173.1022 

.6 

2884.2715 

190.381 

.2 

2393-I452 

I73-4I63 

•7 

2893.7984 

190.695 

•3 

2401.8239 

I73-7305 

.8 

2903.3411 

191.005 

•4 

2410.5183 

174.0446 

•9 

2912.8994 

I9I.32J 

•5 

2419.2283 

174.3588 

61 

2922.4734 

191.637 

.6 
•7 

2427.9541 
2436.6957 

174.673 
174.9871 

.1 

.2 

2932.0631 
2941.6686 

I9I.95I 
I92.26C 

.8 

2445.4529 

I75.3OI3 

.3 

2951.2897 

I92.58C 

•9 

2454.2258 

I75-6I54 

•4 

2960.9266 

192.894 

56 

2463.0144 

I75.92Q6 

•5 

2970.5791 

I93.20S 

.1 

2471.8187 

176.2438 

.6 

2980.2474 

I93-522 

.2 

2480.6388 

176.5579 

•7 

2989.9314 

193.83^ 

•3 

2489.4745 

176.8721 

.8 

2999.6311 

I94.I5C 

•4 

2498.326 

I77.I862 

•9 

3009.3465 

194.46  = 

.5 

2507.1931 

177.5004 

62 

3019.0776 

194.779 

.6 

2516.076 

177.8146 

.1 

3028.8244 

195.09: 

.7 

2524.9736 

178.1287 

.2 

3038.5869 

195.40^ 

.8 
•9 

2533.8889 
2542.8189 

178.4429 
178.757 

-3 

•4 

3048.3652 
3058.1591 

195.72] 
196.03  = 

57 

2551.7646 

179.0712 

•5 

3067.9687 

106.35 

.1 

2560.726 

I79-3854 

.6 

3077.7941 

196.664 

.2 

2569-703I 

179.6995 

•7 

3087.6341 

196.97? 

•3 

2578.696 

180.0137 

.8 

3097.4919 

197.292 

•4 

2587.7045 

180.3278 

•9 

3107.3644 

•5 

2596.7287 

180.642 

63 

3117.2526 

I97-92C 

.6 

2605.7687 

180.9562 

.1 

3127.1565 

i98.23v 

•7 

2614.8244 

l8l.27O3 

.2 

3137.0761 

i98.54< 

.8 

2623.8957 

181.5845 

.3 

3147.0114 

198.86; 

-9 

2632.9828 

I8l.8o86 

•4 

3156.9624 

199.17- 

58 

2642.0856 

182.2128 

-5 

3166.9291 

199.49 

.1 

2651.2041 

182.527 

.6 

3176.9116 

199.80^ 

.2 

2660.3383 

I82.84II 

•7 

3186.9097 

200.  IK 

•3 

2669.4882 

183.1553 

.8 

3196.9236 

200.43/ 

«4 

2678.6538 

183.4694 

•9 

3206.9531 

200.  74* 

'.6 

2687.8351 
2697.0322 

183.7836 
184.0978 

64 
.1 

3216.9984 
3227.0594 

201  .06; 

201.  37< 

•7 

2706.2449 

184-4119 

.2 

3237.1361 

20I.6gj( 

.8 

27J5-4734 

184.7261 

•3 

3247.2284 

202.00, 

•9 

2724.7175 

I85.O4O2 

•4 

202.3  1  < 

59 

2733-9774 

185.3544 

•5 

3267.4603 

202  .63, 

.1 

2743-253 

185.6686 

.6 

3277.5999 

202.94 

.2 

2752.5443 

185.9827 

«7 

3287.7551 

203.26 

•3 

2761.8512 

186.2969 

.8 

3297.9261 

203.57. 

•4 

2771.1739 

I86.6II 

-9 

3308.1127 

203.88< 

•5 

2780.5123 

186.9252 

65 

3318.315 

.6 

2789.8665 

187.2394 

.1 

3328.5331 

204.5  1  < 

•7 

2799.2363 

187.5535 

.2 

3338.7668 

204.83 

.8 

2808.6218 

187.8677 

.3 

3349.0163 

205.  1  4< 

•9 

2818.0231 

I88.l8l8 

4 

3359.2815 

205.  4cx 

60 

2827.44 

188.496 

-5 

3369.5623 

205.77, 

.1 

2836.8727 

I88.8I02 

.6 

3379'8589 

2o6.o8< 

.2 

2846.321 

189.1243 

•7 

3390.1712 

206.40, 

•3 

2855.7851 

189  4385 

.8 

3400.4993 

206.71 

•4 

2865.2649 

189.7526 

•9 

3410.843 

207.03 

AREAS   AND   CIRCUMFERENCES    OF    CIRCLES.         249 


DlAM. 

ARIA. 

ClRCUM. 

DlAH. 

AREA. 

CntctiM. 

66 

3421.2024 

207.3456 

•5 

40I5.l6lI 

224.6244 

.1 

343I-5775 

207.6598 

.6 

4026.4002 

224.9386 

.2 

3441.9684 

207.9739 

.7 

4037.655 

225.2527 

•3 

3452.3749 

208.2881 

.8 

4048.9255 

225.5669 

-4 

3462.7972 

208.6022 

•9 

4060.2117 

225.881 

3473'235i 

208.9164 

72 

4071.5136 

226.1952 

.6 

3483.6888 

209.2306 

.1 

4082.8312 

226.5094 

•7 

3494.1582 

209.5447 

.2 

4094.1645 

226.8235 

.8 

3504.6433 

209.8589 

-3 

4105.5136 

227.1377 

•9 

3515.1441 

210.173 

•4 

4116.8783 

227.4518 

67 

3525.6606 

2IO.4872 

•5 

4128.2587 

227.766 

.1 

3536.1928 

2IO.8OI4 

.6 

4J39-655 

228.0802 

.2 

3546.7407 

2II.II55 

•7 

4151.0668 

228.3943 

•3 

3557-3044 

211.4297 

.8 

4162.4943 

228.7085 

•4 

3567.8837 

211.7438 

•9 

4I73-9376 

229.0226 

-5 

3578.4787 

2I2.O58 

73 

4185.3966 

229.3368 

.6 
-7 

3589.0895 
3599.716 

212.3722 
212.6863 

.1 

.2 

4196.8713 
4208.3617 

229.651 
2299651 

.8 
•9 

3610.3581 
3621.016 

213.0005 
213.3146 

•3 
•4 

4219.8678 
4231.3896 

230.2793 
230.5934 

68 

3631.6896 

213.0288 

-5 

4242.9271 

2309076 

.1 

3642.3789 

213.943 

.6 

4254.4804 

231.2218 

.2 

3653-0839 

214.2571 

•7 

4266.0493 

231  5359 

•3 

3663.805 

214.5713 

.8 

4277.634 

231.8501 

•4 

3674-54I 

214-8854 

•9 

4289.2343 

232.1642 

'.6 
•7 

3685.2931 
3696.061 
3706.8445 

215.1996 
215.5138 
215.8279 

74 
.1 

.2 

4300.8504 
4312.4822 
4324.1297 

232.4784 
232.7926 
233.1067 

.8 
•9 

3717.6438 
3728.4587 

2I6.I42I 
216.4562 

•3 
•4 

4335.7928 
4347-47I7 

233.4209 
233-735 

69 

3739.2894 

216.7704 

-5 

4359.1663 

234.0492 

.1 

3750.1358 

217.0846 

.6 

4370.8767 

234.3634 

.2 

3760.9979 

2X7.3987 

•7 

4382.6027 

234-6775 

•3 

3771.8756 

217.7129 

.8 

4394-3444 

234.9917 

-4 

3782.7691 

218.027 

•9 

4406.1019 

235-3058 

3793.6783 

218.3412 

7er 

4417  875 

235.62 

.6 

3804.6033 

218.6554 

lO 

4429.6639 

-7 

3815.5439 

218.9695 

.2 

4441.4684 

236.2483 

.8 

3826.5002 

219.2837 

•3 

4453.2887 

236.5625 

•9 

3837.4722 

219.5978 

•4 

4465.1247 

236.8766 

70 

3848.46 

219.912 

•5 

4476.9763 

237.1908 

,i 

3859.4635 

22O.2262 

.6 

4488.8437 

237-505 

.2 

3870.4826 

220.5403 

•7 

4500.7268 

237.8191 

•3 

3881.5175 

220.8545 

.8 

4512.6257 

238.1333 

•4 

3892.5681 

221.1686 

•9 

4524.5402 

238.4474 

•5 

3903.6343 

221.4828 

76 

4536.4704 

238.7616 

.6 

3914.7163 

221.797 

.1 

4548.4163 

239.0758 

•7 

3925.814 

222.IIII 

.2 

4560.378 

239-3899 

.8 

3936.9275 

222.4253 

•3 

4572.3553 

239.7041 

•9 

3948.9566 

222.7394 

•4 

4584.3484 

240.0182 

71 

3959.2014 

223.0536 

•5 

4596.3571 

240.3324 

.1 

3970.3619 

223.3678 

.6 

4608.3816 

240.6466 

.2 

3981.5382 

223.6819 

•7 

4620.4218 

240.9607 

•3 

3992.7301 

223.9961 

.8 

4632.4777 

241.2749 

•4 

4003.9378 

224.3IO2 

•9 

4644.5493 

241.589 

AREAS   AND    CIRCUMFERENCES    OF    CIRCLES. 


DlAM. 

ARIA. 

ClRCUM. 

DtAM. 

AREA. 

ClRCUM. 

77 

4656.6366 

241.9032 

•5 

5345.6287 

259.182 

.1 

4668.7396 

242.2174 

.6 

5358.5957 

259.4962 

.2 

4680.8583     242.5315 

•7 

5371.5784 

259.8103 

.3 

4692.9928     242.8457 

.8 

5384.5767 

200.1245 

•4 

4705.1429   I   243.1598 

•9 

5397.5908 

260.4386 

-5 

4717.3087     243.474 

83 

5410.6206 

200.7528 

.6 

4729.4903 

243.7882 

.1 

5423.6661 

261.067 

•7 

4741.6876 

244.1023 

.2 

5436.7273 

26l.38ll 

.8 

4753.9005 

244.4165 

•3 

54498042 

261.6953 

•9 

4766.1292 

244.7306 

•4 

5462.8968 

262.0094 

78 

4778.3736 

245.0448 

•5 

5476.0051 

262.3236 

.1 

4790.6337 

245-359 

.6 

5489.1292 

262.6378 

.2 

4802.9095 

245.6731 

•7 

5502.2689 

262.9519 

•3 

4815.201 

245.9873 

.8 

5515.4244 

263.2661 

•4 

4827.5082 

246.3014 

•9 

5528.5955 

263.5802 

•5 
.6 

4839.8311 
4852.1698 

246.6156 
246.9298 

84 
.1 

5541.7824 
5554-985 

263.8944 
264.2086 

•7 

4864.5241 

247.2439 

.2 

5568.2033 

264.5227 

.8 

4876.8942 

247.5581 

•3 

5581.4372 

264.8369 

•9 

4889.2799 

247.8722 

•4 

5594  6869 

265.151 

79 

4901.6814 

248.1864 

•5 

5607.9523 

265.4652 

.1 

4914.0986 

248.5006 

.6 

5621.2335 

265.7794 

.2 

4926.5315 

248.8147 

•7 

5634-5303 

266.0935 

•3 

4938.98 

249.1289 

.8 

5647-8428 

266.4077 

•4 

4951.4443 

249.443 

-9 

5661.1711 

266.7218 

•5 

4963.9243 

249.7572 

85 

5674.515 

267.036 

.6 
•7 

4976.4201 
4988.9315 

250.0714 
250.3855 

.1 

.2 

5687.8747 
57OI-2«5 

267.35O2 
267.6643 

.8 

5001.4586 

250.6997 

•3 

5714.6411 

267.9785 

•9 

5014.0015 

251.0138 

•4 

5728.0479 

268.2926 

80 

5026.56 

251.328 

•5 

5741.4703 

268.6o68 

.1 

5039.1343 

251.6422 

.6 

5754.9085 

268.921 

.2 

5051.7242 

25  J"  9563 

•7 

5768.3624 

209.2351 

•3 

5064.3299 

252.2705 

.8 

5781.8321 

269.5493 

•4 

5076.9513 

252.5846 

•9 

5795-3I74 

269.8634 

•5 

5089.5883 

252.8988 

86 

5808.8184 

270.1776 

.6 

5102.2411 

253-2I3 

.1 

5822.3351 

270.4918 

•7 

5114.9096 

253-527I 

.2 

5835.8676 

270.8059 

.8 

5127.5939 

253.8413 

•3 

5849.4157 

27I.I2OI 

«9 

5140.2938 

254.1554 

•4 

5862.9796 

271.4342 

81 

5153.0094 

254.4696 

•5 

5876-5591 

271.7484 

.1 

5165.7407 

254.7838 

.6 

5890.1544 

272.0626 

.2 

5178.4878 

255.0979 

•7 

5903.7654 

272.3767 

-3 

5191.2505 

255.4121 

.8 

5917.3921 

272.6909 

•4 

5204.0289 

255.7262 

-9 

5931.0345 

273.005 

.6 

5216.8231 
5229.633 

256.0404 
256.3546 

87 
.1 

5944.6926 
5958.3644 

273.3192 

•7 

5242.4586 

256.6687 

.2 

5972.0559 

2739475 

.8 

5255.2999 

256.9829 

•3 

5985.7612 

274.2617 

•9 

5268.1569 

257.297 

•4 

5999.4821 

274.5758 

82 

5281.0296 

257.6lI2 

•5 

6013.2187 

274.89 

.1 

5293.918 

257.9254 

.6 

6026.9711 

275.2042 

.2 

5306.8221 

258.2395 

•7 

6040.7392 

275.5183 

•3 

53I9-742 

258.5537 

.8 

6054.5229 

275  8325 

•4 

5332-6775 

258.8678 

•9 

6068.3224 

276.1466 

AEEAS   AND    CIRCUMFERENCES   OF   CIRCLES. 


251 


DlAM. 

AREA. 

ClKCUM. 

DlAM. 

AREA. 

ClECUM. 

88 

6082.1376 

276.4608 

•5 

6866.1631 

293.7396 

.1 

6095.9685 

276.775 

.6 

6880.858 

294.0538 

.2 

6109.8151 

277.0891 

•7 

6895.5685 

294.3679 

.3 

6123.6774 

277-4033 

.8 

6910.2948 

294.6821 

•4 

6I37-5554 

277.7174 

•9 

6925.0367 

294.9962 

.6 

6151.4491 
6165.3586 

278.0316 
278.3458 

$4 
.1 

6939-7944 
6954.5678 

295.3104 
295.6246 

i 

6179.2837 
6193.2246 

278.6599 
278.9741 

.2 

•3 

6969-3569 
6984.1616 

295.9387 
296.2529 

-9 

6207.1811 

279.2882 

•4 

6998.9821 

296.567 

89 

6221.1534 

279.6024 

•5 

7013.8183 

296.8812 

.1 

6235  1414 

279.9166 

.6 

7028.6703 

297.1954 

.2 

6249  1451 

280.2307 

-7 

7043-5379 

297-5095 

•3 

6263.1644 

280.5449 

.8 

7058.4212 

2978237 

•4 

6277.1995 

280.859 

•9 

7073.3203 

208.1378 

.6 

6291.2503 
6305  -3l69 

28I.I732 
281.4874 

95 
.1 

7088.235 
7103.1655 

298.452 
298.7662 

•7 

63I9-399I 

28I.80I5 

.2 

299.0803 

.8 
•9 

6333497 
6347.6107 

282.1157 
282.4298 

•3 
•4 

7I33-0735 
7148.0511 

299-3945 
299.7086 

9° 

6361.74 

282.744 

•5 

7163.0443 

300.0228 

.1 

6375.8851 

283.0582 

.6 

7178.0533 

300.337 

.2 

6390.0458 

283.3723 

•7 

7193.078 

300.6511 

•3 

6404.2223 

283.6865 

.8 

7208.1185 

300.9653 

•4 

64184144 

284.0006 

•9 

7223.1746 

301.2794 

•5 
.6 

6432.6223 
6446.8459 

284.3148 
284.629 

96 
.1 

7238.2464 
72533339 

301.5936 
301.9078 

!s 

•9 

6461.0852 
6475.3403 
6489.61  1 

284.9431 
285.2573 
285.5714 

.2 

•3 
•4 

7268.4372 
7283.5561 
7298.6908 

302.2219 
302.5361 
302.8502 

91 

6503.8974 

285.8856 

•5 

7313.8411 

303.1644 

.1 

6518.1995 

286.1998 

.6 

7329.0072 

303.4786 

.2 

6532.5174 

286.5139 

•7 

7344.189 

303.7927 

•3 

6546.8509 

286.8281 

.8 

7359-3865 

3041009 

•4 

6561.2002 

287.1422 

•9 

7374-5997 

304.421 

•7 

65755651 
6589.9458 
6604.3422 

287.4564 
287.7706 
288.0847 

97 
.1 

.2 

7389.8286 
7405.0732 
7420.3335 

304-7352 
305.0494 
305.3635 

.8 

6618.7543 

288.3989 

•3 

7435.6096 

305-6777 

•9 

6633.1821 

288.713 

•4 

7450.9013 

305.9918 

92 

6647.6256 

289.O272 

•5 

7466.2087 

306.306 

.1 

6662.0848 

289.3414 

.6 

7481.5319 

306.6202 

.2 

6676.5598 

289.6555 

•7 

7496.8708 

306.9343 

•3 

6691.0504 

289.9697 

.8 

7512.2253 

307.2485 

•4 

6705-5567 

290.2838 

•9 

7527-5956 

307.5626 

•5 

6720.0787 

290.598 

98 

7542.9816 

307.8768 

.6 

6734.6165 

29O.9I2I 

.1 

7558.3833 

308.J9I 

•7 

674917 

291.2263 

.2 

7573.8007 

308.5051 

.8 

6763.7391 

291.5405 

•3 

7589.2338 

3O8.8l93 

•9 

6778  324 

291.8546 

•4 

7604.6826 

309.1334 

93 

6792.9246 

292.1088 

•5 

7620.1471 

309.4476 

.1 

6807.5409 

292.483 

.6 

7635.6274 

.2 

6822.1729 

292.7971 

•  7 

7651.1233 

310.0759 

•3 

6836.8206  ,  293.1113 

.8 

7666.635 

310.3901 

•4 

6851.484     293.4254 

•9 

7682.1623 

310.7042 

252        AREAS   AND   CIRCUMFERENCES   OF    CIRCLES. 


DlAM. 

ABBA. 

ClRCUM. 

DlAM. 

AREA. 

CTRCUM. 

99 
,i 

.2 

•3 
•4 

7697.7054 
7713.2642 
7728.8337 
7744.4288 
7760.0347 

31I.OI84 
311.3326 
311.6467 
311.9009 
312.275 

.6 

'.8 
•9 

7775.6563 
7791.2937 
7806.9467 
7822.6154 
7838.2999 

312.5892 
312.9034 
3I3-2I75 
3I353I7 
313.8458 

To  Compute  Area  or  Circumference  of  a  Diameter  greater 
than,    any    in    preceding    Table. 

See  Rules,  pages  235-6  and  241-2. 

Or,  If  Diameter  exceeds  100  and  is  less  than  1001. 

Put  a  decimal  point,  and  take  out  area  or  circumference  as  for  a  Whole 
Number  by  removing  decimal  point,  if  for  an  area,  two  places  to  right ,  and 
if  for  a  circumference,  one  place. 

EXAMPLE. — What  is  area  and  what  circumference  of  a  circle  967  feet  in  diame- 
ter? 

Area  of  96.7  is  7344.189;  hence,  for  967  it  is  734  418.9;  and  circumference  of  96.7 
is  303.7927,  and  for  967  it  is  3037.927 

To  Compute  Area  and  Circumference  of  a  Circle  by  JJog- 
aritlims. 

See  Rules,  pages  236,  242. 

.A^reas   and.   Circumferences   of  Circles. 

FROM  i  TO  50  FEET  (advancing  by  an  Inch). 
OR,  FROM  i  TO  50  INCHES  (advancing  by  a  Twelfth). 


DlAM. 

AREA. 

ClRCUM. 

DlAM. 

AREA. 

ClKCUM. 

Feet. 

Feet. 

Feet. 

Feet. 

i/fc 

.7854 

3.1416 

3ft- 

7.0686 

9.4248 

I 

.9217 

3-4034 

I 

7.4668 

9,6866 

2 

1.069 

3.6652 

2 

7.8758 

9.9484 

3 

1.2272 

3.927 

3 

8.2958 

IO.2IO2 

4 

1.303 

4.1888 

4 

8.7267 

10.472 

5 

L5763 

4.4506 

5 

9.1685 

10.7338 

6 

1.7671 

4.7124 

6 

9.62II 

10.9956 

7 

1.969 

4.9742 

7 

10.0848 

11.2574 

8 

2.1817 

5.236 

8 

IO-5593 

11.5192 

9 

2.4053 

5.4978 

9 

11.0447 

II.78I 

10 

2.6398 

5.7596 

10 

11.541 

12.0428 

ii 

2.8853 

6.O2I4 

ii 

12.0483 

12.3046 

2  ft. 

3.1416 

6.2832 

4.A 

12.5664 

12.5664 

I 

3.4088 

6545 

i 

T30955 

12.8282 

2 

3.687 

6.8068 

2 

T3"6354 

13.09 

3 

3.9761 

7.0686 

3 

14.1863 

I3.35I8 

4 

4.2761 

7.3304 

4 

14.7481 

13.6136 

5 

4.5869 

7.5922 

5 

15.3208 

I3-8754 

6 

4.9087 

7.854 

6 

I5-9043 

14.1372 

7 

5.24I5 

8.1158 

7 

16.4989 

14.499 

8 

5.5852 

8.3776 

8 

17.1043 

14.6608 

9 

5.9396 

8.6394 

9 

17.7206 

14.9226 

10 

6.305 

8.9012 

10 

18.3478 

15.1844 

ii 

6.6814 

9.163 

ii 

18.9859 

15.4462 

AKEAS   AND    CIKCUMFEEEXCES   OF   CIRCLES. 


253 


DiAM 

ARIA. 

ClRCVH. 

DlAM. 

AREA. 

ClRCUM. 

Feet. 

Feet. 

Fr3t. 

Feet. 

5A 

I9-635 

I5-708 

6 

70.8823 

29.8452 

I 

20.2949 

15-9698 

7 

72.1314 

30.107 

2 

20.9658 

16.2316 

8 

73«39I3 

30.3688 

3 

21.6476 

16.4934 

9 

74.6621 

30.6306 

4 

22.3403 

16.7552 

10 

75-9439 

30.8924 

5 

23-0439 

17.017 

ii 

77-2365 

31.1543 

6 

8 
9 

10 

ii 

23-7583 
24.4837 
25.22 

25.9673 
26.7254 
27.4944 

17.2788 
17.5406 
17.8024 
18.0642 
18.326 
18.5878 

10  ft. 

i 

2 

3 

4 
5 

78.54 
79-8545 
81.1798 
82.5161 
83-8633 
85.2214 

3I-4I6 
31.6778 
3I-9396 
32.2014 
32.4632 
32.725 

3ft. 

28.2744 

18.8496 

6 

86.5903 

32.9868 

1 

29.0653 

I9.III4 

7 

87.9703 

33.2486 

2 

29.867 

I9-3732 

8 

89.3611 

'  33-5I04 

3 

30.6797 

19.635 

9 

90.7628 

33-7722 

4 

3I-5033 

19.8968 

10 

92.1754 

34-034 

5 

32.3378 

20.1586 

ii 

93-599 

34-2958 

6 

I 

9 

10 

ii 

33-I83I 
34-0394 
34.9067 
35.7848 
36.6738 
37-5738 

20.4204 
20.6822 
20.944 
21.2058 
21.4676 
21.7294 

ii  ft. 

i 

2 

3 
4 
5 

95-0334 
96.4787 

97-935 
99.4022 
100.8803 
102.3693 

34.5576 
34.8194 
35.o8l2 

35-343 
35-6048 
35-8666 

7A 

38.4846 

21.9912 

6 

103.8691 

36.1284 

I 

39.4064 

22.253 

7 

105.38 

36.3902 

2 

40.339 

22.5148 

8 

106.9017 

36.652 

3 

41.2826 

22.7766 

9 

108.4343 

36.9138 

4 

42.2371 

23.0384 

10 

109.9778 

37-1756 

5 

43.2025 

23.3002 

ii 

111.5323 

374374 

6 

7 

8 

44.1787 
45.1659 

46.164^ 

23.562 
23.8238 
24.0856 

12  ft. 

I 
2 

113.0976 
114.6739 
116.261 

37-6992 
37.o6i 
38.2228 

9 

10 

ii 

47-I73I 
48.193 
49.2238 

24-3474 
24.6092 
24.871 

3 
4 

5 

117.8591 
1  19.468 
121.088 

38.4846 
38.7464 
39.0082 

9ft. 

50.2656 

25.1328 

6 

122.7187 

39-27 

I 

51-3183 

25.3946 

7 

124.3605 

39-53I8 

2 

52.3818     25.6564 

8 

126.0131 

39-7936 

3 

534563 

25.9182 

9 

127.6766 

40.0554 

4 

54.5417 

26.18 

10 

129.351 

40.3172 

5 

55-638 

26.4418 

ii 

131.0366 

40.579 

6 

8 

9 
10 
ii 

56.7451 
57-8632 

58-9923 
60.1322 
61.283 
62.4448 

26.7036 

26.054 
27.2272 
27.489 
27.7508 
28.0126 

13  ft- 

i 

2 

3 
4 
5 

132.7326 
134.4398 
136.1578 
137.8868 
139.6267 
141.3774 

40.8408 
41.1026 
41.3644 
41.6262 
41.888 
42.1498 

w* 

63.6174 

28.2744 

6 

I43-I39I 

42.4116 

i 

64.801 

28.5362 

7 

144.9117 

42.6734 

2 

65.9954 

28.798 

8 

146.6953 

42.9352 

3 

67.2008 

29.0598 

9 

148.4897 

43-197 

4 

68.417 

29.3216 

10 

150.295 

43.4588 

5 

69.6442 

29.5834 

ii 

152-1113 

43-7206 

254         AREAS    AND    CIRCUMFERENCES    OP   CIRCLES. 


DlAM. 

AKKA. 

ClRCUM. 

DlAM. 

AREA. 

ClKCUM. 

Feet. 

Feet. 

Feet. 

Feet. 

14  ft. 

I53-9384 

43.9824 

6 

268.8031 

58.1196 

I 

155.7764 

44.2442 

7 

271.2302 

58.3814 

2 

157.6254 

44.506 

8 

273.6683 

58.6432 

3 

I59-4853 

44.7678 

9 

276.1172 

58.905 

4 

161.3561 

45.0296 

10 

278.577 

59.1668 

5 

163.2378 

45.2914 

ii 

281.0477 

59.4286 

6 

8 
9 

165.1303 
167.0338 

I70-8736 

45-5532 
45-8I5 
46.0768 
46.3386 

19  ft. 

i 

2 

283.5294 
286.0219 
288.5255 

59.6904 
59-9522 
60.214 

10 

ii 

172.8098 
I74-7569 

46.6004 
46.8622 

4 

5 

293.5651 
296.IOI2 

60.7376 
60.9994 

15  ^ 

176.715 

47.124 

6 

298.6483 

6l.20I2 

i 

178.684 

7 

301.2064 

61.523 

2 

180.6638 

47.6476 

8 

303-7753 

61.7848 

3 

182.6546 

47-9094 

9 

306.3551 

62.0466 

4 

184.6563 

48.1712 

10 

308.9458 

62.3084 

5 

186.6689 

48.433 

ii 

3U-5475 

62.5702 

6 
7 

188.6924 
190.7267 

48.6948 
48.9566 

20  ft. 

I 

314.16 
316.7834 

62.832 
63.0938 

8 
9 

10 

ii 

192.7721 
194.8283 
196.8954 
198.9734 

49.2184 
49.4802 
49-742 
50.0038 

2 

3 
4 
5 

319.4178 
322.0631 

324.7193 
327.3864 

63.3556 
63.6174 
63.8792 
64.141 

i6ft. 

20I.0624 

50.2656 

6 

330.0643 

64.4028 

i 

203.1622 

50.5274 

7 

332.7532 

64.6646 

2 

205.273 

50.7892 

8 

335-4531 

64.9264 

3 

207.3947 

9 

338.1638 

65.1882 

4 

209.5273 

51.3128 

10 

340.8854 

6545 

5 

211.6707 

5L5746 

ii 

343.6l8 

65.7118 

6 

8 
9 

10 

ii 

213.8252 
215.9904 
218.1667 
220.3538 
222.5518 
224.7607 

51.8364 
52.0982 
52.36 
52.6218 
52-8836 

53-I454 

21  ft. 

I 
2 

3 
4 
5 

346.3614 

349-  "57 
351.881 

357.4442 
360.2422 

65-9736 
66.2354 
66.4972 
66.759 
67.0208 
67.2826 

17  A 

226.9806 

53-4072 

6 

363-0511 

67.5444 

i 

229.2113 

53-669 

7 

365.8709 

67.8062 

2 

23I-453 

53-9308 

8 

368.7017 

68.068 

3 

233-7056 

54.1926 

9 

371-5433 

68.3298 

4 

235.9691 

54-4544 

10 

374-3958 

68.5916 

5 

238.2434 

54.7162 

ii 

377.2592 

68.8534 

6 

8 
9 

240.5287 
242.8249 
245.1321 
247.4501 

55-5oi6 
55-7634 

22ft. 

I 
2 

3 

380.1336 
383.0188 

385915 
388.8221 

69.1152 

69.377 
69.6388 
69.9006 

19 

249.779 

56.0252 

7QI  74. 

7O.l62J 

II 

252.II88 

56.287 

5 

394.6689 

70.4242 

'8  ft. 

254.4696 

56.5488 

6 

397.6087 

70.686 

i 

256.8312 

56.8106 

7 

400.5594 

70.9478 

2 

259.2038 

57.0724 

8 

403.5211 

71.2096 

3 

261-5873 

57-3342 

9 

406.4936 

7T«47J4 

4 

263.9817 

57-596 

10 

409-477 

5 

266.3869      57-8578 

ii 

412.4713 

71-995 

AREAS    AND    CIRCUMFERENCES    OF    CIRCLES. 


255 


DIAM.       AREA. 

ClROCM. 

DIAM. 

AREA. 

ClRCUM. 

Feet. 

Feet. 

Feet. 

Feet. 

23  A    4I5-4766 

72.2568 

6 

593-9587 

86.394 

I     418.4927 

72.5186 

7 

597.5639 

86.6558 

2 

421.5198 

72.7804 

8 

601.18 

86.9176 

3 

424.5578 

73.0422 

9 

604.8071 

87.1794 

4 

427.6067 

73'304 

10 

608.445 

87.4412 

5 

430.6664 

73.5658 

II 

612.0938 

87.703 

6 

433-7371 
436.8187 

73.8276 
740894 

28ft. 

I 

6i5-7536 
619.4242 

87.9648 
88.2266 

8 

439-9J 

74-3512 

2 

623.1058 

88.4884 

9 

10 

ii 

443.0147 
446.129 
449.2542 

74-6I3 
74.8748 
75-I366 

3 
4 
5 

626.7983 
630.5016 
634.2159 

88.7502 
89.012 
89.2738 

24ft. 

452.3904 

75.3984 

6 

637.9411 

89.5356 

I 

455-5374 

75.6602 

7 

641.6772 

89-7974 

2 

458.6954 

75.922 

8 

645-4243 

90.0592 

3 

461.8643 

76.1838 

9 

649.1822 

90.321 

4 

465.044 

76.4456 

10 

652.951 

90.5828 

5 

468.2347 

76.7074 

ii 

656-7307 

90.8446 

6 

8 
9 

10 

ii 

47I-4363 
474.6488 
477  8723 
481.1066 
484.3518 
487.6076 

76.9692 

77.231 
77.4928 
77-7546 
78.0164 
78.2782 

29  A 

i 

2 

3 
4 
5 

660.5214 
664.3229 
668.1354 
671.9588 

675.7931 
679.6382 

91.1064 
91.3682 
91.63 
91.8918 
92.1536 
92.4154 

25  A 

490.875 

78.54 

6 

683.4943 

92.6772 

i 

494.1529 

78.8018 

7 

687.3613 

92.939 

2 

497.4418 

79.0636 

8 

691.2393 

93.2008 

3 

500.7416 

79-3254 

9 

695.1281 

93.4626 

4 

504.0523 

79.5872 

10 

699.0278 

937244 

r 

507-3738 

79.849 

ii 

702.9384 

93-9862 

6 

9 
10 
ii 

.510.7063 
514.0485 
517.404 
520.7693 
524.1454 
527.5324 

80.1108 
80.3726 
80.6344 
80.8962 
81.158 
81.4198 

30  A 

i 

2 

3 
4 
5 

706.86 
710.7924 
7I4.7358 
718.6901 
722.6553 
726.6313 

94.248 
94.5098 
94.7716 
95.0334 
95.2952 
95.557 

26ft. 

530-9304 

816816 

6 

730.6183 

958188 

I 

534-3397 

81.9434 

7 

734.6162 

96.0806 

2 

537-759 

82.2052 

8 

738-625! 

96.3424 

3 
4 
5 

541.1897 

544-6313 
548.0837 

82.467 
82.7288 
82.9906 

9 

10 

ii 

742.6448 
746.6754 
750  7164 

96.6042 
96.866 
97.1278 

6 

1 

9 

10 

ii 

27  A 

55I.547I 
555.0214 
558.5066 
562.0028 
565.5098 
569.0277 

572.5566 

83-2524 
83.5142 
83-776 
84.0378 
84.2996 
84.5614 

84.8232 

3'/<. 

I 

2 

3 

4 

754.7694 
758-8327 
762907 
766.9922 
771.0883 
775.1952 
779-3I3I 

97.3896 
97.6514 
97.9132 
98.175 
98.4368 
98.6986 
98.9604 

i 

576.0963 

85.085 

7 

783.4419 

99-2222 

2 

579.6467 

85.3468 

8 

787.5817 

99.484 

3 

4 

583.2086 
586.781 

85.6086 
85.8704 

9 

10 

79^7323 
795.8938 

99-7458 
100.0076 

5 

590.364^ 

86.1322 

II 

800.0662 

100.2694 

256        AREAS   AND    CIRCUMFERENCES    OF    CIRCLES. 


DlAM. 

AREA. 

ClRCtTM. 

DlAM. 

AREA. 

ClRCtTM. 

Feet. 

Feet. 

Feet. 

Feet. 

J2.A 

804.2496 

100.5312 

6 

1046.3491 

114.6684 

I 

808.4439 

100.793 

7 

I05I.I324 

114.9302 

2 

812.649 

101.0548 

8 

1055.9266 

II5.I92 

3 

816.8651 

101.3166 

9 

1060.7318 

II5-4538 

4 

821.092 

101.5784 

10 

1065.5478 

II5-7I56 

5 

825.3299 

101.8402 

ii 

1070.3747 

115.9774 

6 

8 
9 

10 

ii 

829.5787 
833-8384 
838.1091 
842.3906 
846.683 
850.9863 

102.102 
102.3638 
102.6256 
102.8874 
103.1492 
103-411 

37  A 

i 

2 

3 

4 

5 

1075.2126 
1080.0613 
1084.921 
1089.7916 
1094.6731 
10995654 

116.2392 
Il6.5OI 
116.7628 
117.0246 
117.2864 
117.5482 

33  A 

85S-3006 

103.6728 

6 

1  104.4687 

II7.8I 

i 

859.6257 

103.9346 

7 

1109.3829 

Il8.07l8 

2 

863.9618 

104.1964 

8 

III4.308 

118.3336 

3 

868.3088 

104.4582 

9 

1119.2441 

1185954 

4 

872.6667 

104.72 

10 

1124.191 

1188572 

5 

877-0354 

104.9818 

ii 

1129.1489 

II9.II9 

6 

8 
9 

10 

ii 

881.4151 
885.8057 
890.2073 
894.6197 
899.043 
903.4772 

105.2436 

105.5054 
105.7672 
IO6.O29 
106.2908 
106.5526 

3»A 

i 

2 

3 
4 
5 

II34.II76 
II39.0972 
1144.0878 
11490893 
II54.IOI7 
II59.I249 

II93808 
119.6426 
II99044 
1  2O  l662 
120.428 
120  6898 

34  .A 

907.9224 

106.8144 

6 

1164.1591 

I20.95I6 

i 

912.3784 

107.0762 

7 

1169.2042 

I2I.2I34 

2 

916.8454 

107.338 

8 

1174.2603 

121.4758 

3 

921.3233 

107.5998 

9 

1179.3272 

121.737 

4 

925.812 

I07.86l6 

10 

1184.405 

121.9988 

5 

930.3117 

108.1234 

ii 

1189.4937 

122.2606 

6 

8 

9 
10 
ii 

934.8223 

939-3439 
943.8763 
948.4196 
952.9738 
957-5392 

108.3852 
108.647 
108.9088 
109.1706 
109.4324 
109.6942 

39A 

i 

2 

3 
4 

5 

1194.5934 
1199.7039 
1204.8254 
1209.9578 
I2I5.IOI 
1220.2552 

122.5224 
122.7848 
123.046 
123.3078 
123.5696 
123.8314 

35  A 

962.115 

109.956 

6 

1225.4203 

124.0932 

i 

966.7019 

110.2178 

7 

1230.5063 

124-355 

2 

971.2998 

110.4796 

8 

I235-7833 

I24.6l68 

3 

975.9086 

110.7414 

9 

1240.9811 

124.8786 

4 

980.5287 

III.0032 

10 

1246.1898 

125.1404 

5 

985.1588 

III.265 

ii 

1251.4094 

125.4022 

6 

989.8005 
994-4527 

III.5268 
III.7886 

40  ft. 

i 

1256.64 
I26l.88l4 

125.664 
125.9258 

8 

999.116 

112.0504 

2 

1267.1338 

126.1876 

9 

10 

ii 

1003.7903 
10084754 
1013.1714 

II2.3I22 
112.574 
112.8358 

3 

4 
5 

1272.3971 
1277.6712 
1282.9563 

126.4494 
I26.7II2 
126.973 

16  A 

1017.8784 

113.0976 

6 

1288.2523 

127.2348 

i 

1022.5962 

"3-3594 

7 

1293.5592 

127.4966 

a 

1027.325 

113.6212 

8 

1298.877 

127.7584 

3 

1032.0647 

113.883 

9 

1304.2058 

128  0202 

4 

1036.8153 

114.1448 

10 

I309-5454 

128.282 

5 

1041.5767 

114.4066 

ii 

1314.8959 

I28.5438 

AREAS   AND    CIRCUMFERENCES   OF   CIRCLES.        257 


DlAli. 

AREA. 

ClECCM. 

DlAM. 

ABBA. 

ClRCUM. 

Feet. 

Feet. 

Feet. 

Feet. 

41  ft. 

1320.2574 

128.8056 

6 

1625.9743 

142.9428 

I 

1325.6297 

129.0674 

7 

1631.9357 

143.2046 

2 

i33I-OI3 

129.3292 

8 

1637.9081 

143.4664 

3 

1336.4072 

129.591 

9 

1643.8913 

143.7282 

4 

1341.8123 

129.8528 

10 

1649.8854 

143-99 

5 

1347.2282 

130.1146 

ii 

1655.8904 

144.2518 

6 

1352.6551 

130.3764 

46A 

1661.9064 

144.5136 

7 

1358.0929 

130.6382 

i 

1667.9332 

144-7754 

8 

1363-5416 

I30.9 

2 

1673.971 

145.0372 

9 

1369.0013 

I3I.l6l8 

3 

1680.0197 

145.299 

10 

1374.4718 

131.4236 

4 

1686.0792 

145.5608 

ii 

I379-9532 

131.6854 

5 

1692.1497 

145.8226 

42  y*. 

i 

2 

3 
4 

I385-4456 
1390.9488 
1396.463 
1401.9881 
1407.5241 

131.9472 
132.209 
132.4708 
132.7326 
132.9944 

6 

8 
9 

10 

1698.2311 
1704.3195 
1710.4267 
1716.5408 
1722.6658 

146.0844 
146.3462 
146.608 
146.8698 
147.1316 

5 
6 

14130709 
1418.6287 

133.2562 
I33-5I8 

ii 
47  A 

1728.8017 
1734.9486 

147-3934 
147.6552 

7 

1424.1974 

I33-7798 

I 

1741.1063 

147917 

8 
9 

10 

1429.777 
1435-3676 
1440.969 

134.0416 

I34-3034 
I34-5652 

2 

3 
4 

I747-275 
I753-4546 
I759-645I 

148.1788 
148.4406 
148.7024 

ii 

1446.5813 

134.827 

5 

1765.8464 

148.9642 

43  A 

i 

2 

3 

4 

6 

8 
9 

10 

ii 

1452.2046 
1457.8387 
1463.4838 
1469.1398 
1474.8066 
1480.4844 
1486.1731 
1491.8717 
I497-5833 
I503-3047 
1509-037 
1514.7802 

1350888 
I35-3506 
135.6124 
135.8742 
136.136 
136.3978 
1366596 
136.9214 
137.1832 

137-445 
137.7068 
137.9686 

6 

8 
9 

10 

ii 

48A 

i 

2 

3 

4 

1772.0587 
1778.2819 
1784.516 
1790.7611 
1797.017 
1803.2838 
1809.5616 
1815.8502 
1822.1498 
1828.4603 
1834.7817 
1841.1139 
1847.4571 

149.226 
149.4878 
149.7496 
150.0114 
150.2732 
150.535 
150.7968 
151.0586 
151.3204 
151.5822 

151.844 
152.1058 
152.3676 

44  ft. 

1520.5344 

138.2304 

7 

I853.8II2 

152.6294 

i 

1526.2994 

138.4922 

8 

1860.1763 

152.8912 

2 

1532.0754 

138.754 

9 

1866.5522 

153-153 

3 

1537.8623 

139.0158 

10 

1872.939 

153.4148 

4 

1543.66 

139.2776 

ii 

I879-3367 

153.6766 

5 

1549.4687 

139-5394 

49  A 

1885.7454 

1539384 

6 

1555-2883 

139.8012 

i 

1892.1649 

154.2002 

7 

1561.1188 

140.063 

2 

1898.5954 

154462 

8 

1566.9603 

140.3248 

3 

19050368 

154.7238 

9 

1572.8126 

140.5866 

4 

1911.4897 

154.9856 

10 

1578.6756 

140.8484 

5 

1917.9522 

155.2474 

ii 

1584.5499 

I4I.II02 

6 

1924.4263 

155.5092 

45  ft- 

I590-435 

141.372 

7 

1930.9113 

I55-77I 

i 

1596.3309 

141.6338 

8 

I937-4073 

156.0328 

2 

1602.2378 

141.8956 

9 

1943.9142 

156.2946 

3 

1608.1556 

142.1574 

10 

1950.4318 

156.5564 

4 

1614.0843 

142.4192 

ii 

1956.9604 

156.8182 

5 

1620.0238 

I42.68I 

So  ft. 

1963-5 

157-08 

258 


SIDES    OF    SQUARES   EQUAL   TO    AREAS. 


Sides   of  Squares— equal   in.   .A^rea   to   a   Circle. 

Diameter  from  i  to  100. 


Diaro. 

Side  of  Sq. 

Diam. 

Side  of  Sq. 

Diam. 

Side  of  Sq. 

Diam. 

I 

.8862 

14 

12.4072 

27 

23.9281 

40 

X 

.1078 

H 

12.6287 

K 

24.1497 

y± 

.3293 

H 

12.8503 

24.3712 

y* 

% 

.5509 

H 

13.0718 

% 

24.5928 

M 

2 

.7724 

15 

!3.2934 

28 

24.8144 

41 

^ 

•994 

# 

13-515 

M 

25.0359 

X 

2.2156 

n 

13.7365 

Y2 

25-2575 

.  X 

% 

2.4371 

» 

i3.958i 

% 

25-479 

% 

3 

2.6587 

16 

14.1796 

29 

25.7006 

42 

K 

2.8802 

3^ 

14.4012 

% 

25.9221 

y± 

3.1018 

S 

14.6227 

K. 

26.1437 

% 

2i 

3-3233 

» 

14.8443 

n 

26.3653 

% 

4 

3-5449 

17 

15.0659 

30 

26.5868 

43 

X 

3-7665 

15.2874 

X 

26.8084 

K 

X 

3-988 

M 

15-509 

H 

27.0299 

X 

H 

4.2096 

% 

15.7305 

H 

27-25I5 

H 

5 

4-43" 

18 

15-9521 

31 

27-473 

44 

% 

4.6527 

i 

16.1736 

H 

27.6946 

% 

K 

4.8742 

16.3952 

y* 

27.9161 

% 

% 

5-0958 

% 

16.6168 

% 

28.1377 

% 

6 

5-3*74 

19 

16.8383 

32 

28.3593 

45 

3€ 

5.5389 

3^ 

17.0599 

& 

28.5808 

k 

M 

5.7605 

3^ 

17.2814 

y* 

28.8024 

x 

% 

5-982 

% 

17-503 

% 

29.0239 

% 

1 

6.2036 

20 

17.7245 

33 

29-2455 

46 

% 

6.4251 

y± 

17.9461 

% 

29.467 

% 

X 

6.6467 

y* 

18.1677 

y* 

29.6886 

% 

% 

6.8683 

% 

18.3892 

% 

29.9102 

M 

8 

7.0898 

21 

18.6108 

34 

30.1317 

47 

% 

7.3H4 

% 

18.8323 

& 

30.3533 

¥ 

% 

7.5329 

% 

19.0539 

X 

30.5748 

% 

7-7545 

% 

19.2754 

% 

30.7964 

% 

9 

7.976 

22 

19.497 

35 

31.0179 

48 

X 

8.1976 

^ 

19.7185 

y± 

3L2395 

k 

K 

8.4192 

K 

19.9401 

y* 

31.4611 

^ 

% 

8.6407 

% 

20.1617 

% 

31.6826 

% 

10 

8.8623 

23 

20.3832 

36 

31.9042 

49 

k 

9.0838 

| 

20.6048 

y± 

32.1257 

k 

9.3054 

20.8263 

y* 

32.3473 

H 

% 

9-5269 

M 

21.0479 

% 

32.5688 

% 

II 

9.7485 

24 

21.2694 

37 

32-7904 

50 

k 

9-97 

3£ 

21.491 

y± 

33.0112 

3^ 

% 

10.1916 

i 

21.7126 

% 

33.2335 

B 

% 

10.4132 

% 

21.9341 

% 

33-4551 

x 

12 

10.6347 

25 

22.1557 

38 

33.6766 

51 

* 

10.8563 

y± 

22.3772 

* 

33-8982 

» 

11.0778 

y* 

22.5988 

34.H97 

% 

11.2994 

% 

22.8203 

M 

34.3413 

^i 

13 

11.5209 

26 

23.0419 

39 

34.5628 

52 

^ 

11.7425 

x 

23.2634 

3€ 

34.7884 

K 

M 

11.9641 

K 

23-485 

^ 

35.006 

^ 

% 

12.1856 

% 

23.7066 

M 

35-2275 

% 

SIDES   OF   SQUARES   EQUAL  TO   AEEAS. 


259 


Side  of  Sq.  | 

Diam. 

Side  of  Sq. 

Diam. 

Side  of  Sq. 

Diam. 

Side  of  Sq. 

46.97 

65 

57.6047 

77 

68.2395 

89 

78.8742 

47.1916 

/£ 

57.8263 

X 

68461 

M 

79.0957 

47-4I31 

%" 

58.0479 

% 

68.6826 

79-3  J  73 

M 

58.2694 

% 

68.9041 

M 

79.5389 

47.8562 
48.0778 
48.2994 
48.5209 

66 

58.491 
58.7125 
58.9341 
59.1556 

78 
% 

69.1257 

69-3473 
69.5688 

69.7904 

if 

79.7604 
79-982 
80.2035 
80.4251 

48.7425 
48.964 

6\ 

59-3772 

79 

7O.OI  19 
70.2335 

80.6467 

49.1856 

X 

59.8203 

% 

70455 

M 

80.8682 

49.4071 

60.0419 

% 

70.6766 

81.0898 

49.6287 

68  * 

60.2634 

80 

70.8981 

A 

81.3113 

49.8503 

/^ 

60.485 

/^ 

71.1197 

92 

81.5329 

50.0718 

K 

60.7065 

M 

7L34I3 

8i.7544 

50.2934 

% 

60.9281 

M 

71.5628 

i 

81.976 

50.5149 

69 

61.1497 

81 

71.7844 

82.1975 

50.7365 
50.958 

X 

61.3712 
61.5928 

i 

72-0059 
72.2275 

93i 

82.4191 
82.6407 

51.1796 

X 

61.8143 

/i 

72.4491 

i^ 

82.8622 

51.4012 

70 

62.0359 

82 

72.6706 

M 

83.0838 

51.6227 

5I.8443 
52.0658  j 

52.2874  I 

7i% 

62.2574 
62.479 
62.7006 
62.9221 

i 

H 
83 

72.8921 
73."37 
73-3353 
73-5568 

94 

83-3053 
83.5269 
83.7484 

Si  O7 

52.5089 

k 

63.1437 

/4 

oj.y/ 

52.7305 

B 

63.3652 

M 

73-9999 

95 

84.1916 

52.9521 

63.5868 

« 

74.2215 

% 

84.4131 

53.1736 

72 

63-8083 

84 

74-4431 

% 

^•6347 

53-3952 

64.0299 

74.6647 

A 

84.8562 

53.6167 

% 

64.2514 

M 

74.8862 

96 

85.0778 

53-8383 

X 

64.4730 

g 

75.1077 

85-2993 

54.0598 

73 

64.6946 

85 

75.3293 

g 

85.5209 

54.2814 

64.9161 

75.5508 

M 

85-7425 

54-503  - 
54.7245 

1 

65.I377 
65.3592 

1 

75.7724 
75-9934 

97 

85.9646 
86.185 

54.946i 

74 

65.5808 

86 

76.2155 

M 

86.4071 

55-1676 

k 

65.8023 

% 

76.4371 

86.6289 

55-3892 
55-6107 

H 

66.0239 
66.2455 

X 
K 

76.6586 
76.8802 

9»4 

86.8502 

O_  f^—iQ 

55.8323 
56.0538 
56.2754 

1 

66.467 

66.6886 
66.9104 

\ 

X 

77.1017 
77.3233 
77-5449 

1 

57.O7IO 
87.2933 
87.5449 

56497 

67.1317 

77.7664 

99 

87.7364 

56.7185 
56.9401 

^ 

67.3532 
67.5748 

88 

77.988 
78.2095 

K 

87.958 
88.1796 

57.1616 

xl 

67.7964 

x^ 

78.4316 

M 

88.40X1 

57-3832 

% 

68.0179 

% 

78.6526 

100 

88.6227 

-A.pplicati.on   of  Ta~ble. 
To  A.sc«rtain.  a  Sq.xi.are  tliat  lias  same  ^Vrea  as  a  Q-iven. 

Circle. 

ILLUS.— If  side  of  a  square  that  has  same  area  as  a  circle  of  73.25  ins.  is  required. 
By  Table  of  Areas,  page  233,  opposite  to  73.25  is  4214.11;  and  in  this  table  is 
64.9161,  which  is  side  of  a  square  having  same  area  as  a  circle  of  that  diameter. 


26o 


LENGTHS    OF   CIRCULAR   ARCS. 


Lengths    of  Circular  .A^rcs,  Tip   to   a    Semicircle. 

Diameter  of  a  Circle  =  i,  and  divided  into  1000  equal  Parts. 


H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

.1 

1.02645 

-15 

1.05896 

.2 

.10348 

•25 

.15912 

•3 

1.22495 

.IOI 

1.02698 

.151 

1.05973 

.201 

.10447 

.251 

.16033 

.301 

1.22635 

.102 

1.02752 

.152 

1.06051 

.202 

.10548 

.252 

.16157 

.302 

1.227  76 

.103 

1.02806 

-153 

1.0613 

.203 

.1065 

.253 

.162  79 

.^03 

1.22918 

.104 

1.0286 

.154 

1.06209 

.204 

.10752 

•254 

.16402 

•304 

1.23061 

.105 

^.02914 

.155 

1.06288 

•205 

•10855 

•255 

.16526 

•305 

1.23205 

.106 

i  .029  7 

.156 

1.06368 

.206 

•  10958 

.256 

.16649 

.306 

1-23349 

.107 

1.03026 

-57 

1.06449 

.207 

.11062 

"257 

.16774 

-307 

1.23494 

.108 

1.03082 

.158 

1-0653 

.208 

.11165 

.258 

.16899 

.308 

1.23636 

.109 

1-03^39 

•159 

1.06611 

.209 

.11269 

.259 

.17024 

•309 

1.2378 

.11 

i.  0310 

.16 

1.06693 

.21 

.H374 

.26 

.1715 

•31 

1.23925 

.III 

1.03254 

.161 

1.06775 

.211 

.11479 

.261 

•I7275 

•3" 

1.2407 

.112 

1.03312 

.162 

1.06858 

.212 

.11584 

.262 

.17401 

.312 

1.242  16 

•"3 

1.03371 

.163 

1.06941 

•213 

.11692 

.263 

•17527 

.313 

1.2436 

.114 

1-0343 

.164 

1.07025 

.214 

.11796 

.264 

•17655 

.314 

1.24506 

•  US 

1.0349 

.165 

1.07109 

•  215 

.11904 

.265 

.17784 

.315 

1.24654 

.116 

1.03551 

.166 

1.07194 

.2l6 

.12011 

.266 

.17912 

.316 

1.24801 

.IT? 

1.03611 

.167 

1.07279 

.217 

.121  l8 

.267 

.1804 

•317 

1.24946 

.118 

1.03672 

.168 

1.07365 

.218 

.12225 

.268 

.18162 

.318 

1.25095 

.119 

1-03734 

.169 

1.07451 

.219 

.12334 

.269 

.18294 

.319 

1.25243 

.12 

1.03797 

.17 

1.07537 

.22 

.12445 

.27 

.18428 

•32 

I-25391 

.121 

1.0386 

.171 

1.07624 

.221 

.12556 

.271 

•18557 

.321 

1-25539 

.122 

1.03923 

.172 

1.07711 

.222 

.12663 

.272 

.18688 

.322 

1.25686 

.123 

1.03987 

•J73 

1.07799 

.223 

.12774 

•273 

.18819 

•323 

1.25836 

.124 

1.04051 

.174 

1.07888 

.224 

.12885 

.274 

.18969 

•324 

1.25987 

•125 

1.041  16 

.175 

1.07977 

.225 

.12997 

•275 

.19082 

.325 

1.26137 

.126 

1.04181 

.176 

1.08066 

.220 

.13108 

.276 

.19214 

.326 

1.26286 

.127 

1.04247 

.177 

1.08156 

.227 

.13219 

.277 

•!9345 

•327 

1.26437 

.128 

1-04313 

.1.78 

1.08246 

.228 

•I333I 

.278 

.19477 

•328 

1.26588 

.129 

1.0438 

.179 

1-08337 

.229 

.13444 

.279 

.1961 

•329 

1.2674 

•J3 

1.04447 

.18 

1.08428 

•23 

«I3557 

.28 

•J9743 

•33 

1.26892 

•131 

1.04515 

.181 

1.08519 

.231 

.13671 

.281 

.19887 

.331 

1.27044 

.132 

1.04584 

.182 

i.  086  1  1 

.232 

.13786 

.282 

.20011 

.332 

1.27196 

•133 

1.04652 

.183 

1.08704 

•233 

•13903 

•283 

.201  46 

•333 

1.27349 

.134 

1.04722 

.184 

1.08797 

•234 

.1402 

.284 

.202  82 

•334 

1.27502 

.135 

1.04792 

.185 

1.0889 

•235 

.14136 

•285 

.204  19 

•335 

1.27656 

.136 

1.04862 

.186 

1.08984 

.236 

.14247 

.286 

.20558 

.336 

1.2781 

•137 

1.04932 

.187 

1.09079 

•237 

•14363 

.287 

.20696 

•337 

1.27964 

.138 

1.05003 

.188 

1.09174 

.238 

.1448 

.288 

.20828 

•338 

1.281  18 

.139 

1-05075 

.189 

1.09269 

•239 

•14597 

.289 

.20967 

•339 

1.28273 

.14 

1.05147 

.19 

1.09365 

.24 

.14714 

.29 

.21202 

•34 

1.28428 

.141 

1.0522 

.191 

1.09461 

.241 

.14831 

.291 

.21239 

•341 

1.28583 

.142 

1.05293 

.192 

1-09557 

.242 

.14949 

.292 

.21381 

.342 

1.28739 

.143 

1.05367 

.193 

1.09654 

•243 

.15067 

•293 

.2152 

•343 

1.28895 

.144 

1.05441 

.194 

1.09752 

.244 

.15186 

.294 

.21658 

•344 

1.29052 

•145 

1.05516 

•195 

1.0985 

.245 

.15308 

.295 

.21794 

•345 

1.29209 

.146 

I-0559l 

.196 

1.09949 

.246 

.15429 

.296 

.21926 

•346 

1.29366 

.147 

1.05667 

.197 

1.10048 

.247 

•15549 

.297 

.22061 

•347 

1.29523 

.148 

1-05743 

.198 

1.10147 

.248 

•1567 

.298 

.222O3 

•348 

1.29681 

.149 

1.058  19 

.199 

1.10247 

•249 

.15791 

.299 

.22347 

•349 

1.29839 

LENGTHS   OF   CIRCULAR   ARCS. 


261 


H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

•35 

1.29997 

•38 

1.34899 

.41 

1.40077 

•44 

L455I2 

•47 

I.5"85 

.351 

1.30156 

.381 

1.35068 

.411 

1.40254 

.441 

1.45697 

.471 

L5I378 

•352 

L303I5 

.382 

L35237 

.412 

1.40432 

.442 

1.45883 

.472 

•353 

1.30474 

.383 

1.35406 

.413  i  1.4061 

•443 

1.46069 

-473 

1.51764 

•354 

1.30634 

•384    1-35575 

.414  1  1.40788 

•444 

1.46255 

•474 

1.51958 

•355 

1.30794 

•385    1-35744 

•4*5 

1.40966 

•445 

1.46441 

•475 

I.52I  52 

.356 

1.30954 

•386 

I-359I4 

.416 

1.41145 

.446 

1.46628 

.476 

I-  523  46 

•357 

•387 

1.36084 

.417 

1.41324 

•447 

1.46815 

•477 

^525  41 

•358 

1.31276 

.388 

1.36254 

.418    1.41503 

.448 

1.47002 

.478 

L527  36 

359 

L3I437 

•389 

1.36425 

.419 

1.41682 

•449 

1.47189 

•479 

I-5293I 

n 

I.3I599 
1.31761 

•39 
•391 

1.36596 
1.36767 

.42 
.421 

1.41861 
1.42041 

•45 
•451 

1-47377 
L47565 

.48 
.481 

1.53126 
1.53322 

.362 

1.31923 

•392 

1.36939 

.422 

1  .422  22 

•452 

1  -477  53 

«2 

I«535  J8 

.363 

1.32086 

•393 

1.37111 

•423 

1.42402 

•453 

1.47942 

•483 

J-537  J4 

•364 

.365 
.366 

1.32249 
L324I3 
L32577 

•394 

-3?l 
•396 

1.37283 

1-37455 
1.37628 

.424 

•425 
.426 

1.42583 
1.42764 
1.42945 

•454 
•455 
.456 

1.48131 
1.4832 
1.48509 

•485 
.486 

I-539I 
1.54106 
L54302 

.367 

1.32741 
1.32905 
1.33069 

•397 
•398 
•399 

1.37801 

1  -379  74 
1.38148 

.427 
.428 
.429 

I.43I27 
L43309 
I.4349I 

•457 
•458 
•459 

1.48699 
1.48889 
1.49079 

.488 
.489 
•49 

1.54499 
1.54696 
1.54893 

•37 

I-33234 

•4 

1.38322 

•43 

1.43673 

.46 

1.49269 

.491 

1.55288 

•371 

1-33399 

.401 

1.38496 

•431 

1.43856 

.461 

1.4946 

.492 

1.55486 

•372 

1-33564 

.402 

1.38671 

•432 

1.44039 

.462 

1.49651 

•493 

1-55685 

•373 

1-3373 

•403 

1.38846! 

•433 

1.44222 

•463 

1.49842 

•494 

I.55854 

•374 

1.33896 

.404 

1.39021 

•434 

1.44405 

.464 

1.50033 

•495 

1.56083 

•375 

1.34063 

•405 

1.39196 

•435 

1.44589 

•465 

1.50224 

•496 

1.56282 

•376 

1.34229 

.406 

1-39372; 

.436   1.44773 

466 

1.50416 

•497 

1.56481 

•377 

I.34396 

.407 

i  .395  48  j 

.437    1-44957 

•467 

1.50608 

.498 

1.5668 

•378 

1.34563 

.408 

I-39724 

•438 

L45I42 

.468 

1.508 

•499 

1.56879 

•379 

I-3473I 

.409 

1-399      I 

•439 

I-45327 

•469 

1.50992 

•5 

1.57079 

To  Ascertain  Length,  of  an  -A.ro  of  a  Circle  "by  pre- 
ceding Table. 

RULE.— Divide  height  by  base,  find  quotient  in  column  of  heights,  take 
length  for  that  height  opposite  to  it  in  next  column  on  the  right  hand. 
Multiply  length  thus  obtained  by  base  of  arc,  and  product  will  give  length. 

EXAMPLE.— What  is  length  of  an  arc  of  a  circle,  base  or  span  of  it  being  100  feet, 
and  height  25? 

25-7- too  =  .25;  and  .25,  per  table,  =  1.15912,  length  of  base,  which,  multiplied  by 
loo  —  u$.gi2feet. 

When,  in  division  of  a  height  by  base,  the  quotient  has  a  remainder  after 
third  place  of  decimals,  and  great  accuracy  is  required. 

RULE.— Take  length  for  first  three  figures,  subtract  it  from  next  following 
length ;  multiply  remainder  by  this  fractional  remainder,  add  product  to 
first  length,  and  sum  will  give  length  for  whole  quotient. 

EXAMPLE.— What  is  length  of  an  arc  of  a  circle,  base  of  which  is  35  feet,  and 
height  or  versed  sine  8  feet? 

8-^35  =  .228  571 4;  tabular  length  for  .228  =  1.13331,  and  for  .229  =  1.  13444 
the  difference  between  which  is  .001 13.  Then  .5714  X  .001 13  =  .000645  682. 

Hence  .228         =1.13331, 
and         .0005714=  .000645682 

i-i33955682~,  the  sum  by  which  base  of 
arc  is  to  be  multiplied ;  and  1.133  955  682  X  35  =  39.688  45  feet. 


262 


LENGTHS    OF    CIRCULAR    ARCS. 


Lengths   of  Circular  .A.rcs   from   1°   to   18O°. 

(Radius  =  i.) 


Degrees. 

Length. 

Degrees. 

Length. 

Degrees. 

Length. 

Degrees.  |   Length. 

I 

.0174 

46 

.8028 

91 

1.5882 

I36 

2.3736 

2 

•0349 

47 

.8203 

92 

1.6057 

137 

2.3911 

3 

.0524 

48 

•8377 

93 

1.6231 

138 

2.4085 

4 

.0698 

49 

.8552 

94 

1.6406 

T  AeRr 

139 

2.426 

5 
6 
7 

.0873 

•1047 

.1222 

50 
52 

.8727 
.8901 
.9076 

96 
97 
98 

I.O5OI 

1.6755 
1.693 
I.7IO4 

140 
142 

2-4435 
2.4609 
2.4784 

8 

•T396 

53 

.925 

99 

1.7279 

J43 

2.4958 

.TCT7I 

54 

•9424 

144 

2-5133 

'  '  Cl  / 

55 

•9599 

100 

1-7453 

145 

2-5307 

10 

•1745 

56 

•9774 

101 

1.7628 

146 

2.5482 

ii 

.192 

57 

•9948 

102 

1.7802 

147 

2.5656 

12 

.2094 

58 

1.0123 

103 

1.7977 

148 

2.5831 

13 

.2269 

59 

1.0297 

104 

1.8151 

149 

2.6005 

15 

16 

17 

18 

•2443 
.26l8 
.2792 
.2967 

•33l6 

60 
61 
62 

1.0472 
1.0646 
1.0821 
1.0995 
1.117 

105 
106 
107 
108 
109 

1.8326 
1.85 
1.8675 
1.8849 
1.9024 

150 

152 
153 
154 

2.618 

2-6354 
2.6529 
2.6703 
2.6878 

65 

I-I345 

no 

1.9199 

155 

2-7053 

21 
22 

•3491 
•3665 
•384 

66 

67 
68 

1.1519 
1.1694 
1.1868 

III 

112 

"3 

1-9373 
1.9548 
1.9722 

156 
157 
158 

2.7227 
2.7402 
2.7576 

23 
24 

.4014 
.4180 

69 

1.2043 

114 

1.9897 

159 

2.7751 

25 
26 
27 
28 
29 

^  "^ 
•4363 
•4538 
.4712 
.4887 
.5061 

70 

72 
73 
74 

1.2217 
1.2392 
1.2566 
1.2741 
1.2915 

"5 
116 
117 
118 
119 

2.0071 
2.0246 
2.042 

2.0595 
2.0769 

160 
161 
162 
163 
164 

2.7925 
2.81 

2.8274 
2.8449 
2.8623 

3° 

.5236 

75 

1.309 

120 

2.0944 

165 

2.8798 

•541 

76 

1.3264 

121 

2.1118 

166 

2.8972 

32 

.5585 

77 

1-3439 

122 

2.1293 

167 

2.9147 

33 

•5759 

78 

1-3613 

123 

2.1467 

168 

2.9321 

34 

•5934 

79 

1.3788 

124 

2.1642 

169 

2.9496 

35 
36 
37 
38 
39 

.6109 
.6283 
.6458 
.6632 
.6807 

80 
81 
82 
83 
84 

1.3963 
I-4I37 
1.4312 
1.4486 
1.4661 

125 
126 
I27 
128 
I29 

2.1817 
2.1991 
2.2166 
2.2304 
2.2515 

170 
171 
172 

173 
174 

2.967 
2.9845 
3.002 
3.0194 
3.0369 

40 

.6981 

85 

1-4835 

I30 

2.2689 

3-0543 

41 

•7156 

86 

1.501 

2.2864 

^76 

3-0718 

42 

•733 

87 

1.5184 

I32 

2.3038 

177 

3.0892 

43 

•7505 

88 

1.5359 

133 

2.3213 

178 

3.1067 

44 

.7679 

89 

1-5533 

134 

2.3387 

179 

3.1241 

45 

•7854 

90 

1.5708 

135 

2.3562 

180 

3.1416 

To   Ascertain    Length,   of  a   Circular   Arc   "by   Table. 
RULE. — From  column  opposite  to  degrees  of  arc,  take  length,  and  multii 
ply  it  by  radius  of  circle. 

EXAMPLE. — Number  of  degrees  in  an  arc  are  45°,  and  diameter  of  circle  5  feet. 
Then  .7854  tab.  length  X  5-7-  2  =  1.363$  feet.  ' 


LENGTHS    OF   ELLIPTIC   ABCS. 

Lengths   of  Elliptic   -A^rcs. 

Up  to  a  Semi-ellipse. 
Transverse  Diameter  =.  i,  and  divided  into  1000  equal  Parts. 


263 


H'ght. 

Length. 

H'ghL 

LMgth. 

H'ght. 

Length. 

H'ght 

Length. 

H'ght. 

Length. 

tl 

1.04202 

•15 

.0933 

.2 

I.I50I4 

•25 

1.21136 

•3 

1.27669 

.101 

1.04262 

.151 

.09448 

.2OI 

•I5I3I 

.251 

1.21263 

.301 

1.27803 

.102 

1.04362 

.152 

•09558 

.202 

.15248 

.252 

1.2139 

.302 

1.27937 

.103 

1.04462 

•153 

.09669 

.203 

.15366 

•253 

I.2I5I7 

•303 

1.28071 

.104 

1.04562 

•154 

.0978 

.204 

.15484 

•254 

1.21644 

•304 

1.28205 

.105 

1.04662 

•155 

.09891 

.205 

.15602 

•255 

1.21772 

•305 

1.28339 

.106 

1.04762 

.156 

.10002 

.206 

•1572 

.256 

I.2I9 

.306 

1.28474 

.107 

1.04862 

•!57 

.101  13 

.207 

•15838 

•257 

I.22O28 

•307 

1.28609 

.108 

1.04962 

.158 

.10224 

.208 

•15957 

.258 

1.22156 

.308 

1.28744 

.109 

1.05063 

•159 

•10335 

.209 

.16076 

•259 

1.22284 

•309 

1.28879 

.11 

1.05164 

.16 

.10447 

.21 

.16196 

.26 

1.22412 

.31 

I.290I4 

.III 

1.05265 

.161 

.1056 

.211 

.16316 

.261 

1.22541 

•3" 

1.29149 

.112 

1.05366 

.162 

.10672 

.212 

.16436 

.262 

1.2267 

.312 

1.29285 

•"3 

1.05467 

.163 

.10784 

.213 

•16557 

•263 

1.22799 

.313 

1.29421 

.114 

1.05568 

.164 

.10896 

.214 

.16678 

.264 

1.22928 

•3*4 

1-29557 

•115 

1.05669 

.165 

.11008 

.215 

.16799 

.265 

I-23057 

•315 

1.29603 

.116 

1-0577 

.166 

.1112 

.216 

.1692 

.266 

1.23186 

.316 

1.29829 

.117 

1.05872 

.167 

.11232 

.217 

.17041 

.267 

I-233I5 

•317 

1.29965 

.118 

1.05974 

.168 

•"344 

.218 

.17163 

.268 

1-23445 

.318 

I.30I  02 

.119 

1.06076 

.169 

.11456 

.219 

•17285 

.269 

1-23575 

.319 

1.30239 

.12 

1.06178 

•i? 

.11569 

.22 

.17407 

.27 

1-23705 

•32 

1.30376 

.121 

1.0628 

.171 

.11682 

.221 

•17529 

.271 

I-23835 

.321 

I-305I3 

.122 

1.06382 

.172 

•"795 

.222 

.17651 

.272 

1.23966 

.322 

1.3065 

.123 

1.06484 

•173 

.11908 

.223 

•17774 

•273 

1.24097 

•323 

1.30787 

.124 

1.06586 

.174 

.12021 

.224 

.17897 

.274 

1.24228 

•324 

1.30924 

.125 

1.06689 

•175 

.12134 

.225 

.1802 

•275 

1-24359 

•325 

I.3I06I 

.126 

1.06792 

.176 

.12247 

.226 

.18143 

.276 

1.2448 

.326 

1.31198 

.127 

1.06895 

.177 

.1236 

.227 

.18266 

•277 

1.24612 

•327 

L3I335 

,128 

1.06998 

.178 

•12473 

.228 

.1839 

.278 

1.24744 

.328 

I.3I472 

.129 

I.0700I 

.179 

.12586 

.229 

.18514 

•279 

1.24876 

•329 

I.3l6l 

•13 

1  .072  04 

.18 

.12699 

•23 

.18638 

.28 

I.250I 

•33 

I.3I748 

•131 

1.07308 

.181 

.12813 

.231 

.18762 

.281 

1.25142 

•331 

I.3I886 

.132 

1.07412 

.182 

.12927 

.232 

.188-86 

.282 

1.25274 

•332 

1.32024 

•133 

1.07516 

.183 

.13041 

•233 

.1901 

•283 

1.25406 

•333 

I.32I62 

•134 

1.07621 

.184 

«  13*55 

•234 

.19134 

.284 

I-25538 

•334 

L323 

•135 

1.07726 

.185 

.13269 

•235 

.19258 

.285 

1-2567 

•335 

1.32438 

.I36 

1.07831 

.186 

•13383 

.236 

.19382 

.286 

1.25803 

•336 

1.32576 

•137 

1.07937 

.187 

•13497 

*J  T    7  1 

•237 

.19506 

.287 

1.25936 

•337 

L327I5 

.I38 

1.08043 

.188 

.I36ll 

.238 

.1963 

.288 

1.26069 

-338 

1.32854 

•139 

1.08149 

.189 

.13726 

•239 

•19755 

.289 

1.26202 

•339 

L32993 

.14 

1.08255 

.19 

.13841 

.24 

.1988 

.29 

1-26335 

•34 

L33I32 

.141 

1.08362 

.191 

«I3956 

.241 

.20005 

.291 

1.26468 

•341 

1.33272 

.142 

1.08469 

.192 

.14071 

.242 

•2013 

.292 

1.26601 

•342 

I-334I2 

.143 

1.08576 

•193 

.14186 

•243 

.20255 

•293 

1.26734 

•343 

1-33552 

.144 

1.08684 

.194 

.14301 

•244 

.2038 

.294 

1.26867 

•344 

1.33692 

•MS 

1.08792 

•195 

.14416 

•245 

.20506 

•295 

1.27 

•345 

L33833 

.146 

1.08901 

.196 

•i453i 

.246 

.20632 

.206 

1.27133 

•346 

1  -339  74 

.147 

I.090I 

.197 

.14646 

.247 

•20758 

•297 

1.27267 

-347 

i-34i  15 

.148 

1.091  19 

.198 

.14762 

.248 

.20884 

.208 

1.27401 

-348 

1.34256 

•J49 

T.09228 

.199 

.14888 

.249 

.2101 

•»99 

L27535 

•349 

1-34397 

264 


LENGTHS    OF   ELLIPTIC    ARCS. 


H'ght. 

Length. 

H'ght 

Length. 

H'ght 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

•35 

1-34539 

•405 

I.42533 

.46 

1.50842 

.515 

1.59408 

•57 

1.68036 

•351 

1.34681 

.406 

1.42681 

.461 

1.50996 

•  5l6 

I.59564 

•571 

1.68195 

•352 

1-34823 

•407 

1.42829 

.462 

I.5II5 

.517 

1.5972 

.572 

1.68354 

•353 

1-34965 

.408 

1.42977 

•463 

L5I304 

.518 

1.59876 

•573 

1.685  13 

•354 

1.35108 

.409 

L43I25 

•464 

I-5I458 

.519 

1.60032 

•574 

1.68672 

•355 

1-35251 

.41 

1.43273 

•465 

1.51612 

.52 

1.  60188 

•575 

1.68831 

•356 

1-35394 

.411 

1.43421 

.466 

I.5I766 

.521 

1.60344 

•576 

1.6899 

•357 

1-35537 

.412 

1.43569 

•467 

1.5192 

.522 

1.605 

•577 

1.69149 

•358 

1.3568 

•413 

1.43718 

.468 

1.52074 

.523 

1.60656 

•578 

1.69308 

•359 

1-35823 

.414 

1.43867 

.469 

1.52229 

•524 

1.  608  12 

•579 

1.69467 

•36 

i-35967 

•415 

1.44016 

•47 

1.52384 

.525 

1.60968 

•58 

1.69626 

•361 

1.361  ii 

.416 

1.44165 

.471 

1.52539 

.526 

I.6II24 

•581 

1.69785 

.362 

1-36255 

.417 

1.443  J4 

.472 

1.52691 

•527 

I.6I28 

•582 

1.69945 

•363 

1  -363  99 

.418 

1.44463 

•473 

1.52849 

.528 

1.61436 

•583 

1.70105 

•364 

I-36543 

.419 

1.44613 

•474 

1.53004 

•529 

1.61592 

.584 

1.70264 

•365 

1.36688 

.42 

1.44763 

•475 

I-53I  59 

•53 

1.61748 

•585 

1.70424 

•366 

1.36833 

.421 

1.44913 

•476 

i-533I4 

•531 

1.61904 

.586 

1.70584 

.367 

1.36978 

.422 

1.45064 

•477 

1.53469 

.S32 

1.0206 

.587 

L70745 

.368 

1-37123 

.423 

1.4.  52  14. 

•478 

1.53625 

•533 

I.622I6 

.588 

1.70905 

•369 

1.37268 

.424    1.45364 

•479 

i.5378i 

•534 

1.62372 

•589 

1.71065 

•37 

I.374I4 

.425    1.45515 

.48 

1  -539  37 

•535 

1.62528 

•59 

1.71225 

•371 

1.37662 

.426  '  1.45665 

.481 

I-54093 

.536 

1.62684 

•591 

1.71286 

•372 

1.37708 

•427 

I.458I5 

.482 

1.54249 

•537 

1.6284 

•592 

1.71546 

•373 

I.37854 

.428 

1.45966 

•483 

1-54405: 

.538 

1.62996 

j-593 

1.71707 

•374 

1.38 

•429 

1.46167 

•484 

i.5456i 

•539 

1.63152 

•594 

1.71868 

•375 

1.38146 

•43 

1.46268 

•485 

1.54718 

•54 

1.63300 

•595 

1.72029 

•376 

1.38292 

•431 

1.46419 

.486 

1.54875 

•541 

x-            f 

1.63465 

•596 

1.7219 

•377 

1.38439 

.432 

1.4657 

.487 

1-55032 

.542 

1,636  23 

•597 

I-7235 

•378 

1-38585 

•433 

1.46721 

.488 

1-55189 

•543 

1.6378 

•598 

1.72511 

•379 

1-38732 

•434 

1.46872 

.489 

I.55346 

•544 

1.63937 

•599 

1.72672 

•38 

1-38879 

•435 

1.47023 

•49 

I.55503 

-545 

1.64094 

.6 

1.72833 

•381 

1.39024 

•436 

1.47174 

.491 

1.5566 

•546 

1.64251 

.601 

1.72994 

.382 

1.39169 

•437 

1.47326 

.492 

1.55817 

•547 

1.64408 

.602 

L73I55 

•383 

I-393I4 

•438 

1.47478 

•493 

1-55974 

.548 

1.64565 

.603 

I-733I6 

•384 

1-39459 

•439 

M763 

•494 

1.56131 

•549 

1.64722 

.604 

1-73477 

•385 

1.39605 

•44 

1.47782 

-495 

1.56289 

•55 

1.64879 

.605 

1.73638 

.386 
•387 

I-3975I 
1.39897 

.441 
.442 

1-47934 
1.48086 

•496 
•497 

1.56447 
1.56605 

•551 
•552 

1.65036 
1.65193 

.606 
.607 

1-73799 
I.7396 

388 

1.40043 

•443 

1.48238 

.498 

1.56763 

•553 

1.6535 

.608 

1.74121 

•389 

1.40189 

•444 

1.48391 

•499 

1.56921 

•554 

1.65507 

.609 

1.74283 

•39 

L40335 

•445 

1.48544 

•5 

1.57089 

•555 

1.65665 

.61 

1.74444 

•391 

1.40481 

•446 

1.48697 

.501 

I.57234 

•556 

1.65823 

.611 

1.74605 

•392 

1.40627 

•447 

1-4885 

•502 

I.57389 

•557 

1.65981 

.612 

1.74767 

•393 

1.40773 

.448 

1.49003 

•503 

1-57544 

•558 

1.66139 

-613 

1.74929 

•394 

1.40919 

•449 

I-49I57 

•504 

1.57699 

•559 

1.66297 

.614 

1.75091 

•395 

1.41065 

•45 

1.49311 

•505 

I-57854 

•56 

1.66455 

.615 

1-75252 

•396 

1.41211 

•45  ! 

1.49465 

•506 

1.58009 

-561 

1.66613 

.616 

i«754I4 

•397 

I-4I357 

•452 

1.49618 

•507 

1.58164 

•562 

1.66771 

.617 

I-75576 

•398 

1.41504 

•453 

1.49771 

-508 

1-583  J9 

.563 

1.66929 

.618 

I-75738 

•399 

1.41651 

•454 

1.49924 

•509 

1.58474 

.564 

1.67087 

.619 

1-759 

•4 

1.41798 

•455 

1.50077 

•5i 

1.58629 

•565 

1.67245 

.620 

1.76062 

.401 

1.41945 

•456 

1.5023 

•5" 

1-58784 

-566 

1.67403    .621 

1.76224 

.402 

i  .420  92 

•457 

1-50383 

.512 

1.5894 

-567 

1.67561 

.622 

1.76386 

•403 

1.42239 

•458 

1-50536 

•513 

1.59096 

.568 

1.67719 

.623 

1.76548 

.404 

1.42386 

•459 

1.50689 

.514 

1.59252 

•569 

1.67877; 

.624 

1.7671 

LENGTHS   OP  ELLIPTIC   ARCS. 


265 


H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

.625 

1.76872 

.68 

1.85874 

-735 

L95059 

•79 

2.04462 

.845 

2-i4I55 

.626 

1.77034 

.681 

1.86039 

-736 

1.95228 

.791 

2.04635 

.846 

2.14334 

.627 

1.77197 

.682 

1.86205 

•737 

1-95397 

.792 

2.04809 

.847 

2.14513 

.628 

!«77359 

•683 

1.8637 

.738 

1.95566 

•793 

2-04983 

.848 

2.14692 

.629 

1.77521 

.684 

1-86535 

•739 

1-95735 

•794 

2.05157 

•849 

2.14871 

•63 
4631 

1.77684 
1.77847 

.685 
.686 

1.867 

1.86866 

•74 
•741 

1.95994 
1.96074 

•795   2.05331 
•796  2.05505 

•85 
.851 

2.1505 
2.15229 

.632 
•633 
.634 

1.78009 
1.781  72 
1.78335 

.687 
.688 
.689 

1.87031 
1.87196 
1.87362 

.742 
•743 
•744 

1.96244 
1.96414 
1.96583 

•797 
.798 
•799 

2.05079 
2-05853 
206027 

.852 
•853 
•854 

2.15409 
2.15589 
2.1577 

•635 

1.78498 

.69 

1.87527 

•745 

i.o6753 

.8 

2.O02O2 

.855 

2.1595 

.636 

1.7866 

.691 

1.87693 

.746 

1.96923 

.801 

2.06377 

.856 

2.1613 

.637 

1.78823 

.692 

1.87859 

•747 

1.97093 

.802 

2.065  52 

•857 

2.16309 

.638 

1.78986 

.693 

1.88024 

•748 

1.97262 

.803 

2.06727 

•858 

2.16489 

•639 

1.79149 

.694 

1.8819 

•749 

1.97432 

.804 

2.06901 

•859 

2.16668 

.64 

1.79312 

•695 

1.88356 

•75 

1.97602 

•805 

2.070  76 

.86 

2.16848 

.641 

1-79475 

.696 

1.88522 

•751 

1.97772 

.806 

2.07251 

.861 

2.17028 

'f*3 

1.70038 
1.79801 

.697 
.698 

1.88688 
1.88854 

•752 
•753 

1-97943 
1.981  13 

.807 
.808 

2.07427 
2.07602 

.862 
.863 

2.17209 
2.17389 

•644 

1.79964 

.699 

1.8902 

•754 

1.98283 

.809 

2.07777 

.864 

2.1757 

•645 

1.80127 

•7 

1.89186 

•755 

1.08453 

.81 

2.07953 

.865 

2.17751 

.646 

1.8029 

.701 

1.89352 

•756 

1.98623 

.811 

2.081  28 

.866 

2.17932 

.647 

1.80454 

.702 

1.89519 

•757 

1.98794 

.812 

2.08304 

.867 

2.18113 

.648 

1.80617 

•703 

1.89685 

•758 

1.98964 

•813 

2.0848 

.868 

2.18294 

•649 

1.8078 

•704 

1.89851 

•759 

1.99134 

.814  12.08656 

.869 

2.18475 

.65 

1.80943 

•705 

1.90017 

.76 

1.99305    -815    2.08832 

•87 

2.18656 

.651 

1.81107 

.706 

1.901  84 

.761 

1.994  76  |-8i6   2.09008 

.871 

2.18837 

.652 

1.81271 

.707 

1-9035 

.762 

i.99647||.8i7   2.09198 

.872 

2.19018 

•653 

i.8i435 

.708 

1.905  17 

•763 

1.99818 

.818   2.0936 

.873     2,102 

•654 

1.81599 

.709 

1.90684 

.764 

1.99989 

.819 

2.09536    .874 

2.19382 

•655 

1.81763 

.71 

1.90852 

•765 

2.0016 

.82 

2.097I2H.875 

2.19564 

.656 

1.81928 

.711 

1.91019 

.766 

2.00331 

.821 

2.09888     .876 

2.19746 

•657 

1.82091 

.712 

1.91187 

•767 

2.005  02 

1  .822 

2.10065 

•877 

2.19928 

•658 

1.82255 

.713 

L9I355 

.768 

2.00673 

•823 

2.10242 

.878 

2.201  I 

•659 

1.82419 

.714 

1-91523 

.769 

2.00844 

.824 

2.IO4  19 

.879 

2.20292 

.66 

1.82583 

•715 

1.91691 

•77 

2.OIOI6      -825 

2.10596 

.88 

2.204  74 

.661 

1.82747 

.716 

1.91859 

.771 

2.0II87 

.826 

2.10773 

.881 

2.20656 

,662 

1.82911 

.717 

1.92027 

•772 

2.01359 

.827 

2.1095 

.882 

2.20839 

•663 

1-83075 

.718 

1.92195 

•773 

2.0I53I 

.828 

2.III27 

.883 

2.21022 

.664 

1.8324 

.719 

1.92363 

•774 

2.01702      -829 

2.11304 

.884 

2.2I2O5 

•665 

1.83404 

.72 

1-92531 

•775 

2.018  74  11  .83 

2.11481 

.885 

2.21388 

.666 

1.83568 

.721 

1.927 

.776 

202045    -831 

2.11659 

.886 

2.2I57I 

.667 

I.83733 

.722 

1.92868 

-777 

2.02217    .832 

2.11837 

.887 

2.21754 

.668 

1.83897 

•723 

1.93036 

.778 

2.02389 

-833 

2.I2OI5 

.888 

2.21937 

,669 

1.84061 

.724 

1.93204 

•779 

2.02561 

•834 

2.12193 

.889 

2.2212 

.67 

1.84226 

•725 

193373 

.78 

2.02733 

•835 

2.12371 

.89 

2.22303 

.671 

1.84391 

.726 

I9354I 

.781 

2.02907 

.836 

2.12549 

.891 

2.22486 

.672 

1.84556 

•727 

1  937  I 

.782 

2.0308 

.837 

2.12727 

.892 

2.22O7 

.673 

1.8472 

.728 

1.93878 

-783 

2.03252 

.838 

2.12905 

.893 

2.22854 

.674 

1.84885 

.729 

1.94046 

.784 

2.03425    -839 

2.13083 

.894 

2.23038 

•675 

1.8505 

•73 

1.94215 

.785 

2.03598!  .84 

2.13261 

.895 

2.23222 

.676 

1.85215 

•731 

I-94383 

.786  |  2.037  71 

.841 

2.13439 

.896 

2.23406 

•677 

I.85379 

•732 

1.94552 

•787 

2.03944 

.842 

2.13618 

.897 

2.2359 

.678 

I-85544 

•733 

1.94721 

.788 

2.04117 

.843 

2.13797 

.898 

2-237  74 

•67911.85709 

•734 

1.9489 

•789 

2.0429    1  .844 

2.13976 

•899 

2.23958 

Z 

LENGTHS   OF   ELLIPTIC    ARCS. 


(Tght. 

Length, 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

H'ght. 

Length. 

•9 

2.241  42 

.92 

2.27803 

•94 

2.31479 

.96 

2.35241 

.98 

2.39055 

.901 

2.24325 

.921 

2.27987 

.941 

2.31666 

.061 

2.35431 

.081 

2.39247 

.902 

2.24508 

.922 

2.281  7 

.942 

2.31852' 

.962 

2.35621 

.082 

2-39439 

«9°3 

2.24691 

•923 

2.28354 

•943 

2.32038 

•963 

2.3581 

.983 

2.39631 

.904 

2.248  74 

.924 

2.28537 

•944 

2.322  24 

.964 

2.36 

.984 

2.39823 

•9°5 

2.25057 

•925 

2.2872 

•945 

2.32411 

.965 

2.36191 

.985 

2.400  16 

.906 

2.2524 

.926 

2.28903 

.946 

2.32598 

.966 

2.36381 

.086 

2.40208 

.907    2.25423 

.927 

2.29086 

•947 

2.32785 

.967 

2.305  71 

.087 

2.404 

.908   2.25606 

.928 

2.292  7 

.948 

2.32972 

.968 

2.367  62 

.088 

2.40592 

.909   2.25789 

.929 

2-29453 

•949 

2.3316 

.969 

2.36952 

.989 

2.407  84 

.99 

2.409  76 

.91 

2.25972 

•93 

2.29636 

•95 

2.33348 

•97 

2.371  43 

.991 

2.41169 

.911 

2.261  55 

•931 

2.298  2 

•951 

2-33537 

.971 

2-37334 

.992 

2.41362 

.912 

2.26338 

•932 

2.30004 

•952 

2.337  26 

•972 

2.37525 

•993 

2.41556 

•9i3 

2.265  21 

•933 

2.301  88 

•953 

2.33915 

•973 

2.37716 

•994 

2.41749 

.914 

2.26704 

•934 

2.30373 

•954 

2.34104 

•974 

2.37908 

•995 

2.41943 

-9*5 

2.26888 

•935 

2.30557 

•955 

2.34293 

•975 

2.381 

.996 

2.421  36 

.916 

2.27071 

•936 

2.30741 

•956 

2.34483 

.976 

2.382  91 

•997 

2.42329 

.917 

2.27254 

•937 

2.30926 

•957 

2.34073 

•977 

2.38482 

•998 

2.425  22 

.918 

2.27437 

•938 

2.31111 

•958 

2.34862 

.978 

2.38673 

•999 

2.42715 

.919 

2.2762 

•939 

2.31295 

•959 

2.35051 

•979 

2.38864 

i. 

2.42908 

To   Ascertain.    Length    of  an   Elliptic   Arc    (right   Semi- 
Ellipse)   "by  preceding   Table. 

RULE. — Divide  height  by  base,  find  quotient  in  column  of  heights,  and 
take  length  for  that  height  from  next  right-hand  column.  Multiply  length 
thus  obtained  by  base  of  arc,  and  product  will  give  length. 

EXAMPLE.— What  is  length  of  arc  of  a  semi-ellipse,  base  being  70  feet,  and  height 
30. 10  feet? 

30. 10  -7-70  =  .43;  and  43,  per  table,  =  1.462  68. 
Then  1.462  68  X  70  =  102. 3876  feet 

When  Curve  is  not  that  of  a  right  Semi-Ellipse,  Height  being  half  of  Trans- 
verse Diameter. 

RULE. — Divide  half  base  by  twice  height,  then  proceed  as  in  preceding 
example ;  multiply  tabular  length  by  twice  height,  and  product  will  give 
length. 

EXAMPLE.— What  is  length  of  arc  of  a  semi-ellipse,  height  being  35  feet,  and  base 
60  feet? 

60  -4-  2  =  30,  and  30  -•-  35  x  2  =  .  428,  tabular  length  of  which  =  1.459  66. 
Then  1.45966  X  35X2  =  102.1762  feet. 

When,  in  Division  of  a  Height  by  Base,  Quotient  has  a  Remainder  after 

third  Place  of  Decimals,  and  great  A  ccuracy  is  required, 
RULE. — Take  length  for  first  three  figures,  subtract  it  from  next  following 
length;  multiply  remainder  by  this  fractional  remainder,  add  product  to 
first  length,  and  sum  will  give  length  for  whole  quotient. 

EXAMPLE.  —  What  is  length  of  an  arc  of  a  semi-ellipse,  base  being  171.1  feet  and 
height  125  feet? 

171.3-7-2  =  85.65,  and  125  X  2  =  250.      85. 65  -7-250:=.  3426  ;  tabular  length  for 
,342  =  1.334 12> and  f°r  -343  =  '-335  52,  the  difference  between  which  is  .0014. 
Then  .  6  x.  0014  =  .0084. 

Hence,  .342  =1.33412 
.0006=  .0084 


1.342  52,  the  sum  by  which  base  of  arc 
vstobe  multiplied;  and  1.34252  X  171- 3  =  229. 973676/0*. 


AKEAS    OF    SEGMENTS    OF    A    CIKCLE.  267 

A.reas   of*  Segments   of  a   Circle. 

The  Diameter  of  a  Circle  =  i,  and  divided  into  1000  equal  Parts. 


Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

1!  Versed 
Seg.  Area,   j   Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

.001 

.00004 

.052 

.01556 

.103 

.04269 

•154 

.07675 

•205 

."584 

.OO2 

.00012 

•053 

.Ol6oi 

.104 

•0433 

•155 

.07747 

.206 

.11665 

.003 

.O0022 

•054 

.01646 

.105 

.04391 

•  156 

.0782 

.207 

.11746 

.004 

.00034 

.055 

.01691 

.106 

.04452 

•157 

.07892 

.208 

.11827 

-005 

.00047 

.056 

•01737 

.107 

•045  14 

.158 

.07965 

.209 

.11908 

.006 

.00002 

•057 

.01783 

.108 

•045  75 

•159 

.08038 

.21 

.1199 

.007 

.00078 

.058 

.0183 

.109 

.04638 

.16 

.O8l  II 

.211 

.12071 

.008 

.00095 

•059 

.01877 

.11 

•047 

.161 

.08185 

.212 

•I2i  53 

.009 

.00113 

.06 

.019  24 

.III 

.04763 

.162 

.08258 

.213 

•  12235 

.01 

•00133 

.061 

.01972 

.112 

.04826 

.163 

.08332 

.214 

.12317 

.Oil 

•00153 

.062 

.02O2 

•"3 

.04889 

.164 

.08406 

.215 

.12309 

.012 

.00175 

.063 

.02068 

.114 

•04953 

.165 

.0848 

.216 

.12481 

.013 

.00197 

.064 

.O2I  17 

•US 

.050  16 

.166 

.08554 

.217 

•  12563 

.014 

.0022 

•065 

.021  65 

.116 

.0508 

.167 

.08629 

.218 

.12646 

.015 

.00244 

.066 

.02215 

.117 

•05145 

.168 

.08704 

.219 

.12728 

.Ol6 

.00268 

.067 

.02265 

.118 

.05209 

.169 

.08779 

.22 

.12811 

.017 

.O0294 

.068 

.02315 

.II9 

.05274 

•17 

.08853 

.221 

.12894 

.018 

.0032 

.069 

.02366 

.12 

.05338 

.171 

.08929 

.222 

.12977 

.OI9 

.00347 

.07 

.02417 

.121 

.05404 

.172 

.09004 

•223 

.1306 

.02 

•00375 

.071 

.02468 

.122 

.05469 

•173 

.0908 

.224 

•I3I44 

.021 

.00403 

.072 

.025  19 

.123 

.05534 

.174 

•09155 

.225 

.13227 

.022 

.00432 

•073 

.02571 

.124 

.056 

•175 

.09231 

.226 

•133" 

.023 

.00462 

.074 

.02624 

.125 

.05666 

.176 

.09307 

.227 

•13394 

.024 

.00492 

•075 

.02676 

.126 

•05733 

.177 

.09384 

.228 

•13478 

.025 

.00523 

.076 

.02729 

.127 

•05799 

.178 

.0946 

.229 

•13562 

.O26 

'00555 

.077 

.02782 

.128 

.05866 

.179 

•09537 

.23 

.13646 

.027  |   .00587 

.078 

.02835 

.129 

•05933 

.18 

.09613 

.231 

•I373I 

.028      .006  19 

.079 

.02889 

•13 

.06 

.181 

.0969 

.232 

•138  15 

.029 

.00653 

.c3 

.02943 

•131 

.06067 

.182 

•09767 

.233 

•139 

•03 

.00686 

.081    .02997 

.132 

•06135 

•  183 

•09845 

•234 

.13984 

•031 

.00721 

.082 

.03052 

•133 

.06203 

.184 

.09922 

.14069 

.032 

.00756 

.083 

.03107 

•134 

.06271 

•  185 

.1 

•236 

.14*54 

•033 

.00791 

.084 

.031  62 

•135 

.06339 

.186 

.10077 

•237 

.14239 

•034 

.00827 

.085 

.032  1  8 

.136 

.06407 

.187 

•IOI55 

.238 

.14324 

•035 

.00864 

.086 

.032  74 

•137 

.06476 

.188 

.10233 

.239 

.14409 

.036 

.009OI 

.087 

•0333 

.138 

•06545 

.189 

.103  12 

.24 

.14494 

•037 

.00938 

.088 

•03387 

•139 

.06614 

.19 

.1039 

.241 

.1458 

.038 

.00976 

.089 

.03444 

.14 

.06683 

.191 

.10468 

.242 

.14665 

•039 

.01015 

.09 

•03501 

.141 

•06753 

.192 

•10547 

.243 

.14751 

.04 

.01054 

.091 

•03558 

.142 

.06822 

•193 

.10626 

.244 

.14837 

.041 

.01093 

.092 

.036  16 

•143 

.06892 

.194 

.10705 

.245 

.14923 

.042 

.OH33 

•093 

.03674 

.144 

.06962 

•195 

.10784 

.246 

.15009 

•043 

.01173 

.094 

.03732 

•145 

•07033 

.I96 

.10864 

.247 

.15095 

.044 

.01214 

•095 

•0379 

.146 

.071  03 

.197 

.10943 

.248 

.15182 

•045 

.01255 

.096 

.03849 

.147 

.071  74 

.198 

.11023 

.249 

.15268 

.046 

.01297 

.097 

.03908 

.148 

.07245 

.199 

.11102 

•25 

.15355 

.047 

•01339 

.098 

.03968 

.149 

.073  16 

.2 

.11182 

.251 

.15441 

.048 

.01382 

.099 

.04027 

•15 

.07387 

.201 

.11262 

.252 

•15528 

.049 

.01425 

.1 

.040  87 

•151 

.07459 

.202 

•"343 

.253 

•15615 

•05 

.01468 

.101 

.041  48 

.152 

.07531 

.203 

.11423 

.254 

.15702 

.051 

.015  12 

.102 

.04208 

•153 

.07603 

.204 

."503 

.255 

•15789 

268 


AREAS   OF   SEGMENTS   OF   A   CIECLE. 


Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

Versed 
Sine. 

Seg.  Area. 

^56" 

.15876 

•305 

.202  76 

•354 

.2488 

•403 

•29631 

•452 

•344  77 

•257 

.15964 

.306 

.20368 

•355 

.24976 

.404 

.29729 

•453 

•345  77 

.258 

.16051 

•307 

.2046 

•356 

.25071 

•405 

.298  27 

•454 

•346  76 

•259 

.16139 

.308 

•20553 

•357 

.25167 

.406 

.29925 

•455 

•347  76 

.26 

.16226 

•309 

.20645 

•358 

•25263 

.407 

.30024 

•456 

•34875 

.261 

.16314 

•31 

.20738 

•359 

•25359 

.408 

.301  22 

•457 

•34975 

.262 

.16402 

•311 

.2083 

•36 

•25455 

.409 

.3022 

•458 

•35075 

.263 

.1649 

.312 

.20923 

.361 

•25551 

.41 

•30319 

•459 

.351  74 

.264 

•  16578 

•3i3 

.21015 

•362 

.25647 

.411 

•30417 

.46 

•352  74 

.265 

.16666 

•314 

.2IIO8 

•363 

•25743. 

.412 

•30515 

.461 

•353  74 

.266 

•16755 

•3i5 

.2I2OI 

•364 

•25839 

•413 

.30614 

.462 

•354  74 

.267 

.16844 

.316 

.21294 

•365 

•25936 

.414 

.307  12 

•463 

•355  73 

.268 

.16931 

•317 

•21387 

.366 

.26032 

•415 

.308ll 

•464 

•356  73 

.269 

.1702 

•318 

.2148 

•367 

.261  28 

.416 

.30909 

•465 

•357  73 

.27 

.17109 

•3i9 

•21573 

•368 

.262  25 

.417 

.31008 

.466 

•35872 

.271 

.17197 

•32 

.21667 

•369 

.26321 

.418 

.31107 

.467 

•35972 

.272 

.17287 

.321 

.2176 

•37 

.264  1  8 

.419 

•31205 

.468 

•36072 

•273 

.17376 

.322 

.21853 

•371 

.265  14 

.42 

•31304 

•469 

.361  72 

.274 

•17465 

•323 

.21947 

•372 

.26611 

.421 

•3!403 

•47 

.362  72 

•275 

•17554 

•324 

.22O4 

•373 

.26708 

.422 

•31502 

•47  1 

•363  7i 

.276 

•17643 

•325 

.221  34 

•374 

.26804 

•423 

•3i6 

•472 

•36471 

.277 

•17733 

•326 

,22228 

•375 

.26901 

.424 

.31699 

•473 

•365  7i 

.278 

.17822 

•327 

.22321 

.376 

.26998 

•425 

•31798 

•474 

.36671 

.279 

.17912 

•328 

.22415 

•377 

•27095 

.426 

•31897 

•475 

.36771 

.28 

.I8OO2 

•329 

.22509 

•378 

.27192 

•427 

.31996 

•476 

•36871 

.281 

.18092 

•33 

.22603 

•379 

.27289 

.428 

•32095 

•477 

•36971 

.282 

.18182 

•331 

.22697 

•38 

.27386 

.429 

.32194 

•478 

.37071 

•283 

.18272 

•332 

.22791 

•381 

.27483 

•43 

.32293 

•479 

•3717 

.284 

.18361 

•333 

.22886 

•382 

.27580 

•431 

•32391 

.48 

•3727 

.285 

.18452 

•334 

.2298 

•383 

.27677 

•432 

•3249 

.481 

•3737 

.286 

.18542 

•335 

.23074 

•384 

•277  75 

•433 

•3259 

.482 

•3747 

.287 

•18633 

•336 

.23169 

.385 

.27872 

•434 

.32689 

•483 

•3757 

.288 

•18723 

•337 

.23263 

•386 

.27969 

•435 

.32788 

•484 

•3767 

.289 

.18814 

•338 

.23358 

.387 

.28067 

•436 

.32887 

•485 

•3777 

.29 

.18905 

•339 

•23453 

.388 

.281  64 

«437 

.32987 

.486 

•3787 

.291 

.18995 

•34 

•23547 

•389 

.28262 

•438 

.33086 

•487 

•3797 

.292 

.19086 

•341 

.23642 

•39 

•28359 

•439 

•33185 

.488 

•3807 

•293 

.191  77 

•342 

•23737 

•391 

•28457 

•44 

•33284 

.489 

•3817 

.294 

.19268 

•343 

.23832 

•392 

•28554 

.441 

.33384 

•49 

.3827 

•295 

.1936 

•344 

.23927 

•393 

.28652 

.442 

•33483 

.491 

•3837 

.296 

•I945I 

•345 

.240  22 

•394 

•2875 

•443 

.33582 

.492 

•3847 

.297 

.19542 

•346 

.241  I7 

•395 

.28848 

•444 

.33682 

•493 

•3857 

.298 

.19634 

•347 

.242  12 

•396 

.28945 

•445 

.33781 

•494 

.3867 

.299 

•19725 

•348 

.24307 

•397 

.29043 

.446 

.3388 

•495 

.3877 

•3 

.19817 

•349 

.24403 

•398 

.291  41 

•447 

•3398 

.496 

.3887 

.301 

.19908 

•35 

.24498 

•399 

.29239 

•448 

•34079 

•497 

•3897 

.302 

.2 

•351 

•24593 

•4 

.29337 

•449 

•341  79 

.498 

•3907 

.303 

.20O92 

•352 

.24689 

.401 

•29435 

•45 

.34278 

•499 

•3917 

.304 

.201  84 

•353 

.24784 

.402 

•29533 

•451 

•34378 

•5 

•3927 

To  Compnte  A.rea  of  a  Segment  of  a  Circle  "by  preceding 

Tatole. 

RULE.—- Divide  height  or  versed  sine  by  diameter  of  circle ;  find  quotient  in 
column  of  versed  sines.  Take  area  for  versed  sine  opposite  to  it  in  next  col- 
umn on  right  hand,  multiply  it  by  square  of  diameter,  and  it  will  give  area. 


AREAS   OF   ZONES   OF   A   CIRCLE. 


269 


EXAMPLE.  —  Required  area  of  a  segment  of  a  circle,  its  height  being  10  feet  and 
diameter  of  circle  50. 

10  -r-  50  =  .  2,  and  .  2,  per  table,  •=  .  1  1  1  82  ;  then  .  1  1  1  82  x  so2  =  279.  55  feet 

When,  in  Division  of  a  Height  by  Base,  Quotient  has  a  Remainder  after 

Third  Place  of  Decimals,  and  great  Accuracy  is  required. 
RULE.  —  Take  area  for  first  three  figures,  subtract  it  from  next  following 
area,  multiply  remainder  by  said  fraction,  add  product  to  first  area,  and 
sum  will  give  area  for  whole  quotient. 

EXAMPLE.—  What  is  area  of  a  segment  of  a  circle,  diameter  of  which  is  10  feet,  and 
height  of  it  1.575? 

i.  575  -4-  10  =  .1575  ;  tabular  area  for  .  157  =  .078  92,  and  for  .  158  =  .079  65,  the  dif- 
ference between  which  is  .00073. 
Then  .  5  X  -ooo  73  =  .000  365. 
Hence,  .157  =.07892 

.0005  =  .ooo  365 

.079  285,  sum  by  which  square  of  diameter 
0f  circle  is  to  be  multiplied  ;  and  .079  285  X  io2  =  j.g 


of  Zones   of  a   Circle. 

The  Diameter  of  a  Circle  =  i,  <md  divided  into  1000  equal  Parts. 


H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area. 

.001 

.OOI 

•035 

•03497 

.069 

.06878 

.103 

.IO227 

•137 

.13527 

.002 

.002 

.036 

•03597 

.07 

.06977 

.104 

.10325 

.138 

.13623 

.003 

.003 

•037 

.03697 

.071 

.070  76 

.105 

.10422 

•139 

.13719 

.004 

.004 

.038 

.03796 

.072 

•071  75 

.106 

.1052 

.14 

.13815 

.005 

.005 

•039 

.03896 

•073 

.072  74 

.107 

.10618 

.141 

.13911 

.006 

.006 

.04 

.03996 

.074 

•07373 

.108 

.10715 

.142 

.14007 

.007 

.007 

.041 

.04095 

•075 

.07472 

.109 

.10813 

•143 

.14103 

.008 

.008 

.042 

.04195 

.076 

•0755 

.11 

.IO9II 

.144 

.14198 

.009 

.009 

•043 

.04295 

.077 

.07669 

.III 

.11008 

•145 

.14294 

.01 

.01 

.044 

.04394 

.078 

.07768 

.112 

,III06 

.146 

•1439 

.Oil 

.Oil 

•045 

.04494 

.079 

.07867 

•113 

.11203 

.147 

.14485 

.012 

.012 

.046 

•04593 

.08 

.07966 

.114 

•"3 

.148 

.14581 

.013 

.013 

.047 

.04693 

.O8l 

.08064 

•"5 

.11398 

.149 

.14677 

.014 

.014 

.048 

•04793 

.082 

.08163 

.116 

•"495 

•15 

.14773 

.015 

.015 

.049 

.048  92 

.083 

.08262 

.117 

.11592 

.151 

.14867 

.Ol6 

.Ol6 

•05 

.04992 

.084 

.0836 

.118 

.1169 

.152 

.14962 

.017 

.017 

.051 

.05091 

•085 

.08459 

.119 

.  1787 

•153 

.15058 

.018 

.018 

.052 

.0519 

.086 

•08557 

.12 

.  1884 

.154 

.15153 

.OIQ 

.019 

•053 

.0529 

.087 

.08656 

.121 

.  1981 

.155 

.15248 

.02 

.02 

•054 

.05389 

.088 

•08754 

.122 

.  2078 

.156 

•15343 

.021 

.O2I 

•055 

.05489 

.089 

•08853 

.123 

•  2175 

•157 

.15438 

.022 

.022 

.056 

.05588 

.09 

.08951 

.124 

.12272 

.158 

•15533 

.023 

.023 

•057 

.05688 

.091 

.0905 

.125 

.12369 

•159 

.15628 

.024 

.024 

.058 

•05787 

.092 

.09148 

.126 

.12469 

.16 

•I3723 

.025 

.025 

•°59 

.05886 

•093 

.09246 

.127 

.12562 

.l6l 

.15817 

.026 

.02599 

.06 

.05986 

.094 

•09344 

.128 

.12659 

.102 

.15912 

.027 

.02699 

.061 

.06085 

•095 

.09443 

.129 

•12755 

.163 

.16006 

.028 

.02799 

.062 

.06184 

.096 

•0954 

•J3 

.12852 

.164 

.16101 

.029 

.02898 

.063 

.06283 

.097 

.09639 

.131 

.12949 

.165 

.16195 

•03 

.02998 

.064 

.06382 

.098 

.09737 

.132 

•13045 

.166 

.1629 

.031 

.03098 

.065 

.06482 

.099 

•09835 

.133 

.13141 

.167 

.16384 

.032 

.03198 

.066 

.0658 

.1 

•09933 

•134 

.13238 

.168 

.16478 

•033 

.03298 

.067 

.0668 

.IOI 

.10031 

•135 

•13334 

.169 

.16572 

•034 

•03397 

,068 

.0678 

.IO2 

.101  29 

.136 

•1343 

I.I7 

.16667 

z* 


2/0 


AKEAS    OF   ZONES    OF    A    CIRCLE. 


H'ght. 

Area. 

H'ght. 

Are*. 

H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area. 

.171 

.16761 

.226 

.21805 

.281 

.26541 

•336 

.30864 

•39i 

•34632 

.172 

.16855 

.227 

.21894 

.282 

.26624 

•337 

•30938 

•392 

•34694 

•173 

.16948 

.228 

.21983 

.283 

.26706 

•338 

.31012 

•393 

•34756 

.174 

.17042 

.229 

.220  72 

.284 

.26789 

•339 

.31085 

•394 

.34818 

•175 

.17136 

•23 

.22l6l 

.285 

.26871 

•34 

•3ir59 

•395 

•34879 

-I76 

.1723 

.231 

.2225 

.286 

•26953 

•341 

.31232 

•396 

-3494 

.177 

•17323 

.232 

•22335 

.287 

•27035 

•342 

•31305 

•397 

•35001 

.178 

.17417 

•233 

.22427 

.288 

.27117 

•343 

•31378 

•398 

.35062 

.179 

•I75i 

•234 

•225  I5 

.289 

.27199 

•344 

•3i45 

•395 

.35122 

.18 

.17603 

•235 

.22604 

.29 

.2728 

•345 

•31523 

•4 

•351  82 

.181 

.17697 

.236 

.22692 

.291 

.27362 

•346 

•31595 

.401 

•35242 

.182 

.1779 

•237 

.2278 

.292 

•27443 

•347 

.31667 

.402 

•35302 

•I83 

.17883 

238 

.22868 

•293 

.27524 

•348 

.31739 

•403 

•353  61 

.184 

.17976 

•239 

.22956 

.294 

.27605 

•349 

.31811 

•404 

•3542 

.185 

.18069 

.24 

.23044 

•295 

.27686 

•35 

.31882 

•405 

•354  79 

.186 

.18162 

.241 

.231  31 

.20 

.27766 

•351 

•31954 

.406 

•35538 

.187 

.18254 

.242 

.232  19 

.297 

.27847 

•352 

.32025 

.407 

•35596 

.188 

•18347 

•243 

.23306 

.208 

.27927 

•353 

.32096 

.408 

.35654 

.189 

.1844 

.244 

•23394 

•299 

.28007 

•354 

.321  67 

.409 

•357" 

.19 

.18532 

•245 

.23481 

•3 

.28088 

•355 

.32237 

.41 

•35769 

.191 

.18625 

.246 

.23568 

.301 

.281  67 

•356 

.32307 

.411 

.35826 

.192 

.18717 

.247 

•23655 

.302 

.28247 

•357 

•32377 

.412 

•35883 

•193 

.18809 

.248 

.23742 

•3°3 

.28327 

.358 

.32447 

•4J3 

•35939 

.194 

.18902 

.249 

.23829 

.304 

.28406 

•359 

.32517 

.414 

•35995 

•195 

.18994 

•25 

•239  15 

•305 

.28486 

•36 

•32587 

•415 

.36051 

.196 

.19086 

.251 

.24002 

.306 

.28565 

.361 

.32656 

.416 

.361  07 

.197 

.191  78 

.252 

.24089 

.307 

.28644 

.362 

•32725 

.417 

.361  62 

.198 

.1927 

•253 

•241  75 

.308 

.28723 

•363 

•32794 

.418 

.36217 

.199 

•19361 

•254 

.242  61 

•309 

.28801 

•364 

.32862 

.419 

.36272 

.2 

•19453 

•255 

•24347 

•3i 

.2888 

•365 

•32931 

.42 

.36326 

.2OI 

•19545 

.256 

•24433 

•3ii 

.28958 

.366 

•32999 

.421 

•3638 

.202 

.19636 

•257 

•245  19 

.312 

.29036 

•367 

•33067 

.422 

.36434 

.203 

.19728 

.258 

.246  04 

•3!3 

.291  15 

.368 

.33135 

•423 

.36488 

.204 

.19819 

•259 

.2469 

•314 

.291  92 

•369 

•33203 

•424 

•36541 

.205 

.1991 

.26 

•247  75 

•3i5 

.2927 

•37 

.3327 

•425 

.36594 

.206 

.20001 

26l 

.248  61 

.316 

.29348 

•371 

•33337 

.426 

.36646 

.207 

.20092 

.262 

.249  46 

•317 

.29425 

•372 

•33404 

•427 

.36698 

.208 

.201  83 

.263 

.25021 

•318 

.29502 

•373 

•33471 

.428 

i  -3675 

.209 

.202  74 

.264 

.251  16 

•319 

.2958 

•374 

•33537 

•429 

.36802 

.21 

.20365 

•265 

.25201 

•32 

.29656 

•375 

•33604 

•43 

•36853 

.211 

.20456 

.266 

.25285 

.321 

•29733 

•376 

.3367 

•431 

.36904 

.212 

.20546 

.267 

•2537 

.322 

.2981 

•377 

•33735 

.432 

.36954 

.213 

.20637 

.268 

•25455 

•323 

.29886 

•378 

•338oi 

•433 

•37005 

.214 

.207  27 

.269 

•25539 

•324 

.29962 

•379 

.33866 

•434 

•37054 

.215 

.208  18 

.27 

.25623 

•325 

.30039 

•38 

•33931 

•435 

•37J04 

.2l6 

.20908 

.271 

.25707 

.326 

.301  I4 

.381 

•33996 

•436 

.371  53 

.217 

.20998 

.272 

•25791 

•327 

.3019 

.382 

.34061 

•437 

.37202 

.218 

.21088 

.273 

•25875 

•328 

.30266 

•383 

•34125 

•438 

•3725 

.219 

.21178 

.274 

•25959 

•329 

•30341 

•384 

.3419 

•439 

•37298 

.22 

.21268 

•275 

.26043 

•33 

.304  16 

•385 

.34253 

•44 

•37346 

.221 

.21358 

.276 

.261  26 

.331 

.30491 

•386 

•343  1  7 

.441 

•37393 

.222 

.21447 

.277 

.26209 

•332 

.30566 

•387 

•3438 

•442 

•3744 

.223 

•21537 

.278 

.26293 

•333 

.30641 

.388 

•34444 

•443 

•37487 

.224 

.21626 

.279 

.26376 

•334 

•307  15 

•389 

•34507 

•444 

37533 

•225 

.21716 

.28 

.26459 

•335 

.3079 

•39 

•34509 

•445 

•375  79 

AREAS   OF   ZONES    OP   A    CIRCLE. 


271 


H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area. 

H'ght. 

Area.      M  H'ght. 

Area. 

.446 

.37624 

•457 

.38096 

.468 

•385  14 

*H79 

.38867       .49 

.39137 

•447 

.37669 

•458 

.381  37 

.469 

•38549 

.48 

.38895 

I  .491 

•391  56 

.448 

•377  14 

•459 

.381  77 

•47 

.38583 

.481 

•38923 

.492 

.391  75 

•449 

•37758 

.46 

.382  16 

.471 

.38617 

.482 

.3895 

•493 

.391  92 

•45 

.37802 

.461 

.38255 

•472 

.3865 

•483 

.38976 

•494 

.39208 

•451 

•37845 

.462 

.38294 

•473 

.38683 

.484 

.39001 

•495 

.39223 

•452 

.37888 

•463 

•38332 

•474 

.387  15 

•485 

.39026 

.496 

.39236 

•453 

-379  3  * 

.464 

•38369 

•475 

.38747 

.486 

.3905      |  -497 

.39248 

•454 

•37973 

•465 

.38406 

.476 

.38778 

.487 

.39073 

•49» 

•39258 

•455 

.38014 

.466 

.38443 

•477 

.38808 

.488 

.39095 

•499 

.39266 

.456  |  .38056 

.467 

.384  79 

.478 

.38838 

.489 

•391  17 

•5 

.392  7 

This  Table  is  computed  only  for  Zones,  longest  Chord  of  which  is  Diam- 
eter. 

To    Coxnptite   A_rea   of  a   Zone   "by   preceding   Ta"ble, 

When  Zone  is  I^ess  than  a  Semicircle, 

RULE. — Divide  height  by  diameter,  find  quotient  in  column  of  heights. 
Take  area  for  height  opposite  to  it  in  next  column  on  right  hand,  multiply 
it  by  square  of  diameter,  and  product  will  give  area  of  zone. 
EXAMPLE.— Required  area  of  a  Zone,  diameter  of  which  is  50,  and  its  height  15. 

15 -=-50  =  . 3;  and  .3,  as  per  table,  =  .28088. 
Hence  .28088  X  so2  =  702. 2  area. 

When  Zone  is  Greater  than  a  Semicircle. 

RULE. — Take  height  on  each  side  of  diameter  of  circle,  and  ascertain,  by 
preceding  Rule,  their  respective  areas ;  add  areas  of  these  two  portions  to- 
gether, and  sum  will  give  area. 

EXAMPLE.  — Required  area  of  a  zone,  diameter  of  circle  being  50,  and  heights  of 
zone  on  each  side  of  diameter  of  circle  20  and  15. 

20-7-50  =  . 4;  .4,  as  per  table,  —  .35182;  and  .35182  X  502  =  879.55. 

15-:- 50  =  . 3;  .•$,  as  per  table,  =.28088;  and  .28088  X  so2 =702. 2. 
Hence  879.554-702.2  =  1581.75  area. 

When,  in  Division  of  a  Height  by  Chord,  Quotient  has  a  Remainder  after 
Third  Place  of  Decimals,  and  great  Accuracy  is  required. 

RULE. — Take  area  for  first  three  figures,  subtract  it  from  the  next  follow- 
ing area,  multiply  remainder  by  said  fraction,  and  add  product  to  first  area ; 
sum  will  give  area  for  whole  quotient. 

EXAMPLE.  —What  is  area  of  a  zone  of  a  circle,  greater  chord  being  100  feet,  and 
breadth  of  it  14  feet  3  ins.? 

14  feet  3  ins.  =14.25,  and  14.25-^-100  =  .1425;  tabular  length  for  .142  =  .140  07, 
and  for  .143  =  .141 03,  difference  between  which  is  .00096. 
Then  .5  X  .000  96  =  .ooo  48.     Hence. 142   =.14007 
.0005  =.00048 

.  140  55,  sum  by  which  square  of  greater 
chord  is  to  be  multiplied;  and  .14055  X  ioo7^  1405.5/6^. 


272 


SQUARES,   CUBES,   AND   ROOTS. 


Squares,   Cubes,   and   Square   and.   Cube   Roots, 

i  to  1600. 


NUMBER. 

SQUARE. 

-'CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

I 

I 

I 

I 

I 

2 

4 

8 

1.4142136 

1.259921 

3 

9 

27 

1.7320508 

1.4422496 

4 

16 

64 

2 

1.587401  I 

5 

25 

125 

2.236068 

1.7099759 

6 

36 

216 

2.4494897 

1.817  1206 

7 

49 

343 

2.6457513 

1.9129312 

8 

64 

512 

2.828  427  I 

2 

9 

81 

729 

3 

2.080  083  7 

10 

100 

1000 

3.1622777 

2.1544347 

ii 

121 

i33i 

3.3166248 

2.223  980  i 

12 

144 

i  728 

3.464  101  6 

2.2894286 

J3 

169 

2197 

3-6055513 

2.3513347 

14 

106 

2744 

3.741  657  4 

2.4101422 

15 

225 

3375 

3.8729833 

2.466212  i 

16 

2  36 

4096 

4 

2.5198421 

J7 

289 

49J3 

4.1231056 

2.571  2816 

18 

324 

5832 

4.2426407 

2.620  741  4 

19 

361 

6859 

4.3585989 

2.668  401  6 

20 

400 

8000 

4.472  136 

2.7144177 

21 

441 

9261 

4-5825757 

2.7589243 

22 

484 

10648 

4.6904158 

2.8020393 

23 

529 

12  167 

4-795  831  5 

2.843  867 

24 

576 

13824 

4.8989795 

2.884  499  I 

25 

625 

15625 

5 

2.9240177 

26 

676 

17576 

5.0990195 

2.962496 

27 

729 

19683 

5.1961524 

3 

28 

784 

21952 

5.2915026 

3.0365889 

29 

841 

24389 

5.3851648 

3.0723168 

30 

900 

2700O 

5.4772256 

3.1072325 

31 

961 

29791 

5.5677644 

3.141  3806 

32 

1024 

32768 

5.6568542 

3.1748021 

33 

1089 

35937 

5.7445626 

3-2075343 

34 

1156 

39304 

5.8309519 

3.2396118 

35 

1225 

42875 

5.9160798 

3.2710663 

36 

1296 

46656 

6 

3.3019272 

37 

1369 

50653 

6.082  762  5 

3.332  221  8 

38 

I444 

54872 

6.164414 

3-36i  975  4 

39 

1521 

59319 

6.244998 

3.3912114 

40 

I6OO 

64000 

6.3245553 

3.4199519 

4i 

1681 

68921 

6.403  124  2 

3.4482172 

42 

1764 

74088 

6.480  740  7 

3.4760266 

43 

1849 

79507 

6.5574385 

3-503  39s  i 

44 

1936 

85184 

6.6332496 

3-5303483 

45 

2025 

91  125 

6.7082039 

3-5568933 

46 

21  16 

97336 

6.78233 

3.5830479 

47 

2209 

103  823 

6.855  654  6 

3.608  826  i 

48 

2304 

110592 

6.928  203  2 

3.634  241  i 

49 

2401 

117649 

7 

3.6593057 

5° 

25°°i 

125000 

7.0710678 

3.6840314 

51 

20  01 

*3*v5' 

y.i/H  4^°  4 

3.7084298 

52 

2704 

140608 

7.2II  1026 

3.732511  i 

53 

2809 

148877 

7.28^1099 

3.7562858 

54 

2916 

157  464 

7.3484692 

3.7797631 

SQUAKES,  CUBES,  AND  EOOT8. 


273 


NUMBER. 

StUABB. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

55 

3025 

166375 

7.^61985 

3.8029525 

56 

3136 

175616 

7.4833148 

3.825  862  4 

57 

3249 

185  193 

7.5498344 

3.848  501  I 

58 

3364 

195  112 

7-6i5  773  i 

3.8708766 

59 

3481 

205379 

7.681  145  7 

3.892  996  5 

60 

3600 

2l6oOO 

7.745  966  7 

3.9148676 

61 

3721 

226981 

7.8102497 

3.9364972 

62 

3844 

238328 

7.8740079 

3.9578915 

63 

3969 

250047 

7-9372539 

3.9790571 

64 

4096 

262  144 

8 

4 

65 

4225 

274625 

8.062  257  7 

4.020  725  6 

66 

4356 

287496 

8.1240384 

4.041  240  i 

67 

4489 

300763 

8.1853528 

4.061  548 

68 

4624 

314432 

8.2462113 

4.081  655  i 

69 

4761 

328509 

8.306  623  9 

4.101  566  i 

70 

4900 

343000 

8.3666003 

4.1212853 

?i 

5041 

3579" 

8.426  149  8 

4.1408178 

72 

5184 

373248 

8.485  281  4 

4.1601676 

73 

5329 

389017 

8.5440037 

4.179339 

74 

5476 

405  224 

8.602  325  3 

4.1983364 

75 

5625 

421  875 

8.660  254 

4.2171633 

76 

5776 

438976 

8.7177979 

4.2358236 

77 

5929 

456533 

8.7749644 

4.254321 

78 

6084 

474552 

8.831  7609 

4.272  658  6 

79 

6241 

493039 

8.888  194  4 

4.2908404 

80 

6400 

5I2OOO 

8.9442719 

4.3088695 

81 

6561 

531  441 

9 

4.3267487 

82 

6724 

551368 

9-055386I 

4.3444815 

83 

6889 

57I  787 

9.1104336 

4.3620707 

84 

7056 

592  704 

9.165  151  4 

4.3795I9I 

85 

7225 

614  125 

9.2195445 

4.3968296 

86 

7396 

636056 

9.2736185 

4.4140049 

87 

7569 

658503 

9-327379I 

4.431  047  6 

88 

7744 

681472 

9.3808315 

4.4479602 

89 

7921 

704969 

9.4339811 

4.464  745  i 

90 

8100 

7290OO 

9.486833 

4.481  404  7 

91 

8281 

753571 

9-539392 

4.4979414 

92 

8464 

778688 

9.591  663 

4.5I43574 

93 

8649 

804357 

9.643  650  8 

4-5306549 

94 

8836 

830584 

9-6953597 

4-5468359 

95 

9025 

857375 

9.7467943 

4.562  902  6 

96 

92  16 

884  736 

9-797959 

4-578857 

97 

9409 

912673 

9.848  857  8 

4.5947009 

98 

9604 

941  192 

9.8994949 

4.610  436  3 

99 

9801 

970209 

9.9498744 

4.626065 

IOO 

IOO  00 

1000  OOO 

10 

4.6415888 

101 

IO2OI 

1  030  301 

10.049  875  6 

4.6570095 

102 

10404 

i  061  208 

10.0995049 

4.672  328  7 

103 

10609 

1  092  727 

10.1488916 

4.6875482 

104 

I08I6 

1  124864 

10.198039 

4.7026694 

105 

I  1025 

1157625 

10.246  950  8 

4.717694 

106 

I  1236 

1  191  016 

10.295  630  i 

4.7326235 

107 

II449 

i  225  043 

10.3440804 

4-7474594 

108 

I  1664 

1259712 

10.392  304  8 

4.7622032 

109 

1  1881 

i  295  029 

10.440  306  5 

4.7768562 

no 

I2IOO 

1331000 

10.488  088  5 

4.7914199 

274 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBER. 

S<JUAB«. 

CUBB. 

SQUARE  ROOT. 

CUBE  ROOT. 

Ill 

12321 

I  367  631 

10.535  653  8 

4.805  895  5 

112 

12544 

I  404  928 

10.583  005  2 

4.820  284  5 

"3 

12769 

I  442  897 

10.630  145  8 

4.8345881 

114 

12996 

i  481  544 

10.6770783 

4.848  807  6 

"5 

13225 

i  520  875 

10.7238053 

4.862  944  2 

116 

13456 

1560896 

10.7703296 

4.876099 

117 

13689 

i  601  613 

10.8166538 

4.8909732 

118 

13924 

1643032 

10.862  780  5 

4.904  868  i 

119 

i  41  61 

i  685  159 

10.008712  i 

4.918  684  7 

120 

14400 

i  728000 

10.9544512 

4.932  424  2 

121 

14641 

i  771  561 

ii 

4.9460874 

122 

14884 

i  815  848 

11.045361 

4.9596757 

I23 

15129 

i  860  867 

11.0005365 

4.9731898 

124 

15376 

i  906624 

11.1355287 

4.986631 

125 

15625 

1953125 

11.1803399 

5 

126 

15876 

2000376 

11.2249722 

5.0132979 

I27 

161  29 

2048383 

11.2694277 

5.026  525  7 

128 

16384 

2097  152 

11.3137085 

5.0396842 

I29 

16641 

2  146  689 

11.3578167 

5.0527743 

130 

16900 

2  197000 

11.4017543 

5-065  797 

131 

i  7161 

2  248  091 

11.4455231 

5-078  753  i 

I32 

17424 

2  299968 

11.4891253 

5.091  643  4 

133 

17689 

2352637 

11.5325626 

5.1044687 

134 

17956 

2  406  104 

"•5758369 

5.1172299 

135 

18225 

2460375 

11.61895 

5.1299278 

136 

18496 

2515456 

11.661  0038 

5.1425632 

137 

18769 

2571353 

11.7046999 

5.1551367 

138 

19044 

2  628  072 

11.7473401 

5.1676493 

139 

19321 

2  685  619 

11.7898261 

5.1801015 

140 

19600 

2744000 

11.8321596 

5.1924941 

I4I 

19881 

2803221 

11.8743421 

5.2048279 

142 

20164 

2863288 

11.9163753 

5.2171034 

H3 

20449 

2  924  207 

11.9582607 

5.2293215 

144 

20736 

2  985  984 

12 

5.241  482  8 

145 

21025 

3048625 

12.0415946 

5.2535879 

146 

21316 

3112136 

12.083046 

5.2656374 

147 

2  1609 

3176523 

12.1243557 

5.2776321 

148 

2  1904 

3  241  792 

12.1655251 

5.2895725 

149 

22201 

3307949 

12.2065556 

5.3014592 

ISO 

22500 

3375000 

12.2474487 

5.3132928 

T5i 

22801 

3442951 

12.288205  7 

5.325  074 

152 

23104 

3511808 

12.328828 

5.3368033 

153 

23409 

3581577 

12.3693169 

5.3484812 

154 

23716 

3  652  264 

12.4096736 

5.3601084 

155 

24025 

3723875 

12.4498996 

5.3716854 

156 

24336 

3796416 

12.489996 

5.3832126 

157 

24649 

3869893 

12.5299641 

5.3946907 

158 

24964 

3944312 

12.5698051 

5.406  I2C2 

*59 

25281 

4019679 

12.6095202 

54I750I5 

160 

25600 

4096000 

12.649  no  6 

5.4288352 

161 

25921 

4173281 

12.6885775 

5.440  121  8 

162 

26244 

4251528 

12.727922  i 

5.4513618 

163 

26569 

4330747 

12.7671453 

5.4625556 

164 

26896 

4  410  944 

12.8062485 

5-473  703  7 

165 

27225 

4492125 

12.8452326 

5.4848066 

166 

27556 

4  574  296 

12.8840987 

5-4958647 

SQUAEES,  CUBES,  AND  ROOTS. 


275 


NUMBER.            SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

l67 

27889 

4657463 

12.922  848 

5.5068784 

168 

28224 

4  741  632 

12.961  481  4 

5.5178484 

169 

28561 

4826809 

13 

5.5287748 

170 

28900 

4913000 

13.038  404  8 

5-5396583 

171 

29241 

50002II 

13.0766968 

5.5504991 

172 

29584 

5088448 

13.114877 

5.561  297  8 

J73 

29929 

5  I777I7 

13.1529464 

5.5720546 

*74 

30276 

5268024 

13.190906 

5.5827702 

i?5 

30625 

5359375 

13.2287566 

5-593  444  7 

176 

30976 

5  45  1  776 

13.2664992 

5.6040787 

177 

3J329 

5  545  233 

I3-304  134  7 

5.6146724 

178 

31684 

5  639  752 

13.341  664  i 

5.6252263 

179 

32041 

5  735  339 

I3-3790882 

5.6357408 

180 

32400 

5832000 

13.4164079 

5.6462162 

181 

32761 

5  929  741 

13-453624 

5.6566528 

182 

33I24 

6028568 

I3-4907376 

5.667051  i 

183 

33489 

6128487 

I3-5277493 

5.6774114 

184 

33856 

6  229  504 

13.564  66 

5.687  734 

185 

34225 

6331625 

13.6014705 

5.6980192 

1  86 

3450 

6434856 

13.638  181  7 

5.7082675 

187 

34969 

6539203 

13.6747943 

5.7184791 

188 

35344 

6  644  672 

13.7113092 

5.7286543 

189 

35721 

6751269 

13-747  727  i 

5-7387936 

190 

36100 

6859000 

13.7840488 

5.7488971 

191 

36481 

6967871 

13.820  275 

5-7589652 

192 

36864 

7077888 

13.8564065 

5.7689982 

193 

37249 

7189057 

13.89244 

5.7789966 

194 

37636 

7301384 

13.9283883 

5.7889604 

195 

38025 

7  414  875 

13.964  24 

5.79889 

196 

38416 

7  529  S36 

14 

5.808  785  7 

197 

38809 

7645373 

14.035  668  8 

5.8186479 

198 

39204 

7762392 

14.0712473 

5.8284767 

199 

39601 

7880599 

14.106736 

5.8382725 

200 

40000 

8000000 

14.1421356 

5.8480355 

201 

40401 

8  120601 

14.1774469 

5-857  766 

202 

40804 

8  242  408 

14.2126704 

5.8674643 

203 

41209 

8365427 

14.2478068 

5.8771307 

204 

41616 

8489664 

14.282  856  9 

5.8867653 

205 

42025 

8615125 

14.317821  1 

5.8963685 

206 

42436 

8  741  816 

14.352  700  1 

5.9059406 

207 

42849 

8869743 

I4-3874946 

5.9154817 

208 

43264 

8998912 

14.422  205  i 

5.9249921 

209 

43681 

9129329 

14.4568323 

5-9344721 

2IO 

44100 

9261000 

14.491  376  7 

5-943922 

211 

44521 

9393931 

14-525839 

5-953  34i  8 

212 

44944 

9528128 

14.5602198 

5.962  732 

2I3 

45369 

9663597 

14.5945195 

5.9720926 

214 

45796 

9800344 

14.628  738  8 

5.981  424 

215 

46225 

9938375 

14.662  878  3 

5.9907264 

216 

46656 

10077696 

14.6969385 

6 

217 

47089 

10218313 

14.7309199 

6.009  245 

218 

47524 

10360232 

14.764  823  i 

6.018  461  7 

219 

47961 

Jo  503  459 

14.7986486 

6.027  650  2 

2  2O 

48400 

10  648  ooo 

14.832397 

6.036  810  7 

221 

48841 

10  793  861 

14.8660687 

6.045  943  5 

222 

49284 

10  941  048 

14.8996644 

6.055  048  9 

276 


SQUARES,   CUBES,   AND    ROOTS. 


NUMBBR. 

SQUARE. 

CUBB. 

SQUARE  ROOT. 

CUBB  ROOT. 

223 

49729 

II089567 

14.9331845 

6.064  127 

224 

50176 

II239424 

14.9666295 

6.0731779 

225 

50625 

II390625 

15 

6.082  202 

226 

51076 

II543I76 

15.0332964 

6.0911994 

227 

51529 

II097083 

15.0665192 

6.100  1702 

228 

5^4 

II852352 

15.0996689 

6.109  TI4  7 

229 

52441 

12008989 

15.132  746 

6.1180332 

230 

52900 

12  I6700O 

I5-I657509 

6.1269257 

23I 

53361 

12326391 

15.1986842 

6.1357924 

232 

53824 

12487168 

15.2315462 

6.1446337 

233 

54289 

12649337 

I5-2643375 

6.1534455 

234 

54756 

12812904 

15.2970585 

6.162240  i 

235 

55225 

12977875 

I5-329  709  7 

6.1710058 

236 

55606 

13  144  256 

15.362  291  5 

6.179  7466 

-   237 

56169 

I33I2053 

15.3948043 

6.1884628 

238 

56644 

13481  272 

15.4272486 

6.1971544 

239 

57I2I 

13651919 

15.4596248 

6.2058218 

240 

57600 

13824000 

i5-49I9334- 

6.214465 

241 

58081 

I399752I 

15.5241747 

6.223,084  3 

242 

58564 

14172488 

I5-5563492 

6.231  679  7 

243 

59°49 

14348907 

15.5884573 

6.240251  5 

244 

59536 

14  526  784 

15.6204994 

6.248  799  8 

245 

60025 

I4706I25 

15.6524758 

6.2573248 

246 

60516 

14886936 

15.6843871 

6.2658266 

247 

6  1009 

T5009223 

15.7162336 

6.2743054 

248 

61504 

15252992 

15.7480157 

6.282  761  3 

249 

62001 

15438249 

I5'7797338 

6.291  1946 

250 

62500 

15  625  ooo 

15.8113883 

6.2996053 

251 

63001 

15813251 

15.8429795 

6-307  993  5 

252 

63504 

16  003  008 

15.8745079 

6-3163596 

253 

64009 

16194277 

I5-9059737 

6-324  703  5 

254 

64516 

16  387  064 

15-937  377  5 

6.3330256 

255 

65025 

16581375 

15.9687194 

6-341  325  7 

256 

65536 

16  777  216 

16 

6.349  604  2 

257 

66049 

16974593 

16.031  21  ^  5 

6-357  861  i 

258 

66564 

I7I735I2 

16.062  378  4 

6.3660968 

259 

67081 

17373979 

16.0934769 

6.3743111 

260 

67600 

17576000 

16.1245155 

6.382  504  3 

261 

68121 

17779581 

16.1554944 

6.3906765 

262 

68644 

17984728 

16.1864141 

6.3988279 

263 

69169 

18  191  447 

16.2172747 

6-406  958  5 

264 

69696 

18399744 

16.248  076  8 

6.4150687 

265 

70225 

18609625 

16.2788206 

6.4231583 

266 

70756 

18821096 

16.309  506  4 

6.431  227  6 

267 

71289 

19034163 

16.3401346 

6.439  276  7 

268 

71824 

19  248  832 

16.370  705  5 

6-447  305  7 

269 

72361 

19465109 

16.401  2195 

6-4553148 

270 

72900 

19  683  ooo 

16.431  676  7 

6-463  304  i 

271 

73441 

19902511 

16.4620776 

6-4712736 

272 

73984 

20123648 

16,492  422  5 

6-479  223  6 

273 

74529 

20346417 

16.5227116 

6-487  154  i 

274 

75076 

20570824 

16.552  945  4 

6.495  065  3 

275 

75625 

20796875 

16.583  124 

6-502  957  2 

276 

761  76 

21  024  576 

16.6132477 

6-51083 

277 

76729 

21253933 

16643317 

6-5186839 

278 

77284 

21484952 

16.678  332 

6.5265189 

SQUARES,  CUBES,  AND  ROOTS. 


FUMBIB, 

SQUARE. 

CUBB.                        SQL-  A  RE  ROOT. 

CUBE  ROOT. 

279 

77841 

21717639 

16.7032931 

6.5343351 

280 

78400 

21952000 

16.7332005 

6.542  132  6 

281 

78961 

22  188041 

16.7630546 

6.5499116 

282 

79524 

22  425  768 

16.792  855  6 

6-557  672  2 

283 

80089 

22665187 

16.822  603  8 

6.5654144 

284 

80656 

22906304 

16.8522995 

6.573  138  5 

285 

81225 

23  149  125 

16.881  943 

6.5808443 

286 

81796 

23393656 

16.9115345 

6-5885323 

287 

82369 

23639903 

16.9410743 

6.596  202  3 

288 

82944 

23887872 

16.970  562  7 

6.603  854  5 

289 

83521 

24  137  569 

17 

6.61  1  489 

290 

84100 

24389000 

17.0293864 

6.619  106 

291 

84681 

24642  171 

17.0587221 

6.626  705  4 

292 

85264 

24  897  088 

17.0880075 

6.6342874 

293 

85849 

25  153  757 

17.1172428 

6.641  852  2 

294 

86436 

25412184 

17.1464282 

6.6493998 

295 

87025 

25672375 

I7-I75564 

6-656  930  2 

296 

87616 

25934336 

17.2046505 

6.664  443  7 

297 

88209 

26  198  073 

17.2336879 

6.6719403 

298 

88804 

26463592 

17.2626765 

6.67942 

299 

89401 

26730899 

17.2916165 

6.6868831 

300 

9OOOO 

27000000 

17.3205081 

6.6943295 

301 

90601 

27270901 

17-3493516 

6.7017593 

302 

91204 

27543608 

I7-378I472 

6.7091729 

303 

91809 

27818127 

17.4068952 

6.71657 

304 

92416 

28094464 

I7-4355958 

6.7239508 

305 

93025 

28372625 

17.4642492 

6.73I3I55 

306 

93636 

28  652  616 

17.4928557 

6.7386641 

307 

94249 

28934443 

17.5214155 

6.7459967 

308 

94864 

29218112 

17.5499288 

6.7533134 

309 

95481 

29503629 

I7-5783958 

6.7606143 

310 

96lOO 

29  791  ooo 

17.6068169 

6.7678095 

&* 

96721 

30080231 

I7-635  i92  i 

6.775  169 

312 

97344 

30371328 

17.6635217 

6.7824229 

313 

97969 

30664297 

17.691  806 

6.7896613 

314 

98596 

30959144 

17.7200451 

6.7968844 

315 

99225 

31  255  875 

17-7482393 

6.804  092  i 

316 

99856 

31554496 

17.7763888 

6.8112847 

317 

100489 

31855013 

17.8044938 

6.818462 

318 

10  1  1  24 

32157432 

17.8325545 

6.825  624  2 

3i9 

101761 

32461759 

17.860571  i 

6.832  771  4 

320 

102400 

32768000 

17.8885438 

6.8399037 

321 

10  30  41 

33076  161 

17.9164729 

6.8470213 

322 

103684 

33386248 

I7-9443584 

6.854  124 

323 

104329 

33698267 

17.9722008 

6.861  212 

324 

10  49  76 

34012224 

18 

6.8682855 

325 

10  56  25 

34328125 

18.0277564 

6.8753443 

326 

10  62  76 

34645976 

18.055  470  1 

6.8823888 

327 

106929 

34965783 

18.083  T4i  3 

6.8894188 

328 

10  75  84 

35287552 

18.1107703 

6.8964345 

329 

108241 

35  61  1  289 

18.1383571 

6-9034359 

330 

108900 

35  937  ooo 

18.165902  i 

6.9104232 

33i 

10  95  61 

36264691 

18.1934054 

6.9173964 

332 

1  1  02  24 

36594368 

18.2208672 

6.9243556 

333 

110889 

36926037 

18.248  287  6 

6.9313088 

334 

ii  1556 

37  259  704 

18.2756669 

6.938  232  i 

AA 

278 


SQUARES,  CUBES,  AND  ROOTS. 


NUMBER. 

SQUARE.                         CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

335 

II  2225 

37  595  375 

18.303  005  2 

6.9451496 

336 

112896 

37933056 

18.3303028 

6.9520533 

337 

H3569 

38  272  753 

18.3575598 

6.9589434 

338 

114244 

38614472 

18.3847763 

6.9658198 

339 

II  4921 

38  958  219 

18.411  9526 

6.972  682  6 

340 

II  5600 

39304000 

18.4390889 

6.9795321 

34i 

Il628l 

39651821 

18.466  185  3 

6.9863681 

342 

II  6964 

40  ooi  688 

18.493  242 

6.9931906 

343 

117649 

40353607 

18.520  259  2 

7 

344 

118336 

40  707  584 

18.547  237 

7.006  796  2 

345 

H902S 

41063625 

18.5741756 

7.0135791 

346 

1197  16 

41  421  736 

18.6010752 

7.020349 

347 

120409 

41  781  923 

18.627  936 

7.027  105  8 

348 

12  1104 

42  144  192 

18.654  758  1 

7.033  849  7 

349 

12  1801 

42  508  549 

18.681  541  7 

7.0405806 

350 

122500 

42  875  ooo 

18.7082869 

7.047  298  7 

351 

123201 

43243551 

18.734994 

7.0540041 

352 

123904 

43614208 

18.761  663 

7.0606967 

353 

124609 

43986977 

18.7882942 

7.067  376  7 

354 

125316 

44361864 

18.8148877 

7.074044 

355 

126025 

44738875 

18.841  443  7 

7.0806988 

356 

126736 

45118016 

18.8679623 

7.087  341  i 

357 

127449 

45409293 

18.8944436 

7.0939709 

358 

12  8l  64 

45882  712 

18.9208879 

7.1005885 

359 

128881 

46  268  279 

18.9472953 

7.1071937 

360 

129600 

46  656  ooo 

18.973666 

7.1137866 

36i 

130321 

47045831 

!9 

7.1203674 

362 

131044 

47  437  928 

19.026  297  6 

7.126936 

363 

131769 

47  832  147 

19-0525589 

7.1334925 

364 

132496 

48  228  544 

19.078784 

7.140037 

365 

133225 

48627125 

19.1049732 

7.1465695 

366 

133956 

49027896 

19.131  1265 

7.1530901 

367 

134689 

49430863 

19.1572441 

7.1595988 

368 

13  54  24 

49836032 

19.1833261 

7.1660957 

369 

1361  61 

50243409 

19.2093727 

7.1725805 

370 

136900 

50653000 

J9  235  384  i 

7.1790544 

37i 

137641 

51064811 

19.2613603 

7.1855162 

372 

138384 

51478848 

19.2873015 

7.1919663 

373 

139129 

51895117 

19.3132079 

7.198405 

374 

139876 

52313624 

I9-3390796 

7.204  832  2 

375 

14  06  25 

52734375 

19.3649167 

7.21  1  247  c, 

376 

14  13  76 

53157376 

19.3907194 

7.2176522 

377 

1421  29 

53582633 

19.4164878 

7.224045 

378 

14  28  84 

54010152 

19.442  222  I 

7.230  426  8 

379 

143641 

54439939 

19.4679223 

7.236  797  2 

380 

144400 

54872000 

19.493  588  7 

7.243  156  5 

38i 

1451  61 

553o634i 

19.5192213 

7.249  504  5 

382 

145924 

55  742  968 

19.5448203 

7.255  841  5 

383 

146689 

56  181  887 

I9-5703858 

7.262  167  5 

384 

147456 

56  623  104 

19.5959179 

7.268  482  4 

385 

14  82  25 

57066625 

19.621  4169 

7.2747864 

386 

148996 

57512456 

19.6468827 

7.2810794 

387 

149769 

57960603 

19.672  315  6 

7.287  361  7 

388 

150544 

58411072 

19.6977156 

7-293  633 

389 

151321 

58863869 

19.7230829 

7.2998936 

390 

152100 

59319000 

19.7484177 

7.306  143  6 

SQUARES,  CUBES,  AND  BOOTS. 


279 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CCBK  ROOT. 

391 

152881 

59776471 

I9-7737I99 

7.3123828 

392 

I53664 

60236288 

19.7989899 

7.3186114 

393 

154449 

60698457 

19.824  227  6 

7.3248295 

394 

155236 

61  162  984 

19.8494332 

7-33  1  036  9 

395 

156025 

61  629  875 

19.8746069 

7.3372339 

396 

156816 

62  099  136 

19.8997487 

7.3434205 

397 

157609 

62  570  773 

19.924  858  8 

7.3495966 

398 

158404 

63044792 

19.9499373 

7-355  762  4 

399 

I5920I 

63521199 

19.9749844 

7.3619178 

400 

160000 

64000000 

20 

7.368063 

401 

160801 

64481  201 

20.024  984  4 

7.3741979 

402 

16  1604 

64964808 

20.0499377 

7.3803227 

403 

162409 

65450827 

20.074  859  9 

7.3864373 

404 

1632  16 

65939264 

20.0997512 

7.392  541  8 

405 

16  40  25 

66430125 

20.124611  8 

7.3986363 

406 

164836 

66923416 

20.1494417 

7.4047206 

407 

16  56  49 

67  419  143 

20.174241 

7.410  795 

408 

16  64  64 

67917312 

20.1990099 

7.4168595 

409 

16  72  81 

68417929 

20.223  748  4 

7.4229142 

410 

168100 

68921000 

20.248  456  7 

7.4289589 

411 

168921 

69426531 

20.2731349 

7'4349938 

412 

169744 

69934528 

20.297  783  1 

7.4410189 

4i3 

170569 

70444997 

20.322  401  4 

7.4470342 

414 

171396 

70957944 

20.3469899 

74530399 

4i5 

172225 

7M73375 

20.3715488 

7.4590359 

416 

173056 

7I  991  206 

20.396  078  i 

7.4650223 

417 

173889 

725U7I3 

20.420  577  9 

7.4709991 

418 

174724 

73034632 

20.4450483 

7.4769664 

419 

i7556i 

73560059 

20.4694895 

7.482  924  2 

420 

176400 

74088000 

20.4939015 

7.488  872  4 

421 

177241 

74618461 

20.5182845 

7.4948113 

422 

178084 

75  151  448 

20.542  638  6 

7.5007406 

423 

178929 

7568607 

20.566  963  8 

7.5066607 

424 

179776 

76  225  024 

20.5912603 

7.5125715 

425 

180625 

76  765  625 

20.615  528  i 

7.518473 

426 

18  14  76 

77  308  776 

20.6397674 

7.5243652 

427 

18  23  29 

77854483 

20.6639783 

7.5302482 

428 

183184 

78402752 

20.6881609 

7.536  122  I 

429 

184041 

78953589 

20.7123152 

7.541  986  7 

430 

184900 

79507000 

20.7364414 

7.5478423 

431 

185761 

80062991 

20.7605395 

7-5536888 

432 

186624 

80621568 

20.7846097 

7.5595263 

433 

187489 

81  182  737 

20.808  652 

7-5653548 

434 

188356 

81  746  504 

20.832  666  7 

7.57II743 

435 

189225 

82312875 

20.8566536 

7.5760849 

436 

190096 

82  881  856 

20.880613 

7.5827865 

437 

190969 

83453453 

20.904  545 

7.5885793 

438 

191844 

84027672 

20.9284495 

7.5943633 

439 

192721 

84604519 

20.952  326  8 

7.600  138  5 

440 

193600 

85  184000 

20.976177 

7.6059049 

441 

194481 

85  766  121 

21 

7.6116626 

442 

195364 

86350888 

21.023  796 

7.6174116 

443 

19  62  49 

86938307 

21.0475652 

7.623  151  9 

444 

197136 

87528384 

21.0713075 

7.6288837 

445 

198025 

88  121  125 

21.0950231 

7.634  606  7 

446 

198916 

88716536 

2I.II87I2  I 

7.6403213 

280 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  Roo 

447 

199809 

89314623 

21.1423745 

7.646  o: 

448 

2O  07  04 

89915392 

2I.l66oi05 

7.651  7; 

449 

20l6oi 

90518849 

21.189620  I 

7-0574 

450 

202500 

91  125000 

21.2132034 

7.663  cx 

45i 

20  34  01 

9I73385I 

21.2367606 

7.668  7< 

452 

20  43  04 

92  345  408 

21.260291  6 

7-6744. 

453 

205209 

92959677 

21.283  796  7 

7.6800? 

454 

2061  16 

93576664 

21.3072758 

7-685  7. 

455 

20  70  25 

94196375 

21.330729 

7.6913 

456 

20  79  36 

94818816 

21.3541565 

7.697  cx 

457 

20  88  49 

95443993 

21.3775583 

7.7026: 

458 

20  97  64 

96071  912 

21.4009346 

7.708  2v 

459 

210681 

96702579 

21.4242853 

7-7*3  8< 

460 

21  l6oO 

97336000 

21.4476106 

7.7194, 

461 

21  2521 

97972181 

21.4709106 

7-725ov 

462 

213444 

98611  128 

21.4941853 

7.7306 

463 

214369 

99252847 

21.5174348 

7.736  i* 

464 

215296 

99897344 

21.5406592 

7-741  7v 

465 

21  62  25 

100  544  625 

21.5638587 

7-7473 

466 

21  7156 

101  194  696 

21.5870331 

7-752  8< 

467 

218089 

101  847  563 

21.6101828 

7-7584< 

468 

21  9024 

102  503  232 

21.6333077 

7-7639: 

469 

21  9961 

103  161  709 

21.6564078 

7.769  4< 

470 

220900 

103  823  ooo 

21.6794834 

7-774  9* 

471 

22  1841 

104487  in 

21.7025344 

7-78o4< 

472 

22  27  84 

105154048 

21.725561 

7-785  9< 

473 

22  37  29 

105823817 

21.7485632 

7.791  4^ 

474 

22  46  76 

1  06  496  424 

21.771541  i 

7.7969- 

475 

22  56  25 

107  171  875 

21.7944947 

7.802  4^ 

476 

22  65  76 

107850176 

21.8174242 

7.8079: 

477 

22  75  29 

108531333 

21.8403297 

7-8i33* 

478 

22  84  84 

109215352 

21.863211  i 

7.818  & 

479 

229441 

109  902  239 

21.8860686 

7.824  2c 

480 

230400 

110592000 

21.9089023 

7.829  7; 

481 

231361 

in  284641 

21.931  7122 

7.835  i< 

482 

23  23  24 

111980168 

21.9544984 

7.840  50 

483 

233289 

112678587 

21.977261 

7.84601 

484 

234256 

"3379904 

22 

7.851  42 

485 

23  52  25 

114084125 

22.0227155 

7.856-82 

486 

236196 

114791256 

22.045  407  7 

7.862  22 

487 

237169 

115501303 

22.0680765 

7.867  6l 

488 

238144 

116214272 

22.090722 

7.872  oc 

489 

239I2I 

116930  169 

22.1133444 

7.878  3<: 

490 

240IOO 

117649000 

22.1359436 

7-883  72 

491 

241081 

118370771 

22.1585198 

7.88909 

492 

24  2O  64 

1  19  095  488 

22.181073 

7.894  44 

493 

243049 

119823157 

22.2036033 

7.899  75 

494 

244036 

120553784 

22.2261108 

7.905  12 

495 

245025 

121287375 

22.2485955 

7.91045 

496 

246016 

122023936 

22.2710575 

7.915  78 

497 

247009 

122763473 

22.2934968 

7.921  05 

498 

248004 

123505992 

22.3159136 

7.92640 

499 

249001 

124251499 

22.3383079 

7-931  71 

500 

250000 

125000000 

22.3606798 

7-937  oc 

5oi 

25  iooi 

125  751  501 

22.3830293 

7-942  29 

502 

25  20  04 

126  506  008 

22.405  356  5 

7-947  57 

SQUAKES,  CUBES,  AND  BOOTS. 


281 


NUMBKH. 

SQUAEE. 

CUBE. 

SQUARE  ROOT.                 CUBB  ROOT. 

503 

253009 

127263527 

22.427  661  5 

7.952  847  7 

5°4 

25  40  16 

128024064 

22.4499443 

7.9581144 

5°5 

255025 

128  787  625 

22.472  205  I 

7-9633743 

506 

256036 

129554216 

22.494  443  8 

7.968627  i 

5°7 

257049 

130323843 

22.5166605 

7-973  873  i 

508 

258064 

131096512 

22.5388553 

7.9791122 

509 

259081 

131  872  229 

22.561  0283 

7-984  344  4 

5jo 

260100 

132651000 

22.5831796 

7.9895697 

511 

26ll  21 

I3343283I 

22.605  309  1 

7-9947883 

512 

262144 

134217728 

22.627417 

8 

513 

263169 

T35005097 

22.6495033 

8.005  204  9 

5H 

264196 

135796744 

22.671  568  1 

8.010  403  2 

5*5 

26  52  25 

136590875 

22.6936114 

8.0155946 

5i6 

26  62  56 

137388096 

22.7156334 

8.0207794 

51? 

267289 

138188413 

22.737634 

8.025  957  4 

5i8 

268324 

138991832 

22.7596134 

8.031  128  7 

5^9 

269361 

139798359 

22.781  571  5 

8.036  293  5 

520 

27O4OO 

140608000 

22.803  5°8  5 

8.041  451  5 

521 

27I44I 

141  420  761 

22.825  424  4 

8.046603 

522 

27  24  84 

142  236  648 

22.8473193 

8.0517479 

523 

273529 

143055667 

22.8691933 

8.0568862 

524 

274576 

143  877  824 

22.8910463 

8.062018 

525 

275625 

144  703  125 

22.9128785 

8.067  143  2 

526 

276676 

145  531  576 

22.9346899 

8.072  262 

S2? 

27  77  29 

146363183 

22.9564806 

8.077  374  3 

528 

278784 

147  197  952 

22.978  250  6 

8.082  48 

529 

279841 

148035889 

23 

8.0875794 

530 

280900 

148877000 

23.021  7289 

8.092  672  3 

S31 

281961 

149  721  291 

23.043  437  2 

8.0977585 

532 

28  30  24 

150568768 

23.065  1252 

8.102839 

533 

284089 

151419437 

23.086  792  8 

8.107  912  S 

534 

285156 

152273304 

23.10844 

8.112980: 

535 

28  62  25 

I53!30375 

23.130067 

8.118041^ 

536 

287296 

153990656 

23-1516738 

8.1230965 

537 

288369 

154854153 

23.1732605 

8.128  144  \ 

538 

289444 

155720872 

23.194827 

8.133187 

539 

29  05  21 

156590819 

23.2163735 

8.138223 

540 

291600 

157464000 

23.2379001 

8.  143  252  c 

54i 

29  26  81 

158340421 

23.259  406  7 

8.148276^ 

542 

293764 

159220088 

23.280  893  5 

8.153  293  < 

543 

294849 

160  103  007 

23.3023604 

8.158305^ 

544 

295936 

160  989  184 

23.3238076 

8.1633105 

545 

29  70  25 

161  878  625 

23-345  235  i 

8.168309: 

546 

29  81  16 

162  771  336 

23.3666429 

8.173302 

547 

299209 

163667323 

23-388  031  1 

8.178  288  i 

548 

300304 

164  566  592 

23.4093998 

8.183  269  ^ 

549 

301401 

165  469  149 

23.430  749 

8.1882441 

55<=> 

302500 

166375000 

23.452  078  8 

8.193212' 

551 

303601 

167  284  151 

23-4733892 

8.198175: 

552 

304704 

168196608 

23.494  680  2 

8.203  131  < 

553 

305809 

169112377 

23-515952 

8.208  082  1 

554 

306916 

170031464 

23-5372046 

8.213027 

555 

308025 

170953875 

23-558438 

8.217965- 

556 

309136 

171  879616 

23-5796522 

8.222  898  . 

557 

310249 

172808693 

23.600  847  4 

8.227  825  <. 

558           311364 

1  73  74  1  1  1  2             23.622  023  6 

8.232746^ 

A  A* 

282 


SQUARES,   CUBES,   AND    ROOTS. 


NUMBER 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT- 

559 

31  24  81 

174676879 

23.6431808 

8.237  661  4 

560 

313600 

175616000 

23.6643191 

8.2425706 

56i 

31  47  21 

176558481 

23-6854386 

8.247  474 

562 

315844 

1  77  504  328 

23.7065392 

8.2523715 

563 

316969 

178453547 

23.727621 

8.257  263  3 

564 

318096 

179406144 

23.7486842 

8.262  149  2 

565 

31  92  25 

180362125 

23.7697286 

8.2670294 

566 

320356 

181  321  496 

23.7907545 

8.2719039 

567 

32  14  89 

182  284  263 

23.811  7618 

8.2767726 

568 

32  26  24 

183250432 

23.832  750  6 

8.281  625  5 

569 

323761 

184220009 

23-8537209 

8.2864928 

570 

324900 

185  193  ooo 

23.874  672  8 

8.2913444 

57i 

326041 

186169411 

23.8956063 

8.201903 

572 

32  71  84 

187  149  248 

23.9165215 

8.301  030  4 

573 

328329 

188132517 

23.9374184 

8.305  865  i 

574 

329476 

189119224 

23.9582971 

8.3106941 

575 

330625 

190109375 

23.9791576 

8.3I55I75 

576 

33  17  76 

191  102  976 

24 

8.3203353 

577 

332929 

I92I00033 

24.020  824  3 

8.325  147  5 

578 

334084 

193  ioo  552 

24.041  6306 

8.3299542 

579 

33  52  41 

194  104  539 

24.0624188 

8-3347553 

580 

336400 

195  1  12  000 

24.083  189  i 

8.3395509 

58i 

337561 

196122941 

24.1039416 

8.344  34i 

582 

338724 

197  137  368 

24.1246762 

8.3491256 

583 
584 

339889 
34  10  56 

198  155  287 
199176704 

24.1453929 
24.1660919 

8-353  9°4  7 
8.3586784 

585 

342225 

200  201  625 

24.1867732 

8.3634466 

586 

34330 

2O  I  230  056 

24.2074369 

8.3682095 

587 

344569 

202  262  003 

24.228  082  9 

8.3729668 

588 

345744 

203297472 

24.2487113 

8.3777188 

589 

346921 

204336469 

24.2693222 

8.3824653 

590 

348100 

205379000 

24.2899156 

8.3872065 

59i 

349281 

206425071 

24.3104916 

8.391  942  3 

592 

350464 

207  474  688 

24-331  050  1 

8.3966729 

593 

351649 

208  527  857 

24-35I  591  3 

8.401  398  i 

594 

352836 

209  584  584 

24.3721152 

8.406118 

595 

35  40  25 

210644875 

24.392  621  8 

8.410  832  6 

596 

35  52  16 

211708736 

24.4131112 

8.4155419 

597 

356409 

212  776  173 

24-4335834 

8.420  246 

598 

357604 

213  847  192 

24-4540385 

8.424  944  8 

599 

3588oi 

214921799 

24.474  476  5 

8.429  638  3 

600 

360000 

216000000 

24.494  897  4 

8.434  326  7 

601 

361201 

217081801 

24-5  1  5  301  3 

8.439  °°9  8 

602 

36  24  04 

218  167  208 

24-5356883 

8.4436877 

603 

363609 

219256227 

24-5560583 

8.4483605 

604 

36  48  16 

220  348  864 

24.5764115 

8.4530281 

605 

366025 

221  445  125 

24.5967478 

8.4576906 

606 

36  72  36 

222  545  016 

24.6170673 

8.462  347  9 

607 

368449 

223648543 

24-637  37 

8.467 

608 

369664 

224  755  712 

24.657656 

8.471  647  i 

609 

370881 

225  866  529 

24.6779254 

8.476  289  2 

610 

372100 

226981000 

24.6981781 

8.480  926  I 

611 

373321 

228099131 

24.718414  2 

8.4855579 

612 

374544 

229  220  928 

24.7386338 

8.4901848 

613 

375769 

230346397 

24-7588368 

8.4948065 

614 

376996 

231475544 

34.7790234 

8.4994233 

SQUARES,  CUBES,  AND  ROOTS. 


283 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

615 

37  82  25 

232608375 

24.7991935 

8.504035 

616 

37  94  56 

233744896 

24.8193473 

8.508641  7 

617 

380689 

234885113 

24.8394847 

8.5132435 

618 

38  19  24 

236029032 

24.859  605  8 

8.5178403 

619 

383161 

237176659 

24.8797106 

8.522  432  I 

620 

384400 

238  328  ooo 

24.8997992 

8.5270189 

621 

385641 

239  48306  1 

24.9198716 

8.5316009 

622 

386884 

240641  848 

24.9399278 

8.536178 

623 

388129 

241804367 

24.9599679 

8.5407501 

624 

389376 

242  970  624 

24.979992 

8.5453173 

625 

390625 

244140625 

25 

8.5498797 

626 

39  18  76 

245134376 

25.019992 

8.5544372 

627 

393129 

246491883 

25.0399681 

8.5589899 

628 

394384 

247  673  152 

25-0599282 

8.5635377 

629 

395641 

248858189 

25.0798724 

8.5680807 

630 

396900 

250  047  ooo 

25.0998008 

8.5726189 

631 

398161 

251239591 

25.1197134 

8.5771523 

632 

399424 

252435968 

25.1396102 

8.5816809 

633 

400689 

253636137 

25.1594913 

8.5862047 

634 

40  19  56 

254  840  104 

25-I793566 

8.5907238 

635 

403225 

256047875 

25.1992063 

8.595238 

636 

404496 

257259456 

25.2190404 

8.5997476 

637 

405769 

258474853 

25.2388589 

8.604  252  5 

638 

407044 

259694072 

25.2586619 

8.6087526 

639 

408321 

260917  119 

25.2784493 

8.613  248 

640 

409600 

262  144000 

25.298  221  3 

8.6177388 

641 

410881 

263374721 

25-3I79778 

8.622  224  8 

642 

41  21  64 

264  609  288 

25.3377189 

8.6267063 

643 

413449 

265  847  707 

25-3574447 

8.631  183 

644 

4i  47  36 

267  089  984 

25-377  155  I 

8.6356551 

645 

41  6025 

268336125 

25-3968502 

8.6401226 

646 

4i  73  16 

269585136 

25.4165301 

8.644  585  5 

647 

41  8609 

270840023 

25.436  194  7 

8.6490437 

648 

419904 

272097792 

25.4558441 

8.6534974 

949 

42  12  01 

273359549 

25-4754784 

8.6579465 

650 

422500 

274625000 

25.4950976 

8.662  391  i 

651 

42  38  oi 

275894451 

25.5147016 

8.666831 

652 

425104 

277167808 

25-534  290  7 

8.6712665 

653 

426409 

278445077 

255538647 

8.6756974 

654 

42  77  16 

279726264 

25-5734237 

8.6801237 

655 

429025 

281011375 

25.5929678 

8.6845456 

656 

430336 

282300416 

25.6124969 

8.688963 

657 

431649 

283593393 

25.6320112 

8.6933759 

658 

432964 

284890312 

25.6515107 

8.6977843 

659 

434281 

28^191  179 

25.6709953 

8.7021882 

660 

43  56  oo 

287  496  ooo 

25.6904652 

8.7065877 

661 

436921 

288804781 

25.7099203 

8.7109827 

662 

43  82  44 

290  117528 

25.7293607 

8-7J53734 

663 

439569 

291  434  247 

25.7487864 

8.7197596 

664 

440896 

292754944 

25.768  197  5 

8.724  141  4 

665 

442225 

294079625 

25-7875939 

8.7285187 

666 

443556 

295  408  296 

258069758 

8.7328918 

667 

444889 

296740963 

25.8263431 

8.7372604 

668 

446224 

298077632 

25.845696 

8.7416246 

669 

44756i 

299418309 

25.865  034  3 

8.7450846 

670 

448900 

300  763  ooo 

25-8843582             8.7503401 

284 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

67I 

45  02  41 

302  III  711 

25.9036677 

8.7546913 

672 

45  15  84 

3°3  404  448 

25.9229628 

8.7590383 

673 

452929 

304821  217 

25.942  243  5 

8.7633809 

674 

454276 

306182024 

25.96151 

8.7677192 

675 

45  56  25 

307546875 

25.980  762  i 

8.7720532 

676 

456976 

308915776 

26 

8.776383 

677 

458329 

310288733 

26.019  223  7 

8.7807084 

678 

459684 

311665752 

26.038  433  i 

8.7850296 

679 

461041 

313046839 

26.057  628  4 

8.7893466 

680 

462400 

314432000 

26.0768096 

8.7936593 

681 

463761 

315821241 

26.0959767 

8.7979679 

682 

465124 

317214568 

26.115  1297 

8.8022721 

683 

466489 

318611987 

26.1342687 

8.8065722 

684 

46  78  56 

320  013  504 

26.1533937 

8.8108681 

685 

469225 

321419125 

26.1725047 

8.8151598 

686 

470596 

322  828  856 

26.191601  7 

8.8194474 

687 

471969 

324242703 

26.2106848 

8.823  730  7 

688 

473344 

325660672 

26.229  754  i 

8.8280099 

689 

474721 

327082769 

26.2488095 

8.832  285 

690 

476100 

328509000 

26.267851  i 

8.8365559 

691 

477481 

329939371 

26.2868789 

8.8408227 

692 

478864 

331373888 

26,305  892  9 

8.8450854 

693 

48  02  49 

332812557 

26.324  893  2 

8.849,344 

694 

481636 

334  255  384 

26.343  879  7 

8.8535985 

695 

483025 

335702375 

26.362  852  7 

8.8578489 

696 

484416 

337153536 

26.381  8119 

8.862  095  2 

697 

485809 

338608873 

26.4007576 

8.8663375 

698 

487204 

340  068  392 

26.4196896 

8-8705757 

699 

488601 

341532099 

26.4386081 

8.8748099 

700 

490000 

343000000 

26.4575131 

8.87904 

701 

49  14  01 

344472101 

26.476  404  6 

8.883  266  I 

702 

49  28  04 

345948408 

26.495  282  6 

8.887  488  2 

703 

494209 

347  428  927 

26.5141472 

8.8917063 

704 

49  56  16 

348913664 

26.5329983 

8.895  920  4 

705 

497025 

350402625 

26.551  836  i 

8.9001304 

706 

498436 

351895816 

26.5706605 

8.9043366 

707 

499849 

353393243 

26.5894716 

8.9085387 

708 

501264 

354894912 

26.608  269  4 

8.9127369 

709 

502681 

356400829 

26.6270539 

8.9169311 

710 

504100 

357911000 

26.645  825  2 

8.921  1214 

711 

505521 

359425431 

26.6645833 

8.925  307  8 

712 

506944 

360944128 

26.6833281 

8.929  490  2 

713 

508369 

362  467  097 

26.7020598 

8.9336687 

7J4 

509796 

303994344 

26.7207784 

8.9378433 

7J5 

51  1225 

365  525  875 

26.7394839 

8.942  014 

716 

512656 

367  061  696 

26.7581763 

8.946  1809 

717 

514089 

368601813 

26.7768557 

8.9503438 

718 

5J5524 

370146232 

26.795  522 

8.9545029 

719 

516961 

371694959 

26.814  J754 

8.9586581 

720 

518400 

373248000 

26.8328157 

8.9628095 

721 

519841 

374  805  361 

26.8514432 

8.966957 

722 

521284 

376367048 

26.8700577 

8.971  1007 

723 

52  27  29 

377933067 

26.8886593 

8.975  240  6 

724 

52  41  76 

379503424 

26.907  248  i 

8.9793766 

725 

52  56  25 

381  078  125 

26.925  824 

8.9835089 

726 

52  70  76 

382657  176 

26.9443872 

8-9876373 

SQUARES,  CUBES,  AND  ROOTS. 


If  UMBER. 

SQUABB. 

CCBK. 

SQUARE  ROOT. 

CCBB  ROOT. 

727 

52  85  29 

384  240  583 

26.9629375 

8.991  762 

728 

529984 

385828352 

26.981  475  i 

8.9958829 

729 

53*441 

387420489 

27 

9 

73° 

532900 

389017000 

27.0185122 

9.0041134 

731 

53436i 

3906I789I 

27.0370117 

9.OO8  222  9 

732 

53  58  24 

392  223  168 

27.055  498  5 

9.0123288 

7.33 

537289 

393832837 

27-073  972  7 

9.0164309 

734 

538756 

395446904 

27.092  434  4 

9.0205293 

735 

540225 

397065375 

27.1108834 

9.024  623  9 

736 

541696 

398688256 

27.1293199 

9.0287149 

737 

543l69 

400315553 

27.1477439 

9.032  802  I 

738 

544644 

401  947  272 

27.1661554 

9.036  885  7 

739 

5461  21 

403583419 

27.1845544 

9.0409655 

740 

547600 

405  224  ooo 

27.202941 

9.045  041  7 

741 

549081 

406869021 

27.2213152 

9.0491142 

742 

550564 

408518488 

27.2396769 

9.053  183  i 

743 

552049 

410172407 

27.2580263 

9.057  248  2 

744 

55  35  36 

411830784 

27.2763634 

9.061  309  8 

745 

55  50  25 

413493625 

27.294688  i 

9-o65  367  7 

746 

55  65  16 

415160936 

27.3130006 

9.069  422 

747 

558009 

416832723. 

27-3313007 

9.0734726 

748 

559504 

418  508  992 

27-349  588  7 

9.0775197 

749 

561001 

420  189  749 

27.3678644 

9.081  563  1 

750 

562500 

421  875000 

27.3861279 

9.085  603 

75  * 

564001 

423564751 

27.4043792 

9.089  639  2 

752 

565504 

425259008 

27.4226184 

9.0936719 

753 

567009 

426957777 

27.4408455 

9  097  701 

754 

568516 

428  661  064 

27.4590604 

9.101  7265 

755 

570025 

430368875 

27.4772633 

9.1057485 

756 

571536 

432081  216 

27-495  454  2 

9.1097669 

757 

*  57  30  49 

433798093 

27-513633 

9.1137818 

758 

57  45  64 

4355J95I2 

27-53I  7998 

9.1177931 

759 

576081 

437  245  479 

27-5499546 

9.121  801 

760 

577600 

438976000 

27.5680975 

9.1258053 

761 

579121 

440711081 

27.5862284 

9.1298061 

762 

580644 

442450728 

27.6043475 

9.1338034 

763 

582169 

444194947 

27.6224546 

9.1377971 

764 

583696 

445  943  744 

27.6405499 

9.1417874 

765 

585225 

447697125 

27.6586334 

9-H57742 

766 

586756 

449  455  096 

27.676705 

9^497576 

767 

58  82  89 

451217663 

27.6947648 

9-1537375 

768 

589824 

452  984  832 

27.7128129 

9-I577I39 

769 

59i36i 

454756609 

27.7308492 

9.161  6869 

770 

592900 

456533000 

27.7488739 

9.1656565 

771 

594441 

458314011 

27.7668868 

9.1696225 

772 

595984 

460  009  648 

27.784888 

9-  J  73  585  2 

773 

59  75  29 

461  889917 

27.8028775 

9-1775445 

774 

599076 

463  684  824 

27.8208555 

9.1815003 

775 

600625 

465  484  375 

27.838  821  8 

9.1854527 

776 

6021  76 

467288576 

27.8567766 

9.1894018 

777 

603729 

469097433 

27.8747197 

9-1933474 

778 

605284 

470910952 

27.8926514 

9.1972897 

779 

606841 

472  729  139 

27.9105715 

9.201  2286 

780 

608400 

474552000 

27.928  480  i 

9.205  164  i 

781 

609961 

476379541 

27.9463772 

9.2090962 

782 

61  15  24 

478211768 

27.964  262  9 

9.213025 

286 


SQUARES,    CUBES,    AND    ROOTS. 


NUMBER.            SQUABS. 

CUBE. 

SQUARE  ROOT. 

783 

61  30  89 

480048687 

27.982  137  2 

784 

6  1  46  56 

481890304 

28 

785 

61  62  25 

483736625 

28.0178515 

786 

617796 

485587656 

28.0356915 

787 

619369 

487  443  403 

28.0535203 

788 

620944 

489303872 

28.0713377 

789 

62  25  21 

491  169069 

28.089  143  8 

790 

624100 

493039000 

28.1069386 

791 

625681 

494913671 

28.124  722  2 

792 

62  72  64 

496793088 

28.1424946 

793 

62  88  49 

498677257 

28.1602557 

794 

630436 

500566184 

28.1780056 

795 

63  20  25 

502459875 

28.1957444 

796 

63  36-16 

504358336 

28.213472 

797 

635209 

506  261  573 

28.231  1884 

798 

63  68  04 

508169592 

28.2488938 

799 

63  84  01 

510082399 

28.2665881 

800 

640000 

512000000 

28.284271  2 

801 

64  16  01 

513922401 

28.3019434 

802 

64  32  04 

515849608 

28.3196045 

803 

644809 

517781627 

28.3372546 

804 

64  64  16 

519718464 

28.3548938 

805 

64  80  25 

521660125 

28.3725219 

806 

649636 

523606616 

28.390  139  I 

807 

651249 

525  557  943 

28.407  745  4 

808 

65  28  64 

527514112 

28.4253408 

809 

654481 

529475129 

28.4429253 

810 

656100 

531  441  ooo 

28.460  498  9 

811 

65  77  21 

533  411  73i 

28.478061  7 

812 

659344 

535387328 

28.4956137 

813 

660969 

537367797 

28.5131549 

814 

662596 

539353144 

28.530  685  2 

8i5 

664225 

541343375 

28.548  204  8 

816 

665856 

543338496 

28.565  713  7 

817 

667489 

545338513 

28.5832119 

818 

6691  24 

547343432 

28.6006993 

819 

67  07  61 

549353259 

28.618176 

820 

672400 

551368000 

28.635  642  i 

821 

67  40  41 

55338766i 

28.653  097  6 

822 

675684 

555412248 

28.670  542  4 

823 

67  73  29 

557  441  767 

28.6879766 

824 

678976 

559476224 

28.7054002 

825 

680625 

561515625 

28.7228132 

826 

68  22  76 

563559976 

28.7402157 

827 

683929 

565609283 

28.7576077 

828 

685584 

567  663  552 

28.7749891 

829 

68  72  41 

569  722  789 

28.7923601 

830 

688900 

571787000 

28.8097206 

831 

690561 

573856191 

28.8270706 

832 

692224 

575930368 

28.8444102 

833 

693889 

578009537 

28.861  739  4 

834 

695556 

580093704 

28  879058  2 

835 

697225 

582182875 

28.8963666 

836 

698896 

584277056 

289136646 

837 

700569 

586376253 

28.9309523 

838 

702244 

588  480  472 

28.948  229  7 

CUBE  ROOT. 


SQUARES,  CUBES,  AND  ROOTS. 


287 


SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

703921 

590589719 

28.965  496  7 

9.431  642  3 

705600 

592704000 

28.982  753  5 

9-43538 

70  72  81 

59482332! 

29 

9.4391307 

708964 

596947688 

29.0172363 

9.4428704 

710649 

599077107 

29.034  462  3 

9.446  607  2 

712336 

601211584 

29  051  678  I 

9.450341 

714025 

603351125 

29.0688837 

9.4540719 

71  57  16 

605495736 

29.086  079  1 

9-4577999 

717409 

607645423 

29.1032644 

9.4615249 

71  91  04 

609800192 

29.1204396 

9-465  247 

720801 

611960049 

29.1376046 

9.468  966  i 

722500 

614  125000 

29-  1  54  759  5 

9.4726824 

72  42  01 

616295051 

29.1719043 

9-4763957 

725904 

618  470  208 

29.189039 

9.480  1  06  i 

727609 

620650477 

29.206  163  7 

9.4838136 

729316 

622835864 

29.2232784 

9.4875182 

73*025 

625026375 

29.240383 

9.491  22 

73  27  3° 

627  222  Ol6 

29.2574777 

9.4949188 

734449 

629422793 

29.2745623 

9.4986147 

736164 

631  628  712 

29.291  637 

9.502  307  8 

737881 

633839779 

29.308  701  8 

9.505  998 

739600 

636056000 

29-3257566 

9.5096854 

74I32I 

638277381 

29.342  801  5 

9-5I33699 

743°44 

640  503  928 

29-3598365 

9-5I705I5 

744769 

642735647 

29.3768616 

9.5207303 

746496 

644972544 

29-3938769 

9.5244063 

748225 

647214625 

29.410  882  3 

9.5280794 

749956 

649  461  896 

29.427  877  9 

9-531  749  7 

751689 

651714363 

29.4448637 

9-5354I72 

75  34  24 

653972032 

29.461  839  7 

9.5390818 

75  5i  61 

656234909 

29.4788059 

9.542  743  7 

756900 

658  503  ooo 

29.495  762  4 

9.5464027 

758641 

660776311 

29.512  709  i 

9-5500589 

760384 

663054848 

29.5296461 

9-5537J23 

7621  29 

665338617 

29-5465734 

9-557363 

763876 

667  627  624 

29563491 

9.5610108 

765625 

669921875 

29.5803989 

95646559 

76  73  76 

672  221  376 

29.597  297  2 

9.5682982 

769129 

674  526  133 

29614  1858 

9-57  1  937  7 

770884 

676836152 

29.631  064  8 

9-5755745 

772641 

679  !5J  439 

29.647  934  2 

95792085 

774400 

681  472000 

29.6647939 

9.582  839  7 

77  61  61 

683  797  841 

29,681  644  2 

9.586  468  2 

777924 

686128968 

29.698  484  8 

9.5900937 

779689 

688465387 

29-7153159 

95937^9 

781456 

690  807  104 

29.7321375 

9-597  337  3 

783225 

693  154  125 

29.7489496 

9.6009548 

784996 

695  506  456 

29-765  752  1 

9.604  569  6 

786769 

697  864  103 

29.7825452 

9.608  181  7 

788544 

700  227  072 

29.7993289 

9.611  791  i 

790321 

702595369 

29.816  103 

9-6i53977 

7921  oo 

704969000 

29.832  867  8 

9.619001  7 

793881 

707  347  97i 

29.8496231 

9.622  603 

795664 

709  732  288 

29.866369 

9.626  201  6 

797449 

712121957 

29.883  105  6 

9.629  797  5 

799236 

714516984 

29.8998328 

0-6333907 

288 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBER. 

SQUARE. 

CUBK.                |          SQUARE  ROOT. 

CUBB  ROOT. 

895 

801025 

716917375                 29.9165506 

9.6369812 

896 

802816 

719323136 

29-933  259  i 

9.640  569 

897 

804609 

721734273 

29.9499583 

9.644154: 

898 

806404 

724  150  792 

29.966  648  i 

9.647  736  ' 

899 

808201 

726572699 

29.9833287 

9.651  316  (. 

900 

SlOOOO 

729000000 

.    3° 

9.654893* 

901 

81  1801 

73I  432  7OI 

30.016662 

9.658  468  i 

902 

81  3604 

733870808 

30.033  314  8 

9.662  040  v 

9°3 

815409 

7363*4327 

30.0499584 

9.  665  609  i 

904 

81  72  16 

738  763  264 

30.066  592  8 

9.669176: 

9°5 

819025 

741217625 

30.0832179 

9.672740. 

906 

82  08  36 

743677416 

30.0998339 

9.676  301 

907 

82  26  49 

746  142  643 

30.1164407 

9.679  860  < 

908 

824464 

748613312 

30.1330383 

9.683  416  ( 

909 

82  62  81 

751089429 

30.1496269 

9.686  970 

910 

828100 

753571000 

30.1662063 

9.690521 

911 

829921 

756  058  031 

30.1827765 

9.694069 

912 

831744 

758550528 

30.1993377 

9.697615 

9*3 

833569 

761  048  497 

30.2158899 

9.701  158 

914 

835396 

763551944 

30.2324329 

9.704698 

9i5 

83  72  25 

766060875 

30.2489669 

9.708  236 

916 

839056 

768  575  296 

30.2654919 

9.711772 

917 

84  08  89 

771095213 

30.282  007  9 

9-7J5305 

918 

84  27  24 

773620632 

30.2985148 

9-718835 

919 

84  45  61 

776i5i559 

30.3150128 

9.722363 

920 

846400 

778  688  ooo 

30.331  501  8 

9.725888 

921 

848241 

781  229961 

30.3479818 

9.729410 

922 

850084 

783777448 

30.3644529 

9-732  930 

923 

851929 

786330467 

30.380915  i 

9.736448 

924 

853776 

788889024 

30-3973683 

9-739963 

925 

855625 

79M53I25 

30.4138127 

9-743475 

926 

85  74  76 

794022776 

30.430  248  i 

9.746985 

927 

859329 

796  597  983 

30.4466747 

9.750493 

928 

861184 

799178752 

30.4630924 

9-753997 

929 

86  30  41 

801  765  089 

30.4795013 

9-757  5oo 

930 

864900 

804357000 

30.495  901  4 

9.761  ooo 

93  i 

866761 

806954491 

30.5122926 

9.764497 

932 

868624 

809557568 

30.528675 

9.767992 

933 

87  04  89 

812  166237 

30.545  048  7 

9.771  484 

934 

872356 

814  780  504 

30.5614136 

9-774974 

935 

874225 

817400375 

30.577  769  7 

9.778461 

936 

876096 

820  025  856 

30.5941171 

9.781  946 

937 

877969 

822  656  953 

30.6104557 

9.785428 

938 

879844 

825  293  672 

30.626  785  7 

9.788908 

939 

881721 

827936019 

30.643  1069 

9.792386 

940 

883600 

830  584  ooo 

30.6594194 

9.795  861 

941 

885481 

833237621 

30.6757233 

9-799333 

942 

887364 

835896888 

30.6920185 

9.802  803 

943 

88  92  49 

838  561  807 

30.708  305  i 

9.806271 

944 

891136 

841  232  384 

30.724583 

9.809  736 

945 

89  ..025 

843  908  625 

30.7408523 

9.813  198 

946 

89  49  16 

846  590  536 

30.757113 

9.816659 

947 

896809 

849278123 

30.7733651 

9.820  u6< 

948 

89  87  04 

851971392 

30.7896086 

9-823572, 

949 

900601 

854670349 

30.805  843  6 

9.827025 

950 

902500 

857375ooo 

30.822  07 

9'830475 

SQUARES,  CUBES,  AND  BOOTS. 


289 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

951 

904401 

860085351 

30.8382879 

9.8339238 

952 

906304 

862  801  408 

30.8544972 

9.8373095 

953 

908209 

865523177 

30.870  698  I 

9.840812  7 

954 

91  01  16 

868250664 

30.8868904 

9.8442536 

955 

91  20  25 

870983875 

30.903  074  3 

9.847  692 

956 

9*3936 

873  722  816 

30.919  247  7 

9.851  128 

957 

915849 

876467493 

30.9354i66 

9.8545617 

958 

91  77  64 

879217912 

30.951  575  i 

9.8579929 

959 

919681 

881974079 

30.07  725  i 

9.861  421  8 

960 

92  1600 

884  736  ooo 

30.9838668 

9.8648483 

01 

92  35  21 

887503681 

31 

9.868  272  4 

02 

925444 

890277  128 

31.0161248 

9.8716941 

963 

927369 

893056347 

31.0322413 

9.875"35 

964 

92920 

.-895841344 

-31.0483494 

9.8785305 

05 

93  J2  25 

898632125 

31.0644491 

9.881  945  i 

06 

933156 

901  428  60 

31.0805405 

9-885  357  4 

07 

935089 

904231063 

31.006236 

9.8887673 

968 

93  70  24 

907039232 

31.1126984 

9.8921749 

969 

938961 

909853209 

31.1287648 

9.895  580  i 

970 

940900 

912673000 

31.144823 

9-898983 

971 

942841 

915498611 

31.1608729 

9.9023835 

972 

944784 

918330048 

31.1769145 

9.905  ?8i  7 

973 

946729 

921167317 

31.1929479 

9.9091776 

974 

94-8676 

924  oio  424 

31.2089731 

9.9125712 

975 

950625 

926859375 

31.22499 

9.9159624 

976 

95  25  76 

929714176 

31.2409987 

9-9I935I3 

977 

954529 

932574833 

31.2569992 

9.9227379 

978 

956484 

935441352 

31.2729915 

9.926  122  2 

979 

958441 

938313739 

31.2889757 

9.9295042 

980 

00400 

941  192  ooo 

31.304951? 

9.9328839 

981 

962361 

944076141 

31.3209195 

9.9362613 

982 

04324 

946966168 

31.3368792 

9.9396363 

983 

06289 

949  862  087 

31-3528308 

9.9430092 

984 

08256 

952763904 

31.3687743 

9.9463797 

985 

970225 

955671625 

31.3847097 

9.9497479 

986 

97210 

958  585  256 

31.4006369 

9.953II38 

987 

974169 

01504803 

31.4165561 

9.9564775 

988 

976i44 

04430272 

31.4324673 

9.9598389 

989 

9781  21 

07361669 

31.4483704 

9.03  198  i 

990 

98OIOO 

970299000 

31.4642654 

9.9665549 

991 

982081 

973242271 

31.4801525 

9.9699095 

992 

984064 

976  191  488 

31400315 

9-973  261  9 

993 

986049 

979146657 

31.5119025 

9.976612 

994 

988036 

982  107  784 

31.5277655 

9-9799599 

995 

990025 

985074875 

31.5436206 

9.9833055 

90 

992016 

988047936 

31.5594677 

9.9866488 

997 

994009 

991026973 

31.5753068 

9.98999 

998 

996004 

994011992 

31.591  138 

9.9933289 

999 

998001 

997002999 

31.  606  0i  3 

9-9966656 

1000 

I  OOQOOO 

1000  000  OOO 

31.6227766 

10 

1001 

I  00  20  01 

1003003001 

31.638584 

IO.O03  322  2 

1002 

i  oo  40  04 

1006012008 

31.6543836 

10.006  662  a 

1003 

1006009 

1009027027 

31.6701752 

10.009  989  g 

1004 

I0o8oi6 

1012048064 

31.685959 

10.0133155 

1005 

IOI0025 

1015075125       31.7017349 

10.016  638  5 

1006 

I  OI  20  36 

1018108216     i     31-717503 

10.0199601 

BB 

2QO 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBER. 

SQUARE. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

1007 

i  01  4049 

i  021  147  343 

31.7332633 

10.023  279  I 

1008 

i  01  6064 

i  024  192  512 

31.7490157 

10.026  595  8 

lOOQ 

I0l8o8l 

i  027  243  729 

31.7647603 

10.0299104 

1010 

I  02  01  00 

1  030301  ooo 

31.7804972 

10.033  222  8 

IOII 

I  O2  21  21 

1033364331 

31.796226  2 

10.036533 

IOI2 

I024I44 

1036433728 

31.8119474 

10.039  841 

1013 

I  02  6l  69 

1039509197 

31.8276609 

10.043  ^69 

1014 

I  O2  8l  96 

i  042  590  744 

31.8433666 

10.046  450  6 

1015 

I  03  02  25 

1045678375 

31.8590646 

10.049  752  I 

1016 

I  03  22  56 

i  048  772  096 

31.8747549 

10.053051  4 

1017 

I  03  42  89 

i  051  871  913 

31.890437.4 

10.056  348  5 

1018 

J  03  63  24 

1054977832 

31.9061123 

0.0596435 

1019 

1038361 

i  058  089  859 

31.9217794 

0.062  936  4 

IO2I 

n^ojoo 

1042441 

i  064  332  261 

—  3'  -937*138  8 
31.9530906 

0.066  227  i 
0.0695156 

IO22 

i  04  44  84 

i  067  462  648 

31.9687347 

0.072  802 

1023 

i  04  65  29 

1070599167 

31.9843712 

0.076  086  3 

1024 

1  04  85  76 

1073741824 

32 

10.079  368  4 

1025 

i  05  06  25 

i  076  890  625 

32.0156212 

10.082  648  4 

1026 

i  05  26  76 

1080045576 

32.031  234  8 

10.085  926  2 

1027 

1  05  47  29 

1083206683 

32.046  840  7 

10.089  2O1  9 

1028 

i  05  67  84 

1086373952 

32.062  439  i 

10.092  475  5 

1029 

1058841 

1089547389 

32.078  029  8 

10.0957469 

1030 

1060900 

1  092  727  ooo 

32.0936131 

10.0990163 

1031 

i  062961 

1095912791 

32.1091887 

10.1022835 

1032 

1  06  50  24 

1  099  104  768 

32.1247568 

10.1055487 

1033 

i  06  70  89 

1102302937 

32.1403173 

10.  10881  1  7 

J034 

10691  56 

1  105  507  304 

32.1558704 

10.1120726 

1035 

I  07  12  25 

1  108717875 

32.1714159 

10.1155314 

1036 

1073206 

i  1  1  1  934  656 

32.1869539 

10.1185882 

103? 

1075369 

1115157653 

32.2024844 

10.  121  8428 

1038 

1077444 

i  118386872 

32.2180074 

10.126095.3 

1039 

1  07  95  21 

i  121622319 

32.2335229 

10.1283457 

1040 

1081600 

1  124864000 

32.249031 

10.1315941 

1041 

1083681 

1  128  in  921 

32.2645316 

10.1348403 

1042 

1  08  57  64 

1131366088 

32.2800248 

10.1380845 

1043 

1087849 

1  134  626  507 

32.2955105 

IO.T4I  3266 

1044 

i  08  99  36 

1137893184 

323109888 

10.1445667 

1045 

1  09  20  25 

I  141  166  125 

32.3264598 

10.1478047 

1046 

10941  16 

1144445336 

32.3419233 

IO.I5I  0406 

1047 

i  096209 

1147730823 

32-3S73794 

10.1542744 

1048 

1  09  83  04 

1151022592 

32.372  828  i 

10.1575062 

1049 

i  100401 

I  154320649 

32.3882695 

IO.I007359 

1050 

1  10  25  oo 

1  157625000 

32.403  703  5 

10.1639636 

1051 

1  104601 

1  160935651 

32.419  130  1 

10.1671893 

1052 

1  10  67  04 

i  164  252  608 

32.4345495 

10.1704129 

1053 

1  108809 

1167575  877 

32.4499615 

10.1736344 

1054 

i  ii  09  16 

1170905464 

32.4653662 

10.1768539 

1055 

1  113025 

1174241375 

32.480  763  5 

10.1800714 

1056 

1  115136 

1177583616 

32.4961536 

10.1832868 

1057 

1  117249 

i  180932193 

32.5115364 

IO.I865002 

1058 

1119364 

I  184287  112 

32.5269119 

10.1897116 

1059 

I  12  1481 

i  187  648  379 

32.542  280  2 

10.1929209 

1060 

I  123600 

i  191016000 

32.5576412 

10.1961283 

1061 

I  I2572I 

1194389981 

32.5729949 

10.1993336 

1062 

1127844 

1197770328 

32.5883415 

10.2025369 

SQUARES,   CUBES,   AND   BOOTS. 


29I 


NUMBKE. 

SqUABK. 

CUBE. 

SQUARE  ROOT. 

CUBE  ROOT. 

1063 

I  129969 

I  201  157047 

32.6036807 

10.205  738  2 

1064 

I  132096 

I  204  550  144 

32.6190129 

10.208  937  5 

1065 

I  134225 

i  207  949  625 

32.6343377 

10.2121347 

1066 

I  I36356 

1211355496 

32.6496554 

10.21533 

1067 

1138489 

I  214  767  763 

32.6649659 

10.2185233 

1068 

I  140624 

1218186432 

32.6802693 

10.221  7146 

1069 

I  142761 

I  221  6ll  509 

32-695  565  4 

10.2249039 

IO7O 

i  1  4  49  oo 

i  225  043  ooo 

32.7108544 

I0.22809I  2 

1071 

1147041 

I  228480911 

32.7261363 

IO.23I  2766 

1072 

i  14  91  84 

I  231  925  248 

32.7414111 

10.2344599 

1073 

1151329 

1235376017 

32.7566787 

10.2376413 

1074 

i  15  34  76 

I  238  833  224 

32.7719392 

10.240  820  7 

1075 

1  15  56  25 

1242296875 

32.7871926 

10.2439981 

1076 

1157776 

1245766976 

32.8024389 

10.2471735 

1077 

i  I59929 

1249243533 

32.8176782 

10.250347 

1078 

1  16  20  84 

I  252  726  552 

32.8329103 

10.2535186 

1079 

1  164241 

1256216039 

32.8481354 

10.256  688  i 

I080 

i  166400 

I  259  7I2OOO 

32.8633535 

10.2598557 

1081 

1  168561 

I  263214441 

32.8785644 

10.2630213 

1082 

1  170724 

I  266  723  368 

32.8937684 

10.266  185 

1083 

1172889 

I  270  238  787 

32.9089653 

10.2693467 

1084 

H75056 

I  273  760  704 

32.9241553 

10.2725065 

1085 

1177225 

I  277  289  125 

32.9393382 

10.2756644 

1086 

i  179396 

I  280  824  056 

32.9545141 

10.2788203 

1087 

i  181569 

1284365503 

32.969683 

10.2819743 

1088 

1183744 

1287913472 

32.984845 

10.285  1264 

1089 

i  185921 

I  29I  467  969 

33 

10.288  276  5 

1090 

1  188100 

i  295  029  ooo 

33.015  148 

10.291  424  7 

1091 

1  190281 

1298596571 

33.030  289  1 

10.2945709 

1092 

1  192464 

1302  170688 

33.0454233 

10.2977153 

1093 

1  194649 

1  305  75i  357 

'  33-0605505 

10.3008577 

1094 

1  196836 

1309338584 

33-0756708 

10.3039982 

1095 

1  199025 

1312932375 

33.0907842 

10.307  136  8 

1096 

I  20  12  l6 

1316532736 

33.1058907 

10.310  273  5 

1097 

1203409 

1320139673 

33.1209903 

10.3134083 

1098 

I  205604 

1  323  753  192 

33.136083 

10.316541  1 

1099 

I  20  78  OI 

i  327  373  299 

33.1511689 

10.3196721 

IIOO 

I2IOOOO 

1331000000 

33.1662479 

10.322  801  2 

1  1  01 

I  21  22  OI 

I33463330I 

33-18132 

10.3259284 

IIO2 

I  214404 

1338273208 

33.1963853 

10.3290537 

1103 

I  216609 

1341919727 

33.2114438 

10.332  177 

1104 

i  21  88  16 

1  345  572  864 

33.2266955 

10.3352985 

1105 

I  22  IO  25 

1  349  232  625 

33.2415403 

10.3384181 

1106 

I  22  32  36 

1352899016 

33-2565783 

10.341  535  8 

1107 

1225449 

1356572043 

33.2716095 

10.344651? 

1108 

I  22  76  64 

1360251712 

33.2866339 

10.347  765  7 

1109 

I  22  98  8l 

i  363  938  029 

33-301  651  6 

10.3508778 

I  IIO 

I  23  21  OO 

1  367  631  ooo 

33.3166625 

10.353988 

mi 

I23432I 

i  37  J  330  631 

33.3316666 

10.3570964 

III2 

1236544 

1375036928 

33.346664 

10.360  202  Q 

III3 

I  23  87  69 

1378749897 

33.3616546 

10.363  307  6 

III4 

I  240996 

i  382  469  544 

33.3766385 

10.3664103 

mS 

I  24  32  25 

i  386  195  875 

33.391  615  7 

10.3695113 

1116 

i  24  54  56 

1389928896 

33.406  586  2 

10.3726103 

1117 

i  24  76  89 

1393668613 

33.4215499 

!0.375  707  6 

m8 

1249924 

1  397  4J5  °3a 

33436b07 

10.378  803 

292 


SQUARES,    CUBES,    AND    ROOTS. 


NUMBER.  } 

SQUARE. 

CtTBH. 

SVTARB  ROOT. 

CUBK  ROOT. 

IIIQ 

i  25  21  61 

i  401  168  159 

33-45I4573 

10.3818965 

1  120 

1254400 

i  404  928  ooo 

33.466401  I 

10.384  988  2 

II2I 

1256641 

i  408  694  561 

3348l  338  I 

10.388  078  I 

1122 

i  25  88  84 

1412467848 

33.4962684 

10.391  166  i 

1123 

i  26  1  1  29 

i  416  247  867 

33.511  1921 

10.3942523 

1124 

1263376 

1  420  034  624 

33.5261092 

10.3973366 

1125 

i  26  56  25 

1423828125 

33.5410196 

10.4004192 

1126 

i  26  78  76 

1  427  628  376 

33-5559234 

10.4034999 

1127 

1270129 

i  431  435  383 

33.5708206 

10.4065787 

1128 

i  27  23  84 

i  435  249  152 

33-585  7"  2 

10.4096557 

1129 

i  274641 

1439069689 

33.6005952 

10.412  73,1 

II3O 

i  276900 

1  442  897  ooo 

33.6154726 

10.415  8044 

II3I 

i  279161 

i  446  731  091 

33.6303434 

10.418  876 

1132 

i  28  14  24 

.145057108 

33.645  207  7 

10.421  945  8 

"33 

i  28  36  89 

1454419637 

33.6600653 

10.4250138 

"34 

i  28  59  56 

i  458  274  104 

33.674  916  5 

10.428  08 

"35 

i  28  82  25 

1462135375 

33.689  761 

10.431  1443 

1136 

i  29  04  96 

i  466  003  456 

33.7045991 

10.4342069 

"37 

i  292769 

1469878353 

33.7194306 

10.437  267  7 

1138 

1295044 

1473760072 

33.7342556 

10.440  326  7 

"39 

i  29  73  21 

1477648619 

33.7490741 

10.4433839 

1140 

i  299600 

1481  544000 

33.7638860 

10.4464393 

1141 

1301881 

I  485  446  221 

33.7786915 

10.4494929 

1142 

1304164 

M89355288 

33-7934905 

10.4525448 

"43 

1306449 

i  493  271  207 

33.808283 

10.4555948 

1144 

1308736 

1497193984 

33.8230691 

10  458  643  i 

"45 

131  1025 

1501  123625 

33.8378486 

10.461  6896 

1146 

i  31  33  16 

i  505  060  136 

33.8526218 

10.4647343 

"47 

1315609 

1509603523 

33.8673884 

10.4677773 

1148 

1317904 

1  512  953  792 

33.882  148  7 

10.4708185 

"49 

i  32  02  01 

i  516910949 

33.8969025 

104738579 

1150 

i  32  25  oo 

1520875000 

33.9116499 

10.4768955 

"5i 

i  32  48  oi 

1524845951 

33.9263909 

10.4799314 

"52 

1  32  71  04 

i  528  823  808 

33.9411255 

10.482  965  6 

"53 

1329409 

1532808577 

33-955  853  7 

10.485  998 

"54 

1331716 

i  536  800  264 

33-9705755 

10.489  028  6 

"55 

i  33  40  25 

1540798875 

33.985  291 

10.4920575 

1156 

1336336 

i  544  804  416 

34 

10.4950847 

"57 

i  33  86  49 

1548816893 

34.014  702  7 

10.4981101 

1158 

1340964 

1552836312 

34.029399 

10.501  133  7 

"59 

1343281 

i  556  862  679 

34.044089 

10.504  155  6 

1160 

1345600 

i  560  896  ooo 

34.058  772  7 

10.5071757 

1161 

1347921 

i  564  936  281 

34.0734501 

10.5101942 

1162 

i  35  °2  44 

1568983528 

34.088  121  I 

10.5132109 

1163 

i  35  25  69 

1573037747 

34.1027858 

10.5162259 

1164 

i  35  48  96 

1577098944 

34.1174442 

10.5192391 

1165 

i  35  72  25 

i  581  167  125 

34.1320963 

10.5222506 

1166 

i  35  95  56 

i  585  242  296 

34,1467422 

10.5252604 

1167 

1361889 

1589324463 

34.1613817 

10.528  268  5 

168 

i  36  42  24 

15934*3632 

34.176015 

10.5312749 

169 

i  36  65  61 

1597509809 

34.190642 

10.534  279  5 

170 

i  36  89  oo 

1601613000 

34.205  262  7 

10.537  282  5 

171 

1371241 

1605723211 

34.2198773 

10.540  283  7 

172 

1  37  35  84 

i  609  840  448 

34.2344855 

10.543  283  2 

"73 

1  37  59  29 

1613964717 

34.2490875 

10.546  281 

"74 

1378276 

1618096024 

34.263  683  4 

10.5492771 

SQUARES,   CUBES,   AND   BOOTS. 


293 


NCMBBB. 

SQUARE. 

CUBB. 

SQUARE  ROOT. 

CCBB  ROOT. 

"75 

1380625 

1622234375 

34.278273 

10.552  271  5 

I76 

I  38  29  76 

1626379776 

34.2928564 

10.555  264  2 

177 

1385329 

1630532233 

34-3074336 

10.5582552 

I78 

1387684 

163469X752 

34.3220046 

10.5612445 

179 

1390041 

1638858339 

34-3365694 

10.564  232  2 

180 

i  39  24  oo 

I  643  032000 

34.351  128  I 

10.567218  I 

181 

1394761 

I  647  212  741 

34.3656805 

10.5702024 

182 

1397124 

1651400568 

34.380  226  8 

10.5731849 

183 

1399489 

1655595487 

34-394  767 

10.576  165  8 

1184 

1  40  18  56 

i  659  797  504 

34.409301  i 

10.5791449 

1185 

i  40  42  25 

i  664  006*625 

34.4238289 

10.5821225 

1186 

i  40  65  96 

i  668  222  856 

34.4383507 

10.5850983 

1187 

1408969 

1  672  446  203 

34.4528663 

10.588  072  5 

1188 

1411344 

1  676  676  672 

34-467  375  9 

10.591  045 

1189 

1413721 

1680914269 

34.4818793 

10.5940158 

1190 

i  41  61  oo 

1685159000 

344963766 

10.596985 

1191 

1418481 

1689410871 

34.5108678 

10.5999525 

1192 

1420864 

1693669888 

34.525353 

10.6029184 

"93 

i  42  32  49 

1697936057 

34-5398321 

10.605  882  6 

1194 

i  42  56  36 

I  702  209  384 

34-5543051 

10.608  845  i 

ii95 

i  42  80  25 

1706489875 

34.568  772 

10.611  806 

1196 

1430416 

1710777536 

34-5832329 

IO.6l4  765  2 

1197 

1432809 

1715072373 

34.5976879 

10.617  722  8 

1198 

i  43  52  04 

1719374392 

34.612  1366 

10.620  678  8 

1199 

1437601 

1723683599 

34.6265794 

10.6236331 

1200 

1440000 

I  728000000 

34.641  016  2 

10.626  585  7 

1  201 

1442401 

1732323601 

34.6554469 

10.629  536  7 

1202 

1444804 

1  736654408 

34.6698716 

10.632  486 

1203 

1447209 

1740992427 

34.6842904 

10.6354338 

1204 

1449616 

i  745  337  664 

34.698  703  1 

10.6383799 

1205 

1452025 

1749690125 

34.7131099 

10.6413244 

I2O6 

1454436 

i  754  049  816 

34.7275107 

10.644  267  2 

1207 

1456849 

1758416743 

34.741  905  5 

10.647  208  5 

1208 

i  45  92  64 

i  762790912 

34.7562944 

10.650  148 

I2O9 

1461681 

1767172329 

34.7706773 

10.653  086 

I2IO 

i  46  41  oo 

i  771561000 

34-7850543 

10.656  022  3 

I2II 

1466521 

17759569^ 

34-7994253 

10.658957 

1212 

1468944 

1780360128 

34.8137904 

10.661  890  2 

1213 

1471369 

i  784  770  597 

34.8281495 

10.664821  7 

1214 

147370 

1789188344 

34.842  502  8 

10.667  751  6 

1215 

1476225 

I7936i3375 

34.8568501 

10.6706799 

1216 

i  47  86  56 

1798045696 

34.871  191  5 

10.6736066 

1217 

1  48  10  89 

1  802  485  313 

34.885  527  i 

10.6765317 

1218 

1483524 

i  806  932  232 

34.8998567 

10.6794552 

1219 

1485961 

1811386459 

34.9141805 

10.6823771 

1220 

1  48  84  oo 

1815848000 

34.9284984 

106852973 

1221 

1490841 

i  820  316  861 

34.9428104 

10.688  216 

1222 

1493284 

i  824  793  048 

34.9571166 

10.691  133  i 

1223 

M95729 

i  829  276  567 

34.9714169 

10.694  048  6 

1224 

i  49  81  76 

1833767424 

34.9857114 

10.6969625 

1225 

1500625 

i  838  265  625 

35 

10.699  874  8 

1226 

1503076 

i  842  771  176 

35.0142828 

10.7027855 

I227 

1505529 

i  847  284  083 

35.0285598 

10.7056947 

1228 

i  50  79  84 

1851  804352 

35.0428309 

10.7086023 

1229 

1510441 

1856331989 

35.0570963 

10.7115083 

1230 

1512900 

i  860  867  ooo 

35.07i.3558 

10.7144127 

BB* 

294 


SQUARES^  CUBES,  AND  BOOTS. 


NUMBER. 

SQUARE. 

CUBE.                        SQUARE  ROOT. 

CUBK  ROOT. 

1231 

I5I536I 

1865409391 

35.0856096 

10.7173155 

1232 

I5I7824 

1869959168 

35.0998575 

10.7202168 

1233 

I  52  02  89 

1874516337 

35.1140997 

10.7231165 

1234 

I  52  27  56 

I  879  080  904 

35.1283361 

10.7260146 

1235 

I  52  52  25 

1883652875 

35.1425568 

10.728911  2 

1236 

1527696 

i  888  232  256 

35-I5679I7 

10.731  8062 

1237 

1530169 

1892819053 

35.1710108 

10.7346997 

1238 

1532644 

i  897  413  272 

35.1852242 

10.737  591  6 

1239 

i  53  51  21 

1902014919 

35.1994318 

10.740481  9 

1240 

1537600 

1  906  624  ooo 

35-2I36337 

10.7433707 

1241 

i  540081 

1  911  240521 

35.2278299 

10.7462579 

1242 

i  54  25  64 

1915864488 

35.2420204 

10.7491436 

1243 

1545049 

1920495907 

35.2562051 

10.7520277 

1244 

1547536 

1  925  134  784 

35.2703842 

10.7549103 

1245 

i  55  oo  25 

1  929  781  125 

35.284  557  5 

10.7577913 

1246 

i  55  25  16 

1934434936 

35.298  725  2 

10.7606708 

1247 

1555009 

1939096223 

35.3128872 

I0-  763  548  8 

1248 

i  55  75  04 

i  943  764  992 

35.3270435 

10.766425  2 

1249 

i  56  oo  01 

1948441249 

35.341  194  i 

10.7093001 

1250 

i  56  25  oo 

1953125000 

35-355  339  i 

10.7721735 

1251 

1565001 

i  957816251 

35.3694784 

10.7750453 

1252 

i  56  75  04 

1962515008 

35.383612 

10.7779156 

1253 

i  570009 

1967221  277 

35-397  74 

10.7807843 

1254 

i  57  25  16 

1971935064 

35.4118624 

10.783651  6 

1255 

1575025 

1976656375 

35.4259792 

10.7865173 

1256 

1577536 

i  981  385  216 

35.4400903 

10.7893815 

1257 

1580049 

i  986  121  593 

35-454  195  8 

10.792  244  i 

1258 

i  58  25  64 

1990865512 

35.4682957 

10.7951053 

1259 

1585081 

1995616979 

35.48239 

10.7979649 

1260 

i  58  76  oo 

2000376000 

35.4964787 

10.800  823 

1261 

1590121 

2005  142581 

35.5105618 

10.803  679  7 

1262 

1592644 

2009916728 

35-5246393 

10.806  534  8 

1263 

1  59  5i  69 

2014698447 

35.5387113 

10.8093884 

1264 

1597696 

2019487744 

35.5527777 

10.8122404 

1265 

1  60  02  25 

2  O24  284  625 

35.5668385 

10.8150909 

1266 

1602756 

2  O29  089  096 

35.5808937 

10.81794 

1267 

1  60  52  89 

2033901  163 

35-5949434 

10.820  787  6 

1268 

1  60  78  24 

2  038  720  832 

35.6089876 

10.8236336 

1269 

1  61  03  61 

2043548109 

35.6230262 

10.826  478  2 

1270 

161  2900 

2  048  383  OOO 

35.6370593 

10.8293213 

1271 

1  61  5441 

20532255II 

35.6510869 

10.832  162  9 

1272 

1  61  79  84 

2  058  075  648 

35.665  109 

10.835  003 

1273 

i  62  05  29 

2062933417 

35.679  125  5 

10.837841  6 

1274 

i  62  30  76 

2  067  798  824 

35-6931366 

10.8406788 

1275 

i  62  56  25 

2072671875 

35.707  142  i 

10.8435144 

1276 

i  62  81  76 

2077552576 

35.7211422 

10.846  348  5 

1277 

1630729 

2  082  440  933 

35-735  136  7 

10.849  J8i  2 

1278 

i  63  32  84 

2087336952 

35.7491258 

10.8520125 

1279 

1635841 

2  092  240  639 

35.7631095 

10.854  842  2 

1280 

i  63  84  oo 

2097  152000 

35.7770876 

10-8576704 

I28l 

1640961 

2  102  071  041 

35.7910603 

10.860  497  2 

1282 

1643524 

2  106  997  768 

35.805  027  6 

10.863  322  5 

1283 

1  64  60  89 

2  III932  187 

35.8189894 

10-866  146  4 

1284 

i  64  86  56 

2116874304 

35.8329457 

10.8689687 

1285 

i  65  12  25 

2  121  824  125 

35.8468966 

10-871  789  7 

,'286 

1653796 

2  126781656 

35.8608421 

10.874609  i 

SQUARES,  CUBES,  AND  BOOTS. 


295 


SQUARB. 

CUBB. 

SCIUARH  ROOT. 

CUBE  ROOT. 

1656369 

2I3I746003 

35.874  782  2 

10.877427  I 

1658944 

2  136  719  872 

35.8887169 

10.880  243  6 

1661521 

2  141  700  569 

35.002  646  I 

10.8830587 

16641  oo 

2  146  689  000 

35.9165699 

10.8858723 

l66668l 

2151685  171 

35.9304884 

10.8886845 

1  66  92  64 

2156689088 

35.9444015 

10.891  495  2 

167  1849 

2  161  700  757 

35.9583092 

10.894  304  4 

1674436 

2  166  720  184 

35.9722115 

10.8971123 

1  67  70  25 

2171747375 

35.9861084 

10.8999186 

i  6796  16 

2176782336 

36 

10.902  723  5 

1682209 

2  181  825  073 

36.0138862 

10.905  526  9 

i  68  48  04 

2186875592 

36.027  767  I 

10.908329 

1687401 

2191933899 

36.041  642  6 

10.911  1296 

1690000 

2  197000000 

36.0555128 

10.913  928  7 

i  69  26  01 

2  202  073  901 

36.0693776 

10.916  726  5 

i  69  52  04 

2  207  155  608 

36.0832371 

10.9195228 

1697809 

2212  245  127 

36.0970913 

10.9223177 

i  700416 

2217342464 

36.  1  10  940  2 

10.925  in  i 

1703025 

2  222  447  625 

36.1247837 

10.9279031 

1  70  56  36 

2227560616 

36.138622 

10.9306937 

i  70  82  49 

2  232  681  443 

36.152455 

10.933  482  9 

1710864 

2237810112 

36.1662826 

10.9362706 

1713481 

2  242  946  629 

36.180105 

10.939  OS6  9 

i  716100 

2  248  091  OOO 

36.1939221 

10.941  841  8 

1718721 

2253243231 

36.207  734 

10.9446253 

1721344 

2  258  403  328 

36.221  5406 

10.9475074 

1723969 

2263571297 

36.2353419 

10.950  1  88 

i  72  65  96 

2268747144 

36.249  137  9 

10.9529673 

i  72  92  25 

2273930875 

36.262  928  7 

10.955  745  * 

1731856 

2  279  122  496 

36.2767143 

10.9585215 

1734489 

2  284  322  013 

36.2904946 

10.961  296  5 

i  73  71  24 

2289529432 

36.3042697 

10.964070  i 

1739761 

2  294  744  759 

36.3180396 

10.966  842  3 

1742400 

2299968000 

36.3318042 

10.9696131 

1745041 

2  305  199  161 

36.345  563  7 

10.972  382  5 

1747684 

2  310  438  248 

36.3593179 

10.9751505 

1750329 

2315685267 

36.373067 

10.9779171 

1752976 

2320940224 

36.3868108 

10.980  682  3 

1755625 

2326203125 

36.4005494 

10.9834462 

1758276 

2  331  473976 

36.414  282  9 

10.986  208  6 

i  76  09  29 

2336752783 

36.428  01  1  2 

10.9889696 

1763584 

2342039552 

36.4417343 

10.9917293 

i  766241 

2347334289 

36.4554523 

10.9944876 

i  768900 

2  352  637  ooo 

36.469  165 

10.997  244  5 

1771561 

2357947691 

36.482  872  7 

ii 

1774224 

2363266368 

36.4965752 

11.0027541 

1776889 

2368593037 

36.5102725 

11.0055069 

1  77  95  56 

2373927704 

36.5239647 

11.0082583 

i  78  22  25 

2379270375 

36.5376518 

n.oii  0082 

i  78  48  96 

2384621056 

36.5513338 

11.0137569 

1787569 

2389979753 

36.5650106 

11.016504  i 

1790244 

2395346472 

36.5786823 

11.01925 

i  792921 

2400721  219 

36.5923489 

11.0219945 

1795600 

2  406  104  000 

36.6060104 

11.0247377 

i  798281 

2411494821 

36.6196668 

11.0274795 

18009647 

2416893688 

36.6333181 

11.0302199 

296 


SQUARES,  CUBES,  AND  BOOTS. 


NUMBKR. 

SQUARE.                          CUBE. 

SQUARE  ROOT. 

CUB«  ROOT. 

1343 

1803649 

2  422  3OO  607 

36.646  964  4 

11.032959 

1344 

1806336 

2427715584 

36.660  605  6 

11.0356967 

1345 

I  809025 

2433138625 

36.674  241  6 

11.038433 

1346 

i  81  17  16 

2  438  569  736 

36.687  872  6 

11.041  168 

1347 

i  81  44  09 

2444008923 

36.7014986 

11.043901  7 

1348 

i  81  71  04 

2449456192 

36.7151195 

11.0466339 

1349 

1819801 

2454911549 

36.7287353 

11.0493649 

1350 

i  82  25  oo 

2  460  375  ooo 

36.7423461 

11.0520945 

1351 

i  82  52  01 

2465846551 

36.7559519 

11.054822  7 

1352 

1827904 

2471326208 

36.7695526 

11.0575497 

1353 

1830609 

2476813977 

36.7831483 

11.060275  2 

1354 

1833316 

2482309864 

36.796739 

11.0629994 

1355 

1836025 

2487813875 

36.8103246 

II.o657222 

1356 

1838736 

2493326016 

36.823  905  3 

11.0684437 

1357 

i  84  14  49 

2  498  846  293 

36.8374809 

11.071  1639 

1358 

i  84  41  64 

2504374712 

36.8510515 

11.0738828 

1359 

1846881 

2509911279 

36.8646172 

11.0766003 

1360 

1849600 

2  5i5  456  ooo 

36.8781778 

11.0793165 

I36l 

I  85  23  21 

2521008881 

36.891  733  5 

11.082031  4 

1362 

1855044 

2  526  569  928 

36.905  284  2 

11.0847449 

1363 

1857769 

2532I39I47 

36.9188299 

11.0874571 

*364 

1860496 

2537710544 

36.9323706 

11.090  1679 

1365 

i  86  32  25 

2543302125 

36.9459064 

11.0928775 

1366 

1865956 

2548895896 

36.9594372 

11.0955857 

1367 

1  86  86  89 

2554497863 

36.9729631 

11.0982926 

1368 

1  87  14  24 

2  560  108  032 

36.986484 

11.1009982 

1369 

1874161 

2565726409 

37 

11.1037025 

1370 

1876900 

2571353000 

37.013511 

11.106405  4 

J37i 

1879641 

2576987811 

37.0270172 

11.109107 

1372 

1882384 

2582630848 

37.0405184 

ii.  in  8073 

1373 

1885129 

2588282117 

37.0540146 

11.1145064 

1374 

i  88  78  76 

2593941624 

37.067  506 

11.117204  i 

1375 

1890625 

2599609375 

37.0899924 

11.1199004 

1376 

1893376 

2605285376 

37.094474 

11.1225955 

1377 

i  89  61  29 

2610969-633 

37.1079506 

11.1252893 

1378 

1898884 

2  6l6662  152 

37.1214224 

11.127981  7 

1379 

1,901641 

2622362939 

37.1348893 

11.1306729 

1380 

1  90  44«oo 

2  628  072  OOO 

37.1483512 

11.1333628 

1381 

1  9071*61 

263378934! 

37.1618084 

11.1360514 

1382 

1909924 

2-639514968 

37.1752606 

11.1387386 

1383 

191.2689 

2-645248887 

37.1887079 

11.1414246 

1384 

1  9I%54  56 

2650-991  104 

37.2021505 

11.1441093 

1385 

i  91  82  25 

2  656  741  625 

37.2155881 

1  1  .  146  792  6 

1386 

1920996 

2  662  50O  456 

37.2290209 

11.1494747 

1387 

1923769 

2  668  267  603 

37.2424489 

11.1521555 

1388 

1926544 

2  674  043  072 

37.255  872 

11.154835 

1389 

1929321 

2  679  826  869 

37.2692903 

"•IS7SI33 

1390 

1932100 

2685619000 

37.282  703  7 

11.1601903 

1391 

1  93  48  81 

2691419471 

37.2961124 

11.162865  9 

1392 

1937664 

2  697  228  288 

37.3095162 

11.1655403 

1393 

1940449 

2703045457 

37.3229152 

11.1682134 

1394 

1943236 

2  708  870  984 

37-3363094 

11.1708852 

J395 

1  94  60  25 

2714704875 

37.3496988 

".1735558 

1396 

1948816 

2  720  547  136 

37.3630834 

11.176225 

1397 

1951609 

2726397773 

37.3764632 

11.178893 

J398 

1954404 

2  732  256  792 

37.3898382 

11.1815598 

SQUARES,  CUBES,  AND  ROOTS. 


NfMBKR, 

SVARK. 

CCBB. 

SQUARE  ROOT. 

CUBB  ROOT. 

1399 

I9S720I 

2  738  I24  199 

37.403  208  4 

11.1842252 

I4OO 

1960000 

2744000OOO 

37-4I65738 

11.1868894 

1401 

i  96  28  01 

2  749  884  201 

37.4299345 

11.1895523 

1402 

I  96  56  04 

27557768o8 

37.4432904 

11.1922139 

1403 

1968409 

2  761  677  827 

37.4566416 

11.1948743 

1404 

I  97  12  16 

2  767  587  264 

37.469988 

II«I975334 

1405 

1974025 

2773505125 

37.4833296 

11.200  191  3 

1406 

i  97  68  36 

277943I4IO 

37.4966665 

11.2028479 

1407 

1979649 

2  785  366  143 

37.5099987 

11.2055032 

1408 

1982464 

279I3093I2 

37.5233261 

11.208  1573 

1409 

1985281 

2  797  260  929 

37.5366487 

11.2108101 

I4IO 

1988100 

2803221000 

37.5499667 

11.213461  7 

I4II 

1990921 

2809189531 

37-5632799 

II.2IOII2 

1412 

1993744 

2815166528 

37.5765885 

11.218761  I 

1413 

1996569 

2821  151997 

37.5898922 

11.2214089 

1414 

1999396 

2827145944 

37.603  191  3 

11.2240054 

MIS 

2  OO  22  25 

2  833  148  375 

37.6164857 

11.2267007 

I4l6 

2  OO  50  56 

2  839  159  296 

37.629  775  4 

11.2293448 

1417 

2007889 

2845178713 

37.6430604 

11.2319876 

I4l8 

2  OI  07  24 

2851  206632 

37.6563407 

11.2346292 

1419 

20I356I 

2857243059 

37.6696164 

11.2372696 

1420 

2OI  6400 

2863288000 

37.6828874 

11.2399087 

1421 

2OI  9241 

2869341461 

376961536 

11.2425465 

1422 

2  02  20  84 

2  875  403  448 

37.7094153 

II.245I83I 

1423 

2  02  49  29 

2881473967 

37.722672  2 

11.2478185 

1424 

2  02  77  76 

2887553024 

37.7359245 

11.2504527 

1425 

2  03  06  25 

2893640625 

37.7491722 

11.2530856 

1426 

2033476 

2899736.776 

37.7624152 

11.2557173 

1427 

2  03  63  29 

2  005  841  483 

37-7756535               11.2583478 

1428 

2  03  91  84 

29II954752 

37.7888873 

11.200977 

1429 

2  O4  2O  41 

2918076589 

37.802  1163 

11.263605 

143° 

2044900 

2  924  2O7  OOO 

37.8153408 

11.266231  8 

1431 

2047761 

2930345991 

37.8285606 

11.2688573 

1432 

2  O5  00  24 

2936493568 

37.8417759 

11.271481  6 

1433 

2053489 

2942649737 

37.8549864 

11.2741047 

1434 

2  05  63  56 

2948814504 

37.8681924 

11.2767266 

1435 

2  05  92  25 

2954987875 

37.8813938 

11.2793472 

1436 

2062096 

2  O6l  169  856 

37.8945906 

11.2819666 

1437 

2064969 

2967360453 

37.9077828 

11.2845849 

1438 

2067844 

2973559672 

37.9209704 

11.287201  9 

1439 

2  O7  07  2  1 

2979767519 

37-934I535 

11.289817  7 

1440 

2  07  36  00 

2  985  984  000 

37-9473319 

11.2924323 

1441 

2076481 

2992  2O9  121 

37.960  505  8 

11.2950457 

1442 

2079364 

2  998  442  888 

37-973  675  * 

11.2976579 

1443 

2  O8  22  49 

3004685307 

37.9868398 

11.3002688 

1444 

2085136 

3  oio  936  384 

38 

11.3028786 

1445 

2  08  80  25 

3017196125 

38.0131556 

11.3054871 

1446 

20909  16 

3023464536 

38.026  306  7 

11.3080945 

1447 

2  09  38  09 

3029741623 

38.0394532 

11.3107006 

1448 

2  09  67  04 

3036027392 

38.052  595  2 

"•3133056 

1449 

209060I 

3042321849 

38.065  732  6 

11.3159094 

1450 

2  10  25  00 

3  048  625  ooo 

38.0788655 

11.3185119 

1451 

2  10  54  01 

3054936851 

38.0919939 

11.321  1132 

1452 

2  10  83  04 

3061257408 

38.1051178 

n-3237I34 

M53 

2  II  1209 

3067586677 

38.1182371 

11.3263124 

1454 

2  1141  1  6 

3073924664 

38.1313519 

11.3289102 

298 


SQUARES,  CUBES,  AND  ROOTS. 


NUMBBB. 

SQUARE. 

CUBB. 

SQUARE  ROOT. 

CUBE  ROOT. 

1455 

211  7025 

3080271375 

38.1444622 

11.3315067 

1456 

2119936 

3086-626816 

38.1575681 

11.3341022 

1457 

2  122849 

3092990993 

38.1706693 

11.3366964 

1458 

2125764 

3099363912 

38.1837662 

11.3392894 

1459 

2I2868I 

3  ^S  745  579 

38.1968585 

11.3418813 

1460 

2  13  I60O 

3112  136000 

38.2099463 

11.3444719 

1461 

2I3452I 

3118535181 

38.2230297 

11.3470614 

1462 

2*37444 

3124943128 

38.2361085 

11.3496497 

1463 

2  14  03  69 

3I3I359847 

38.249  182  9 

11.3522368 

1464 

2  14  32  96 

3137785344 

38.262  252  9 

11.3548227 

1465 

2  14  62  25 

3144219625 

38.2753184 

11-3574075 

1466 

2  14  91  56 

3150662696 

38.2883794 

"•359991  I 

1467 

2152089 

3157114563 

38.301  436 

11.3625735 

1468 

2155024 

3163575232 

38.3144881 

II-365I547 

1469 

2157961 

3170044709 

38.3275358 

II-3677347 

1470 

2  160900 

3*76  523  ooo 

38.340579 

"•3703136 

1471 

2163841 

3183010111 

38-3536178 

11.372891  4 

1472 

2166784 

3189506048 

38.3666522 

II-3754679 

1473 

2  16  97  29 

3196010817 

38.3796821 

11.3780433 

1474 

2  172676 

3202524424 

38.3927076 

11.3806175 

1475 

2175625 

3209046875 

38.405  728  7 

11.3831906 

1476 

2178576 

3215578176 

38.4187454 

11.3857625 

M77 

2181529 

3222118333 

38.431  757  7 

"•3883332 

1478 

2  18  44  84 

3228667352 

38.444  765  6 

11.3909028 

1479 

2187441 

3235225239 

38.457  769  i 

11.3934712 

1480 

2  190400 

3241792000 

38.470  768  i 

11.3960384 

1481 

2193361 

3248367641 

38.483  762  7 

11.3986045 

1482 

2196324 

3254952168 

38.496  753 

11.401  1695 

1483 

2  19  92  89 

3261545587 

38-509  739 

11.4037332 

1484 

2  20  22  56 

3268147904 

38.522  720  6 

11.4062959 

1485 

2205225 

3274759125 

38.535  697  7 

11.4088574 

1486 

2208196 

3  281  379  256 

38.5486705 

11.4114177 

1487 

221  IIO9 

3288008303 

38.5616389 

11.4139769 

1488 

2214144 

3  294  646  272 

38.574603 

11.4165349 

1489 

221  7121 

3301293169 

38.587  562  7 

11.4190918 

1490 

2220IOO 

3307949000 

38.6005181 

1  1  .420  647  6 

1491 

2223081 

3314613771 

38.6134691 

11.4242022 

1492 

2226064 

3321287488 

38.6264158 

11.4267556 

1493 

2  22  90  49 

3327970157 

38-6393582 

11.4293079 

1494 

2  23  20  36 

3334661  784 

38.652  296  2 

11.4318591 

1495 

2235025 

3341362375 

38.6652299 

11.4344092 

1496 

2  23  80  16 

3348071936 

38.6781593 

11.4369581 

1497 

2  24  10  09 

3354790473 

38.6910843 

11.4395059 

1498 

2  24  40  04 

3361517992 

38.704005 

11.4420525 

1499 

2  24  70  OI 

3368254499 

38.7169214 

11.444598 

1500 

2  25OOOO 

3375000000 

38.7298335 

11.4471424 

1501 

2  25  30  01 

338i7545oi 

38.742  741  2 

11.4496857 

1502 

2  25  OO  04 

3388518008 

38.7556447 

11.4522278 

1503 

2259009 

3395290527 

38.7685439 

11.4547688 

1504 

2  26  20  16 

3  402  072  064 

38.7814389 

11.4573087 

1505 

2  26  50  25 

3  408  862  625 

38.7943294 

11.4598474 

1506 

2268036 

3415662216 

38.807  2158 

11.462385 

1507 

2  27  10  49 

3422470843 

38.820  097  8 

11.4649215 

1508 

2  27  40  64 

3429-288512 

38.832  975  7 

1  1  .467  456  8 

1509 

2  27  70  81 

3436115229 

38.845  849  i 

11.4699911 

1510 

2280100 

3442951000 

38.8587184 

11.4725243 

SQUARES,  CUBES,  AND  ROOTS. 


299 


DUMBER. 

SQUARE.                         CUBE. 

SQUARE  ROOT. 

CUBK  ROOT. 

15" 

22831  21 

3  449  795  831 

38.8715834 

11.4750562 

1512          2286144 

3456649728     1 

38.8844442              11.4775871 

I5I3 

2289169 

3463512697 

38.8973006              11.4801169 

1514 

2299196 

3470384744 

38.9101529 

11.4826455 

2  29  52  25 

3477265875 

38.923  ooo  9 

11.485  1731 

1516 

2  29  82  56 

3484156096 

38.935  844  7 

11.4876995 

J5J7 

2301289 

3491055413 

38.9486841 

11.4902249 

1518 

2304324 

3497963832 

38.9615194 

11.4927491 

1519 

2307361 

3  504  88  1  359 

38.9743505 

11.4952722 

1520 

2  31  04  00 

3511808000 

38.9871774 

11.4977942 

1521 

2313441 

3518743761 

39 

11.5003151 

1522 

2316484 

3525688648 

39.0128184 

11.5028348 

1523 

2  31  95  29 

3532642667 

39.0256326 

".5053535 

1524 

2  32  25  76 

3539605824 

39.0384426 

11.507871  I 

1525 

2325625 

3  546  578  125 

39.0512483 

11.5103876 

1526 

2328676 

3  553  559  576 

39.0640499 

11.512903 

J527 

2331729 

3560558183 

39.0768473 

11.5154173 

1528 

2  33  47  84 

3567549952 

39.0896406 

11.5179305 

1529 

2337841 

3574558889 

39.1024296 

11.5204425 

*530 

2340900 

3581577000 

39.1152144 

11.5229535 

1531 

2343961 

3  588  604  291 

39.1279951 

".5254634 

1532 

2  34  70  24 

3  595  640  768 

39.1407716 

11.5279722 

J533 

2350089 

3  602  686  437 

39-J535439 

".5304799 

1534 

2  35  3i  56 

3609741304 

39.166312 

11.5329865 

1535 

2356225 

3616805375 

39.179076 

"•535492 

1536 

2359296 

3623878656 

39.1918359 

"•5379965 

J537 

2362369 

3630961  153 

39.2045915 

11.5404998 

1538 

2365444 

3  638  052  872 

39.2173431 

11.5430021 

J539 

2368521 

3645153819 

39.2300905 

".5455033 

1540 

237  1600 

3652264000 

39.242  833  7 

11.5480034 

2374681 

3659383421 

11.5505025 

1542 

2377764 

3666512088 

39-268307  8 

11.5530004 

1543 

2380849 

3673650007 

39.281  038  7 

"•5554973 

1544 

2383936 

3680797184 

39.2937654 

"-557993I 

t545 

2387025 

3687953625 

39.306488 

11.5604878 

1546 

23901  16 

3695119336 

39.3192065 

11.5629815 

1547          2393209 

3702294323 

39.3319208 

11.565474 

1548          2396304 

3709478592 

39.344631  i 

11.5679655 

1549 

2399401 

3716672149 

39-3573373 

".5704559 

1550 

2  40  25  OO 

3  723  875  ooo 

39.3700394 

11.5729453 

2  40  56  OI 

3731087151 

39-3827373 

"•5754336 

1552 

2  40  87  04 

3738308608 

39-3954312 

11.5779208 

1553 

241  1809 

3  745  539  377 

39.408  121 

11.5804069 

1554 

2  41  49  16 

3752779464 

39.4208067 

11.5828919 

1555 

2418025 

3760028875 

39.4334883 

"•5853759 

1556 

2421136 

3  767  287  616 

39.446  165  8 

11.5878588 

1557 

2424249 

3  774  555  693 

39.4588393 

11.5903407 

1558 

2427364 

3781833112 

39.471  508  7 

11.5928215 

1559 

2  43  04  81 

3789119879 

39.484174 

"•5953013 

1560 

2433600 

3  796  416  ooo 

39.496  835  3 

"•5977799 

1561 

2436721 

3  803  721  481 

39.5094925 

11.6002576 

1562 

2439844 

3811036328 

39.522  145  7 

11.6027342 

1563 

2442969 

3818360547 

39-5347948 

11.6052097 

1564 

2446096 

3  825  641  144 

39-5474399 

11.6076841 

1565 

2449225 

3833037125 

39.5600809 

11.6101575 

1566 

2452356 

3840389496 

39.5727179 

11.6126292 

300 


SQUAKES,  CUBES,  AND  ROOTS. 


NUMBER. 

SQUABB. 

CUBB. 

SQUARE  ROOT. 

CUBE  ROOT. 

1567 

2455489 

3847751263 

39-585  350  8 

11.615  IQI  2 

1568 

2  45  86  24 

3  855  122  432 

39-597  979  7 

11.6175715 

1569 

246  1761 

3  862  503  009 

39.6106046 

11.620040  7 

1570 

2  46  49  oo 

3  869  893  ooo 

39.623  225  5 

11.6225088 

1571 

2  46  80  41 

3877292411 

39.635  842  4 

11.6249759 

I572 

2  47  1  1  84 

3  884  701  248 

39.648  455  2 

11.627442 

T573 

2  47  43  29 

3892119517 

39.661  064 

11.629907 

1574 

2  47  74  76 

3899547224 

39.673  668  8 

11.632371 

1575 

2  48  06  25 

3  906  984  375 

39.686  269  6 

11.6348339 

1576 

2  48  37  76 

3914430976 

39.6988665 

11.6372957 

1577 

2  48  69  29 

3921887033 

39.7114593 

11.6397566 

1578 

2  49  oo  84 

3929352552 

39.7240481 

11.642  2164 

1579 

2493241 

3936827539 

39.736  632  9 

11.6446751 

1580 

2  49  64  oo 

3944312000 

39.7492138 

11.6471329 

1581 

2  49  95  61 

3951805941 

39.7617907 

11.6495895 

1582 

2  50  27  24 

3959309368 

39-7743636 

11.6520452 

1583 

2  50  58  89 

3  966  822  287 

39.7869325 

11.6544998 

1584 

2  50  90  56 

3  974  344  704 

39-7994975 

11.6569534 

1585 

2  SI  22  25 

3981876625 

39.8120585 

11.6594059 

1586 

2515390 

3  989  418  056 

39.8246155 

11.661  8574 

1587 

2  51  85  69 

3996969003 

39.8371686 

11.6643079 

1588 

2521744 

4004529472 

39.8497177 

11.6667574 

1589 

2  52  49  21 

4012099469 

39.862  262  8 

1  1  .669  205  8 

1590 

2  52  81  oo 

4  019  679  ooo 

39.874  804 

11.6716532 

I59i 

253  1281 

4027268071 

39.887  341  3 

11.6740996 

159? 

2534464 

4  034  866  688 

39.899  874  7 

11.6765449 

1593 

2537649 

4  042  474  857 

39.912  404  i 

11.6789892 

1594 

2  54  08  36 

4  050  092  584 

39.924  929  5 

11.6814325 

1595 

2544025 

4057719875 

39-937451  i 

11.6838748 

150 

2  54  72  16 

4065356736 

39.9499687 

11.686316  i 

1597 

2  55  04  09 

4073003173 

399^24824 

11.6887563 

1598 

2  55  36  04 

4  080  659  192 

39-974  992  2 

11.691  1955 

1599 

2  55  68  01 

4088324799 

39.987  498 

11.6936337 

1600 

2560000 

4096000000 

40 

11.6960709 

Uses  of  preceding  table  may  be  extended  by  aid  of  following  Rules,  tc 
Compute  Square  or  Cube  of  a  higher  Number  than  is  contained  in  it. 

To    Compute    Sq.ta.are. 

When  Number  is  an  Odd  Number. 

RULE. —Take  the  two  numbers  nearest  to  each  other,  which,  added  together, 
make  that  sum ;  then  from  sum  of  squares  of  these  two  numbers,  multiplied  by  2> 
subtract  i,  and  remainder  will  give  result. 

To    Compute    Square    or    Cxi"be. 

When  Number  is  divisible  by  a  Number  without  leaving  a  Remainder. 
RULE. — If  number  exceed  by  2,  3,  or  any  other  number  of  times,  any  numb^i 
contained  in  table,  multiply  square  or  cube  of  that  number  in  table  by  square  of  a 
j.  etc.,  and  product  will  give  result. 
EXAMPLE.— Required  square  of  1700. 

1700  is  10  times  170,  and  square  of  170  is  2  8900. 
Then,  2  89  oo  X  io2  =  2  89  oo  oo. 
a.-— What  is  cube  of  2400? 

2400  is  twice  1200,  and  cube  of  1200  is  1 728000000. 
Then  1728  ooo  ooo  x  23  =  13824000000. 


SQUABES,  CUBES,  AND  BOOTS.          JO  I 

EXAMPLE.  —What  is  square  of  1745? 

Two  nearest  numbers  are  {  g7^  1  =  1745. 
Then,  per  table,    88«: 


i  52  25  13  X  2  =r  3  045  026  —  i  =  3  04  50  25. 

To  Compute  Sq.uare  or  Cube  Root  of  a  high.er  Number 
th.au   is    contained,    in    Table. 

When  Number  is  divisible  by  4  or  8  without  leaving  a  Remainder. 

RULE.—  Divide  number  by  4  or  8  respectively,  as  square  or  cube  root  is  required; 
take  root  of  quotient  in  table,  multiply  it  by  2,  and  product  will  give  root  required. 

EXAMPLE.—  What  are  square  and  cube  roots  of  3200? 

3200  -4-  4  =  800,  and  3200  H-  8  =  400. 

Then,  square  root  for  800,  per  table,  is  28.  28  42  71  2,  which,  being  X  2  =  56.  56  85  42  4 
root. 
Cube  root  for  400,  per  table,  is  7.  368  063,  which,  being  x  2  =  14.  736  126  root. 

When  the  Root  (which  is  taken  as  Number)  does  not  exceed  1600. 

The  Numbers  in  table  are  roots  of  squares  or  cubes,  which  are  to  be  taken 
as  numbers. 

ILLUSTRATION.—  Square  root  of  6400  is  80,  and  cube  root  of  51200x3  is  80. 

When  a  Number  has  Three  or  more  Ciphers  at  its  right  hand. 

RULE.—  Point  off'number  into  periods  of  two  or  three  figures  each,  according  as 
square  or  cube  root  is  required,  until  remaining  figures  come  within  limits  of  table; 
then  take  root  for  these  figures,  and  remove  decimal  point  one  figure  for  every  pe- 
riod pointed  off. 

EXAMPLE.  —  What  are  square  or  cube  roots  of  1  500  ooo? 

1  500000  =  150,  remaining  figure,  square  root  of  which=  12.  247  45;  hence  1224.745, 
square  root. 

1500000=1500,  remaining  figures,  cube  root  of  which  =  11.447  14  ;  hence 
£14.4714,  c^&e  root. 

To   Ascertain   Cube   Root   of*  any  Number  over  16OO. 

RULE.—  Find  by  table  nearest  cube  to  number  given,  and  term  it  assumed  cube; 
multiply  it  and  given  number  respectively  by  2  ;  to  product  of  assumed  cube  add 
given  number,  and  to  product  of  given  number  add  assumed  cube. 

Then,  as  sum  of  assumed  cube  is  to  sum  of  given  number,  so  is  root  of  assumed 
cube  to  root  of  given  number. 

EXAMPLE.  —  What  is  cube  root  of  224809? 

By  table,  nearest  cube  is  216000,  and  its  root  is  60. 

216  ooo  X  2  -f-  224  809  =  656  809, 
And  224  809  X  2  -}-  216  ooo  =  665  618. 
Then  656809  :  665618  ::  60  :  60.804+,  r°°t- 

To  Ascertain  Square   or  Cube  Root   of  a  Number  con» 
sisting   of  Integers    and.    Decimals. 

RULE.—  Multiply  difference  between  root  of  integer  part  and  root  of  next  higher 
integer  by  decimal,  and  add  product  to  root  of  integer  given;  the  sum  will  give  root 
of  number  required. 

This  is  correct  for  Square  root  to  three  places  of  decimals,  and  for  Cube  root  to  seven. 
Co 


3<D2  SQUARES,   CUBES,  AND    BOOTS. 

EXAMPLE.— What  is  square  root  of  53.75,  and  cube  root  of  843.75  ? 


V54      =7-3484 
V53      =7-28oi 

#844       =9-4503 
#843       =9-4466 

.0037 
•75 

.0683 
•75 

.051  225 
V53      =7-2801 

.002775 
3/843       =9.4466 

V53-75  =  7-33'325                                        #843.75  =  9.449375 

When  the  Square  or  Cube  Root  is  required  for  Numbers  not  exceeding  Roots 
given  in  Table. 

Numbers  in  table  are  squares  and  cubes  of  roots. 

RULE.  —  Find,  by  table,  in  column  of  numbers  that  number  representing  figures 
of  integer  and  decimals  for  which  root  is  required,  and  point  it  oft'  decimally  by 
nlar.es  of  ->  nr  o  figures  as  snnare  or  ouhft  root,  is  rennired-  and  ormosit.fi  t,o  it,    in 

column  of  roots,  take  root  and  point  off  i  or  2  additional  places  of  decimals  to  those 
in  root,  as  square  or  cube  root  is  required,  and  result  is  root  required. 

EXAMPLE  i.— What  are  square  roots  of  .15,  1.50,  and  15.00? 
In  table,  15  has  for  its  root  3.87  298;  hence  .38  7298  =  square  root  for  .15. 
150  has  for  its  root  12. 24  74  5 ;  hence  i.  22  47  45  =  square  root  for  1.50. 
1500  has  for  its  root  38.72  98 ;  hence  3.87  29  8  =  square  root  for  15. 

2. —What  are  cube  roots  of  .15, 1.50,  and  15.00? 

Add  a  cipher  to  each,  to  give  the  numbers  three  places  of  figures,  as  .150, 1.500, 
and  15.000. 

In  table  150  has  for  its  root  5.3133;  hence  .531 33  =  cube  root  o/.is. 

1500  has  for  its  root  11.447 ;  hence  1.1447  =  cu^  root  0/1.50. 

15  has  for  its  root  2.4662;  and  15.000,  by  addition  of  3  places  of  figures,  has 
24.662 ;  hence  2.4662  =  cube  root  of 15.00. 

To  Ascertain.  Square  or  Cu."be   Roots  of*  Decimals  alone. 

RULE. — Point  off  number  from  decimal  point  into  periods  of  two  or  three  figures 
each,  as  square  or  cube  root  is  required.  Ascertain  from  table  or  by  calculation 
root  of  number  corresponding  to  decimal  given,  the  same  being  read  off  by  remov- 
ing the  decimal  point  one  place  to  left  for  every  period  of  2  figures  if  square  root  is 
required,  and  one  place  for  every  period  of  3  figures  if  cube  root  is  required. 

EXAMPLE.— What  are  square  and  cube  roots  of  .810,  .081,  and  .0081  ? 
.810,   when  pointed  off  =  .8i,     and  ^/.Bi     =.9. 
.081,       "          "        "  =  .081,     "    V-08!   =.2846. 
.0081,      "          "        "  =  .oo8i,  "    V-°°8i=.o9. 

.810,  when  pointed  off  =  .810,  and -^.810  =93217. 
.081,  "  "  "  =.o8i,  "  ^.081  =.43267. 
.0081,  "  "  "  =.0081,  "  -^.0081  =  .200 83. 

To  Compute  4th    Root  of*  a   CT  umber* 

RULE.— Take  square  root  of  its  square  root. 
EXAMPLE.— What  is  the  •$/  of  1600? 

^1600  =  40,  and  V4°=6-3245553* 

To   Compute   6th.   Root  of  a   IST 
RULE.— Take  cube  root  of  its  square  root. 
EXAMPLE.— What  is  the  $  of  441  ? 

V44i  =  21,  and  ^21  =  2.7  589  243. 


FOURTH    AND   FIFTH   POWEBS   OF   KUMBEES. 


303 


4th.   and.  5th.   IPowers   of  Nnnibers. 

From  i  to  150. 


Number. 

4th  Power. 

5th  Power. 

Number. 

4th  Power. 

5th  Power. 

»  :'.4-»^,t 

I 

i 

64 

16777  216 

i  073  741  824 

2 

16 

32 

65 

17850625 

i  160290625 

3 

81 

243 

66 

18974736 

i  252  332  576 

4 

256 

1024 

67 

20I5I  121 

i  350  125  107 

5 

625 

3I25 

68 

21  381  376 

1  453  933  568 

6 

i  296 

7  776 

69 

22667  I2I 

1564031349 

7 

2401 

16807 

70 

24  oio  ooo 

i  680  700  ooo 

8 

4096 

32768 

7i 

25  411  681 

1804229351 

9 

6561 

59°49 

72 

26  873  856 

1934917632 

10 

IOOOO 

IOOOOO 

73 

28398241 

2073071593 

ii 

14641 

161051 

74 

29  986  576 

2219006624 

12 

20736 

248  832 

75 

31640625 

2373046875 

'3 

28561 

371  293 

76 

33  362  176 

2  535  525  376 

»4 

38416 

537  824 

77 

35153041 

2706784157 

15 

50625 

759375 

78 

37015056 

2887174368 

16 

65  536 

1  048  576 

79 

38950081 

3077056399 

X7 

83521 

1419857 

80 

40960000 

3276800000 

18 

104976 

i  889  568 

81 

43046721 

3486784401 

J9 

130321 

2476099 

82 

45212176 

3  707  398  432 

20 

160000 

3200000 

83 

47  458  321 

3  939  040  643 

21 

194481 

4084  101 

84 

49787136 

4182119424 

22 

234  256 

5153632 

8s 

52200625 

4437053125 

23 

279841 

6436343 

86 

54  708  016 

4  704  270  176 

24 

331  776 

7962624 

87 

57  289  761 

4984209207 

25 

390625 

9765625 

88 

59969536 

5277319168 

26 

456  976 

11881376 

89 

62  742  241 

5584059449 

27 

531  441 

14348907 

9° 

65610000 

5904900000 

28 

614656 

17210368 

91 

68574961 

6  240  321  451 

29 

707  281 

20511  149 

92 

71639296 

6590815232 

30 

810000 

24300000 

93 

74  805  201 

6956883693 

31 

923521 

28629  15I 

94 

78074896 

7339040224 

32 

1  048  576 

33554432 

95 

81450625 

7  737  809  375 

33 

1185921 

39  '35  393 

96 

84  034  656 

8153726976 

34 

1336336 

45  435  424 

88  529  281 

8587340257 

35 

1  500625 

52521875 

98 

92  236  816 

9039207968 

36 

1679616 

60466176 

99 

96  059  601 

9509900499 

37 

38 

I  874  161 
2085136 

69343957 
79235168 

oo 

01 

100  OOO  OOO 

104060401 

IOOOOOOOOOO 

10510100501 

39 

231344! 

90224199 

02 

108243216 

11*040808032 

4° 

2560000 

102400000 

03 

112550881 

11592740743 

4i 

2  825  761 

115856201 

04 

116985856 

12  166  529024 

42 

3111696 

130691232 

05 

121  550625 

12762815625 

43 

34I880I 

147  008  443 

06 

126  247  696 

13382255776 

44 

3748096 

164916224 

07 

131079601 

14025517307 

45 

4  100625 

184528125 

08 

136048896 

14693280768 

46 

4  477  456 

205  962  976 

09 

141  158  i6i 

15386239549 

47 

4879681 

229345007 

10 

146410000 

1  6  105  looooo 

48  ' 

5308416 

254803968 

II 

151  807041 

16850581551 

49 

5  764  801 

282  475  249 

12 

157  35i  936 

17623416832 

So 
5i 

6250000 
6765201 

312500000 
345025251 

13 
14 

163047361 
168896016 

18424351793 
19  254  145  824 

52 

7311616 

380204032 

15 

174900625 

20113581875 

53 

7890481 

418195493 

16 

181  063936 

21  003416576 

54 

8  503  056 

459165024 

i7 

187388721 

21924480357 

55 

9150625 

503  284  375 

18 

193  877  776 

22  877  577  568 

56 
57 

9  834  496 
10556001 

550731776 
601  692  057 

'9 

20 

200  533  921 
207360000 

23  863  536  599 
24883200000 

58 

11316496 

656  356  768 

21 

214358881 

25937424601 

59 

12117361 

714924299 

22 

221533456 

27027081632 

60 

12960000 

777600000 

23 

228886641 

28153056843 

61 
62 

13845841 
14776336 

844  596  301 
916132832 

24 
25 

236421376 
244140625 

29316250624 
30517578125 

63 

15752961 

992  436  543 

26 

252  047  376 

31757969376 

304 


POWERS    OF    NUMBERS. — RECIPROCALS. 


Number. 

4th  Power. 

5th  Power. 

Number. 

4th  Power. 

5th  Power. 

127 
128 
129 

260  144641 
268  435  456 
276922881 

33038369407 
34  359  738  368 
3572305i649 

139 
140 
141 

37330I64I 
384  160000 
395  254  161 

51  888  844  699 
53782400000 
55730836701 

130 

285610000 

37129300000 

142 

406  586  896 

57735339232 

IS' 

294499921 

38579489651 

143 

418  161  601 

59  797  108  943 

132 

303  595  776 

40074642432 

144 

429981696 

61  917  364224 

133 

312900721 

41615795893 

145 

442050625 

64097340625 

'34 

322417936 

43  204  003  424 

146 

454  371  856 

66  338  290976 

135 

332150625 

44  840  334  375 

147 

466  948  881 

68641485507 

136 

342102016 

46  525  874  176 

148 

479785216 

71008211  968 

»37 

352  275  361 

48261724457 

149 

492  884  401 

73439775749 

138 

362  673  936 

50  049  003  1  68 

150 

506250000 

759375ooooo 

To    Compxite   4th    Power   of  a   Nximber   greater   than    is 
contained,   in    Table. 

RULE.— Ascertain  square  of  number  by  preceding  table  or  by  calculation,  and 
square  it;  product  is  power  required. 
EXAMPLE  —What  is  4tb  power  of  1500? 

i5oo2  —  2250000,  and  2  2  50  ooo2  =  5  062  500000000. 

To   Compnte   Sth    Power   of  a   Number  greater   than   is 
contained,   in    Table. 

RULE.— Ascertain  cube  of  number  by  preceding  table  or  by  calculation,  and  mul- 
tiply it  by  its  square;  product  is  power  required. 

To  Compxxte  4th  and  5th   Powers  by  another  Method. 

RULE. — Reduce  number  by  2  until  it  is  one  contained  within  table.  Take  power 
which  is  required  of  that  number,  and  multiply  it  by  16,  i62,  or  i63  respectively 
for  each  division,  by  2  for  4th  power,  and  by  32,  32%  or  323  respectively  for  each 
division  by  2  for  sth  power. 

EXAMPLE.— What  are  the  4th  and  sth  powers  of  600? 

600  -4-  2  —  300,  and  300  -r-  2  =  150. 

The  4tti  power  of  150,  per  table,  —  506  250000,  wrhich  x  i62,  multiplier  for  a  second 
division  256  =  129600000000,  ^th  power. 

Again,  the  sth  power  of  150  =  75937  500000,  which  X  322,  multiplier  for  a  second 
division  1024  =  77  760000000000  =  power. 

To   Compute   Gth   Power  of  a   Number. 

RULE. — Square  its  cube. 

EXAMPLE.— What  is  the  6th  power  of  2? 

2l2=  64. 

To  Compute  4th    or  Sth  Root   of  a  N"xxmber  per  Table. 

RULE.— Find  in  column  of  4th  and  sth  powers  number  given,  and  number  from 
which  that  power  is  derived  will  give  root  required. 
EXAMPLE.— What  is  the  sth  root  of  3  200000? 

3200000  in  table  is  sth  power  of  20;  hence  20  is  root  required. 


RECIPROCALS. 

Reciprocal  of  a  number  is  quotient  arising  from  dividing  i  by  number;  thus,  re- 
ciprocal  of  2  is  i  -f-  2  = .  5 
Product  of  a  number  and  its  reciprocal  is  always  equal  to  i ;  thus,  2  x  .5  —  r. 

Reciprocal  of  a  vulgar  fraction  is  denominator  divided  by  numerator .  thus,  -  = .  5, 


LOGARITHMS. 

LOGARITHMS. 
Hiogarith-ms  of  Numbers. 

Logarithms  are  a  series  of  numbers  adapted  to  facilitate  the  operation  of 
numerical  computation, 

Addition  being  substituted  for  Multiplication,  Subtraction  for  Division, 
Multiplication  for  Involution,  and  Division  for  Evolution. 

The  Logarithm  of  a  number  is  the  exponent  of  a  power  to  which  10 
must  be  raised  to  give  that  number. 

It  is  not  necessary,  however,  that  the  base  should  be  10,  it  may  be  any  other  num- 
ber; but  Tables  of  Logarithms,  in  common  use,  are  computed  with  10  as  the  base. 
Thus,  Number      100  Log.  =  2,  as  io2  base  and  exponent  =     100. 
"        10000    "     =  4,  "  io4     "       "  "        =10000. 

The  Unit  or  Integral  part  of  a  Logarithm  is  termed  the  Index,  and  the  Decimal 
part  the  Mantissa;  the  sum  of  the  index  and  mantissa  is  the  Logarithm. 

The  Index  of  the  Logarithm  of  any  number,  Integral  or  Mixed,  when  the  base  is  io, 
is  equal  to  the  number  of  digits  to  the  left  of  the  decimal  point  less  i.  From  o  to 
9,  it  is  o;  from  io  to  99,  it  is  i,  and  from  100  to  999,  it  is  2,  etc. 

Thus,  logarithm  of  3304  =  3.51904,  3  being  the  index  and  .51904  the  mantissa. 

The  Index  of  the  Logarithm  of  a  Decimal  Fraction  is  a  negative  number,  and  is 
equal  to  the  number  of  places  which  the  first  significant  figure  of  the  decimal  is  re- 
moved from  the  place  of  units. 

Thus,  index  of  logarithm  .005  is  3  or  —3,  the  first  significant  figure,  5,  being  re- 
moved three  places  from  that  of  units.  The  bar  or  minus  sign  is  placed  over  an 
index  to  indicate  that  this  alone  is  negative,  while  the  decimal  part  is  positive. 

The  Difference  is  the  tabular  difference  between  the  two  nearest  logarithms. 

The  Proportional  Part  is  the  difference  between  the  given  and  the  nearest  less 
tabular  logarithm. 

The  Arithmetical  Complement  of  a  number  is  the  remainder  after  subtracting  it 
from  a  number  consisting  of  i,  with  as  many  ciphers  annexed  as  the  number  has 
integers.  When  the  index  of  a  logarithm  is  less  than  io,  its  complement  is  ascer- 
tained by  subtracting  it  from  io. 

II  Ki  s  t  r  at  i  on  s . 


Number.  Logarithm. 

4743 3-676053 

474-3 2.676053 

47-43 1-676053 

4-743 676  053 


Number.  Logarithm. 

•4743 £-676053 

•04743 2.676053 

.004743 3-676053 


Computation   of*  Negative   Indices. 

To  add  two  Negative  Indices.  Add  them  and  put  the  sum  negative.  As  5  -f-  3  =  8. 
To  add  a  Positive  and  Negative  Index.    Subtract  the  less  from  the  greater,  and 
to  remainder  give  the  positive  or  negative  sign,  according  as  the  positive  or  nega- 
tive index  is  the  greater.    As  6  -\-  2  =;  4,  and  6  +  2  =  4. 
ILLUSTRATION.  —Add  6. 387  57  and  2. 924  59.  6. 387  57 

2.92459 
5.31216 

Here  the  excess  of  i  from  13  in  the  first  decimal  place,  being  positive,  is  carried 
to  the  positive  6,  which  makes  7,  and  7  —  2=5. 

To  Subtract  a  Negative  Index.  Change  its  sign  to  plus  or  positive,  and  then  add 
it  as  in  addition.  _  As  3  from  2,  =  3  -f  2  =  5.  And  5  from  2,  =  5  -}-  2  =  3 ;  also 
3  from  5,  =3  + 5  =  2". 

ILLUSTRATION.  —Subtract  5. 765  52  from  2. 346  74.  2. 346  74 

5-76552 

2.581  22 

Here,  excess  of  i  in  the  first  decimal  place  used  with  the  .3  in  subtracting  the  .8 
from  the  1.3  is  to  be  subtracted  from  the  upper  number  2,  which  makes  it  3;  then 
3  +  5=2-  CG* 


306 


LOGARITHMS. 


To  Subtract  a  Positive  Index.  Change  its  sign  to  negative,  and  then  add  as  in 
addition.  As  2 —  2  =  2-1-2  =  4. 

To  Multiply  a  Negative  Index.  Multiply  the  fractional  parts  by  the  ordinary  rule, 
then  multiply  the  negative  index,  which  will  give  a  negative  product,  and  when  an 
excess  over  10  is  to  be  carried,  subtract  the  less  index  from  the  greater,  and  the  re- 
mainder gives  the  positive  or  negative  index,  according  as  the  positive  or  negative 
index  is  the  greater.  As  2  X  5  =  io,  and  i  to  be  carried  =  9. 

ILLUSTRATION. —Multiply  2.3681  by  2,  and  3.7856  by  6. 

2.3681  3.7856 

2  6 


4.7362  14-7136 

Here  2X2  =  4,  also  3"  X  6  =  18,  with  a  positive  excess  of  4  =  "14. 

To  Divide  a  Negative  Index.  If  index  is  divisible  by  divisor,  without  a  remain- 
der, put  quotient  with  a  negative  sign.  If  negative  exponent  is  not  divisible  by 
divisor,  add  such  a  negative  number  to  it  as  will  make  it  divisible,  and  prefix  an 
equal  positive  integer  to  fractional  part  of  logarithm;  then  divide  increased  nega- 
tive exponent  and  the  other  part  of  logarithm  separately  by  ordinary  rules,  and  for- 
mer quotient,  taken  negatively,  will  be  index  to  fractional  part  of  quotient.  As 
6  -=-  3  =  2.  io-r-3  requires  2  to  be  added  or  2  to  be  subtracted,  to  make  it  divisible 
without  a  remainder,  then  io-f  s^  =  12,72-^-3  =  4,  and  2  (the  sum  subtracted)  -r- 
3  =  .66,  the  quotient  therefore  is  4.66. 

ILLUSTRATION  i.—  Divide  6.324282  by  3. 

6. 324  282  -7-3  =  2. 108  094. 

a.— Divide  14.326745  by  9. 

I4-326745  -f-  9  =  18  -J-  4.326  745  -r-  9  =  2.480749+. 

Here  4  is  added  toTJ,  that  the  sum  ~i8  may  be  divided  by  9,  and  as  4  is  added,  4 
must  be  prefixed  to  the  fractional  part  of  the  logarithm,  and  thus  the  value  of  the 
logarithm  is  unchanged,  for  there  is  added  4,  and  4  =  o,  or  4  is  subtracted  and  4 
added. 

To  Ascertain  Logarithm  of  a  !N~umber  by  Table. 

When  the  Number  is  less  than  101. 

Look  into  first  page  of  table,  and  opposite  to  number  is  its  logarithm  with  its 
index  prefixed. 

ILLUSTRATION.— Opposite  7  is  .845098,  its  logarithm;  hence  70=1.845098,  .7  = 
7. 845  098,  and  .  07  =  2. 845  098. 

When  the  Number  is  between  100  and  1000. 

RULE. — Find  the  given  number  in  left-hand  column  of  table  headed  No.,  and  un- 
der o  in  next  column  is  decimal  part  of  its  logarithm,  to  which,  is  to  be  prefixed  a 
whole  number  for  an  index,  of  i  or  2,  according  as  the  number  consists  of  2  or  3 
figures. 
EXAMPLE.— What  is  logarithm  of  450,  and  what  of  .45  ? 

Log.  450  =  2.653213,  and  of  .45  =  1.653213. 
When  the  Number  is  between  1000  and  io  ooo. 

RULE.— Find  the  three  left-hand  figures  of  the  number  in  the  left-hand  column 
of  the  table  headed  No.,  and  under  the  4th  figure  at  top  of  table  is  the  four  last 
figures  of  the  decimal  part  of  logarithm,  to  which  is  to  be  prefixed  the  proper 
index. 
EXAMPLE.— What  is  logarithm  of  4505,  and  what  of  .04505? 

Log.  4505  =  3.653  695,  and  of  .045  05  =  2.653  695- 


LOGARITHMS.  307 

When  the  Number  consists  of  Five  Figures. 

RULE. — Find  the  logarithm  of  the  number  composed  of  the  first  four  figures  as 
preceding,  then  take  the  tabular  difference  from  the  right-hand  column  under  D 
and  multiply  it  by  the  fifth  figure;  reject  the  right-hand  figure  of  the  product  and 
add  the  other  figures,  which  are,  and  are  termed,  a  proportional  part  to  the  logarithm 
found  as  above,  observing  that  the  right-hand  figure  of  the  proportional  part  is  to 
be  added  to  that  of  the  logarithm,  and  the  rest  in  order. 
EXAMPLE.— Required  logarithm  of  83  407  ? 

NOTE. — When  the  number  consists  of  less  than  4  figures  conceive  a  cipher  an- 
nexed to  make  it  four. 

Log.  of  8340  (83  407)  =  4.921 166 

Tabular  difference  52,  which  x  7  (sth  figure)  =  364  =  364 

4.921 202  4  logarithm. 

The  difference  of  the  numbers  is  nearly  proportionate  to  the  difference  of  their 
logarithms. 

Thus,  difference  between  the  numbers  8340  and  8341,  the  next  in  order,  is  i,  and 
the  difference  between  their  logarithms  or  tabular  difference  is  52. 

The  log.  of  this  i  in  the  4th  place  is  therefore  52.  The  correction  then,  for  the  7 
of  the  sth  place,  which  is  .7  of  i  in  the  4th  place,  is  ascertained  by  the  proportion 
i  :  52  ::  .7  :  36.4. 

The  correction  is  obtained  by  multiplying  the  tabular  difference  by  7,  rejecting 
the  right  hand  figure  of  the  product,  if  the  log.  is  to  be  confined  to  six  decimal 
places. 

When  the  Number  consists  of  any  Number  over  Four  Figures. 
RULE.— Proceed  as  for  four  figures  for  the  first  four,  multiplying  the  tabular  dif- 
ference by  the  excess  of  figures  over  4  and  rejecting  one  right-hand  figure  of  the 
product  for  a  number  of  five  figures,  and  two  for  one  of  six,  and  so  on. 
EXAMPLE  L— Required  logarithm  of  834079? 

Log.  of  8340  (834079)=:  5.921 166 
Tabular  difference  52,  which  X  79=  4108 

5.92120708  logarithm. 
2.— Required  logarithm  of  8340794? 

Log.  of  8340  (8  340 794)  =  6.921 166 

Tab.  diff.  52,  which  x  794  (5th,  6th,  and  jth  figures)  =  41288 

6. 921 207  288  logarithm. 

Or,  Mantissa  of  8340          =  .921166 

u     "          7     (sth  figure)  X  52  tab.  dif.  =         364 
"     "  9    (6th     «    )X52    u     "  =  468 

"     »  4(7th      "    )X52    "     "   =  208 

Log.  with  index  for  7  figures 6.921 207  288 

To   Ascertain.   Logarithm    of  a   Mixecl    :N"xnn"ber. 
RULB.— Take  out  logarithm  of  the  number  as  if  it  were  an  integer  or  whole  num- 
ber, to  which  prefix  the  index  of  the  integral  part  of  the  number. 
EXAMPLE.— What  is  logarithm  of  834.0794? 
Mantissa  of  log.  of  834.0794  =  9  212  073 ;  hence  log.  of  834.0794  =  2.921 207  3. 

To   Ascertain   Logarithm   of*  a  Decimal    Fraction.. 

RULE.— Take  logarithm  from  table  as  if  the  figures  were  all  integers,  and  prefix 
index  as  by  previous  rules. 
EXAMPLE. — Logarithm  of .  1234  =  1.091 305. 

To   Ascertain    Logarithm   of*  a   "Vxilgar    Fraction. 
RULE. — Reduce  the  fraction  to  a  decimal,  and  proceed  as  by  preceding  rule.     Or, 
subtract  logarithm  of  denominator  from  that  of  numerator,  and  the  difference  will 
give  logarithm  required. 
EXAMPLE. — Logarithm  of  jj^? 

^  =  .1875.    Log.  .1875      ="1.273001  logarithm. 
Or,  Log.  3  =  .477121 
1    16  =  1.20412 

7.273001  logarithm. 


308 


LOGARITHMS. 


To  .A-scertain    tne   Nu.m"ber   Corresponding    to   a  Q-iven 
Logarithm. 

When  the  given  or  exact  Logarithm  is  in  the  Table. 

OPERATION. — Opposite  to  first  two  figures  of  logarithm,  neglecting  the  index,  in 
column  o,  look  for  the  remaining  figures  of  the  log.  in  that  column  or  in  any  of  the 
nine  at  the  right  thereof;  the  first  three  figures  of  the  number  will  be  found  at  the 
left  in  column  under  No.,  and  the  fourth  at  top  directly  over  the  log. 

The  number  is  to  be  made  to  correspond  to  index  of  logarithm,  by  pointing  off 
decimals  or  prefixing  ciphers. 

ILLUSTRATION. — What  is  number  corresponding  to  log.  3-963977  ? 

Opposite  to  963977,  in  page  329,  is  920,  and  at  top  of  column  is  4;  hence,  num- 
ber =  9204. 

When  the  given  or  exact  Logarithm  is  not  in  the  Table. 

OPERATION.— Take  the  number  for  the  next  less  logarithm  from  table,  which  will 
give  first  four  figures  of  required  number. 

To  ascertain  the  other  figures,  subtract  the  logarithm  in  table  from  the  given 
logarithm,  add  ciphers,  and  divide  by  the  difference  in  column  D  opposite  the 
logarithm.  Annex  quotient  to  the  four  figures  already  ascertained,  and  place  deci- 
mal point. 

ILLUSTBATION  i.— What  is  number  corresponding  to  log.  5.921 207? 
Given  log.  =  5.921 207 

Next  less  in  table  5.921166  8340 

0=52)4100(78-!- 78 

364  834078 

460 
_4l6 
Hence,  number  =  834  078.  44 

a. — What  is  number  corresponding  to  log.  3.922853? 
Given  log.  =                       3.922853 
Next  less  in  table              3.922829              8372 
D  =  52)  2400  (46  +  46 

£*_  837^6 

320 

3'2  iO 

Hence,  number  =  8372.46.  8 

3VInltiplioation. 

RULE.— Add  together  the  logarithms  of  the  numbers  and  the  sum  will  give  the 
logarithm  of  the  product. 
EXAMPLE  i.— Multiply  345.7  by  2.581. 
Log.  345.7     =2.538699 
2.581=  .411788 

2. 950  487  log.  of  product.    Number  =  892. 251. 
*. — Multiply  .03902,  59.71,  and  .003147. 

Log.  .03902  =2.591287 
••  59.71  =£.776047 
"  .003147  =  3.497897 

3. 865  231  log.  of  product.    Number  =  .007  332 15. 

Division. 

RULE.— From  logarithm  of  dividend  subtract  that  of  divisor,  and  remainder  will 
give  logarithm  of  the  quotient. 
EXAMPLE.— Divide  371.4  by  52.37. 
Log.  371.4  =2.569842 
52.37  =  1.719083 

•  850  7  59  log.  of  guotient.    Number  =  7. 091 85. 


LOGARITHMS.  309 

Rule   of  Three,  or   Proportion. 

RULE.— Add  together  the  logarithms  of  the  second  and  third  terms,  from  their 
sum  subtract  logarithm  of  the  first,  and  the  remainder  will  give  logarithm  of  the 
fourth  term. 

Or,  instead  of  subtracting  logarithm  of  first  term,  add  its  Arithmetical  Comple- 
ment, and  subtract  10  from  its  index. 

EXAMPLE  i. — What  is  fourth  proportional  to  723.4,  .025 19,  and  3574? 
As       723.4         log.  =  _  2.859379 

Is  to         .02519   *'   =2.401228 
So  is  3574  "    =3-553iS5 

i-954383 
First  term  2-859379 

7. 095  004  log.  of  4th  term.    Number  = .  124  453. 

By  Arithmetical  Complement. 

ILLUSTRATION. — As  723.4  log.  =  2.859  379,  Ar.  com.  =  7. 140621 
Is  to  .025 19  u  =  2.401 228 

So  is  3574  "  =  3-553I55 

i  095  004  log.  of  tfh  term. 
Number  =  .124  453. 

2.— If  an  engine  of  67  IP  can  raise  57  600  cube  feet  of  water  in  a  given  time,  what 
HP  is  required  to  raise  8  575  ooo  cube  feet  in  like  time  ? 
Log.  8  575  ooo  =  6.933  234 
67  =  1.826075 
8.759309 
57600  =  4.760422 

3. 998  877  log.  of  4th  term.    Number  =  9974. 4  IP. 

3.  —If  14  men  in  47  days  excavate  5631  cube  yards,  what  time  will  it  require  to 
excavate  47  280  at  same  rate  of  excavation  ?  394. 626  days. 

Involution. 

RULE.— Multiply  logarithm  of  given  number  by  exponent  of  the  power  to  which 
it  is  to  be  raised,  and  the  product  will  give  the  logarithm  of  the  required  power. 
EXAMPLE.— What  is  cube  of  30.71  ? 
Log.  30.71  =  1.48728 

3 

4. 461 84  log.  of  power.    Number = 28  962. 73. 

Evolution. 

RULE.— Divide  logarithm  of  given  number  by  exponent  of  the  root  which  is  to  be 
estracted,  and  quotient  will  give  logarithm  of  required  root. 
EXAMPLE  i.— What  is  cube  root  of  1234? 

Log.  1234  =  3.091315 

Divide  by  3  =  1.030  438  log.  of  root.    Number  =  10. 72601. 
2.— What  is  4th  root  of  .007  654? 

Log.  .007654  =  3.883888 
Divide  by  4  (here  3  -}- 1  +  i)  =  1.470  972  log.  of  root.    Number  =  .295  78. 

To   Ascertain.    Reciprocal   of  a   Number. 

RULE. — Subtract  decimal  of  logarithm  of  the  number  from  .000000;  add  i  to  in- 
dex of  logarithm  and  change  its  sign.    The  result  is  logarithm  of  the  reciprocal 
EXAMPLE. —Required  reciprocal  of  230? 

.000000 
Log.  230  =  2.361728 

3.638  272  =  log.  of  .004  348  reciprocal. 


3io 


LOGARITHMS. 


Simple    Interest. 

RULE. — Add  together  logarithm  of  principal,  rate  per  cent.,  and  time  in  years,  from 
the  sum  subtract  2,  and  the  remainder  will  give  logarithm  of  the  interest. 
EXAMPLE.— What  is  interest  on  $500,  @  6  per  cent.,  for  3  years? 
Log.  500  =  2. 698  97 
6=   .778151 
3=   .477121 
3-954  242 


i.  954  242  log.  of  interest    Number  =  90  dollar 9. 

Compound.    Interest. 

RULE. — Compute  amount  of  $  i  or  £  i,  etc.,  at  the  given  rate  of  interest  for  one 
year  for  the  first  term,  which  is  termed  the  ratio. 

Multiply  logarithm  of  ratio  by  the  time,  add  to  product  logarithm  of  the  principal, 
and  sum  is  logarithm  of  the  amount. 

.Logarithms   of*  Ratios    at  given    Rates   Per    Cent. 


Rate. 

Log.  of  Ratio. 

Rate,     i  Log.  of  Ratio. 

Rate. 

Log.  of  Ratio. 

Rate. 

Log.  of  Ratio, 

I 

.0043214 

3-25 

.013  890  i 

5-5 

.0232525 

7-75 

.0324373 

1.25 

•005  395 

3-5 

0149403 

5-75 

.024  2804 

8 

•0334238 

i-5 

.006466 

3-75 

.0159881 

6 

•0253059 

8.25 

•0344279 

i-75 

.0075344 

4 

•0170333 

6.25 

.0263289 

8-5 

•0354297 

2 

.0086002 

4-25 

.018076  i 

6-5 

.0273496 

8.75 

.0364293 

2.25 

.0096633 

4-5 

.0191163 

6.75 

.0287639 

9 

.0374265 

2-5 

.0107239 

4-75 

.020154 

7 

.0293838 

9.25 

.0384214 

2.75 

.0117818 

5 

.021  1893 

7-25 

•0303973 

9-5 

.0394141 

3 

.0128372 

5-25 

.022  222  1 

7-5 

.031  408  5 

9-75 

.040  404  3 

EXAMPLE.— What  will  $364,  at  6  per  cent,  per  annum,  compounded  yearly,  amount 
to  in  23  years? 
Log.  of  ratio  from  above  table   .025  305  9 

*3 


364 


.5820357 
2.561101 

3.1431367  log.  of  amount.  Number  =  1390. 39  doll 


Miscellaneous    Illustrations. 

i.  What  is  area  and  circumference  of  a  circle  of  21.72  feet  in  diameter? 
Log.  of  21.72      1.336860 


Log.  of  21. 722  =2.673720 
"  «  .7854=1.895091 
"  "  2. 568  81 1  =  370. 54  feet  area. 

Log.  of  21. 72     =1.33686 
'    3.1416=  -497^5 

"     "  j.834  ox  =  68. 236  feet  circum. 

a.  Sides  of  a  triangle  are  564,  373,  and  747  feet;  what  is  its  area? 
Log.  of  sides    564  +  373  +  747  =  2  925  3I2 

"    "  .5  side—  61  =  842  —  564  =  2.444045 

"  .5  side  — 6  =  842— 373  =  2.671 173 

"  .5  side  —  c  =842  —  747  =  1.977724 

2^10.018  254 

Area  =  Number  of      5.009 127  =  102120. 4  feet 

3.— What  is  logarithm  of  83'6? 

Log.     X  3   =  —  X  log.  8  =  3.6  X.  903  09  =  3. 251 124.    Number  =1782.89= 


LOGARITHMS   OF   NUMBBBS. 


Logarithms   of  iN'uxxi'bers. 

From  i  to  10000. 


No. 

Logarithm. 

No.     1     Logarithm. 

No! 

Logarithm. 

No. 

Logarithm. 

1 

.O 

26 

1.414973 

51 

1.70757 

76 

1.880814 

2 

•301  °3 

27 

1.431  364 

52 

1.716003 

77 

1.886491 

3 

•477  "i 

28 

1.447  158 

53 

1.724276 

78 

1.892095 

4 

.60206 

29 

1.462398 

54 

I-73«394 

79 

1.897627 

5 

.69897 

30 

1.477  J2i 

55 

1.740363 

80 

1.90309 

6 

.778  151 

31 

1.491  362 

56 

1.748  188 

81 

1.908485 

7 

.845098 

32 

L505  15 

57 

1-755  875 

82 

1.913814 

8 

.90309 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

•954  243 

34 

I-53I  479 

59 

1.770852 

84 

1.924279 

10 

i 

35 

1.544068 

60 

1.778  151 

85 

1.929419 

11 

1.041  393 

36 

1.556303 

61 

1.78533 

86 

1.934498 

12 

1.079  J8i 

37 

1.568202 

62 

1.792392 

87 

I.9395I9 

13 

I.II3943 

38 

I-579784 

63 

I-79934I 

88 

1.944483 

14 

1.146  128 

39 

1.591065 

64 

1.80618 

89 

1-94939 

15 

1.176091 

40 

1.60206 

65 

1.812913 

90 

1.954243 

16 

1.204  12 

41 

i.  612  784 

66 

1.819544 

91 

1.959041 

i? 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832  509 

93 

1.968483 

19 

1.278  754 

44 

1.643453 

69 

1.838849 

94 

1.973  128 

20 

1.301  03 

45 

1.653213 

70 

1.845098 

95 

1.977  724 

21 

1.322219 

46 

1.662  758 

71 

1.851  258 

96 

1.982  271 

22 

1.342423 

47 

1.672098 

72 

1.857332 

1.986772 

23 

1.361  728 

48 

1.681  241 

73 

1.863323 

98 

1.991  226 

24 

1.380211 

49 

1.690  196 

74 

1.869232 

99 

1.995635 

25 

1-39794 

50 

1.69897 

75 

1.875061 

100 

2 

No. 

\o 

.1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

100 

00-  OOOO 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

346i 

3891 

43« 

161 

oo-  43*21 

4751 

5l8l 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

102 

oo-  86 

9026 

9451 

9876 

— 

— 

425 

102 

01-   — 

— 



—  i- 

03 

0724 

1147 

157 

1993 

2415 

424 

103 

01-  2837 

3259 

368 

41 

4521 

494 

536 

5779 

6i97 

6616 

420 

IO4 

oi-  7033 

7451 

7868 

8284 

87 

9116 

9532 

9947 

— 

— 

417 

IO4 

02-  — 

— 



— 

— 

— 

0361 

0775 

416 

105 

02-  1189 

1603 

2Ol6 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

106 

02-  5306 

5715 

6l25 

6533 

6942 

735 

7757 

8164 

8571 

8978 

408 

107 

02-  9384 

9789 



— 

405 

107 

03-   — 

0195 

06 

1004 

1408 

1812 

2216 

2619 

3021 

404 

108 

03-  3424 

3826 

4227 

4628 

5029 

543 

583 

623 

6629 

7028 

400 

109 

03-  7426 

7825 

8223 

862 

9017 

9414 

9811 

— 

— 

398 

109 

04-   — 

— 



— 

— 

0207 

0602 

0998 

397 

110 

04-  1393 

1787 

2l82 

2576 

2969 

3362 

3755 

4148 

454 

4932 

393 

in 

04-  5323 

5714 

6l05 

6495 

6885 

7275 

7664 

8053 

8442 

883 

389 

112 

04-  9218 

9606 

9993 

— 

388 

112 

05-   — 

— 

038 

0766 

"53 

1538 

1924 

2309 

2694 

386 

"3 

05-  3078 

3463 

3846 

423 

4613 

4996 

5378 

576 

6142 

6524 

383 

114 

05-  6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

383 

114 

06-   

— 

— 

— 

— 

— 

— 

032 

379 

No. 

O 

1  t 

a 

3 

4 

5 

6 

7 

8 

9 

D 

312 


LOGARITHMS   OF  NUMBERS. 


No. 


0123456789 

o6-  0698  1075  1452  1829  2206  ;  2582  2958  3333  3709  4083 

06-  4458  4832  5206  558  5953  6326  6699  7071  7443  7815 

06-  8186  8557  8928  9298  9668  |  —   —   —   —   — 

07-  —   —   —   —   —   0038  0407  0776  1145  1514 

07-  1882  225  2617  2985  3352 

3718  4085  4451  4816  5182 

07-  5547  5912  6276  664  7004 

7368  7731  8094  8457  8819 

07-  9181  9543  9904  —   — 

—   —   —   —   — 

08-  —   —   —  0266  0626 

0987  1347  1707  2067  2426 

08-  2785  3144  3503  3861  4219 

4576  4934  5291  5647  6004 

08-  636  6716  7071  7426  7781 

8136  849  8845  9198  9552 

08-  9905  —   —   —   — 

—   —   —   —   — 

09-  —  0258  0611  0963  1315 

1667  2018  237  2721  3071 

09-  3422  3772  4122  4471  482 

5169  5518  5866  6215  6562 

09-  691  7257  7604  7951  8298 

8644  899  9335  9681  — 

10-  —   —   —   —   — 

—   —   —   —  0026 

10-  0371  0715  1059  1403  1747 

2091  2434  2777  3119  3462 

10-  3804  4146  4487  4828  5169 

551  5851  6191  6531  6871 

io-  721   7549  7888  8227  8565 

8903  9241  9579  9916  — 

jj_  —   —   —   —   — 

—   —   —   —  0253 

ii-  059  0926  1263  1599  1934 

227  2605  294  3275  3609 

"-  3943  4277  4611  4944  5278 

5611  5943  6276  6608  694 

ii-  7271  7603  7934  8265  8595 

8926  9256  9586  9915  — 

12-   

—   —   —   .—  0245 

12-  0574  0903  1231  156   1888 

2216  2544  2871  3198  3525 

12-  3852  4178  4504  483   5156 

5481  5806  6131  6456  6781 

12-  7105  7429  7753  8076  8399 

8722  9045  9368  969   — 

13-  —   —   —   —   — 

—    —    —   0012 

J3-  0334  o^SS  0977  I298  1619 

1939  226  258  29   3219 

*3-  3539  3858  4177  449°  4814 

5i33  545i  5769  °°86  6403 

13-  6721  7037  7354  7671  7987 

8303  8618  8934  9249  9564 

13-  9879  —   —   —   — 

—   —   —   —   — 

14-  —  0194  0508  0822  1136 

145  1763  2076  2389  2702 

J4-  3015  3327  3639  3951  4263 

4574  4885  5196  5507  5818 

14-  6128  6438  6748  7058  7367 

7676  7985  8294  8603  8911 

14-  9219  9527  9835  —   — 

—   —   —   —   — 

15-  —   —   —  0142  0449 

0756  1063  137  1676  1982 

15-  2288  2594  29   3205  351 

3815  412  4424  4728  5032 

15-  5336  564  5943  6246  6549 

6852  7154  7457  7759  8061 

15-  8362  8664  8965  9266  9567 

9868  —   —   —   — 

16-  —   —   —   —   — 

j  —  0168  0469  0769  1068 

16-  1368  1667  1967  2266  2564 

2863  3161  346  3758  4055 

l6~  4353  465  4947  5244  554  1 

5838  6134  643  6726  7022 

16-  7317  7613  7908  8203  8497 

8792  9086  938  9674  9968 

17-  0262  0555  0848  1141  1434 

1726  2019  2311  2603  2895 

17-  3186  3478  3769  406  4351 

4641  4932  5222  5512  5802 

17-  6091  6381  667  6959  7248 

7536  7825  8113  8401  8689 

17-  8977  9264  9552  9839  — 

—   —   —   —   — 

18-  —   —   —   —  0126 

0413  0699  0986  1272  1558 

18-  1844  2129  2415  27   2985 

327  3555  3839  4123  4407 

18-  4691  4975  5259  5542  5825 

6108  6391  6674  6956  7239 

18-  7521  7803  8084  8366  8647 

8928  9209  949  9771  — 

TO-  ______ 

—   —   —   —  0051 

o    r    a    3    4 

56789 

tOGAEITHHS    OF  NUMBEBS. 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

155 

19-  0332 

0612 

0892 

1171 

1451 

173 

201 

2289 

2567 

2846 

279 

156 

19-  3125 

3403 

3681 

3959 

4237 

45H 

4792 

5069 

5346 

5623 

278 

19-  59 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

158 

19-  8657 

8932 

9206 

9481 

9755 

— 

— 

— 

275 

158 

20-   

— 

0029 

0303 

°577 

085 

1124 

274 

159 

20-  1397 

167 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

160  20  412 

439i 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

161  20  6826 

7096 

7365 

7634 

7904 

8i73 

8441 

871 

8979 

9247 

269 

l62   20  9515 

9783 



268 

l62   21  -   — 

0051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

163   21-  2l88 

2454 

272 

2986 

3252 

3783 

4049 

43H 

4579 

266 

164   21   4844 

5109 

5373 

5638 

5902 

6166 

643 

6694 

6957 

7221 

264 

165  21-  7484 

7747 

801 

8273 

8536 

8798 

906 

9323 

9585 

9846 

262 

166  22-  0108 

037 

0631 

0892 

"53 

1414 

1^75 

1936 

2196 

2456 

261 

167   22-  2716 

2976 

3236 

340 

3755 

4015 

4274 

4533 

4792 

259 

168  22  5309 

5568 

5826 

6084 

6342 

66 

6858 

7"5 

7372 

763' 

258 

169   22-  7887 

8144 

84 

8657 

8913 

917 

9426 

9682 

9938 

257 

169   23-   — 

— 

0193 

256 

170  23  0449 

0704 

096 

1215 

147 

1724 

1979 

2234 

2488 

2742 

255 

171  23-  2996 

325 

3504 

3757 

4011 

4264 

4517 

477 

5023 

5276 

253 

172  23-  5528 

578i 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

173  1  23-  8046 

8297 

8548 

8799 

9049 

9299 

955 

98 

251 

173  24-  — 

— 

— 

— 

— 

— 

— 

— 

005 

03 

250 

174  24-  0549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

279 

249 

175 

24-  3038 

3286 

3534 

3782 

403 

4277 

4525 

4772 

5019 

5266 

248 

176 

24-  5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

24-  7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

246 

177 

25-  ~ 

— 

— 

— 

— 

— 

— 

— 

0176 

245 

178 

25-  042 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

261 

243 

179 

25-  2853 

3096 

3338 

358 

3822 

4064 

4306 

4548 

479 

5031 

242 

180 

25-  5273 

55H 

5755 

590 

6237 

6477 

6718 

6958 

7198 

7439 

241 

181 

25-  7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

26-  0071 

031 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

183 

26-  2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

184 

26-  4818 

5054 

529 

5525 

5996 

6232 

6467 

6702 

6937 

235 

185 

26-  7172 

7406 

7641 

7875 

811 

8344 

8578 

8812 

9046 

9279 

234 

186 

26-  9513 

9746 

998 

— 

— 

— 

— 

234 

186 

27-  — 

0213 

0446 

0679 

0912 

"44 

1377 

1609 

233 

187 

27-  1842 

2074 

2306 

2538 

277 

3001 

3233 

3464 

3696 

3927 

232 

188 

27-  4158 

4389 

462 

485 

5081 

53" 

5542 

5772 

6002 

6232 

230 

189 

27-  6462 

6692 

6921 

7151 

738 

7609 

7838 

8067 

8296 

8525 

229 

190 

27-  8754 

8982 

9211 

9439 

9667 

9895 

— 

— 

— 

— 

228 

190 

28-  — 

— 

— 

0123 

0351 

0578 

0806 

228 

191 

28-  1033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192  i  28-  3301 

3527 

3753 

3979 

4205 

443  1 

4656 

4882 

5107 

5332 

226 

193  !  28-  5557 

5782 

6007 

6232 

6456 

6681 

6905 

7*3 

7354 

7578 

225 

194  28-  7802 

8026 

8249 

8473 

8696 

892 

9M3 

9366 

9589 

9812 

223 

/95  29-  0035 

0257 

048 

0702 

0925 

"47 

1369 

1591 

1813 

2034 

222 

196  29-  2256 

2478 

2699 

292 

3363 

3584 

3804 

4025 

4246 

221 

197  29-  4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198  29-  6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

2I9 

199  29-  8853 

9071 

9289 

9507 

9725 

9943 

— 

— 

— 

218 

I99|30-  - 

0161 

0378 

0595 

0813 

218 

No.  1     o 

• 

9. 

3 

4 

f 

6 

7 

8 

9 

~D" 

314 


LOGARITHMS    OF   NUMBERS. 


01234 

56789 

30-  103  I247  J4^4  1681  1898 

2114  2331  2547  2764  298 

30-  3196  3412  3628  3844  4059 

4275  449i  4706  4921  5136 

30-  535i  5566  5781  590  6211 

6425  6639  6854  7068  7282 

30-  7496  771   7924  8137  8351 

8564  8778  8991  9204  9417 

30-  963  9843  —   —   — 

—   —   —   —   — 

31-  —   —  0056  0268  0481 

0693  0906  1118  133  1542 

31-  1754  1966  2177  2389  26 

2812  3023  3234  3445  3656 

31-  3867  4078  4289  4499  471 

492  513  534  555i  576 

31-  597  618  639  6599  6809 

7018  7227  7436  7646  7854 

31-  8063  8272  8481  8689  8898 

9106  9314  9522  973  9938 

32-  0146  0354  0562  0769  0977 

1184  1391  1598  1805  2012 

32-  2219  2426  2633  2839  3°46 

3252  3458  3665  3871  4077 

32-  4282  4488  4694  4899  5105 

531  5516  5721  5926  6131 

32-  6336  6541  6745  695  7155 

7359  7563  7767  7972  8176 

32-  838  8583  8787  8991  9194 

9398  9601  9805  —   — 

33-  —   —   —   —   — 

—    —    —   0008  O2II 

33-  0414  0617  0819  1022  1225 

1427  163   1832  2034  2236 

33-  2438  264  2842  3044  3246 

3447  3649  385  4051  4253 

33-  4454  4655  4856  5057  5257 

5458  5658  5859  6o59  626 

33-  646  666  686  706  726 

7459  7659  7858  8058  8257 

33-  8456  8656  8855  9054  9253 

9451  965  9849  —   — 

34-  —   —   —   —   — 

—   —   —  0047  0246 

34-  0444  0642  0841  1039  1237 

*435  l632  183  2028  2225 

34-  2423  262  2817  3014  3212 

3409  3606  3802  3999  4196 

34-  4392  4589  4785  4981  5178 

5374  557  57°6  59°2  6157 

34-  6353  6549  6744  6939  7^5 

733  7525  772  7915  811 

34-  8305  85   8694  8889  9083 

9278  9472  9666  986   — 

35-  —   —   —   —   — 

—   —   —   —  0054 

35-  0248  0442  0636  0829  1023 

1216  141   1603  1796  1989 

35-  2183  2375  2568  2761  2954 

3*47  3339  3532  3724  39l6 

35-  4108  4301  4493  4685  4876 

5068  526  5452  5643  5834 

35-  6026  6217  6408  6599  679 

6981  7172  7363  7554  7744 

35-  7935  8125  8316  8506  8696 

8886  9076  9266  9456  9646 

35-  9835  —   —   —   — 

—   —   —   —   — 

36-  —  0025  0215  0404  0593 

0783  0972  1161  135  1539 

36-  1728  1917  2105  2294  2482 

2671  2859  3048  3236  3424 

36-  3612  38   3988  4176  4363 

455i  4739  4926  5113  530i 

36-  5488  5675  5862  6049  6236 

6423  66  1  6796  6983  7169 

3°-  7356  7542  7729  79*5  8101 

8287  8473  8659  8845  903 

36-  9216  9401  9587  9772  9958 

—   —   —   —   — 

37-  —   —   —   —   — 

0143  0328  0513  0698  0883 

37-  1068  1253  1437  1622  1806 

1991  2175  236  2544  2728 

37-  2912  3096  328  3464  3647 

3831  4015  4198  4382  4565 

37-  4748  4932  5115  5298  5481 

5664  5846  6029  6212  6394 

37-  6577  6759  6942  7124  7306 

7488  767  7852  8034  8216  ; 

37-  8398  858  8761  8943  9124 

9306  9487  9668  9849  — 

38  _____ 

_____  oo3 

38-  021  1  0392  0573  0754  0934 

1115  1296  1476  1656  1837 

38-  2017  2197  2377  2557  2737   2917  3097  3277  3456  3636 

38-  3815  3995  4*74  4353  4533  4712  489!  5°7  5249  5428 

38-  5606  5785  5964  6142  6321  6499  6677  6856  7034  7212 

38-  739  7568  7746  7923  8101  ;  8279  8456  8634  8811  8989 

01234    56789 

LOGARITHMS    OF   NUMBERS. 


315 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

245 

38-  9166 

9343 

952 

9698 

9875 

— 

— 

— 

— 

— 

177 

245 

39-  — 

— 



— 

— 

0051 

0228 

0405 

0582 

0759 

177 

246 

39-  0935 

III2 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

39-  2697 

2873 

3048 

3224 

34   3575 

3751 

3926 

4101 

4277 

176 

248 

39-  4452 

4627 

4802 

4977 

5152  5326 

5501 

5676 

585 

6025 

175 

249 

39-  6199 

6374 

6548 

6722 

6896  7071 

7245 

7419 

7592 

7766 

174 

250 

39-  794 

8lI4 

8287 

8461 

8634 

8808 

8981 

9*54 

9328 

9501 

173 

25I 

39-  9674 

9847 



— 

— 

— 

— 

*73 

251 

40-  — 

002 

0192 

0365 

0538 

0711 

0883 

1056 

1228 

173 

252 

40-  1401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

40-  3121 

3292 

3464 

3635 

3807 

3978 

4149 

432 

4492 

4663 

171 

254 

40-  4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

637 

171 

255 

40-  654 

67I 

688l 

7051 

7221 

7391 

7501 

773i 

7901 

807 

170 

256 

40-  824 

841 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

257 

40-  9933 



— 

— 

— 

— 

— 

169 

257 

41-  — 

0102 

O27I 

044 

0609 

0777 

0946 

1114 

1283 

1451 

169 

258 

41-  162 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3*32 

168 

259 

4i-  33 

3467 

3635 

3803 

397 

4137 

4305 

4472 

4639 

4806 

167 

260 

4i-  4973 

514 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

41-  6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

797 

8i3S 

166 

262 

41-  8301 

8467 

8633 

8798 

8964 

9129 

9295 

946 

9625 

9791 

165 

263 

41-  9956 

165 

263 

42-  — 

OI2I 

0286 

0451 

0616 

0781 

0945 

in 

1275 

1439 

165 

264 

42-  1604 

1768 

1933 

2097 

2261 

2426 

259 

2754 

2918 

3082 

164 

265 

42-  3246 

341 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

266 

42-  4882 

5045 

5208 

537i 

5534 

5697 

586 

6023 

6186 

6349 

163 

267 

42-  6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

268 

42-  8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

42-  9752 

9914 

— 

— 

— 

— 

— 

— 

— 

162 

269 

43-  — 

0075 

0236 

0398  0559 

072 

0881 

1042 

1203 

161 

270 

43-  1364 

1525 

1685 

1846 

2007  2167 

2328 

2488 

2649 

2809 

161 

271 

43-  2969 

313 

329 

345 

3DI  !  377 

393 

409 

4249 

4409 

160 

272 

43-  4569 

4729 

4888 

5048 

5207  5367 

5526 

5685 

5844 

6004 

159 

273 

43-  6163 

6322 

6481 

664 

6799  6957 

7116 

7275 

7433 

759« 

159 

274 

43-  775i 

7909 

8067 

8226 

8384  i  8542 

8701 

8859 

9017 

9175 

158 

275 

43-  9333 

9491 

9648 

9806 

9964 

— 

— 

— 

— 

— 

iss 

275 

44-  — 

— 

— 

— 

— 

OI22 

0279 

0437 

0594 

0752 

158 

276 

44-  0909 

1066 

1224 

1381 

1538  1695 

1852 

20«9 

2166 

2323 

157 

277 

44-  248 

2637 

2793 

295 

3106  3263 

3419 

3576 

3732 

3889 

157 

278 

44-  4045 

42OI 

4357 

4513 

4669  4825 

4981 

5137 

5293 

5449 

156 

279 

44-  5604 

576 

5915 

6071 

6226  6382 

6537 

6692 

6848 

7003 

J55 

280  44-  7158 

73  r  3 

7468 

7623 

7778  7933 

8088 

8242 

8397 

8552 

155 

281  44-  8706 

8861 

9°I5 

917 

9324  9478 

9633 

9787 

9941 

154 

281  45-  — 

— 

— 

— 

— 



— 

— 

— 

0095 

154 

282  45-  0249 

0403 

0557 

0711 

0865 

IOT8 

1172 

1326 

J479 

1633 

154 

283  i  45-  1786 

194 

2093 

2247 

24   2553 

2706 

2859 

3012 

3165 

153 

284  45-  33i8 

347i 

3624 

3777 

393   4082 

4235 

4387 

454 

4692 

153 

285  45-  4845 

4997 

5i5 

5302 

5454  Sooo 

5758 

591 

6062 

6214 

152 

286  45-  6366 

6518 

667 

6821 

6973  7125 

7276 

7428 

7579 

7731 

152 

287  45-  7882 

8033 

8184 

8336 

8487  8638 

8789 

894 

9091 

9242 

151 

288;  45-9392 

9543 

9694 

9845 

9995   — 

— 

151 

288  46-  — 

—   0146 

0296 

0447 

°597 

0748 

I51 

289  ;  46-  0898 

1048 

1198 

1348 

J499 

1649 

1799 

1948 

2098 

2248 

150 

No.  i      o 

• 

2 

3 

4 

5 

6 

7 

8 

9 

D 

316 


LOGARITHMS    OF    NUMBERS. 


No. 


01234 

56789 

D 

46-  2398  2548  2697  2847  2997 
46-  3893  4042  4I9I  434  449 
46-  5383  5532  568  5829  5977 
46-  6868  7016  7164  7312  746 
46-  8347  8495  8643  879  8938 

3146  3296  3445  3594  3744 
4639  4788  4936  5085  5234 
6126  6274  6423  6571  6719 
7608  7756  7904  8052  82 
9085  9233  938  9527  9675 

150 
149 
149 
148 
148 

46-  9822  9969  —   — 
47-  —   —  0116  0263  041 
47-  1292  1438  1585  1732  1878 
47-  2756  2903  3049  3195  3341 
47-  4216  4362  4508  4653  4799 
47-  5671  5816  5962  6107  6252 

0557  07°4  0851  0998  1145 
2025  2171  2318  2464  261 

3487  3633  3779  3925  4071 
4944  509  5235  5381  5526 
6397  6542  6687  6832  6976 

147 
147 
146 
146 
1,46 

4s 

47-  7121  7266  7411  7555  77 
47-  8566  8711  8855  8999  9143 
48-  0007  0151  0294  0438  0582 
48-  1443  1586  1729  1872  2016 
48-  2874  3016  3159  3302  3445 

7844  7989  8133  8278  8422 
9287  9431  9575  9719  9863 

0725  0869  IOI2  1156  1299 
2159  2302  2445  2588  2731 

3587  373  3872  4015  4i57 

145 
144 
144 
143 
143 

48-  43   4442  4585  4727  4869 
48-  5721  5863  6005  6147  6289 
48-  7138  728  7421  7563  7704 
48-  8551  8692  8833  8974  9114 
48-  9958  —   - 
49-  —  0099  0239  038  052 

Son  5153  5295  5437  5579 
643  6572  6714  6855  6997 
7845  7986  8127  8269  841 
9255  9396  9537  9677  9818 

O66l  080I  0941  I08l  1222 

142 
142 
141 
141 
140 
140 

49-  1362  1502  1642  1782  1922 
49-  276  29   304  3179  3319 
49-  4155  4294  4433  4572  47ii 
49-  5544  5683  5822  596  6099 
49-  693  7068  7206  7344  7483 

2O62  22OI  2341  2481  2621 

3458  3597  3737  3876  4015 
485  4989  5128  5267  5406 
6238  6376  6515  6653  6791 
7621  7759  7897  8035  8173 

140 
139 
139 
139 
138 

49-  8311  8448  8586  8724  8862 
49-  9687  9824  902 

50-  1059  1196  1333  147  1607 
50-  2427  2564  27   2837  2973 
50-  3791  3927  4063  4199  4335 

8999  9137  9275  9412  955 

0374  °5IT  0648  0785  0922 
1744  188  2017  2154  2291 
3109  3246  3382  35x8  3655 
4471  4607  4743  4878  5014 

138 
137 
137 
137 
136 

50-  515  5286  5421  5557  5693 
50-  6505  664  6776  6911  7046 
50-  7856  7991  8126  826  8395 
50-  9203  9337  9471  9606  974 

51-  0545  0679  0813  0947  1081 

5828  5964  6099  6234  637 
7181  7316  7451  7586  7721 
853  8664  8799  8934  9068 

9874  ~   ~ 
—  0009  0143  0277  0411 
1215  1349  1482  1616  175 

136 
135 
135 
134 
134 
134 

51-  1883  2017  2151  2284  2418 
51-  3218  3351  3484  3617  375 
51-  4548  4681  4813  4946  5079 
51-  5874  6006  6139  6271  6403 
51-  7196  7328  746  7592  7724 

2551  2684  2818  2951  3084 
3883  4016  4149  4282  4415 
i  5211  5344  5476  5609  5741 
6535  6668  68   6932  7064 
7855  7987  8119  8251  8382 

133 
133 
133 
132 
132 

51-  8514  8646  8777  8909  904 
51-  9828  9959  —   —   — 
S2-  —   —  009  0221  0353 
52-  1138  1269  14   153  1661 
52-  2444  2575  2705  2835  2966 
52-  3746  3876  4006  4136  4266 

9171  9303  9434  9566  9697 

0484  0615  0745  0876  1007 
1792  1922  2053  2183  2314 
3096  3226  3356  3486  3616 
4396  4526  4656  4785  4915 

130 
130 

01234 

56789 

LOGABITHMS    OP  NUMBERS. 


317 


No. 

1     ° 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D 

335 

52-  5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

621 

129 

336 

52-  6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

337 

"    yv^-' 
52-  763 

7759 

7888 

8016 

8i45 

8274 

8402 

8531 

866 

8788 

129 

338 

52-  8917 

9°45 

9J74 

9302 

943 

9559 

9687 

98i5 

9943 

— 

128 

338 

53-  — 

— 

— 

— 

— 

— 

— 

— 

— 

0072 

128 

339 

53-  02 

0328 

0456 

0584 

0712 

084 

0968 

1096 

1223 

1351 

128 

340 

53-  1479 

1607 

1734 

1862 

199 

2117 

2245 

2372 

25 

2627 

128 

34i 

53-  2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

342 

53-  4026 

4153 

428 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

53-  5294 

542i 

5547 

5674 

58 

5927 

6053 

618 

6306 

6432 

126 

344 

53-  6558 

6685 

6811 

6937 

7063 

7189 

7315 

744i 

7567 

7693 

126 

545 

53-  7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

53-  9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

— 

— 

126 

346 

54-  — 

— 

— 

— 

— 

— 

— 

— 

0079 

0204 

125 

347 

54-  0329 

0455 

058 

0705 

083 

0955 

108 

1205 

133 

1454 

125 

348 

54-  1579 

1704 

1829 

J953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 

54-  2825 

295 

3074 

3199 

3323 

3447 

357i 

3696 

382 

3944 

124 

350 

54-  4068 

4192 

43i6 

444 

4564 

4688 

4812 

4936 

506 

5183 

124 

3Si 

54-  5307 

543i 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

352 

54-  6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

54-  7775 

7898 

8021 

8i44 

8267 

8389 

8512 

8635 

8758 

8881 

123 

354 

54-  9003 

9126 

9249 

937i 

9494 

9616 

9739 

9861 

9984 

— 

123 

354 

55-  — 

— 

— 

— 

— 

— 

0106 

123 

355 

55-  0228 

O35i 

0473 

0595 

0717 

084 

0962 

1084 

1206 

1328 

122 

356 

55-  145 

J572 

1694 

1816 

1938 

206 

2181 

2303 

2425 

2547 

122 

357 

55-  2668 

279 

2911 

3033 

3155 

3276 

3398 

3519 

364 

3762 

121 

358 

55-  3883 

4004 

4126 

4247 

4368 

4489 

461 

473i 

4852 

4973 

121 

359 

55-  5094 

5215 

5336 

5457 

5578 

5699 

582 

594 

6061 

6182 

121 

360 

55-  6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

36i 

55-  7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

55-  8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

55-  99°7 

— 

— 

— 

— 

— 

— 

120 

363 

56-  - 

0326 

0146 

0265 

0385 

0504 

0624 

0743 

0863 

0982 

II9 

364 

56-  uoi 

1221 

134 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

"9 

365 

56-  2293 

2412 

2531 

265 

2769 

2887 

3006 

3125 

3244 

3362 

II9 

366 

56-  3481 

36 

37i8 

3837 

3955 

4074 

4192 

43" 

4429 

4548 

II9 

367 

56-  4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

573 

118 

368 

56-  5848 

5966 

6084 

6202 

632 

6437 

6555 

6673 

6791 

6909 

118 

369 

56-  7026 

7J44 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

56-  8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

914 

9257 

117 

37i 

56-  9374 

9491 

9608 

9725 

9842 

9959 

— 

— 

— 

— 

117 

371 

57-  — 

— 

0076 

0193 

0309 

0426 

117 

372 

57-  0543 

066 

0776 

0893 

101 

1126 

1243 

1359 

1476 

1592 

117 

373 

57-  1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

57-  2872 

2988 

3*04 

322 

3336 

3452 

3568 

3684 

38 

3915 

116 

375 

57-  4031 

4J47 

4263 

4379 

4494 

461 

4726 

4841 

4957 

5072 

116 

376 

57-  5i88 

5303 

5419 

5534 

565 

5765 

588 

590 

6m 

6226 

I15 

377 

57-  6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

"5 

378 

57-  7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

841 

8525 

"5 

379 

57-  8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9  , 

D 

D 

0* 

LOGARITHMS    OF   NUMBEBS. 


No. 

01234 

56789 

380 

57-  9784  9898  —   — 

_   _   —   —   — 

380 

58-   —    0012  0126  0241 

0355  0469  0583  0697  0811 

38i 

58-  0925  1039  1153  1267  1381 

1495  1608  1722  1836  195 

382 

58-  2063  2177  2291  2404  2518  j  2631  2745  2858  2972  3085 

383 

58-  3199  3312  3426  3539  3652 

3765  3879  3992  4*05  4218 

384 

58-  4331  4444  4557  467  4783 

4896  5009  5122  5235  5348 

385 

58-  5461  5574  5686  5799  5912 

6024  6137  625  6362  6475 

386 

58-  6587  67   6812  6925  7037 

7149  7262  7374  7486  7599 

387 

58-  7711  7823  7935  8047  816 

8272  8384  8496  8608  872 

388 

58-  8832  8944  9056  9167  9279 

9391  9503  9°I5  9726  9838 

389 

58-995   —   —   —   — 

—   —   —   —   — 

389 

59-  —  0061  0173  0284  0396 

0507  0619  073  0842  0953 

390 

59-  1065  1176  1287  1399  151 

1621  1732  1843  T955  2066 

39i 

59-  2177  2288  2399  251  2621 

2732  2843  2954  3064  3175 

392 

59-  3286  3397  3508  3618  3729 

384  395  4061  4171  4282 

393 

59-  4393  4503  4614  4724  4834 

4945  5055  5165  5276  5386 

394 

59-  5496  5606  5717  5827  5937 

6047  6l57  6267  6377  6487 

395 

59-  6597  6707  6817  6927  7037 

7146  7256  7366  7476  7586 

396 

59-  7695  7805  7914  8024  8134 

8243  8353  8462  8572  8681 

397 

59-  8791  89   9009  9119  9228 

9337  9446  9556  9605  9774 

398 

59-  9883  9992  —   —   — 

—   —   —   —   — 

398 

60-   —   OIOI  O2I   0319 

0428  0537  0646  0755  0864 

399 

00-  0973  1082  II9I  1299  1408 

1517  1625  1734  1843  1951 

400 

60-  2O6   2169  2277  2386  2494 

2603  2711  2819  2928  3036 

401 

60-  3144  3253  3361  3469  3577 

3686  3794  3902  401  4118 

402 

60-  4226  4334  4442  455  4658 

4766  4874  4982  5089  5197 

403 

60-  5305  5413  5521  5628  5736 

5844  5951  6059  Ol66  6274 

404 

60-  6381  6489  6596  6704  68  n 

6919  7026  7133  7241  7348 

405 

60-  7455  7562  7669  7777  7884 

7991  8098  8205  8312  8419 

406 

60-  8526  8633  874  8847  8954 

9061  9167  9274  9381  9488 

407 

60-  9594  9701  9808  9914  — 

—   —   —   —   — 

407 

61-  —   —   —   —  0021 

0128  0234  0341  0447  0554 

408 

61-  066  0767  0873  0979  1086 

1192  1298  1405  1511  1617 

409 

61-  1723  1829  1936  2042  2148 

2254  236  2466  2572  2678 

410 

61-  2784  289  2996  3102  3207 

33  J3  3419  3525  363  3736 

411 

61-  3842  3947  4053  4159  4264 

437  4475  458  1  4686  4792 

412 

61-  4897  5003  5108  5213  5319 

5424  5529  5634  574  5845 

413 

61-  595  6055  616  6265  637 

6476  6581  6686  679  6895 

414 

61-  7    7105  721   7315  742 

7525  7629  7734  7839  7943 

415 

61-  8048  8153  8257  8362  8466 

8571  8676  878  8884  8989 

416 

61-  9093  9198  9302  9406  9511 

9615  9719  9824  9928  — 

416 

62-  —   —   —   —   — 

—   —   —   —  0032 

4i7 

62-  0136  024  0344  0448  0552 

0656  076  0864  0968  1072 

418 

62-  1176  128  1384  1488  1592 

1695  1799  1903  2007  211 

419 

62-  2214  2318  2421  2525  2628 

2732  2835  2939  3042  3146 

420 

62-  3249  3353  3456  3559  3663 

3766  3869  3973  4076  4179 

421 

62-  4282  4385  4488  4591  4695 

4798  4901  5004  5107  521 

422 

62-  5312  5415  5518  5621  5724 

5827  5929  6032  6135  6238 

423 

62-  634  6443  6546  6648  6751 

6853  6956  7058  7161  7263 

424 

62-  7366  7468  7571  7673  7775 

7878  798  8082  8185  8287 

No. 

01234 

56789 

LOGARITHMS    OF   NUMBERS. 


319 


No. 


o     i     2    3    4 

56789 

62-  8389  8491  8593  8695  8797 
62-  941   9512  9613  9715  9817 
63-  —   —    —   —   — 
63-  0428  053   0631  0733  0835 
63-  1444  1545  1647  1748  1849 
63-  2457  2559  266   2761  2862 

89   9002  9104  9206  9308 
99i9  —   —   —   — 

0021  0123  0224  0326 
0936  1038  1139  1241  1342 
1951  2052  2153  2255  2356 
2963  3064  3165  3266  3367 

63-  3468  3569  367   3771  3872 
63-  4477  4578  4679  4779  488 
63-  5484  5584  5685  5785  5886 
63-  6488  6588  6688  6789  6889 
63-  749  759  769  779  7§9 

3973  4074  4*75  4276  43?6 
4981  5081  5182  5283  5383 
5986  6087  6187  6287  6388 
6989  7089  7189  729  739 
799  809  819  829  8389 

63-  8489  8589  8689  8789  8888 
63-  9486  9586  9686  9785  9885 
64-  - 
64-  0481  0581  068  0779  0879 
64-  1474  1573  1672  1771  1871^ 
64-  2465  2563  2662  2761  280 

8988  9088  9188  9287  9387 
9984  —   —   —   — 
—  0084  0183  0283  0382 
0978  1077  1177  1276  1375 
197  2069  2168  2267  2366 
2959  3058  3156  3255  3354 

64-  3453  355i  365  3749  3847 
64-  4439  4537  4636  4734  4832 
64-  5422  5521  5619  5717  5815 
64-  6404  6502  66  6698  6796 
64-  7383  748i  7579  7676  7774 

3946  4044  4143  4242  434 
4931  5029  5127  5226  5324 
5913  6011  611  6208  6306 
6894  6992  7089  7187  7285 
7872  7969  8067  8165  8262 

64-  836  8458  8555  8653  875 

64-  9335  9432  953  9627  9724 
65-  ----- 
65-  0308  0405  0502  0599  0696 
65-  1278  1375  1472  1569  1666 
65-  2246  2343  244  2536  2633 

8848  8945  9043  914  9237 
9821  9919  —   —   — 

—    —   00l6  0113  021 

0793  089  0987  1084  1181 
1762  1859  X956  2053  215 
273  2826  2923  3019  3116 

65-  3213  3309  3405  3502  3598 
65-  4i77  4273  4369  4465  4562 
65-  5138  5235  533i  5427  5523 
65-  6098  6194  629  6386  6482 
65-  7056  7*52  7247  7343  7438 

3695  379i  3888  3984  408 
4658  4754  485  4946  5042 
5619  5715  581  5906  6002 
6577  6673  6769  6864  696 
7534  7629  7725  782  7916 

65-  Son  8107  8202  8298  8393 
65-  8965  906  9155  925  9346 
65-  9916  —   —   —   — 
66-  —  ooi  i  0106  0201  0296 
66-  0865  096  1055  115   1245 
66-  1813  1907  2002  2096  2191 

8488  8584  8679  8774  887 
9441  9536  9631  9726  9821 

0391  0486  0581  0676  0771 
1339  1434  1529  1623  1718 
2286  238  2475  2569  2663 

66-  2758  2852  2947  3041  3135 
66-  370i  3795  3889  3983  4078 
66-  4642  4736  483  4924  5018 
66-  5581  5675  5769  5862  5956 
66-  6518  6612  6705  6799  6892 

323  3324  3418  3512  3607 
4172  4266  436  4454  4548 
5112  5206  5299  5393  5487 
605  6143  6237  6331  6424 
6986  7079  7173  7266  736 

66-  7453  7546  764  7733  7826 
66-  8386  8479  8572  8665  8759 
66-  9317  941  9503  9596  9689 
67-  ----- 
67-  0246  0339  0431  0524  0617 
67-  1173  1265  1358  1451  1543 

792  8013  8106  8199  8293 
8852  8945  9038  9131  9224 
9782  9875  9967  —   — 
—   —   —  006  0153 
071  0802  0895  0988  108 
1636  1728  1821  1913  2005 

01    a    3    4 

56789 

320 


LOGARITHMS    OF   NUMBERS. 


No. 

01234 

56789 

470 

67-  2098  219  2283  2375  2467 

256   2652  2744  2836  2929  i 

47  1 

67-  3021  3113  3205  3297  339 

3482  3574  3666  3758  385 

472 

67-  3942  4034  4126  4218  431 

4402  4494  4586  4677  4769 

473 

67-  4861  4953  5045  5137  5228 

532  5412  5503  5595  5687 

474 

67-  5778  587  5962  6053  6145 

6236  6328  6419  6511  6602 

475 

67-  6694  6785  6876  6968  7059 

7151  7242  7333  7424  7516 

476 

67-  7607  7698  7789  7881  7972 

8063  8154  8245  8336  8427 

477 

67-  8518  8609  87   8791  8882 

8973  9064  9155  9246  9337 

478 

67-  9428  9519  961  97   9791 

9882  9973  —   —   -- 

478 

og_  

—  0063  0154  0245 

479 

68-  0336  0426  0517  0607  0698 

0789  0879  097  106  1151 

480 

68-  1241  1332  1422  1513  1603 

1693  1784  1874  1964  2o55 

481 

68-  2145  2235  2326  2416  2506 

2596  2686  2777  2867  2957 

482 

68-  3047  3137  3227  3317  3407 

3497  3587  3677  3767  3857 

483 

68-  3947  4037  4127  4217  4307 

439°  4486  4576  4666  4756 

484 

68-  4845  4935  5025  5114  5204 

5294  5383  5473  5563  5652 

485 

68-  5742  5831  5921  601  61 

6189  6279  6368  6458  6547 

486 

68-  6636  6726  6815  6904  6994 

7083  7172  7261  7351  744 

487 

68-  7529  7618  7707  7796  7886 

7975  8064  8153  8242  8331 

488 

68-  842  8509  8598  8687  8776 

8865  8953  9042  9131  922 

489 

68-  9309  9398  9486  9575  9664 

9753  9841  993   —   — 

489 

69  —   -   -   - 

—   —   —  0019  0107 

490 

69-  0196  0285  0373  0462  055 

0639  0728  0816  0905  0993 

491 

69-  1081  117  1258  1347  1435 

1524  1612  17   1789  1877 

492 

69-  1965  2053  2142  223  2318 

2406  2494  2583  2671  2759 

493 

69-  2847  2935  3023  3111  3199 

3287  3375  3463  355i  3639 

494 

69-  3727  3815  3903  399i  4078 

4166  4254  4342  443  4517 

495 

69-  4605  4693  4781  4868  4956 

5044  5131  5219  5307  5394 

496 

69-  5482  5569  5657  5744  5832 

5919  6007  6094  6182  6269 

497 

69-  6356  6444  6531  6618  6706 

6793  688  6968  7055  7142 

498 

69-  7229  7317  7404  7491  7578 

7665  7752  7839  7926  8014 

499 

69-  8101  8188  8275  8362  8449 

8535  8622  8709  8796  8883 

500 

69-  897  9057  9144  9231  9317 

9404  9491  9578  9664  9751 

501 

69-  9838  9924  —   —   — 

501 

70-  —   —  ooi  i  0098  0184 

0271  0358  0444  0531  0617 

502 

70-  0704  079  0877  0963  105 

1136  1222  1309  1395  1482 

503 

70-  1568  1654  1741  1827  1913 

1999  2086  2172  2258  2344 

504 

70-  2431  2517  2603  2689  2775 

286l  2947  3033  3119  3205 

505 

70-  3291  3377  3463  3549  3635 

3721  3807  3893  3979  4065 

506 

70-  4151  4236  4322  4408  4494 

4579  4665  4751  4837  4922 

507 

70-  5008  5094  5179  5265  535 

5436  5522  5607  5693  5778 

508 

70-  5864  5949  6035  612  6206 

6291  6376  6462  6547  6632 

509 

70-  6718  6803  6888  6974  7059 

7144  7229  7315  74   7485 

510 

70-  757  7655  774  7826  7911 

7996  8081  8166  8251  8336 

5ii 

70-  8421  8506  8591  8676  8761 

8846  8931  9015  91   9185 

512 

70-  927  9355  944  9524  9609 

9694  9779  9863  9948  — 

512 

yi-   —    

—   —   —   —  0033 

513 

71-  0117  0202  0287  0371  0456 

054  0625  071  0794  0879 

5H 

71-  0963  1048  1132  1217  1301 

J385  147  J554  l639  J723 

No. 

0      1      234 

56789 

LOGARITHMS   OF   NUMBERS. 


321 


No. 

01234 

56789 

515 

71-  1807  1892  1976  206  2144 

2229  2313  2397  2481  2566 

5i6 

71-  265  2734  2818  2902  2986 

307  3154  3238  3323  3407 

517 

71-  3491  3575  3659  3742  3826 

391  3994  4078  4162  4246 

5i8 

71-  433  4414  4497  4581  4665 

4749  4833  49i6  5    5084 

519 

71-  5167  5251  5335  5418  5502 

5586  5669  5753  5836  593 

520 

71-  6003  6087  617  6254  6337 

6421  6504  6588  6671  6754 

521 

71-  6838  6921  7004  7088  7171 

7254  7338  742i  7504  7587 

522 

71-  7671  7754  7837  792  8003 

8086  8169  8253  8336  8419 

523 

71-  8502  8585  8668  8751  8834 

8917  9    9083  9165  9248 

524 

7i-  933*  94H  9497  958  9663 

9745  9828  9911  9994  — 

524 

72-  —   —   —   —   — 

—    __    —    __   0077 

525 

72-  0159  0242  0325  0407  049 

0573  o655  0738  0821  0903 

526 

72-  0986  1068  1151  1233  1316 

1398  1481  1563  1646  1728 

527 

72-  1811  1893  1975  2058  214 

2222  2305  2387  2469  2552 

528 

72-  2634  2716  2798  2881  2963 

3045  3127  3209  3291  3374 

529 

72-  3456  3538  362  3702  3784 

3866  3948  403  4112  4194 

530 

72-  4276  4358  444  4522  4604 

4685  4767  4849  4931  5013 

53* 

72-  5095  5176  5258  534  5422 

5503  5585  5667  5748  583 

532 

72-  5912  5993  6075  6156  6238 

632  6401  6483  6564  6646 

533 

72-  6727  6809  689  6972  7053 

7134  7216  7297  7379  746 

534 

72-  7541  7623  7704  7785  7866 

7948  8029  811  8191  8273 

535 

72-  8354  8435  8516  8597  8678 

8759  8841  8922  9003  9084 

536 

72-  9165  9246  9327  9408  9489 

957  9651  9732  9813  9893 

537 

72-  9974  —   —   —   — 

—   —   —   —   — 

537 

73-  —  0055  0136  0217  0298 

0378  0459  054  0621  0702 

538 

73-  0782  0863  0944  1024  1105 

1186  1266  1347  1428  1508 

539 

73-  1589  1669  175  183  1911 

1991  2072  2152  2233  2313 

540 

73-  2394  2474  2555  2635  2715 

2796  2876  2956  3037  3117 

54i 

73-  3197  3278  3358  3438  3518 

3598  3679  3759  3839  3919 

542 

73-  3999  4079  4i6  424  432 

44   448  456  464  472 

543 

73-  48   488  496  504  512 

52   5279  5359  5439  55*9 

544 

73-  5599  5679  5759  5838  59*8 

5998  6078  6157  6237  6317 

545 

73-  6397  6476  6556  6635  6715 

6795  6874  6954  7034  7113 

546 

73-  7193  7272  7352  743i  75ii 

759  767  7749  7829  7908 

547 

73-  7987  8067  8146  8225  8305 

8384  8463  8543  8622  8701 

548 

73-  8781  886  8939  9018  9097 

9177  9256  9335  9414  9493 

549 

73-  9572  05i  973i  981  9889 

9968  - 

549 

74-  —   —   —   —   — 

—  0047  0126  0205  0284 

550 

74-  0363  0442  0521  06   0678 

0757  0836  0915  0994  1073 

551 

74-  1152  123  1309  1388  1467 

1546  1624  1703  1782  186 

552  74-  1939  2018  2096  2175  2254 

2332  2411  2489  2568  2647 

553  74-  2725  2804  2882  2961  3039 

3118  3196  3275  3353  3431 

554 

74-  35i  3588  3667  3745  3823. 

3902  398  4058  4136  4215 

555  74-  4293  4371  4449  4528  4606 

4684  4762  484  4919  4997 

556  74-  5075  5153  5231  5309  5387 

5465  5543  5621  5699  5777 

557  74-  5855  5933  6on  6089  6167  6245  6323  6401  6479  6556 

558  74-  6634  6712  679  6868  6945  7023  7101  7179  7256  7334 

559 

74-  7412  7489  7567  7645  7722  78   7878  7955  8033  811 

No. 

01234     56789 

322 


LOGARITHMS    OF    NUMBERS. 


No. 

560 

56i 
562 
562 
563 
564 

01234 

56789 

D 

77 
77 

77 
77 
77 
77 

74-  8188  8266  8343  8421  8498 
74-  8963  904  9118  9195  9272 
74-  9736  9814  9891  9968  — 
75-  —   —   —   —  0045 
75-  0508  0586  0663  074  0817 
75-  1279  1356  1433  151  1587 

8576  8653  8731  8808  8885 
935  9427  9504  9582  9659 

0123  02   0277  0354  0431 

0894  0971  1048  1125  1202 

1664  1741  1818  1895  1972 

565 

566 
567 
568 
569 

75-  2048  2125  2202  2279  2356 
75-  2816  2893  297  3047  3123 
75-  3583  366  3736  3813  3889 
75-  4348  4425  45oi  4578  4^54 
75-  5112  5189  5265  5341  5417 

2433  2509  2586  2663  274 

32   3277  3353  343  35°6 
3966  4042  4119  4195  4272 
473  4807  4883  496  5036 
5494  557  5646  5722  5799 

77 
77 
77 
76 
76 

570 

571 
572 
573 
574 

75-  5875  5951  6o27  6l°3  6l8 
75-  6636  6712  6788  6864  694 
75-  7396  7472  7548  7624  77 
75-  8155  823  8306  8382  8458 
75-  8912  8988  9063  9139  9214 

6256  6332  6408  6484  656 
7016  7092  7168  7244  732 
7775  7851  7927  8003  8079 
8533  8609  8685  8761  8836 
929  9366  9441  9517  9592 

76 
76 
76 
76 
76 

575 

575 
576 
577 
578 
579 

75-  9668  9743  9819  9894  997 
76-  —   —   —   —   — 
76-  0422  0498  0573  0649  0724 
76-  1176  1251  1326  1402  1477 
76-  1928  2003  2078  2153  2228 
76-  2679  2754  2829  2904  2978 

0045  0121  0196  0272  0347 

0799  0875  095  1025  noi 

1552  1627  1702  1778  1853 
2303  2378  2453  2529  2604 

3053  3128  3203  3278  3353 

76 
75 
75 
75 
75 
75 

580 

58i 
582 
583 
584 

76-  3428  3503  3578  3653  3727 
76-  4176  4251  4326  44  4475 
76-  4923  4998  5072  5147  5221 
76-  5669  5743  5818  5892  5966 
76-  6413  6487  6562  6636  671 

3802  3877  3952  4027  4101 
455  4624  4699  4774  4848 
5296  537  5445  552  5594 
6041  6115  619  6264  6338 
6785  6859  6933  7°°7  7°82 

75 
75 
75 
74 
74 

585 
586 
587 
588 
588 
589 

76-  7156  723  7304  7379  7453 
76-  7898  7972  8046  812  8194 
76-  8638  8712  8786  886  8934 

76-  9377  945i  9525  9599  9^73 
77_  _____  _ 

77-  0115  0189  0263  0336  041 

7527  7601  7675  7749  7823 
8268  8342  8416  849  8564 
9008  9082  9156  923  9303 
9746  982  9894  9968  — 
—   —   —   —  0042 
0484  0557  0631  0705  0778 

74 
74 
74 
74 
74 
74 

590 

59i 
592 
593 
594 

77-  0852  0926  0999  1073  1146 
77-  1587  1661  1734  1808  1881 
77-  2322  2395  2468  2542  2615 
77-  3055  3J28  3201  3274  3348 
77-  3786  386  3933  4006  4079 

122   1293  1367  144   1514 
1955  2028  2102  2175  2248 
2688  2762  2835  2908  2981 

3421  3494  3567  364  3713 
4152  4225  4298  4371  4444 

74 
73 
73 
73 
73 

505 

596 
597 
598 
599 

77-  4517  459  4663  4736  4809 
77-  5246  5319  5392  5465  5538 
77-  5974  6047  6l2  6193  6265 
77-  6701  6774  6846  6919  6992 
77-  7427  7499  7572  7644  7717 

4882  4955  5028  51   5173  73 
561  5683  5756  5829  5902  ;  73 
6338  6411  6483  6556  6629  1  73 
7064  7137  7209  7282  7354  73 
7789  7862  7934  8006  8079  1  72 

600 

601 
602 
602 
603 
604 

77-  8151  8224  8296  8368  8441 
77-  8874  8947  9019  9091  9163 
77-  959°  9^  9741  98*3  9885 
78-  —   —   —   —   — 
78-  0317  0389  0461  0533  0605 
78-  1037  1109  1181  1253  1324 

8513  8585  8658  873  8802 
9236  9308  938  9452  9524 
9957  —   —   —   — 
—  0029  oioi  0173  0245 
0677  0749  0821  0893  0965 
1396  1468  154  1612  1684 

72 
I  72 
72 
72 
72 
72 

No. 

01234 

56789 

D 

LOGABITHMS   OF   NUMBERS. 


323 


01234 

56789 

78-  1755  1827  1899  1971  2042 

2114  2186  2258  2329  2401 

78-  2473  2544  2616  2688  2759 

2831  2902  2974  3046  3117 

78-  3189  326  3332  3403  3475 

3546  3618  3689  3761  3832 

78-  3904  3975  4046  4118  4189 

4261  4332  4403  4475  4546 

78-  4617  4689  476  4831  4902 

4974  5°45  5"6  5187  5259 

?8-  533  540i  5472  5543  5615 

5686  5757  5828  5899  597 

78-  6041  6112  6183  6254  6325 

6396  6467  6538  6609  668 

78-  6751  6822  6893  6964  7035 

7106  7177  7243  7319  739 

78-  746  7531  7602  7673  7744 

7815  7885  7956  8027  8098 

78-  8168  8239  831  8381  8451 

8522  8593  8663  8734  8804 

78-  8875  8946  9016  9087  9157 

9228  9299  9369  944  951 

78-  9581  9651  9722  9792  9863 

9933  —   —   —   — 

79-  —   —   —   —   — 

—  0004  0074  0144  0215 

79-  0285  0356  0426  0496  0567 

0637  0707  0778  0848  0918 

79-  0988  1059  1129  1199  1269 

134  141  148  155  162 

79-  1691  1761  1831  1901  1971 

2O4I  21  1  1  2l8l  2252  2322 

79-  2392  2462  2532  2682  2672 

2742  28l2  2882  2952  3022 

79-  3092  3162  3231  3301  3371 

3441  35H  3581  3651  3721 

79-  379  386  393  4    407 

4139  4209  4279  4349  4418 

79-  4488  4558  4627  4697  4767 

4836  4906  4976  5045  5115 

79-  5185  5254  5324  5393  5463 

5532  5602  5672  5741  5811 

79-  588  5949  6019  6088  6158 

6227  6297  6366  6436  6505 

79-  6574  6644  6713  6782  6852 

6921  699  706  7129  7198 

79-  7268  7337  7406  7475  7545 

7614  7683  7752  7821  789 

79-  796  8029  8098  8167  8236 

8305  8374  8443  8513  8582 

79-  8651  872  8789  8858  8927 

8996  9065  9134  9203  9272 

79-  9341  9409  9478  9547  9616 

9685  9754  9823  9892  90i 

80-  0029  0098  0167  0236  0305 

0373  0442  0511  058  0648 

80-  0717  0786  0854  0923  0992 

1061  1129  1198  1266  1335 

80-  1404  1472  1541  1609  1678 

1747  l8l5  1884  1952  2021 

80-  2089  2158  2226  2295  2363 

2432  25  2568  2637  2705 

80-  2774  2842  291  2979  3047 

3116  3184  3252  3321  3389 

80-  3457  3525  3594  3662  373 

3798  3867  3935  4003  4071 

80-  4139  4208  4276  4344  4412 

448  4548  4616  4685  4753 

80-  4821  4889  4957  5025  5093 

5161  5229  5297  5365  5433 

80-  5501  5569  5637  5705  5773 

5841  5908  5976  6044  6112 

80-  618  6248  6316  6384  6451 

6519  6587  6655  6723  679 

80-  6858  6926  6994  7061  7129 

7197  7264  7332  74   7467 

80-  7535  7603  767  7738  7806 

7873  7941  8008  8076  8143 

80-  8211  8279  8346  8414  8481 

8549  8616  8684  8751  8818 

80-  8886  8953  9021  9088  9156 

9223  929  9358  9425  9492 

80-  956  9627  9694  9762  9829 

9896  9964  —   —   — 

81-  —   _   —   _   — 

—   —  0031  0098  0165 

81-  0233  03   0367  0434  0501 

0569  0636  0703  077  0837 

81-  0904  0971  1039  1106  1173 

124  1307  1374  1441  1508 

81-  1575  1642  1709  1776  1843 

191   1977  2O44  2III  2178 

81-  2245  2312  2379  2445  2512 

2579  2646  2713  278   2847 

81-  2913  298  3047  3114  3181 

3247  3314  3381  3448  3514 

81-  358i  3648  3714  3781  3848 

3914  3981  4048  4114  4l8l 

81-  4248  4314  4381  4447  4514 

458l  4647  4714  478   4847 

81-  4913  498  5046  5113  5179 

5246  5312  5378  5445  5511 

81-  5578  5644  57"  5777  5843 

591  5976  6042  6109  6175 

0-234 

56789 

324 


LOGARITHMS    OF   NUMBERS. 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

655 

81-  6241 

6308 

6374 

644 

6506 

6573 

6639 

6705 

6771 

6838 

66 

656 

81-  6904 

697 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

81-  7565 

7631 

7698 

7764 

783 

7896 

7962 

8028 

8094 

816 

66 

658 

81-  8226 

8292 

8358 

8424 

849 

8556 

8622 

8688 

8754 

882 

66 

659 

81-  8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

81-  9544 

961 

9676 

9741 

9807 

9873 

9939 

— 

— 

— 

66 

660 

82-  — 

— 

— 

— 

— 

— 

0004 

007 

0136 

66 

661 

82-  O20I 

0267 

0333 

0399 

0464 

053 

0595 

0661 

0727 

0792 

66 

662 

82-  0858 

0924 

0989 

1055 

112 

1186 

1251 

*3!7 

1382 

1448 

66 

663 

82-  1514 

J579 

1645 

171 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

82-  2l68 

2233 

2299 

2364 

243 

2495 

256 

2626 

2691 

2756 

65 

665 

82-  2822 

2887 

2952 

3018 

3083 

3H8 

3213 

3279 

3344 

3409 

65 

666 

82-  3474 

3539 

3605 

367 

3735 

38 

3865 

393 

3996 

4061 

65 

667 

82-  4126 

4191 

4256 

4321 

4386 

445i 

45*6 

458i 

4646 

4711 

65 

668 

82-  4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

536i 

65 

669 

82-  5426 

549  r 

5556 

5621 

5686 

575i 

5815 

588 

5945 

601 

65 

670 

82-  6075 

614 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65 

671 

82-  6723 

6787 

6852 

6917 

6981 

7046 

7111 

7J75 

724 

7305 

65 

672 

82-  7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

795i 

65 

673 

82-  8015 

808 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

82-  866 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9i75 

9239 

64 

675 

82-  9304 

9368 

9432 

9497 

956i 

9625 

969 

9754 

9818 

9882 

64 

676  82-  9947 

— 

— 

— 

— 

— 

— 

— 

— 

64 

676  83- 

oon 

0075 

0139 

0204 

0268 

0332 

0396 

046 

0525 

64 

677  83-  0589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

IIO2 

1166 

64 

678 

83-  123 

1294 

1358 

1422 

1486 

155 

1614 

1678 

1742 

1806 

64 

679 

83-  187 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

83-  2509 

2573 

2637 

27 

2764 

2828 

2892 

2956 

302 

3083 

64 

68  1 

83-  3^47 

3211 

3275 

3338 

3402 

3466 

353 

3593 

3657 

3721 

64 

682 

83-  3784 

3848 

3912 

3975 

4039 

4103 

4166 

423 

4294 

4357 

64 

683 

83-  442i 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

83-  5056 

5^2 

5^83 

5247 

53i 

5373 

5437 

55 

5564 

5627 

63 

685 

83-  5691 

5754 

5817 

5881 

5944 

6007 

6071 

6i34 

6197 

6261 

63 

686 

83-  6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

683 

6894 

63 

687 

83-  6957 

702 

7083 

7146 

721 

7273 

7336 

7399 

7462 

7525 

63 

688  83-  7588 

7652 

7715 

7778 

7841 

7904 

7967 

803 

8093 

8156 

63 

689  83-  8219 

8282 

8345 

8408 

8471 

8534 

8597 

866 

8723 

8786 

63 

690  83-  8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

63 

691  83-  9478 

954i 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

63 

691  84-  — 

— 

— 

0043 

63 

692  84  0106 

0169 

0232 

0294 

0357 

042 

0482 

0545 

0608 

0671 

63 

693  ;  84-  0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

694  ;  84-  1359 

1422 

1485 

1547 

161 

1672 

1735 

1797 

186 

1922 

63 

695  84-  1985 

2047 

211 

2172 

2235 

2297 

236 

2422 

2484 

2547 

62 

696  84-  2609 

2672 

^734 

2796 

2859 

2921 

2983 

3046 

3108 

3i7 

62 

697  !  84-  3233 

3295 

3^57 

342 

3482  3544 

3606 

3669 

373i 

3793 

62 

698  84-  3855 

39i8 

39? 

4042 

4104  4166 

4229 

4291 

4353 

4415 

!  62 

699 

84-  4477 

4539 

4601 

4664 

4726  4788 

485 

4912 

4974 

5036 

,  62 

700 

84-  5098 

5i6 

5222 

5284 

5346  i  54o8 

547 

5532 

5594 

5656 

!  62 

701 

84-  57i8 

578 

5842 

5904 

5966  !  6028 

609 

6151 

6213 

6275 

62 

702 

84-  6337 

6399 

6461 

6523 

6585  6646 

6708 

677 

6832 

6894 

62 

703 

84-  6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

75" 

I  62 

704 

84-  7573 

7634 

7696 

7758 

7819  |  7881 

7943 

8004 

8066 

8127 

62 

No. 

• 

1 

2 

3 

4     5 

6 

7 

8 

9 

1  D 

LOGARITHMS   OP   NTTMBEKS. 


325 


No. 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

D 

705 

84-  8189 

8251 

8312 

8374 

8435 

8497 

8559 

862 

£682 

8743 

~62 

706 

84-  8805 

8866 

8928 

8989 

9051 

9112 

9J74 

9235 

9297 

9358 

61 

707 

84-  9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

8fi-  0033 

0095 

0156 

0217 

0279 

034 

0401 

0462 

0524 

0585 

61 

709 

85-  0646 

0707 

0769 

083 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

85-  1258 

132 

T38l 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

61 

711 

85-  187 

I931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

85-  248 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

7T3 

85-  309 

315 

32II 

3272 

3333 

3394 

3455 

3516 

3577 

3637  61 

85-3698 

3759 

382 

3881 

3941  ,  4002 

4063 

4124 

4185 

4245  61 

715 

85-  4306 

4367 

4428 

4488 

4549 

461 

467 

4731 

4792 

4852 

61 

716 

85-  4913 

4974 

5034 

5095 

5216 

5277 

5337 

5398 

5459 

61 

717 

85-  5519 

558 

564 

5701 

5761  '  5822 

5882 

5943 

6003 

6064 

61 

718 

85-  6124 

6185 

6245 

6306 

6366  6427 

6487 

6548 

6608 

6668 

60 

719 

85-  6729 

6789 

685 

691 

697 

7031 

7091 

7152 

7212 

7272 

60 

720 

85-  7332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

60 

721 

85-  7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

722 

85-  8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723  i  85-  9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

724 

85-  9739 

9799 

9859 

9918 

9978 

— 

— 

— 

60 

724 

86-  — 

0038 

0098 

0158 

0218 

0278 

60 

725 

86-  0338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

86-  0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

86-  1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

728 

86-  2131 

2191 

2251 

231 

237 

243 

2489 

2549 

2608 

2668 

60 

729 

86-  2728 

2787 

2847 

2906 

2966 

3025 

3085 

3J44 

3204 

3263 

60 

730 

86-  3323 

3382 

3442 

3501 

3561 

362 

368 

3739 

3799 

3858 

59 

73  l 

86-  3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

86-  4511 

457 

463 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

733 

86-  5104 

5163 

5222 

5282 

534i 

54 

5459 

5519 

5578 

5637 

59 

734 

86-  5696 

5755 

5814 

5874 

5933 

5992 

6051 

611 

6169 

6228 

59 

785 

86-  6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

676 

6819 

59 

736 

86-  6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

735 

7409 

59 

737 

86-  7467 

7526 

7585 

7644 

7703 

7762 

',32  1 

788 

7939 

7998 

59 

738 

86-  8056 

8115 

8i74 

8233 

8292 

835 

8409 

8468 

8527 

8586 

59 

739 

86-  8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

86-  9232 

929 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

976 

59 

741 

86-  9818 

9877 

9935 

9994 

— 

— 

— 

— 

— 

59 

741 

87-  - 

— 

— 

0053 

OIII 

017 

O2Cc 

0287 

0345 

59 

742 

87-  0404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

093 

58 

743 

87-  0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

744 

87-  1573 

1631 

169 

1748 

1806 

1865 

1923 

1981 

204 

2098 

58 

745 

87-  2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

746 

87-  2739 

2797 

2855 

2913 

2972 

303 

3088 

3146 

3204 

3262 

58 

747 

87-  332i 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

87-  3902 

396 

4018 

4076 

4192 

425 

4308 

4366 

4424 

58 

749 

87-  4482 

454 

4598 

4656 

4714 

4772 

483 

4888 

4945 

5003 

58 

750 

87-  5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

58 

751 

87-  564 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

616 

58 

752 

87-  6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

668 

6737 

58 

753 

87-  6795 

6853 

691 

6968 

7026 

7083 

7141 

7199 

7256 

73M 

58 

754 

87-  7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

EE 

326 


LOGARITHMS    OF   NUMBERS. 


01234 

56789 

87-  7947  8004  8062  8119  8177 

8234  8292  8^49  8407  8464 

87-  8522  8579  8637  8694  8752 

8809  8866  8924  8981  9039 

87-  9096  9153  9211  9268  9325 

9383  944  9497  9555  9°12 

87-  9669  9726  9784  9841  9898 

9956  —   —   —   — 

88-  —   —   —   —   — 

—  0013  007  0127  0185 

88-  0242  0299  0356  0413  0471 

0528  0585  0642  0699  0756 

88-  0814  0871  0928  0985  1042 

1099  IJ56  1213  1271  1328 

88-  1385  1442  1499  1556  1613 

167  1727  1784  1841  1898 

88-  1955  2012  2069  2126  2183 

224  2297  2354  2411  2468 

88-  2525  2581  2638  2695  2752 

2809  2866  2923  298  3037 

88-  3093  315  3207  3264  3321 

3377  3434  3491  3548  3605 

88-  3661  3718  3775  3832  3888 

3945  4002  4059  4115  4172 

88-  4229  4285  4342  4399  4455 

4512  4569  4625  4682  4739 

88-  4795  4852  4909  4965  5022 

5078  5i35  5192  5248  530S 

88-  5361  54i8  5474  553i  5587 

5644  57   5757  5813  587 

88-  5926  5983  6039  6096  6152 

6209  6265  6321  6378  6434 

88-  6491  6547  6604  666  6716 

6773  6829  6885  6942  6998 

88-  7054  7111  7167  7223  728 

7336  7392  7449  75O5  7  561 

88-  7617  7674  773  7786  7842 

7898  7955  8011  8067  8123 

88-  8179  8236  8292  8348  8404 

846  8516  8573  8629  8685 

88-  8741  8797  8853  8909  8965 

9021  9077  9134  919  9246 

88-  9302  9358  9414  947  9526 

9582  9638  9694  975  9806 

88-  9862  9918  9974  —   — 

—   —   —   —   — 

gg_  —   —   —  003  0086 

0141  0197  0253  0309  0365 

89-  0421  0477  0533  0589  0645 

07   0756  0812  0868  0924 

89-  098  1035  1091  1147  1203 

1259  1314  137  1426  1482 

89-  1537  1593  1649  1705  176 

1816  1872  1928  1983  2039 

89-  2095  215  2206  2262  2317 

2373  2429  2484  254  2595 

89-  2651  2707  2762  2818  2873 

2929  2985  304  3096  3151 

89-  3207  3262  3318  3373  3429 

3484  354  3595  365*  37°6 

89-  3762  3817  3873  3928  3984 

4039  4094  415  4205  4261 

89-  4316  4371  4427  4482  4538 

4593  4648  4704  4759  4814 

89-  487  4925  498  5036  5091 

5146  5201  5257  5312  5367 

89-  5423  5478  5533  5588  5644 

5699  5754  5809  5864  592 

89-  5975  603  6085  614  6195 

6251  6306  6361  6416  6471 

89-  6526  6581  6636  6692  6747 

6802  6857  6912  6967  7022 

89-  7077  7132  7187  7242  7297 

7352  7407  7462  7517  7572 

89-  7627  7682  7737  7792  7847 

7902  7957  8012  8067  8122 

89-  8176  8231  8286  8341  8396 

8451  8506  8561  8615  867 

89-  8725  878  8835  889  8944 

8999  9°54  9I09  9^4  9218 

89-  9273  9328  9383  9437  9492 

9547  9602  9656  9711  9766 

89-  9821  9875  993  9985  — 

90-  —  •   —   —   —  0039 

0094  0149  0203  0258  0312 

90-  0367  0422  0476  0531  0586 

064  0695  0749  0804  0859 

90-  0913  0968  1022  1077  II3I 

1  1  86  124  1295  1349  1404 

90-  1458  1513  1567  1622  1676 

1731  1785  184  1894  1948 

90-  2OO3  2O57  21  12  2l66  2221 

2275  2329  2384  2438  2492 

90-  2547  2001  2655  271   2764 

2818  2873  2927  2981  3036 

90-  309   3144  3199  3253  3307 

3361  3416  347  3524  3578 

90-  3633  3687  374i  3795  3849 

3904  3958  4012  4066  412 

90-  4174  4229  4283  4337  4391 

4445  4499  4553  4607  4661 

90-  4716  477  4824  4878  4932 

4986  504  5094  5148  5202 

90-  5256  531  5364  54i8  5472 

5526  558  5634  5688  5742 

012^4 

56789 

LOGARITHMS   OF   NUMBERS. 


327 


No. 

01234 

56789 

805 

90-  S796  585  59°4  5958  6012 

6066  6119  6173  6227  6281 

806 

9°-  6335  6389  6443  6497  6551 

6604  6658  6712  6766  682 

807 

90-  6874  6927  6981  7035  7089 

7143  7196  725  7304  7358 

808 

90-  7411  7465  7519  7573  7626 

768  7734  7787  7841  7895 

809 

90-  7949  8002  8056  811  8163 

8217  827  8324  8378  8431 

810 

90-  8485  8539  8592  8646  8699 

8753  8807  886  8914  8967 

811 

90-  9021  9074  9128  9181  9235 

9289  9342  9396  9449  9503 

812 

90-  9556  9609  9663  9716  977 

9823  9877  993  9984  — 

812 

91-  —   —   —   —   — 

—   —   —   —  0037 

813 

91-  0091  0144  0197  0251  0304 

0358  0411  0464  0518  0571 

814 

91-  0624  0678  0731  0784  0838 

0891  0944  0998  1051  1104 

815 

91-  1158  I2II  1264  1317  1371 

1424  1477  153  1584  1637 

816 

91-  169   1743  1797  185   1903 

1956  2009  2063  2116  2169 

817 

91-  2222  2275  2328  2381  2435 

2488  2541  2594  2647  27 

818 

91-  2753  2806  2859  2913  2966 

3019  3072  3125  3178  3231 

819 

91-  3284  3337  339  3443  3496 

3549  3602  3655  3708  3761 

820 

91-  3814  3867  392  3973  4026 

4079  4132  4184  4237  429 

821 

91-  4343  4396  4449  4502  4555 

4608  466  4713  4766  4819 

822 

91-  4872  4925  4977  503  5083 

5136  5189  5241  5294  5347 

823 

9i-  54   5453  5505  5558  5611 

5664  5716  5769  5822  5875 

824 

91-  5927  598  6033  ooSS  6138 

6191  6243  6296  6349  6401 

825 

91-  6454  6507  6559  6612  6664 

6717  677  6822  6875  6927 

826 

91-  698  7033  7085  7138  719 

7243  7295  7348  74   7453 

827 

91-  7506  7558  7611  7663  7716 

7768  782  7873  7925  7978 

828 

91-  803  8083  8135  8188  824 

8293  8345  8397  845  8502 

829 

91-  8555  8607  8659  8712  8764 

8816  8869  8921  8973  9026 

830 

91-  9078  913  9183  9235  9287 

934  9392  9444  940  9549 

831 

91-  9601  9653  9706  9758  981 

9862  9914  9967  —   — 

831 

92-  —   —   —   —   — 

—   —   —  0019  0071 

832 

92-  0123  0176  0228  028  0332 

0384  0436  0489  0541  0593 

833 

92-  0645  0697  0749  0801  0853 

0906  0958  101   1062  1114 

834 

92-  1166  1218  127  1322  1374 

1426  1478  153  1582  1634 

835 

92-  i686v  1738  179  1842  1894 

1946  1998  205   2102  2154 

836 

92-  2206  2258  231  2362  2414 

2466  2518  257   2622  2674 

837 

92-  2725  2777  2829  2881  2933 

2985  3°37  3089  314   3192 

838 

92-  3244  3296  3348  3399  3451 

3503  3555  3607  3658  371 

839 

92-  3762  3814  3865  3917  3969 

4021  4072  4124  4176  4228 

840 

92-  4279  4331  4383  4434  4486 

4538  4589  4641  4693  4744 

841 

92-  4796  4848  4899  4951  5003 

5054  5106  5157  5209  5261 

842 

92-  53i2  5364  5415  5407  55i8 

557  5621  5673  5725  5776 

843 

92-  5828  5879  5931  5982  6034 

6085  6137  6188  624  6291 

844  92-  6342  6394  6445  6497  6548 

66   6651  6702  6754  6805 

845  i  92-  6857  6908  6959  7011  7062 

7114  7165  7216  7268  7319 

846  92-  737  7422  7473  7524  7576 

7627  7678  773  7781  7832 

847  92-  7883  7935  7986  8037  8088 

814  8191  8242  8293  8345 

848  92-  8396  8447  8498  8549  8601 

8652  8703  8754  8805  8857 

849  92-  8908  8959  901  9061  9112 

9163  9215  9266  9317  9368 

850 

92-  9419  947  9521  9572  9623 

9674  9725  9776  9827  9879 

851 

92-  993  998i   —   —   — 

—   —   —   —   — 

851 

93-  —   —  0032  0083  0134 

0185  0236  0287  0338  0389 

852  93-  044  0491  0542  0592  0643 

0694  0745  0796  0847  0898 

853  93-  0949  i    1051  1  102  1153 

1203  1254  1305  1356  1407 

854 

93-  1458  1509  156  161   1661 

1712  1763  1814  1865  1915 

No. 

01234' 

56789! 

328 


LOGARITHMS   OF   NUMBERS. 


01234 

56789 

93-  1966  2017  2068  21  18  2169 

222   2271  2322  2372  2423 

93-  2474  2524  2575  2626  2677 

2727  2778  2829  2879  293 

93-  2981  3031  3082  3133  3183 

3234  3285  3335  3386  3437 

93-  34S7  3538  3589  3639  369 

374  3791  3841  3892  3943 

93-  3993  4044  4094  4145  4195 

4246  4296  4347  4397  4448 

93-  4498  4549  4599  465  47 

4751  4801  4852  4902  4953 

93-  5003  5054  5104  5154  5205 

5255  53o6  5356  5406  5457 

93-  5507  5558  5608  5658  5709 

5759  5809  586  591  50 

93-  6011  6061  6m  6162  6212 

6262  6313  6363  6413  6463 

93-  6514  6564  6614  6665  6715 

6765  6815  6865  6916  6966 

93-  7016  7066  7117  7167  7217 

7267  7317  7367  74i8  7468 

93-  75i8  7568  7618  7668  7718 

7769  7819  7869  7919  7969 

93-  8019  8069  8119  8169  8219 

8269  8319  837  842  847 

93-  852  857  862  867  872 

877  882  887  892  897 

93-  902  907  912  917  922 

927  932  9369  9419  9469 

93-  9519  9569  9619  9669  9719 

9769  9819  9869  9918  9968 

94-  0018  0068  01  18  0168  0218 

0267  0317  0367  0417  0467 

94-  0516  0566  0616  0666  0716 

0765  0815  0865  0915  0964 

94-  1014  1064  1114  1163  1213 

1263  1313  1362  1412  1462 

94-  1511  1561  1611  166  171 

176  1809  1859  1909  1958 

94-  2008  2058  2107  2157  2207 

2256  2306  2355  2405  2455 

94-  2504  2554  2603  2653  2702 

2752  2801  2851  2901  295 

94-  3    3049  3099  3T48  3J98 

3247  3297  3346  3396  3445 

94-  3495  3544  3593  3^43  3^2 

3742  3791  3841  389  3939 

94-  3989  4038  4088  4137  4186 

4236  4285  4335  4384  4433 

94-  4483  4532  4581  4631  468 

4729  4779  4828  4877  4927 

94-  4976  5025  5074  5124  5173 

5222  5272  5321  537  5419 

94-  5469  5518  5567  5616  5665 

5715  5764  5813  5862  5912 

94-  5961  601  6059  6108  6157 

6207  6256  6305  6354  6403 

94-  6452  6501  6551  66   6649 

6698  6747  6796  6845  6894 

94-  6943  6992  7041  709  714 

7189  7238  7287  7336  7385 

94-  7434  7483  7532  758i  763 

7679  7728  7777  7826  7875 

94-  7924  7973  8022  807  8119 

8168  8217  8266  8315  8364 

94-  8413  8462  8511  856  8609 

8657  8706  8755  8804  8853 

94-  8902  8951  8999  9048  9097 

9146  9195  9244  9292  9341 

94-  939  9439  9488  9536  9585 

9634  9683  9731  978  9829 

94-  9878  9926  9975  —   — 

—   —   —   —   — 

95-  —   —   —  0024  0073 

OI2T  017   0219  0267  0316 

95-  0365  0414  0462  0511  056 

0608  0657  0706  0754  0803 

95-  0851  09   0949  0997  1046 

1095  1143  1192  124   1289 

95-  1338  1386  1435  1483  1532 

158   1629  1677  1726  1775 

95-  1823  1872  192  1969  2017 

2066  2114  2163  2211  226 

95-  2308  2356  2405  2453  2502 

255   2599  2647  2696  2744 

95-  2792  2841  2889  2938  2986 

3034  3083  3131  318   3228 

95-  3276  3325  3373  342i  347 

3518  3566  3615  3663  3711 

95-  376  3808  3856  3905  3953 

40OI  4049  4098  4146  4194 

95-  4243  4291  4339  4387  4435 

4484  4532  458   4628  4677 

95-  4725  4773  4821  4869  4918 

4966  5014  5062  511   5158 

95-  5207  5255  5303  5351  5399 

5447  5495  5543  5592  564 

95-  5688  5736  5784  5832  588 

5928  5976  6024  6072  612 

95-  6168  6216  6265  6313  6361 

6409  6457  6505  6553  6601 

o     i     2    3  •  .  4 

56789 

OP  NUMBERS. 


329 


No. 

01234 

5    6-7    8    9 

905 

95-  6649  6697  6745  6793  684 

6888  6936  6984  7032  708 

906 

95-  7128  7176  7224  7272  732 

7368  7416  7464  7512  7559 

907 

95-  7607  7655  7703  775*  7799 

7847  7894  7942  799  8038 

908 

95-  8086  8134  8181  8229  8277 

8325  8373  8421  8468  8516 

909 

95-  8564  8612  8659  8707  8755 

8803  885  8898  8946  8994 

910 

95-  9041  9089  9137  9185  9232 

928  9328  9375  9423  9471 

911 

95-  95i8  9566  9614  9661  9709 

9757  9804  9852  99   9947 

912 

95-  9995  —   —   —   — 

—   —   —   —   — 

912 

96-  —  0042  009  0138  0185 

0233  028  0328  0376  0423 

9*3 

96-  0471  0518  0566  0613  0661 

0709  0756  0804  0851  0899 

914 

96-  0946  0994  1041  1089  1136 

1184  1231  1279  1326  1374 

915 

96-  1421  1469  1516  1563  1611 

1658  1706  1753  1801  1848 

916 

96-  1895  1943  I99  2038  2085 

2132  218  2227  2275  2322 

917 

96-  2369  2417  2464  2511  2559 

2606  2653  2701  2748  2795 

918 

96-  2843  289  2937  2985  3032 

3079  3126  3174  3221  3268 

919 

96-  33i6  3363  34i  3457  3504 

3552  3599  3646  3693  374i 

920 

96-  3788  3835  3882  3929  3977 

4024  4071  4118  4165  4212 

921 

96-  426  4307  4354  4401  4448 

4495  4542  459  4^37  4684 

922 

96-  4731  4778  4825  4872  49*9 

4966  5013  5061  5108  5155 

923 

96-  5202  5249  5296  5343  539 

5437  5484  553i  5578  5625 

924 

96-  5672  5719  5766  5813  586 

59°7  5954  6001  6048  6095 

925 

96-  6142  6189  6236  6283  6329 

6376  6423  647  6517  6564 

926 

96-  6611  6658  6705  6752  6799 

6845  6892  6939  6986  7033 

927 

96-708  7  1  27  7  1  73  722  7267 

7314  7361  7408  7454  7501 

928 

0-  7548  7595  7642  7688  7735 

7782  7829  7875  7922  7969 

929 

96-  8016  8662  8109  8156  8203 

8249  8296  8343  839  8436 

930 

96-  8483  853  8576  8623  867 

8716  8763  881  8856  8903 

93i 

96-  895  8996  9043  909  9136 

9183  9229  9276  9323  9369 

932 

96-  9416  9463  9509  9556  9602 

9649  9695  9742  9789  9835 

933 

96-  9882  9928  9975  —   — 

—   —   —   —   — 

933 

97-   OO2I  OOO8 

0114  0161  0207  0254  03 

934 

97-  0347  0393  °44  0486  0533 

0579  0626  0672  0719  0765 

935 

97-  0812  0858  0904  0951  0997 

1044  109  1137  1183  1229 

936 

97-  1276  1322  1369  1415  1461 

J5o8  1554  1601  1647  l693 

937 

97-  174  1786  1832  1879  1925 

1971  2018  2064  211  2157 

938 

97-  2203  2249  2295  2342  2388 

2434  2481  2527  2573  2619 

939 

97-  2666  2712  2758  2804  2851 

2897  2943  2989  3035  3082 

940 

97-  3128  3174  322  3266  3313 

3359  3405  345i  3497  3543 

94i 

97-  359  3636  3682  3728  3774 

382  3866  3913  3959  4005 

942 

07-  4051  4097  4i43  4189  4235 

4281  4327  4374  442  4466 

943  1  97-  4512  4558  4604  465  4696 

4742  4788  4834  488  4926 

9*4 

97-  4972  5018  5064  511  5156 

5202  5248  5294  534  5386 

945 

97-  5432  5478  5524  557  5616 

5662  5707  5753  5799  5845 

946 

97-  5891  5937  5983  6029  6075 

6121  6167  6212  6258  6304 

947  97-  635  6396  6442  6488  6533 

6579  6625  6671  6717  6763 

948  •  97-  6808  6854  69   6946  6992 

7037  7083  7129  7175  722 

949  97-  7266  7312  7358  7403  7449 

7495  7541  7586  7632  7678 

950  Q7-  7724  7769  7815  786i  7906 

7952  7998  8043  8089  8135 

951  97-  8181  8226  8272  8317  8363 

8409  8454  85   8546  8591 

952  97-  8637  8683  8728  8774  8819 

8865  8911  8956  9002  9047 

953  97-  9093  9J38  9l84  923  9275 

9321  9366  9412  9457  9503 

954 

1  97-  9548  9594  9639  9685  973 

9776  9821  9867  9912  9958 

No. 

01234 

56789 

EE* 

330 


LOGARITHMS    OF   NUMBERS. 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

955 

98-  0003 

0049 

0094 

014 

0185 

0231 

0276 

0322 

0367 

0412 

45 

956 

98-  0458 

0503 

0549 

0594 

064 

0685 

073 

0776 

0821 

0867  i  45 

957 

98-  0912 

0957 

1003 

1048 

1093 

"39 

1184 

1229 

1275 

132 

45 

958 

98-  1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

959 

98-  1819 

1864 

1909 

1954 

2 

2045 

209 

2135 

2181 

2226 

45 

960 

98-  2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

45 

961 

98-  2723 

2769 

2814 

2859 

2904 

2949 

2994 

304 

3085 

313 

45 

962 

98-  3175 

322 

3265 

331 

3356 

340i 

3446 

3491 

3536 

358i 

45 

963 

98-  3626 

3671 

37l6 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

964 

98-  4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

965 

98-  4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

966 

98-  4977 

5022 

5067 

5112 

5*57 

5202 

5247 

5292 

5337 

5382 

45 

967 

98-  5426 

5471 

55l6 

SS^I 

5606 

5651 

5696 

5741 

5786 

583 

45 

968 

98-  5875 

592 

5965 

601 

6055 

61 

6144 

6189 

6234 

6279 

45 

969 

98-  6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

98-  6772 

6817 

686l 

6906 

6951 

6996 

704 

7085 

713 

7175 

45 

97  1 

98-  7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

972 

98-  7666 

7711 

7756 

78 

7845 

789 

7934 

7979 

8024 

8068 

45 

973 

98-  8113 

8i57 

8202 

8247 

8291 

8336 

8381 

8425 

847 

8514 

45 

974 

98-  8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

896 

45 

975 

98-  9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

976 

98-  945 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

985 

44 

977 

98-  9895 

9939 

9983 

— 

— 

— 

— 

— 

— 

—  i  44 

977 

99-  — 

0028 

0072 

0117 

0161 

0206 

025 

0294  44 

978 

99-  0339 

0383 

0428 

0472 

0516 

0561 

0605 

065 

0694 

0738  44 

979 

99-  0783 

0827 

0871 

0916 

096 

1004 

1049 

1093 

"37 

1182  i  44 

980 

99-  1226 

127 

1315 

1359 

1403 

1448 

1492 

i536 

158 

1625 

44 

981 

99-  1669 

1713 

1758 

1802 

1846 

189 

1935 

1979 

2023 

2067 

44 

982 

99-  2IH 

2156 

22 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

933 

99-  2554 

2598 

2642 

2686 

273 

2774 

2819 

2863 

2907 

295  1 

44 

984 

99-  2995 

3039 

3083 

3127 

3172 

3216 

326 

3304 

3348 

3392 

44 

985 

99-  3436 

348 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

99-  3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

99-  43!7 

436i 

4405 

4449 

4493 

4537 

458i 

4625 

4669 

4713 

44 

988 

99-  4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

989 

99-  5196 

524 

5284 

5328 

5372 

54i6 

546 

5504 

5547 

5591 

44 

090 

99-  5635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

603 

44 

991 

99-  6074 

6117 

6161 

6205 

6249 

6293 

6337 

638 

6424 

6468 

44 

992 

99-  6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

99-  ^49 

6993 

7037 

708 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

99-  7386 

743 

7474 

7517 

756i 

76o5 

7648 

7692 

7736 

7779 

44 

995 

99-  7823 

7867 

791 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

996 

99-  8259 

8303 

8347 

839 

8434 

8477 

8521 

8564 

8608 

8652 

44 

997 

99-  8695 

8739 

8782 

8826 

8869 

8913 

8956 

9 

9043 

9087 

44 

998 

99-  9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

44 

999 

99-  9565 

9609 

9652 

9696 

9739 

9783 

9826 

987 

9913 

9957 

43 

No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D 

HYPERBOLIC   LOGARITHMS    OF   NUMBERS. 


S3' 


Hyperbolic   Logarithms   of  Numbers. 

From  i. 01  to  30. 

In  following  table,  the  numbers  range  from  i.oi  to  30,  advancing  by  .01, 
up  to  the  whole  number  10 ;  and  thence  by  larger  intervals  up  to  30.  The 
hyperbolic  logarithms  of  numbers,  or  Neperian  logarithms,  as  they  are  some- 
times termed,  are  computed  by  multiplying  the  common  logarithms  of  num- 
bers by  the  constant  multiplier,  2.302  585. 

The  hyperbolic  logarithms  of  numbers  intermediate  between  those  which 
are  given  in  the  table  may  be  readily  obtained  by  interpolating  proportional 
differences. 


No. 

Log. 

No. 

Log. 

No. 

Log.   I 

No. 

Log. 

No. 

Log. 

I.OI 

.0099 

I.4I 

.3436 

1.81 

•5933 

2.21 

•793 

2.61 

•9594 

1.02 

.0198 

1.42 

.3507 

1.82 

.5988 

2.22 

•7975 

2.62 

.9632 

1.03 

.0296 

i-43 

•3577 

1.83 

.6043 

2.23 

.802 

2.63 

.967 

1.04 

.0392 

1.44 

.3646 

1.84 

.6098 

2.24 

.8065 

2.64 

.9708 

1.05 

.0488 

1-45 

.3716 

1.85 

•6152 

2.25 

.8109 

2.65 

.9746 

1.06 

.0583 

1.46 

.3784 

1.86 

.6206 

2.26 

•8i54 

2.66 

•9783 

1.07 

.0677 

1.47 

.3853 

1.87 

.6259 

2.27 

.8198 

2.67 

.9821 

.08 

.077 

1.48 

•392 

1.88 

•63J3 

2.28 

.8242 

2.68 

.9858 

.09 

.0862 

1.49 

.3088 

1.89 

.6366 

2.29 

.8286 

2.69 

•9895 

.1 

•0953 

i-5 

.4055 

1.9 

.6419 

2-3 

.8329 

2.7 

•9933 

.11 

.1044 

i-5i 

.4121 

1.91 

.6471 

2.31 

.8372 

2.71 

.9969 

.12 

•"33 

1.52 

.4187 

1.92 

•6523 

2.32 

.8416 

2.72 

i.  0006 

•13 

.1222 

1-53 

•4253 

i-93 

•6575 

2-33 

•8458 

2-73 

1.0043 

.14 

.131 

i-54 

.4318 

1.94 

.6627 

2-34 

.8502 

2.74 

1.008 

•15 

.1398 

i-55 

.4383 

i-95 

.6678 

2-35 

•8544 

2-75 

i  .0116 

.16 

.1484 

1.56 

•4447 

1.96 

.6729 

2.36 

•8587 

2.76 

1.0152 

-1? 

•157 

i-57 

•45U 

1.97 

.678 

2-37 

.8629 

2.77 

i.  01  88 

.18 

•1655 

1.58 

•4574 

1.98 

.6831 

2.38 

.8671 

2.78 

1.0225 

.19 

.174 

i-59 

-4637 

1.99 

.6881 

2-39 

•8713 

2.79 

1.026 

.2 

.1823 

1.6 

•47 

2 

.6931 

2.4 

•8755 

2.8 

1.0296 

.21 

.1906 

1.61 

.4762 

2.OI 

.6981 

2.41 

.8796 

2.81 

1.0332 

.22 

.1988 

1.62 

.4824 

2.  02 

•7031 

2.42 

.8838 

2.82 

1.0367 

•23 

.207 

1.63 

.4886 

2.03 

.708 

2-43 

.8879 

2.83 

1.0403 

.24 

.2151 

1.64 

•4947 

2.04 

.7129 

2.44 

.892 

2.84 

1.0438 

•25 

.2231 

1.65 

.5008 

2.05 

.7178 

2-45 

.8961 

2.85 

1-0473 

.26 

.2311 

1.66 

.5068. 

2.06 

.7227 

2.46 

.9002 

2.86 

1.0508 

.27 

•239 

1.67 

.5128 

2.07 

•7275 

2.47 

.9042 

2.87 

1-0543 

.28 

.2469 

1.68 

.5188 

2.08 

•7324 

2.48 

•9083 

2.88 

1.0578 

.29 

.2546 

1.69 

•5247 

2.O9 

•7372 

2.49 

.9123 

2.89 

1.0613 

•3 

.2624 

i-7 

•5306 

2.1 

.7419 

2-5 

.9163 

2.9 

1.0647 

1-31 

.27 

1.71 

•5365 

2.  II 

.7467 

2.51 

.9203 

2.91 

1.0682 

1.32 

.2776 

1.72 

•5423 

2.12 

•75*4 

2.52 

•9243 

2.92 

1.0716 

1-33 

.2852 

i-73 

.5481 

2.13 

•7501 

2-53 

.9282 

2-93 

1-075 

1-34 

.2927 

1.74 

•5539 

2.14 

.7608 

2-54 

.9322 

2.94 

1.0784 

1-35 

.3001 

i-7S 

•559° 

2.15 

•7655 

2-55 

.9361 

2-95 

i.  0818 

1.36 

.3075 

1.76 

•5653 

2.16 

.7701 

2.56 

•94 

2.96 

1.0852 

i-37 

.3148 

1.77 

•571 

2.17 

•7747 

2-57 

•9439 

2.97 

1.0886 

1.38 

.3221 

1.78 

.5766 

2.18 

•7793 

2.58 

•9478 

2.98 

1.0919 

i-39 

•3293 

1.79 

.5822 

2  19 

•7839 

2-59 

•9517 

2-99 

1-0953 

1.4 

.3365 

1.8 

.5878 

2.2 

.7885 

2.6 

•9555 

3 

1.0986 

HYPEEBOLIC   LOGARITHMS   OF   NUMBERS. 


Log. 

No. 

Log.   [ 

No. 

Log. 

No.     Log. 

No. 

Log. 

I.IOI9 

3-51 

1.2556 

4-01 

1.3888 

4-51 

1.5063 

5-oi 

1.6114 

I  -1053 

3-52 

1.2585 

4.02 

I.39I3 

4-52 

1.5085 

5-02 

1.6134 

1.  1086 

3-53 

1.2613 

4-03 

I-3938 

4-53 

1.5107 

5.03 

1.6154 

I.IIIQ 

3-54 

1.2641 

4.04 

1.3962 

4-54 

1.5129 

5-04 

1.6174 

I.IISI 

3-55 

1.2669 

4-05 

L39S7 

4-55 

1.5151 

5-05 

1.6194 

1.1184 

3.56 

1.2698 

4.06 

1.4012 

4-56 

I.5I73 

5.o6 

1.6214 

I.I2I7 

3-57 

1.2726 

4.07 

1.4036 

4-57 

LS^S 

5.07  1  1.6233 

1.1249 

3.58 

1.2754 

4.08 

1.4061 

4-58 

1.5217 

5.08 

1.6253 

I.I282 

3-59 

1.2782 

4.09 

1.4085 

4-59 

I-5239 

5-09 

1.6273 

1.1314 

3-6 

1.2809 

4.1 

1.411 

4.6 

1.5261 

5-i 

1.6292 

1.1346 

3-6i 

1.2837 

4.II 

I.4I34 

4.61 

1.5282 

5-n 

1.6312 

I.I378 

3-62 

1.2865 

4.12 

I.4I59 

4.62 

I-5304 

5.12 

1.6332 

I.I4I 

3-63 

1.2892 

4.13 

1.4183 

4-63 

1-5326 

5-13 

I-635I 

I.I442 

3-64 

1.292 

4.14 

1.4207 

4.64 

1-5347 

5.i4 

1.6371 

1.1474 

3-65 

1.2947 

4-15 

1.4231 

4-65 

1-5369 

5-15 

1.639 

I.I506 

3-66 

1-2975 

4.l6 

1.4255 

4.66 

1-539 

5-i6 

1.6409 

I-I537 

3-67 

1.3002 

4.17 

1.4279 

4-67 

1.5412 

5-17 

1  .6429 

1.1569 

3.68 

1.3029 

4.18 

1.4303 

4.68 

1-5433 

5-i8 

1.6448 

1.16 

3-69 

1.3056 

4.19 

1.4327 

4.69 

1-5454 

5-19 

1  .6467 

1.1632 

3-7 

1.3083 

4.2 

L435I 

4-7 

I-5476 

5-2 

1.6487 

1.1663 

3-7i 

I.3II 

4.21 

1-4375 

4.71 

1-5497 

5-21 

1.6506 

1.1694 

3-72 

I.3I37 

4.22 

1.4398 

4.72 

i-55i8 

5-22 

1.6525 

1.1725 

3-73 

1.3164 

4-23 

1.4422 

4-73 

1-5539 

5-23 

1.6514 

1.1756 

3-74 

1.3191 

4-24 

1.4446 

4-74 

I-556 

5-24 

1.6563 

1.1787 

3-75 

1.3218 

4-2S 

1.4469 

4-75 

i.558i 

5-25 

1.6582 

1.1817 

3.76 

1.3244 

4.26 

1-4493 

4.76 

1.5602 

5-26 

I.660I 

1.1848 

3-77 

1.3271 

4.27 

1.4516 

4-77 

1-5623 

5.27 

1.662 

1.1878 

3-78 

1.3297 

4.28 

1-454 

4.78 

1.5644 

5-28 

1.6639 

1.1909 

3-79 

L3324 

4.29 

1-4563 

4-79 

1.5665 

5.29 

1.6658 

I-I939 

3-8 

1-335 

4-3 

1.4586 

48 

1.5686 

5-3 

1.6677 

1.1969 

3-8i 

1.3376 

4-31 

1.4609 

4.81 

I-5707 

5-31 

1.6696 

1.1999 

382 

1.3403 

4.32 

1-4633 

4.82 

1.5728 

5-32 

I.67I5 

1.203 

3-83 

1.3429 

4-33 

1.4656 

4-83 

1.5748 

5-33 

1-6734 

1.206 

3-84 

1-3455 

4-34 

1.4679 

4.84 

1-5769 

5-34 

1.6752 

1.209 

3-85 

1.3481 

4-35 

1.4702 

4-85 

1-579 

5-35 

1.6771 

1.2119 

3-86 

L3507 

4-36 

1.4725 

4.86 

1.581 

5.36 

1.679 

1.2149 

3-87 

1-3533 

4-37 

1.4748 

4-87 

1.5831 

5-37 

1.  6808 

1.2179 

3-88 

1-3558 

4-38 

1-477 

4.88 

1-5851 

5.38 

1.6827 

1.2208 

3.89 

I-3584 

4-39 

1-4793 

4.89 

1.5872 

5-39 

1.6845 

1.2238 

3-9 

1.361 

4-4 

1.4816 

4.9 

1.5892 

5-4 

1.6864 

1.2267 

3-9i 

1-3635 

4.41 

1.4839 

4.91 

I-59I3 

5-41 

1.6882 

1.2296 

3-92 

1.3661 

4-42 

1.4861 

4.92 

1-5933 

5-42 

1.6901 

1.2326 

3-93 

1.3686 

4-43 

1.4884 

4-93 

1-5953 

5-43 

1.6919 

1.2355 

3-94 

1.3712 

4.44 

1.4907 

4.94 

1-5974 

5-44 

1.6938 

1.2384 

3-95 

1-3737 

4-45 

1.4929 

4-95 

1-5994 

5-45 

1.6956 

1.2413 

3-90 

1.3762 

4.46 

i-495i 

4.96 

1.6014 

5-46 

1.6974 

1.2442 

3-97 

1.3788 

4-47 

1.4974 

4-97 

1.6034 

5-47 

1.6993 

1.247 

3-98 

1-3813 

4.48 

1.4996 

4-98 

1.6054 

5-48 

I.7OII 

1.2499 

3-99 

1.3838 

4-49 

1.5019 

4.99 

1.6074 

5-49 

1.7029 

1.2528 

4 

1.3863 

4-5 

1.5041 

5 

1.6094 

5-5 

1.7047 

HYPERBOLIC   LOGARITHMS   OF   NUMBERS. 


333 


No. 

Log. 

No. 

Log. 

No. 

Log-   f 

No.  | 

Log.   1 

No. 

Log. 

5.5i 

1.7066 

6.01 

1-7934 

6.5I 

1-8733 

7.01 

1-9473 

7-51 

2.0162 

5-52 

1.7084 

6.02 

I-795I 

6.52 

1.8749 

7.02 

1.9488 

7-52 

2.0176 

5-53  1 

1.7102 

6.03 

1.7967  :: 

6-53 

1.8764 

7.03 

1.9502 

7-53 

2.0189 

5-54 

1.712 

6.04 

1.7984  j 

6-54 

1.8779 

7.04 

I.95I6 

7-54 

2.0202 

5-55 

1.7138 

6.05 

1.8001 

6.55 

1.8795 

7-05 

1-953 

7-55 

2.0215 

5.56 

1.7156 

6.06 

1.8017 

6.56 

I.88I 

7.06 

1-9544 

7.56 

2.0229 

5-57  ' 

1.7174 

6.07 

1.8034 

6.57 

1.8825 

7.07 

1-9559 

7-57 

2.0242 

5-58 

1.7192 

6.08 

1.805 

6.58 

1.884 

7.08 

1-9573 

7-58 

2.0255 

5-59 

1.721 

6.09 

i.  8066 

6-59 

1.8856 

7.09 

L9587 

7-59 

2.0268 

5-6 

1.7228 

6.1 

1.8083 

6.6 

1.8871 

7-1 

1.9601 

7.6 

2.O28I 

5-6i 

1.7246 

6.ii 

1.8099 

6.61 

1.8886 

7.II 

I.96I5 

7.61 

2.0295 

5-62 

1.7263 

6.12 

1.8116 

6.62 

1.8901 

7.12 

1.9629 

7.62 

2.0308 

5-63 

1.7281 

6.13 

1.8132 

6.63 

1.8916 

7-13 

1.9643 

7-63 

2.032J 

5-64 

1.7299 

6.14 

1.8148 

6.64 

1.8931 

7.14 

1.9657 

7.64 

2.0334 

5-65 

I.73I7 

6.15 

1.8165  i 

6.65 

1.8946 

7-15 

1.9671 

7.65 

2.0347 

566 

1-7334 

6.16 

1.8181 

6.66 

1.8961 

7.l6 

1.0685 

7.66 

2.036 

5-6? 

1-7352 

6.17 

1.8197 

6.67 

1.8976 

7.17 

1.9699 

7.67 

2-0373 

5-68 

1-737 

6.18 

1.8213  ; 

6.68 

1.8991 

7.l8 

I-97I3 

7.68 

2.0386 

5-69 

1.7387 

6.19 

1.8229 

6.69 

1.9006 

7.19 

1.9727 

7.69 

2.0399 

5-7 

I.7405 

6.2 

1.8245  | 

6-7 

1.9021 

7.2 

1.9741 

7-7 

2.0412 

5-7i 

1.7422 

6.21 

1.8262 

6.71 

1.9036 

7.21 

1-9755 

7.71 

2.0425 

5-72 

1.744 

6.22 

1.8278 

6.72 

1.9051 

7.22 

1.9769 

7.72 

2.0438 

5-73 

1-7457 

6.23 

1.8294 

6.73 

1.9066 

7-23 

1.9782 

7-73 

2.0451 

5-74 

1-7475 

6.24 

1.831 

6.74 

1.9081 

7.24 

1.9796 

7-74 

2.0464 

5-75 

1.7492 

6.25 

1.8326 

6.75 

1.9095 

7.25 

1.981 

7-75 

2.0477 

5.76 

1.7509 

6.26 

1.8342 

6.76 

1.911 

7.36 

1.9824 

7.76 

2.049 

5-77 

1.7527 

6.27 

1-8358 

6.77 

1.9125 

7-27 

1.9838 

7-77 

2.0503 

5.78 

1-7544 

6.28 

1-8374 

6.78 

1.914 

7.28 

1.9851 

7.78 

2.0516 

5-79 

1.7561 

i  6.29 

1.839 

6.79 

i.9i55 

7.29 

1.9865 

7-79 

2.O528 

5-8 

1-7579 

6-3 

1.8405 

6.8 

1.9169 

7-3 

1.9879 

7.8 

2.0541 

5-8i 

1.7596 

6.3I 

1.8421 

6.81 

1.9184 

7.3i 

1.9892 

7.81 

2.0554 

5-82 

1.7613 

6.32 

1-8437 

6.82 

1.9199 

7-32 

1.9906 

7.82 

2.0567 

5-83 

1-763 

6-33 

1.8453 

6.83 

1.9213 

7-33 

1.992 

7.83 

2.058 

5-84 

1.7647 

6-34 

1.8469 

6.84 

1.9228 

7-34 

1-9933 

7.84 

2.0592 

5.85 

1.7664 

;6.35 

1.8485 

6.85 

1.9242 

7-35 

1.9947 

7.85 

2.0605 

5-86 

1.7681 

6.36 

1.85 

6.86 

1.9257 

7.36 

1.9961 

7.86 

2.o6l8 

5-87 

1.7699 

6.37 

1.8516 

6.87 

1.9272 

7-37 

1.9974 

7.87 

2.0631 

5-88 

1.7716 

6.38 

1-8532 

6.88 

1.9286 

7.38 

1.9988 

7.88 

2.0643 

5-89 

1-7733 

6.39 

1-8547 

6.89 

1.9301 

7-39 

2.0001 

7.89 

2.0656 

5-9 

1-775 

6.4 

1.8563 

6.9 

i.93i5 

7-4 

2.0015 

7-9 

2.0669 

5.9i 

1.7766 

i  6.41 

1-8579 

6.91 

1-933 

7.41 

2.0O28 

7.91 

2.o68l 

5-92 

1.7783 

6.42 

1.8594 

6.92 

1-9344 

7.42 

2.0042 

7.92 

2.0694 

5-93 

1.78 

6-43 

1.861 

6-93 

t-9359 

7-43 

2.0055 

7-93 

2.0707 

5-94 

1.7817 

6.44 

1.8625 

6.94 

1-9373 

7-44 

2.O009 

7-94 

2.O7I9 

5-95 

1.7834 

6.45 

1.8641 

6-95 

I-9387 

7-45 

2.O082 

7-95 

2.0732 

S.96 

1.7851 

i  6.46 

1.8656 

6.96 

i  .9402 

7.46 

2.0096 

7.96 

2.0744 

5-97 

1.7867 

!  6.47 

1.8672 

6.97 

1.9416 

7-47 

2.OIO9 

7-97 

2.0757 

598 

1.7884 

6.48 

1.8687 

6.98 

1-943 

7.48 

2.0122 

7.98 

2.0769 

5-99 

1.7901 

6.49 

1.8703 

6-99 

1-9445 

7-49 

2.0136 

7-99 

2.0782 

6 

1.7918 

6.5 

1.8718 

7 

1.9459 

7-5 

2.OI49 

1  8 

2.0794 

334 


HYPERBOLIC    LOGARITHMS    OF   NUMBERS. 


Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

2.0807 

8.41 

2.1294 

8.81 

2.1759 

9.21 

2.22O3 

9.61 

2.2628 

2.0819 

8.42 

2.1306 

8.82 

2.177 

9.22 

2.2214 

9.62 

2.2638 

2.0832 

8-43 

2.1318 

8.83 

2.1782 

9-23 

2.2225 

9-03 

2.2649 

2.0844 

8.44 

2.133 

8.84 

2.1793 

9.24 

2.2235 

9.64 

2.2659 

2.0857 

8-45 

2.1342 

8.85 

2.1804 

9-25 

2.2246 

9-05 

2.267 

2.0869 

8.46 

2.1353 

8.86 

2.1815 

9.26 

2.2257 

9.66 

2.268 

2.0882 

8.47 

2.1365 

8.87 

2.1827 

9.27 

2.2268 

9.67 

2.269 

2.0894 

8.48 

2.1377 

8.88 

2.1838 

9.28 

2.2279 

9.68 

2.2701 

2.0906 

8-49 

2.1389 

8.89 

2.1849 

9.29 

2.2289 

9.69 

2.2711 

2.0919 

8.5 

2.1401 

8.9 

2.1861 

9-3 

2.23 

9-7 

2.2721 

2.0931 

8.5I 

2.1412 

8.91 

2.1872 

9-31 

2.2311 

9.71 

2.2732 

2.0943 

8.52 

2.1424 

8.92 

2.1883 

9-32 

2.2322 

972 

2.2742 

2.0956 

8-53 

2.1436 

8-93 

2.1894 

9-33 

2.2332 

973 

2.2752 

2.0968 

8-54 

2.1448 

8.94 

2.1905 

9-34 

2.2343 

974 

2.2762 

2.098 

8-55 

2.1459 

8-95 

2.1917 

9-35 

2.2354 

975 

2.2773 

2.0992 

8.56 

2.1471 

8.96 

2.1928 

9.36 

2.2364 

9.76 

2.2783 

2.1005 

8-57 

2.1483 

8.97 

2.1939 

9-37 

2-2375 

9-77 

22793 

2.IOI7 

8.58 

2.1494 

8.98 

2.195 

9-38 

2.2386 

9.78 

2  2803 

2.IO29 

8-59 

2.1506 

8.99 

2.1961 

9-39 

2.2396 

9-79 

2.2814 

2.IO4I 

8.6 

2.1518 

9 

2.1972 

9.4 

2.2407 

98 

2.2824 

2.1054 

8.61 

2.1529 

9.01 

2.1983 

9.41 

2.2418 

9.81 

2.2834 

2.I066 

8.62 

2.1541 

9.02 

2.1994 

9.42 

2.2428 

!  9.82 

2.2844 

2.1078 

8.63 

2.1552 

9-03 

2.2006 

9-43 

22439 

983 

22854 

2.IO9 

8.64 

2.1564 

9.04 

2.2017 

9-44 

2245 

984 

2.2865 

2.II02 

8.65 

2.1576 

9-05 

2.2028 

9-45 

2.246 

9-85 

2.2875 

2.III4 

8.66 

2.1587 

9.06 

2.2039 

9.46 

2.2471 

9.86 

2.2885 

2.II20 

8.67 

2.1599 

9.07 

2.205 

9-47 

2.2481 

9.87 

2.2893 

2.II38 

8.68 

2.161 

9.08 

2.2061 

948 

2  2492 

!  9.88 

2.2003 

2.II5 

8.69 

2.1622 

9.09 

2.2072 

9-49 

2.2502 

9.89 

2.2913 

2.1163 

8.7 

2.1633 

9.1 

2.2083 

9-5 

2.2513 

9-9 

2.2923 

2.II75 

8.71 

2.1645 

9.11 

2.2094 

9.5i 

2.2523 

9.91 

2.2933 

2.1187 

8.72 

2.1656 

9.12 

2.2105 

9-52 

2-2534 

9.92 

2.294(j 

2.II99 

8-73 

2.1668 

9-i3 

2.  2Il6 

9-53 

2.2544 

9-93 

2.295^ 

2.I2II 

8.74 

2.1679 

9.14 

2.2127 

9-54 

2.2555 

9-94 

2.206« 

2.1223 

8-75 

2.1691 

9-i5 

2.2138 

9-55 

2.2565 

9-95 

2-297<J 

2.1235 

8.76 

2.1702 

9.16 

2.2148 

9-56 

2.2576 

9.96 

2.298^ 

2.1247 

8.77 

2.1713 

9.17 

2.2159 

9-57 

2.2586 

9-97 

2.2996 

2.1258 

8.78 

2.1725 

9.18 

2.217 

9.58 

2.2597 

9.98 

2.3006 

2.127 

8.79 

2.1736 

9.19 

2.2l8l 

9-59 

2.2607 

9.99 

2.3016 

2.1282 

8.8 

2.1748 

9.2 

2.2192 

9.6 

2.26l8 

10 

2.3026 

2.3279 

12.25 

2.5052 

14.25 

2.6567 

17-5 

2.8621 

23 

3-J355 

2.3513 

12.5 

2.5262 

14.5 

2.674 

18 

2.8004 

24 

3.1781 

2-3749 

12.75 

2-5455 

14-75 

2.6913 

18.5 

2.9173 

25 

3-2189 

2.3979 

13 

2.5649 

15 

2.7081 

19 

2.9444 

26 

3-2581 

2.42OI 

13-25 

2.584 

iS-5 

2.7408 

19-5 

2.9703 

27 

3-2958 

2-443 

13-5 

2.0027 

16 

2.7726 

20 

2-9957 

28 

3-3322 

2.4636 

13-75 

2.62II 

16.5 

2.8034 

21 

3-0445 

29 

3.3073 

2.4849 

14 

2.6391 

17 

2.8332 

22 

3-09II 

30 

3.4012 

MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       33$ 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 
3?ar  allelogr  am  s . 

DEFINITION.— Quadrilaterals,  having  their  opposite  sides  parallel 

To  Compute  Area  of*  a  Sqnare,  Rectangle,  Rnomtouis,  o* 
Rhomboid.— Figs.  1,  2,  3,  and   4r. 

RULE.— Multiply  length  by  breadth  or  height. 

Or,  I X  b  =  area,  I  representing  length,  and  b  breadth. 
Fig.  i.  Fig.  3. 

Fig.  4- 


b  6 

EXAMPLE.— Sides  a  &,  b  c,  Fig.  i,  are  5  feet  6  ins. ;  what  is  area? 

5.5X5-5  =  30-  25  square  feet. 

NOTE  i.— Opposite  angles  of  a  Rhombus  and  a  Rhomboid  are  equal 
2.— In  any  parallelogram  the  four  angles  equal  360°. 

3.— Side  of  a  square  multiplied  by  1.52  is  equal  to  side  of  an  equilateral  triangle 
of  equal  area. 

Gf-nomon. 

DEFINITION.  —Space  included  between  the  lines  forming  two  similar  parallel- 
ograms, of  which  smaller  is  inscribed  within  larger,  so  that  one  angle  in  each  is 
common  to  both,  as  shown  by  dotted  lines,  Fig.  i. 

To   Compute   Area   of*  a   03-nomon.— Fig.  1. 

RULE. — Ascertain  areas  of  the  two  parallelograms,  and  subtract  less  from 
greater. 

Or,  a — a' =  area,  a  and  a'  representing  areas. 

EXAMPLE.— Sides  of  a  gnomon  are  10  by  10  and  6  by  6  ins. ;  what  is  its  area? 
10  X  10  =  ioo,  and  6  X  6  =  36.    Then  100  —  36  =  64  square  ins. 

Triangles. 

DEFINITION.— Plain  superficies  having  three  sides  and  angles. 

To   Compute   Area  of  a   Triangle.— Figs.  5,  6,  and.   7. 
RULE.— Multiply  base  by  height,  and  divide  product  by  2. 


Or, 


abxcd 


Or,  —  =  area,  b  representing  base,  and  h  height. 


NOTE  i.— Hypotenuse  of  a  right  angle  is  side  opposite  to  right  angle. 
2.— Perpendicular  height  of  a  triangle  =  twice  its  area  divided  by  its  base. 
3.— Perpendicular  height  of  an  equilateral  triangle  =  a  side  x  .866. 
4. — Side  of  an  equilateral  triangle  x  .658255  =  side  of  a  square  of  equal  area, 
Or  -~  1.3468  =  diameter  of  a  circle  of  equal  area. 


Fig.  5- 


Fig.  6. 


Fig.  7- 


EXAMPLE.  — Base  a  6,  Fig. 
5,  is  4  feet,  and  height  c  b,  6  j 
what  is  area  ? 

4  X  6  =  24,  and  24-7-2=12 
square  feet. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

To  Compute  Area  of  a  Triangle  toy  Length  of  its  Sides.-. 

Trigs.  6  and  7. 

RULE. — From  half  sum  of  the  three  sides  subtract  each  side  separately ; 
then  multiply  half  sum  and  the  three  remainders  continually  together,  and 
take  square  root  of  product. 

Or,  \/(s  —  a)  X  (s  —  b)  X  (s  —  c)  S  =  area,  a,  &,  c  representing  sides,  and  S  half  sum 
of  the  three  sides. 
EXAMPLE.— Sides  of  a  triangle,  Figs.  6  and  7,  are  30, 40,  and  50  feet;  what  is  area? 

3o_L.40_i_50      I20  6°  —  30  =  30) 

- — — — — —  =  —  =  60,  or  half  sum  of  sides.  60  —  40  =  20  >  remainders. 


60  —  50  =  10  ) 
Whence,  30  X  20  X  10  X  60  =  360000,  and  ^360000  =  600  square  feet 

When  all  Sides  are  Equal.    RULE.—  Square  length  of  a  side,  and  multi- 
ply product  by  .433. 

Or,  S2  x  .433  =  area,  S  representing  length  of  a  side. 

To   Compute    Length    of   One    Side    of  a    Right-Angled 
Triangle. 

When  Length  of  the  other  Tiro  Sides  are  given. 
To   Ascertain    Hypotenuse.  —  Fig.  C. 

RULE.  —  Add  together  squares  of  the  two  legs,  and  take  square  root  of 
sum.  _ 


_ 

Or,  Va  62  -f  6  c2  =  hypotenuse.    Or,  Vb2  +  h*. 

EXAMPLE.—  Base,  a  &,  Fig.  5.  is  30  ins.,  and  height,  &  c,  40;  what  is  length  of  hy- 
potenuse ? 

3o2  +  4o2  =  2500,  and  V25oo  =  50  ins. 

To    Ascertain,    other    Leg. 

When  Hypotenuse  and  One  of  the  Legs  are  given.—Fig.  5.  RULE.—  Sub- 
tract square  of  given  leg  from  square  of  hypotenuse,  and  take  square  root 
of  remainder. 


EXAMPLE.—  Base  of  a  triangle,  a  &,  Fig.  5,  is  30  feet,  and  hypotenuse,  a  c,  50; 
what  is  height  of  it? 

5o2  —  so2  =  1600,  and  1/1600  =  40/66*. 

To    Compute    Length    of  a    Side. 

When  Hypotenuse  of  a  Right-angled  Triangle  of  Equal  Sides  alone  is 
given.  —  Fig.  8.     RULE.  —  Divide  hypotenuse  by  1.414213. 

Or,     hyp'     =  length  of  a  side. 
Fig.  8.          -  '-4'4213 

/ 

EXAMPLE.  —  Hypotenuse  a  c  of  a  right-angled  triangle,  Fig.  8,  is 
300  feet;  what  is  length  of  its  sides? 

300-:-  1.  414  213  =  212.  1321  feet. 

a~~         & 

To    Compute    Perpendicular    or    Height    of  a    Triangle. 

When  Base  and  Area  alone  are  given.—  Fig.  9.    RULE.—  Divide  twice 
area  by  its  base.        Or,  za  -4-  6  =  h. 

EXAMPLE.—  Area  of  a  triangle,  Fig.  9,  is  10  feet,  and  length  of  its  base,  a  &,  5; 
what  is  its  perpendicular,  c  d? 

TO  X  2  =  20,  and  20-7-  5  =  4  feet. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       337 

To    Compnte    Perpendicular    or    Height    of  a    Triangle. 

When  Base  and  Two  Sides  are  given.  RULE. — As  base  is  to  sum  of  the 
sides,  so  is  difference  of  sides  to  difference  of  divisions  of  base.  Half  this 
difference  being  added  to  or  subtracted  from  half  base  will  give  the  two  di- 
visions thereof.  Hence,  as  the  sides  and  their  opposite  division  of  base  con- 
stitute a  right-angled  triangle,  the  perpendicular  thereof  is  readily  ascertained 
by  preceding  rules. 

tc  +  o«x  »••»•• 
ba 

=  ad;  whence  -\ 


•zab 

EXAMPLE.— Three  sides  of  a  triangle,  a  b  c,  Fig.  9,  are  9.928, 
8,  and  5  feet;  what  is  length  of  perpendicular  on  longest  side  ? 

As  9.928  :  8  -\-  5  : :  8  <v  5  :  3.928  =  difference  of  divisions  of 
the  base. 

Then  3.928  -•-  2  =  1.964,  which,  added  to  9'92   =  4.964  + 

i.  964  =  6  928  =  length  of  longest  division  of  base. 
\*      u,  u 

Hence,  there  is  a  right-angled  triangle  with  its  base  6  928,  and  its  hypotenuse  8 ; 
consequently,  its  remaining  side  or  perpendicular  is  V(g2—  6. 92$*)  =  4  feet. 

When  any  Two  of  the  D^imensions  of  a  Triangle  and  One  of  the  corresponding 
Dimensions  of  a  similar  Figure  are  given,  and  it  is  required  to  ascertain 
the  other  corresponding  Dimensions  of  the  last  Figure. 
Fig.  10.  Fig.  xi. 

Let  a  b  c,  a'  6'  c',  be  two  similar  triangles,  Figs.  10 
and  ii. 

Then  ab  :bc::  a'b' :  6'c',  or  a'6' :  6'c' ::  ab  :  be. 

NOTE.  —Same  proportion  holds  with  respect  to  the 
similar  lineal  parts  of  any  other  similar  figures,  whether 
plane  or  solid. 

EXAMPLE Shadow  of  a  vertical  stake  4  feet  in  length  was  5  feet;  at  same  time, 

shadow  of  a  tree,  both  on  level  ground,  was  83  feet;  what  was  height  of  tree? 
5  a'  b'  :  4  b'  c' ::  83  a  b  :  66.4  feet. 

To   Compute   Acreage. 

Divide  area  into  convenient  triangles,  and  multiply  base  of  each  triangle 
in  links  by  half  perpendicular  in  links ;  cut  off  5  figures  at  the  right,  remain- 
ing figures  will  give  acres ;  multiply  the  5  figures  so  cut  off  by  4,  and  again 
cut  off  5,  and  remainder  will  give  roods ;  multiply  the  5  by  40,  and  again 
cut  off  5  for  perches. 

Trapezium. 

DEFINITION.— A  Quadrilateral  having  unequal  sides  of  which  no  two  are  parallel. 

To   Compute   Area  of  a   Trapezium.— Fig.  12. 
RULE. — Multiply  diagonal  by  sum  of  the  two  perpendiculars  falling  upon 
it  from  the  opposite  angles,  and  divide  product  by  2. 

Or, a~*~c  =  area. 

Fig.  12.    a 

/  \  ^^^,       EXAMPLE.— Diagonal  d  6,  Fig.  12,  is  125  feet,  and  perpen- 
— , .^fl   diculars  a  and  c  50  and  37 ;  what  is  area? 

125  x  50  -|-  37  =  10  875,  and  10  875  -f-  2  =  5437. 5  square  feet. 
FP 


338      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

When  the  Two  opposite  Angles  are  Supplements  to  each  other,  that  is,  when 
a  Trapezium  can  be  inscribed  in  a  Circle,  the  Sum  of  its  opposite  Angles 
being  equal  to  Two  Right  Angles,  or  180°.  RULE.— From  half  sum  of  the 
four  sides,  subtract  each  side  severally ;  then  multiply  the  four  remainders 
continually  together,  and  take  square  root  of  product. 

EXAMPLE.— In  a  trapezium  the  sides  are  15, 13,  14,  and  12  feet:  its  opposite  an- 
gles being  supplements  to  each  other,  required  its  area. 

XS  + 13  + 14  + 12  =  54,  and  —  =  27. 
27     27     27     27  2 

15     13     f4     12 

12  X  14  X  13  X  15  =  32  760,  and  -^32  760=  180.997  square  feet. 

Trapezoid.. 

DEFINITION.— A  Quadrilateral  with  only  one  pair  of  opposite  sides  parallel 

To   Compute  Area   of  a   Trapezoid.— Fig,  13. 
RULE. — Multiply  sum  of  the  parallel  sides  by  perpendicular  distance  be- 
tween them,  and  divide  product  by  2. 

I  ex  ah      .    s+s'xh 

Or, =  area,  s  and  s'  representing  sides. 

'"  X3'  ?          6         .ft      EXAMPLE.  —Parallel  sides  a  b,  c  d,  Fig.  13,  are  100  and  132 
feet,  and  distance  between  them  62.5  feet;  what  is  area? 
100+132  x  62.5  =  14  500,  and  14  500  -r-  2  =  7250  square 

Polygons. 

DEFINITION. —Plane  figures  having  three  or  more  sides,  and  are  either  regular  or 
irregular,  according  as  their  sides  or  angles  are  equal  or  unequal,  and  they  are  named 
from  the  number  of  their  sides  and  angles. 

Regular  Polygons. 

To   Compute    Area   of*  a   Regular   Polygon.  —  Fig.  14. 
RULE.— Multiply  length  of  a  side  by  perpendicular  distance  to  centre; 
multiply  product  by  number  of  sides,  and  divide  it  by  2. 

>t ce*n  =  area,  n  representing  number  of  sides. 

*&•  *4-       «  EXAMPLE.— What  is  area  of  a  pentagon,  side  a  6,  Fig.  14,  being 

5  feet,  and  distance  c e  4.25  feet? 

5  x  4-  25  X  5  (n)  =  106. 25  =  product  of  length  of  a  side,  dis- 
tance to  centre,  and  number  of  sides. 

Then,  106.25  -f-  2  =  53.125  square  feet. 

To  Compute   Radius  of  a  Circle  that  contains   a  Q-iven 
Polygon. 

When  Length  of  a  Perpendicular  from  Centre  alone  is  given.  RULE. — 
Multiply  distance  from  centre  to  a  side  of  the  polygon,  by  unit  in  column  A 
of  following  Table. 

EXAMPLE. — What  is  radius  of  a  circle  that  contains  a  hexagon,  distance  to  centre 
being  4.33  inches? 

4-33X  1.156  =  5  ins. 

To  Compute  Length  of  a  Side  of  a  Polygon  that  is  con- 
tained   in    a    GHven    Circle. 

When  Radius  of  Circle  is  given.  RULE. — Multiply  radius  of  circle,  by 
unit  in  column  B  of  following  Table. 

EXAMPLE.— What  is  length  of  side  of  a  pentagon  contained  in  a  circle  8.5  feet  in 
diameter? 

8.5-7-2  =  4.25  radius,  and  4.25  x  1.1756  =  5  fiet. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       339 

To   Compute    Radius   of*  a   Circumscribing   Circle. 

When  Length  of  a  Side  is  given.    RULE. — Multiply  length  of  a  side  of  the 
polygon,  by  unit  in  column  C  of  following  Table. 

EXAMPLE.— What  is  radius  of  a  circle  that  will  contain  a  hexagon,  a  side  being  5 
inches? 

5X1  =  5  ins. 

To  Compute   Radius   of  a  Circle   that   can  "be   Inscribed 
in   a   Griven   IPolygon. 

When  Length  of  a  Side  is  given.    RULE. — Multiply  length  of  a  side  of 
polygon,  by  unit  in  column  D  of  following  Table. 

EXAMPLE.— What  is  radius  of  the  circle  that  is  bounded  by  a  hexagon,  its  sides 
being  5  inches? 

5  X- 866  =  4.33  ins. 

To    Compute    A.rea   of*  a   Regular   IPolygon. 

Wlten  Length  of  a  Side  only  is  given.     RULE. — Multiply  square  of  side, 
by  multiplier  opposite  to  term  of  polygon  in  following  Table : 


No.  of 

Sides. 

POLYGON. 

AKEA. 

A. 
Radius  of 
Circumscribed 

B. 

Length  of  a 

C. 

Radios  of 
Circumscrib- 

D. 

Radius  of 
Inscribed 

Circle. 

Side. 

ing  Circle. 

Circle. 

3 

Trigon 

•43301 

i  732 

•5773 

.2887 

4 

Tetragon 

I 

.414 

1.4142 

.7071 

•5 

5 

Pentagon 

1.72048 

.238 

1.1756 

.8506 

.6882 

6 

Hexagon 

2.59808 

.156 

I 

.866 

7 

Heptagon 

3-633  91 

.11 

.8677 

.1524 

•0383 

8 

Octagon 

4.82843 

.083 

•7653 

.3066 

.2071 

9 

Nonagon 

6.18182 

.064 

.684 

.4619 

•3737 

10 

Decagon 

7.69421 

.051 

.618 

.618 

.5388 

ii 

Undecagon 

9-36564 

.042 

•5634 

•7747 

7028 

12 

Dodecagon 

11.196  15 

037 

•5176 

.9319 

.866 

EXAMPLE.  —  What  is  area  of  a  square  (tetragon)  when  length  of  its  sides  is 
7  0710678  inches? 

7.071 067  82  =  50,  and  50  X  i  =  50  square  ins. 

To    Compute   Length  of  a   Side  and    Radii  of  a  Regular 
Polygon. 

When  Area  alone  is  given.  RULE. — Multiply  square  root  of  area  of  poly- 
gon by  multiplier  in  column  E  of  the  following  table  for  length  of  side ;  by 
multiplier  in  column  G  for  radius  of  circumscribing  circle,  and  by  multiplier 
in  column  H  for  radius  of  inscribed  circle  or  perpendicular. 


No.  of 
Sides. 

POLYGON. 

E. 

Length  of 
Side. 

G. 

Radius  of 
Circumscrib- 
ing Circle. 

H. 

Radius  of 
Inscribed 
Circle. 

Angle. 

Angle  of 
Polygon. 

Tangent. 

3 

Trigon 

I.5I97 

.8774 

•4387 

120° 

60° 

•5774 

4 

Tetragon 

I 

.7071 

•  5 

90 

9° 

I 

5 

Pentagon 

.7624 

.6485 

•5247 

72 

1  08 

I-3764 

6 

Hexagon 

.6204 

-6204 

•5373 

60 

120 

1.7321 

7 

Heptagon 

.5246 

.6045 

•5446 

51    25.71' 

128   34.29' 

2.0765 

8 

Octagon 

•4551 

.5946 

•5493 

45 

135 

2.4142 

9 

Nonagon 

.4022 

.588 

•5525 

40 

140 

2-7475 

10 

Decagon 

.3605 

•5833 

•5548 

36 

144 

3-0777 

ii 

Undecagon 

.3268 

•5799 

•5564 

32  43-64' 

147    16.36' 

3-4057 

12 

Dodecagon 

.2989 

•5774 

•5577 

30 

'SO 

3-7321 

EXAMPLE  i.— Area  of  a  square  (tetragon)  is  16  inches;  what  is  length  of  its  side? 

•V/i6  =  4,  and  4X1=4  ins. 

2.— Area  of  an  octagon  is  70.698  yards;  what  is  diameter  of  its  circumscribing 
circle? 

•^70.698  X  .  5946  =  5,  and  5  x  2  =  10  yards. 


34O      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

Additional  Uses  of  foregoing  Table. — 6th  and  ;th  columns  of  table  facilitate  con- 
struction of  these  figures  with  aid  of  a  sector.  Thus,  if  it  is  required  to  describe  an 
octagon,  opposite  to  it  in  column  6th,  is  45;  then,  with  chord  of  60  on  sector  as 
radius,  describe  a  circle,  taking  length  45  on  same  line  of  sector;  mark  this  dis- 
tance off  on  the  circumference,  which,  being  repeated  around  the  circle,  will  give 
points  of  the  sides. 

yth  column  gives  angle  which  any  two  adjoining  sides  of  the  respective  figures 
make  with  each  other;  and  8th  gives  tangent  of  .5  angle  in  column  yth. 

To     Coxxipu.te     Radius     of    Inscribed,     or     Circumscribed. 

Circles. 

When  Radius  of  Circumscribing  Circle  is  given.  RULE. — Multiply  radius 
given  by  unit  in  column  E,  in  following  Table,  opposite  to  term  of  polygon 
for  which  radius  is  required. 

When  Radius  of  Inscribed  Circle  is  given.  RULE.— Multiply  radius  given 
by  unit  in  column  F,  in  following  Table,  opposite  to  term  of  polygon  for 
which  radius  is  required. 

To    Compute   Area. 

When  Radii  of  Inscribed  or  Circumscribing  Circles  are  given.  RULE. — 
Square  radius  given,  and  multiply  it  by  unit  in  columns  G  or  H,  in  following 
Table,  and  opposite  to  term  of  polygon  for  which  area  is  required. 

When  Length  of  a  Side  is  given.  RULE.  —  Square  length  of  side  and 
multiply  it  by  unit  in  column  I,  in  following  Table,  opposite  to  term  of 
polygon  for  which  area  is  required. 

To    Compute   T-jength.   of*  a    Side. 

When  Radius  of  Inscribed  Circle  is  given.  RULE. — Multiply  radius  given 
by  unit  in  column  K,  in  following  Table,  and  opposite  to  term  of  polygon  for 
which  length  is  required. 


E. 

F. 

G. 

H. 

I. 

K. 

Radius  of 

Radius  of 

Area. 

Length  of 

No.  of 
Sides. 

POLYGON. 

Inscribed 
by  Circum- 
scribing 

Circumscrib- 
ing by 
Inscribing 

Area. 
By  Radius 
of  Inscribed 
Circle. 

By  Radius 
of  Circum- 
scribing 

Area. 
By  Length 
of  Side. 

Side. 
By  Radius 
of  Inscribed 

Circle. 

Circle. 

Circle. 

Circle. 

3 

Trigon 

•5 

5-  196  15 

1.29904 

•43301 

3.4641 

4 

Tetragon 

.707  II 

.41421 

4 

2 

i 

2 

I 
I 

9 

Pentagon 
Hexagon 
Heptagon 
Octagon 
Nonagon 

.80902 
.86602 
.90097 
.92388 
.93969 

.23607 

•1547 
.10992 
.08239 
.064  18 

3.63272 
3.4641 
3.37102 
3-3I37I 
3-27573 

2.37764 
2.59808 
2.73641 
2.82842 
2.89254 

1.72048 
2.59808 

3-63391 
4.81843 
6.18282 

1.45308 
I-I547 
•963I5 
.82843 

.727  94 

10 

ii 

13 

Decagon 
Undecagon 
Dodecagon 

.95106 
•95949 
•96593 

.05146 
.04222 
.035  28 

3.2492 
3.22989 
3-21539 

2-93893 

2-97353 
3 

7.69421 
9-36564 
11.19615 

.649  84 
•58725 
•5359 

Regular   Bodies. 

To  Compnte  Surface  or  Linear  Edge  of  Regular  Body. 
RULE.— Multiply  square  of  linear  edge,  or  radius  of  circumscribed  or  in- 
scribed sphere,  by  units  in  following  table,  under  head  of  dimension  used : 


No.  of 
Sides. 

BODY. 

Surface  by 
Linear  Edge. 

Radius  of 
Circumscribed 
Sphere. 

Radius  of 
Inscribed 
Sphere. 

Linear  Edge 
by  Surface. 

I 

Tetrahedron 
Hexahedron 

1.73205 
6 

1.63299 
J-  154  7 

4.  898  98 
2 

•75984 

8 

12 
2O 
EVrii 

Octahedron 
Dodecahedron 
Icosahedron 

3.4641 
20.645  78 
8.66025 

1.41421 
•71364 
1.051  46 

2-44949 
.89806 

1-32317 

•53729 
.22008 
•3398i 

EXAMPLE.— What  is  surface  of  a  hexahedron  or  cube,  having  sides  of  5  inches? 
S2  X  6  =  25  X  6  =  150  square  ins. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       34! 

To    Compute    Linear   Edge. 

When  Surface  alone  ig  given.  RULE.— Multiply  square  root  of  surface, 
by  multiplier  opposite  to  term  of  body  under  head  of  Linear  Edge  by  Sur- 
face in  preceding  Table. 

EXAMPLE.— What  is  linear  edge  of  a  hexahedron,  surface  being  6  inches? 
V6  X  -408 25  =  i  inch. 

When  Radius  of  an  Inscribed  or  Circumscribed  Sphere  is  given.  RULE. — 
Multiply  radius  given,  by  multiplier  opposite  to  term  of  body  in  preceding 
Table,  under  head  of  the  Radius  given. 

EXAMPLE.— Radius  of  circumscribing  sphere  of  a  hexahedron  is  10  inches;  what 
is  its  linear  edge? 

10  X  i.iS47  =  II-547  ins* 

To   Compute   Surface. 

When  Linear  Edge  is  given.  RULE. — Multiply  square  of  edge,  by  multi- 
plier opposite  to  term  of  body  hi  preceding  Table,  under  head  of  Surface. 

EXAMPLE. — Linear  edge  of  a  hexahedron  is  i  inch;  what  is  its  surface? 
•    i2  x  6  =  6  square  ins. 

Irregidar  Polygons. 
DEFINITION.— Figures  with  unequal  sides. 

To    Compute    .A^rea   of   an    Irregular    Polygon.— Figs.  15 
and.  16. 

RULE. — Draw  diagonals  and  per- 
pendiculars, as.  d  /*,  dg,  a,  and  c,  Fig. 
15,  and/d,  g  d,  g\  g  M,  and  i,  o,  r,  and 
*,  Fig.  16,  to  divide  the  figures  into 
triangles  and  quadrilaterals:  ascer- 
tain areas  of  these  separately,  and 
take  their  sum. 

NOTE.— To  ascertain  area  of  mixed  or  compound  figures,  or 
such  as  are  composed  of  rectilineal  and  curvilineal  figures  to- 
gether, compute  areas  of  the  several  figures  of  which  the  whole  is  composed,  then 
add  them  together,  and  the  sum  will  give  area  of  compound  figure.    In  this  manner 
any  irregular  surface  or  field  of  land  may  be  measured  by  dividing  it  into  trapeziums 
and  triangles,  and  computing  area  of  each  separately. 

When  any  Part  of  a  Figure  is  bounded  by  a  Curve  the  Area  may  be  ascer- 
tained as  follows: 

Erect  any  number  of  perpendiculars  upon  base,  at  equal  distances,  and 
ascertain  their  lengths. 

Add  lengths  of  the  perpendiculars  thus  ascertained  together,  and  their 
sum,  divided  by  their  number,  will  give  mean  breadth ;  then  multiply  mean 
breadth  by  length  of  base. 

To  Compute  Area  of  a  Long,  Irregular  Figure.— Fig.  17". 

Fig.  17.  RULE. — Take  mean  breadths  at  several  places,  at  equal 

distances  apart,  as  i,  2,  3,  b  dr  etc. ;  add  them  together, 
divide  their  sum  by  number  of  breadths  for  total  mean 
breadth,  and  multiply  quotient  by  length  of  figure. 

6",  etc. 
— • x  I  =  area. 


342      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

To    Coiio.pu.te    an    Area   "bounded,    toy    a    Curve.  —  3T"ig.  18 

(Simpsons  Rale.) 

OPERATION.— Divide  line  a  b  into  any  number  of  equal  parts, 
D7  perpendiculars  from  base,  as  i,  2,  3,  etc.,  which  will  give 
an  odd  number  of  points  of  division.     Measure  lengths  of 
a    i    2    3    4    5     £  these  perpendiculars  or  ordinates,  and  proceed  as  follows: 

To  sum  of  lengths  of  first  and  last  ordinates,  add  four  times  sum  of  lengths  of  all 
even  numbered  ordinates  and  twice  sum  of  odd;  multiply  their  sum  by  one  third 
of  distance  between  ordinates,  and  product  will  give  area  required. 

ILLUSTRATION.— Water-line  of  a  vessel  has  a  length  of  80  feet,  and  ordinates  o,  i, 
1.2, 1.5,  2,  i. 9,  1.5,  i.i,  and  o,  each  10  feet  apart;  what  is  its  area? 

Ordinatt*. 

Even.  Odd.  Sums, 

i  1.2  first   o 

1.5  2  last  o 

1.9  1.5  even  22 

i.i  odd   9.4 

575X4  =  22.    4.7X2  =  9.4  31- 4  X  10  =  314,  Vfh\ch--r  3  =  104. 66  square  feet 

Circle. 

Diameter  is  a  right  line  drawn  through  its  centre,  bounded  by  its  periphery. 

Radius  is  a  right  line  drawn  from  its  centre  to  its  circumference. 

Circumference  is  assumed  to  be  divided  into  360  equal  parts,  termed  degrees; 
each  degree  is  divided  into  60  parts,  termed  minutes;  each  minute  into  60  parts, 
termed  seconds  ;  and  each  second  into  60  parts,  termed  thirds,  and  so  on. 

To   Compute    Circumference   of  a   Circle. 
RULE. — Multiply  diameter  by  3.1416. 

Or,  as  7  is  to  22,  so  is  diameter  to  circumference. 
Or,  as  113  is  to  355,  &Q  is  diameter  to  circumference. 
EXAMPLE. — Diameter  of  a  circle  is  1.25  inches;  what  is  its  circumference? 
1.25  X  3.1416  =  3.927  ins. 

To   Compute   Diameter   of  a   Circle. 
RULE. — Divide  circumference  by  3.1416. 

Or,  as  22  is  to  7,  so  is  circumference  to  diameter. 
NOTE.— Divide  area  by  .7854,  and  square  root  of  quotient  will  give  diameter  of  circle. 

To    Compute    A_rea   of   a    Circle. 
RULE. — Multiply  square  of  diameter  by  .7854. 

Or,  multiply  square  of  circumference  by  .07958. 
Or,  multiply  half  circumference  by  half  diameter. 
Or,  multiply  square  of  radius  by  3. 1416. 
Or,  p  r2  =  area,  r  representing  radius. 

EXAMPLE. — The  diameter  of  a  circle  is  8  inches;  what  is  the  area  of  it? 
82  =  64,  and  64  X  .7854  =  50.2656  ins. 

Proportions    of  a    Circle,  its    Kc^ual,  Inscribed,  and    Cir» 
cumscrilbed    Sq.uares. 

CIRCLE. 

1.  Diameter          X   .8862)       «id      f      F       ,  qniiarft 

2.  Circumference  X    .2821 }  -          Ol  an  Lquai  b(luare- 

3.  Diameter  X    .7071) 

4.  Circumference  X   .2251  [  =Side  of  Inscribed  Square. 

5.  Area  x  .9003 -i-diam.     ) 

6.  Diameter          X  1-3468    =  Side  of  an  Equilateral  Triangle. 

SQUARE. 

7.  A  Side  X  1.4142  =  Diameter  of  its  Circumscribing  Circle. 

8.  "  X  4.443  =  Circumference  of  its  Circumscribing  Circle. 

9.  u  X  1-128   =  Diameter          ) 

10.       "  X3-545   =  Circumference^  of  an  Equal  Circle. 

u.  Square  inches  X  1.273   =  Circle  inches    ) 

NOTE. — Square  described  within  a  circle  is  one  half  area  of  one  described  without  it. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       343 

To  Compute    Side   of*  C3-reatest   Square   that  can.  "be  In- 
scribed,  in    a    Circle. 

RULE.— Multiply  diameter  by  .7071,  or  take  twice  square  of  radius. 

TJsefUl   Factors. 

In  -wliicn.  p  or  it   represents    Circumference   of  a   Circle. 
Diameter  =  i. 


p=  3. 141 592653  589+ 

2J>  =    6.283185307179-!- 
4^=12.566370614359+ 

%p—  i.  570  796326  794+ 
XP=     -785  398i63  397+ 


%  P=  -392699+ 

TSP=  -261799+ 

?&)P=   -008726+ 

Diameter  =  10. 


VP=    1-772453 

yj=  -797884 

Log.  p  =      .49714987 

%y/p=      .886226-)- 

36  p=  113.097  335+ 


x.  Chord  of  arc  of  semicircle  =10 

2.  Chord  of  half  arc  of  semicircle  =  7.071067 

3.  Versed  sine  of  arc  of  semicircle.  =  5 

4.  Versed  sine  of  half  arc  of  semicircle  =   1.464  466 

5.  Chord  of  half  arc,  of  half  of  arc  of  semicircle  =  3. 826  83 

6.  Half  chord,  of  chord  of  half  arc  =  3-535533 

7.  Length  of  arc  of  semicircle  =15-707963 

8.  Length  of  half  arc  of  semicircle  =  7.853981 

9.  Square  of  chord,  of  half  arc  of  semicircle  (2)  =  50 

10.  Square  root  of  versed  sine  of  half  arc  (4)  =   1.210151 

11.  Square  of  versed  sine  of  half  arc  (4)  =  2.144664 

12.  Square  of  chord  of  half  arc,  of  half  arc  of  semicircle  (5)  =  14.64467 

13.  Square  of  half  chord,  of  chord  of  half  arc  (6)  =  12. 5 

NOTE.— In  all  computations  #  is  taken  at  3.1416,  %  p  at  .7854,  %p  at  .5236;  and 
whenever  the  decimal  figure  next  to  the  one  last  taken  exceeds  5,  one  is  added. 
Thus,  3.141  59  for  four  places  of  decimals  is  taken  as  3.1416. 

To  Compute   Length,   of  an   Arc   of  a  Circle.— Fig.  19. 

When  Number  of  Degrees  and  Radius  are  given.  RULE  i.  —  Multiply 
number  of  degrees  in  the  arc  by  3.1416  times  the  radius,  and  divide  by  180. 

2.— Multiply  radius  of  circle  by  .01745329,  and  product  by  degrees  in 
the  arc. 

If  length  is  required  for  minutes,  multiply  radius  by  .000  290  889 ;  if  for 
seconds,  by  .000004848. 

*9  EXAMPLE  i.— Number  of  degrees  in  an  arc,  o  a  &,  Fig.  19,  are 

90,  and  radius,  o  6,  5  inches;  what  is  length  of  arc? 

90  X  (3.1416  X  5)  =  1413-72,  which-:- 180  =  7.854  ins, 
2.— Radius  of  an  arc  is  10,  and  measure  of  its  angle  44°  30* 
30";  what  is  length  of  arc? 

10  X  .017  453  29  = .  174  532  g,  which  X  44  =  7-679  447  6,  length 
for  44°. 

xo  X  -ooo  290  889  =  .002  908  89,  which  X  30  =  .087  266  7,  length  for  30'. 
xo  X  .000004  848  =  .000048  48,  which  X  30  =  .001 454  4,  length  for  30''. 

Then  7. 679  447  6) 

.087  266  7  >  =  7.768 168  7  ins. 
.0014544) 

Or,  reduce  minutes  and  seconds  to  decimal  of  a  degree,  and  multiply  by  it. 

See  Rule,  page  93.  30'  30"  =  .5083,  and  .1745329  from  above  X  44.5083  = 
7.768163  ins. 


344      MENSURATION   OF  AREAS,  LINES,  AND  SURFACES. 

When  Chord  of  Half  Arc  and  Chord  of  Arc  are  given.  RULE. — From  eight 
times  chord  of  half  arc  subtract  chord  of  arc,  and  one  third  of  remainder  will 
give  length  nearly. 

Or,  —  — ,  c'  representing  chord  of  half  arc,  and  c  chord  of  arc. 

EXAMPLE.— Chord  of  half  arc,  a  c,  Fig.  19,  is  30  inches,  and  chord  of  arc,  a  6,  48; 
what  is  length  of  arc  ? 
30  X  8  =  240  —  8  times  chord  of  half  arc  ;  240  —  48  =  192,  and  192  -4-  3  =  64  ins. 

When  Chord  of  Arc  and  Versed  Sine  of  Arc  are  given.  RULE.  —  Mul- 
tiply square  root  of  sum  of  square  of  chord,  and  four  times  square  of  the 
versed  sine  (equal  to  twice  chord  of  half  arc),  by  ten  times  square  of  versed 
sine ;  divide  this  product  by  sum  of  fifteen  times  square  of  chord  and  thirty- 
three  times  square  of  versed  sine ;  then  add  this  quotient  to  twice  chord  of 
half  arc,*  and  sum  will  give  length  of  arc  very  nearly. 

„.     Vc2 4-  4  v-  sin- 2  X  10  v.  sin.2  . 
Or, -^  cz~± — v  sin  2 1~ 2  c '  v'  ww'  ^Presenting  versed  sine. 

EXAMPLE.  —Chord  of  an  arc  is  80,  and  its  versed  sine,  c  r,  30 ;  what  is  length  of  arc  ? 
8o2  =  6400  =  square  of  chord  ;  3o2  =  900  =  square  of  versed  sine. 

v/(64oo  -{-  900  x  4)  =  ioo  =  square  root  of  square  of  chord  and  four  times  square 
of  versed  sine  =  twice  chord  of  half  arc. 

Then  ioo  X  so2  X  10  =  goo  ooo  =  product  o/io  times  square  of  versed  sine  and  root 
above  obtained. 

And  8o2  X  15  =  96  ooo  =  15  times  square  of  chord. 

3o2  x  33  =  29  700  =  33  times  square  of  versed  sine. 

12570° 

Hence  2°^22  =  7.1599,  and  7.1599  +  100,  or  twice  chord  of  half  arc  =  107.1599 

125700 
length. 

When  Diameter  and  Versed  Sine  are  given.  RULE. — Multiply  twice  chord 
of  half  the  arc  by  10  times  versed  sine ;  divide  product  by  27  times  versed 
sine  subtracted  from  60  times  diameter,  add  quotient  to  twice  chord  of  half 
arc,  and  the  sum  will  give  length  of  arc  very  nearly. 

|00™._ 

6od  —  27  v.  sin. 

EXAMPLE.— Diameter  of  a  circle  is  ioo  feet,  and  versed  sine,  cr,  of  arc  25  ;  what 
is  length  of  arc? 

V25  x  ioo  =  50  =  chord  of  half  arc.    See  Rule,  page  345. 


50  X  2  X  25  x  io  =  25  ooo  =  twice  chord  of  half  arc  by  10  times  versed  sine. 
ioo  X  60  —  25  X  27  =  5325  =  27  times  versed  sine  from  60  times  diameter. 

Then  2500°  =  4.6948,  and  4.6948  -f  50X2  =  104. 6948  feet. 

To    Compute    Chord,   of*  an   Arc. 

When  Chord  of  Half  the  Arc  and  Versed  Sine  are  given.  RULE. — From 
square  of  chord  of  half  arc  subtract  square  of  versed  sine,  and  take  twice 
square  root  of  remainder. 

Or,  V  (c'2  —  v.  sin.  *)X2  =  c. 

EXAMPLE.— Chord  of  half  arc,  a  c,  is  60,  and  versed  sine,  c  r,  36;  what  is  length 
of  chord  of  arc? 

6o2  —  s62  =  2304,  and  v/23°4  X  2  =  96. 

*  Square  root  of  sum  of  square  of  chord  and  four  times  square  of  the  versed  sine  is  equal  to  twicr 
chord  of  half  arc. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       345 

When  Diameter  and  Versed  Sine  are  given.  Multiply  versed  sine  by  2, 
and  subtract  product  from  diameter;  subtract  square  of  remainder  from 
square  of  diameter,  and  take  square  root  of  that  remainder. 

Or,  V  d2  —  (d  —  v.  sin.  x  2)2  =  c. 

EXAMPLE.—  Diameter  of  a  circle  is  100,  and  versed  sine  of  the  arc  is  36;  what  la 
length  of  chord  of  arc  ? 

(100  —  36  X  2)2  —  ioo2  =  92i6,  and  -^9216  —  96. 

To    Compute    Chord,   of  Half  an   Arc. 
When  Chord  of  the  Arc  and  Versed  Sine  are  given.     RULE  i.  —  Divide 
square  root  of  sum  of  square  of  chord  of  the  arc  and  four  times  square  of 
versed  sine  by  two. 

2.  —  Take  square  root  of  sum  of  squares  of  half  chord  of  arc  and  versed 
sine. 


When  Diameter  and  Versed  Sine  are  given.    RULE.  —  Multiply  diameter 
by  versed  sine,  and  take  square  root  of  their  product. 

Or,  Vdx  v.  sin.  =  c'. 
To   Compute   Diameter. 
RULE  i.  —  Divide  square  of  chord  of  half  arc  by  versed  sine. 

Or,  c'2  -i-  v.  sin.  =z  diameter. 

2.  —  Add  square  of  half  chord  of  arc  to  the  square  of  versed  sine,  and  divide 
this  sum  by  versed  sine. 


v.  sin. 

To   Compute  Versed    Sine. 
RULE.—  Divide  square  of  chord  of  half  arc  by  diameter. 

c'2 

Or,  —  =  v.  sin. 
d 

When  Chord  of  the  Arc  and  Diameter  are  given.  RULE.—  From  square 
of  diameter  subtract  square  of  chord,  and  extract  square  root  of  remainder  ; 
subtract  this  root  from  diameter,  and  divide  remainder  by  2. 


When  it  is  greater  than  a  Semidiameter.  RULE. — Proceed  as  before,  but 
add  square  root  of  remainder  (of  squares  of  diameter  and  chord)  to  diam- 
eter, and  halve  the  sum. 


EXAMPLE.— Diameter  of  a  circle  is  100,  and  chord  of  arc  97.9796;  what  is  its  versed 
sine? 

ioo  -j-  Vioo2  —  97. 97962 ioo-j-2o 

2  2 

To  Compute  Ordinate  of  a  Circular  Curve.—  Fig.  2O. 

Vr2  —  x2  —  (r  —  v)  =  ordinate. 
ILLUSTRATION. — Radius  of  circle  5  ins.,  versed  sine 
\    2,  and  distance  x  2 ;  what  is  length  of  ordinate  o  ? 

V52_22  — (5-2)  =  4. 58-3  =  1.58  ins. 


X 


346      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

Sector   of  a   Circle. 

DEFINITION.— A  part  of  a  circle  bounded  by  an  arc  and  two  radii. 

To   Compute   Area  of*  a   Sector   of  a   Circle. 
When  Degrees  in  the  Arc  are  given. — Fig.  21.     RULE. — As  360  is  to  num- 
ber of  degrees  in  a  sector,  so  is  area  of  circle  of  which  sector  is  a  part  to  area 
of  sector. 

da 

Or, =  area,  d  representing  degrees  in  arc,  and  a  area 

36o 

x  /       of  circle. 

\^       /  EXAMPLE.  —  Radius  of  a  circle,  o  a,  Fig.  21,  is  5  ins.,  and 

\^/  number  of  degrees  of  sector,  a  b  o,  is  22°  30' ;  what  is  area  ? 

o  Area  of  a  circle  of  5  ins.  radius  =  78. 54  ins. 

Then,  as  360°  :  22°  30'  ::  78.54  :  4-9°875  *'«». 

When  Length  of  the  Arc,  etc.,  are  given.  RULE.— Multiply  length  of  arc 
by  half  length  of  radius,  and  product  is  area. 

Or,  b  X  r  -r-  2  =  area,  6  representing  arc,  and  r  radius. 

Segment   of  a   Circle. 

DEFINITION.— A  part  of  a  circle  bounded  by  an  arc  and  a  chord. 

To    Conapnte   Area   of  a    Segment   of  a   Circle. 
When  Chord  and  Versed  Sine  of  Arc,  and  Radius  or  Diameter  of  Circle  are 

given. 

When  Segment  is  less  than  a  Semicircle,  as  a  b  c,  Fig.  21.  RULE. — Ascer- 
tain area  of  sector  having  same  arc  as  segment ;  then  ascertain  area  of  tri- 
angle formed  by  chord  of  segment  and  radii  of  sector,  and  take  difference  of 
these  areas. 

NOTE Subtract  versed  sine  from  radius;  multiply  remainder  by  one  half  of 

chord  of  arc,  and  product  will  give  area  of  triangle. 

Or,  a  —  a'  =  area,  a  and  a'  representing  areas  of  sector  and  triangle. 

When  Segment  is  greater  than  a  Semicircle.  RULE. — Ascertain,  by  pre- 
ceding rule,  area  of  lesser  portion  of  circle ;  subtract  it  from  area  of  whole 
circle,  and  remainder  will  give  area. 

Or,  a  —  a'  =  area,  a  and  a'  representing  areas  of  circle  and  lesser  portion. 
See  Table  of  Areas  of  Segments,  page  267. 

Fig.  22.    ,  EXAMPLE.  —  Chord,  a  c,  Fig.  22,  is  14.142;  diameter,  6  e,  is  20 

ins. ;  and  versed  sine,  br,  is  2.929;  what  is  area  of  segment? 
14. 142  -r-  2  =  7.071  =  half  chord  of  arc. 

V7-o7i2-T-2.9292  =  7.654  =  square  root  of  sum  of  squares  of 
half  chord  of  arc  and  versed  sine,  which  is  chord  a  b  of  half  arc 
ab  c. 

By  Rule,  page  344, 

7.654  X  2  X  2.929  X  10  =  448.371  =  fr/nce  chord  of  half  arc  by  10 
times  versed  sine. 


20X60  —  2.929X27  =  1120.917=60  times  diameter  subtracted  from  27  times 
versed  sine. 

Then  448. 371  -4-1120.917  =  .4,  and  .4  added  to  7.654  x  2  (twice  chord  of  half  arc) 
=  15.708  inches,  length  of  arc. 

By  Rule  above,  15.708  X  —  =  78.54  =  toe  arc  multiplied  by  half  length  of  radius, 

=  area  of  sector. 
I0  —  2. 929  =  7. 071  =  versed  sine  subtracted  from  a  radius,  which  is  height  oftri 

angle  a  o  c,  and  7.071  x  *4'142  =  5°  =  area  of  triangle. 
Consequently,  78. 54  —  50  =  28. 54. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       347 

When  the  Chords  of  Arc ^  and  of  half  of  Arc,  and  Versed  Sine  are  given. 
RULE. — To  chord  of  whole  arc  add  chord  of  half  arc  and  one  third  of  it 
more ;  multiply  this  sum  by  versed  sine,  and  this  product,  multiplied  by 
.404  26,  will  give  area  nearly. 

Or,  c  +  c'  H v.  sin.  x  •  404  26  =  area  nearly. 

EXAMPLE.— Chord  of  a  segment,  a  c,  Fig.  22,  is  28  feet;  chord  of  half  arc,  a  6,  is 
1 5 ;  and  versed  sine,  b  r,  6 ;  what  is  area  of  segment  ? 

28  -f- 15  -f- 15  =  chord  of  arc  added  to  chord  of  half  arc  and  one  third  of  it  more. 

48  x  6  =  288  = product  of  above  sum  and  versed  sine.    Hence  288  x  .404  26  =  116.427 
square  feet. 

When  the  Chord  of  Arc  and  Versed  Sine  only  are  given.  RULE. — Ascer- 
tain chord  of  half  arc,  and  proceed  as  before. 

To  Compute  Chord,  and  Height  of*  a  Segment  of  a  Circle. 

When  Area  is  given.  RULE. — Divide  area  by  square  of  diameter  of  circle, 
take  tab.  height  for  area  from  table  of  Areas  of  Segments  of  a  Circle,  p.  267, 
multiply  it  by  diameter,  and  product  will  give  required  height. 

From  diameter  subtract  height,  multiply  remainder  by  height,  take  square 
root  of  product  and  multiply  it  by  2  for  required  chord. 

Or,  J^  =  (tab.  area  for  height)  Xd  =  h,  and  Vd  —  hxh.  X  2  =  c. 
Circular   Measure.    (See  Rule,  page  1 13.) 

Sphere. 

DEFINITION.— A  figure,  surface  of  which  is  at  a  uniform  distance  from  centre. 
To    Compnte    Convex    Surface    of  a    Sphere.— Fig.  S3. 
Fig.  23.  RULE. — Multiply  diameter  by  circumference,  and  prod* 

uct  will  give  surface. 

Or,  4  p  r 2  =  surface.  *    Or,  p  d2  =  surface. 

EXAMPLE.— What  is  convex  surface  of  a  sphere,  Fig.  23,  hav- 
ing a  diameter,  a  6,  of  10  ins? 

10  x  31-416  =  314.16  square  ins. 

Segment   of  a   Sphere. 

DEFINITION. — A  section  of  a  sphere. 

To  Compute  Surface  of  a  Segment  of  a  Sphere.— Fig.  34r. 
RULE. — Multiply  height  by  the  circumference  of  sphere,  and  add  product 
to  the  area  of  base. 

Or,  2  p  r  h  =  cor: :  zx  surface  alone. 

Fig.  24.  EXAMPLE.  —  Height,  b  o,  of  a  segment,  a  b  c,  Fig.  24,  is  36  ins., 

and  diameter,  6  e,  of  sphere  100 ;  what  is  convex  surface,  and 
what  whole  surface  ? 

36  X  ioo  X  3- 1416  =  n  309. 76  =  height  of  segment  multiplied  by 
circumference  of  sphere. 

To  ascertain  area  of  base  ;  diameter  and  versed  siie  being 
given,  diameter  of  base  of  segment,  being  equal  to  chord  of  arc, 
is,  by  Rule,  page  344, 

ioo  —  36  X  2  =  28 ;  Vioo2  —  282  =  96. 

96* X. 7854  =7238. 2464  =  convex  surface,  and  7238.24644-11309.76=18548.0064 
=  convex  surface  added  to  area  of  base  =  square  ins. 

NOTE.— When  convex  surface  of  a  figure  alone  is  required,  area  or  areas  of  base 
or  ends  must  be  omitted. 

*  p  or  IT  represent*  in  this,  and  in  all  cases  where  it  is  used,  ratio  of  circumference  of  a  circle  to  itf 
diameter,  or  3.1416. 


348       MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

When  the  Diameter  of  Base  of  Segment  and  Height  of  it  are  alone  given. 
RULE. — Add  square  of  half  diameter  of  base  to  the  square  of  height ;  divide 
this  sum  by  height,  and  result  will  give  diameter  of  sphere. 

Or,  d-r-2  +  h2-r-h  =  diameter. 

Spherical    Zone   (or   IBVustrum   of  a   Sphere). 

DEFINITION.— The  part  of  a  sphere  included  between  two  parallel  chords. 

To    Compute    Surface   of  a    Spherical    Zone.— Wig.  25* 

Fig.  25. ^  RULE. — Multiply  height  by  the  circumference  of  sphere, 

and  add  product  to  area  of  the  two  ends. 
Or,  h  c  -f-  a  -f-  a'  =  surface. 
Or,  2prJi  =  convex  surface  alone. 

EXAMPLE.  —  Diameter  of  a  sphere,  a  &,  Fig.  25,  from  which  a 
zone,  c  g,  is  cut,  is  25  inches,  and  height,  c  g,  is  8 ;  what  is  convex 
surface  ? 
25  X  3.1416  X  8  =  628. 32  =  height  X  circumference  of  sphere  =  square  ins. 

When  the  Diameter  of  Sphere  is  not  given.  RULE. — Multiply  mean  length 
of  the  two  chords  by  half  their  difference ;  divide  this  product  by  breadth 
of  zone,  and  to  quotient  add  breadth.  To  square  of  this  sum  add  square  of 
lesser  chord,  and  square  root  of  their  sum  will  give  diameter  of  sphere. 


Spheroids    or   Ellipsoids. 

DEFINITION.— Figures  generated  by  the  revolution  of  a  semi- ellipse  about  one  of 
its  diameters. 

When  revolution  is  about  Transverse  diameter  they  are  Prolate,  and  when  it  is 
about  Conjugate  they  are  Oblate. 

To    Compute    Surface    of  a    Spheroid.— Fig.  Q6. 

When  Spheroid  is  Prolate.  RULE.  —  Square  diameters,  and  multiply 
square  root  of  half  their  sum  by  3.1416,  and  this  product  by  conjugate 
diameter. 

/d2  I  d'2 

Fig.  26        c  Or,     / — — —  X3-i4i6xd  — surface,  d  and  d'  represent- 

ing conjugate  and  transverse  diameters. 

r       EXAMPLE.— A  prolate  spheroid,  Fig.  26,  has  diameters,  c  d 
and  a&,  of  io  and  14  inches;  what  is  its  surface? 

io2  -f- 142  =  296  —  sum  of  squares  of  diameters. 
296-7-2  =  148,  and  -v/H^  =  12. 1655  =  square  root  of  half 
sum  of  squares  of  diameters. 
12.1655  X  3-1416  X  10=  382.191  in*.  =  product  of  root  above  obtained  X  3-1416, 
and  by  conjugate  diameter. 

When  Spheroid  is  Oblate.  RULE. — Square  diameters,  and  multiply  square 
root  of  half  their  sum  by  3.1416.  and  this  product  by  transverse  diameter. 

/d2-\-d'2 
Or,     / — -£ —  x  3. 1416  X  d'  =  surface. 

EXAMPLE.— An  oblate  spheroid  has  diameters  of  14  and  io  inches;  what  is  its 
surface  ? 

i22  -f-  io2  =  296  =  sum  of  squares  of  diameters. 

296 -=-2  =  148,  and  -,/ 14%  =  12. 1655  =  square  root  of  half  sum  of  squares  of  di- 
ameter. 

12.1655  X  3.1416  x  14  =  535.0679  in».=product  of  root  above  obtained  X  3.1416, 
and  by  transverse  diameter. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.      349 


To  Compute  Convex   Surface  of  a  Segment   of  a    Sphe- 
roid.—Figs.  27  and   3S. 

RULE. — Square  diameters,  and  take  square  root  of  half  their  sum ;  then, 
as  diameter  from  which  the  segment  is  cut  is  to  this  root,  so  is  the  height 
of  segment  to  proportionate  height  required.  Multiply  product  of  other  di- 
ameter and  3.1416  by  proportionate  height  of  segment,  and  this  last  product 
will  give  surface. 

Yd2 -I- d'2  — 2 

-  x  h  x  d'  or  d  X  3- 1416  =  surface. 


Fig.  27. 


Fig.  *& 


dor  a, 

EXAMPLE.  —  Height,  a  o,  of  a  seg- 
ment, ef,  of  a  prolate  spheroid,  Fig. 
27,  is  4  inches,  diameters  being  10  and 
14;  what  is  convex  surface  of  it? 

Square  root  of  half  sum  of  squares  a[ 

of  diameters,  12.1655. 

Then  14 : 12. 1655 : :  4 :  3. 4758  =  height 
of  segment,  proportionate  to  mean  of 
diameters,  and  10  X  3- 1416  X  3-4758  =  109. 1957  ins. 

2. — Height,  co,  of  a  segment  of  an  oblate  spheroid,  Fig.  28,  is  4  inches,  the  diam 
eters  being  14  and  10;  what  is  convex  surface  of  it?  214.0272  square  ins. 

To    Compute    Convex    Sur/ace    of  a    Frustum    or    Zone 
of  a    Spheroid.— Figs.  39    and    3O. 

RULE. — Proceed  as  by  previous  rule  for  surface  of  a  segment,  and  obtain 
proportionate  height  of  frustum ;  then  multiply  product  of  diameter  par- 
allel to  base  of  frustum  and  3.1416  by  proportionate  height  of  frustum,  and 
it  will  give  surface. 


Fig.  29. 


EXAMPLE.—  Middle  frustum,  o  e,  of 
a  prolate  spheroid,  Fig.  29,  is  6  inch- 
es, diameters  of  spheroid  being  10 
and  14;  what  is  its  convex  surface? 

Mean  diameter,  as  per  preceding 
example,  is  12.1655. 

Diameter  parallel  to  base  of  frus- 
tum is  10. 


Fig.  30. 


Then  14  :  12.1655  : 


;  6  :  5.2138,  and  10  X  3.1416  X  5.2138=1  163.7967  square  ins. 

2.—  Middle  frustum  of  an  oblate  spheroid,  as  o  e,  Fig.  30,  is  2  inches  in  height, 
diameters  of  spheroid,  as  in  preceding  examples,  being  10  and  14  ;  what  is  its  con- 
vex surface?  107.0136  square  ins. 

Circular   Zone. 

DEFINITION.—  A  part  of  a  circle  included  between  two  parallel  chords. 
To   Compute   Area   of  a   Circular   Zone. 
RULE.  —  From  area  of  circle  subtract  areas  of  segments. 
Or,  see  Table  of  Areas  of  Zones,  page  269. 

When  Diameter  of  Circle  is  not  given.  —  Multiply  mean  length  of  the  two 
chords  by  half  their  difference  ;  divide  this  product  by  breadth  of  zone,  and 
to  quotient  add  the  breadth. 

To  square  of  this  sum  add  square  of  lesser  chord,  and  square  root  of  their 
sum  will  give  diameter  of  circle. 

EXAMPLE.—  Greater  chord,  Jig,  is  90  inches;  lesser,  a  c,  is  80;  and  breadth  of  zone, 
ao,  is  72.526;  what  is  its  diameter? 

e  =  8s  x  5  =  4*5,  and  -         +  ,,^8.385. 


Then  A/78.  385  2-f8o2=  V12  544-  2  =  112  =  diameter. 

GQ 


35O      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


Cylinder. 

DEFINITION.— A  figure  formed  by  revolution  of  a  right-angled  parallelogram  around 
one  of  its  sides. 

To    Compute    Sxirface    of  a    Cylinder.— -Fig.  31. 
RULE. — Multiply  length  by  circumference,  and  add  product  to  area  of 
the  two  ends. 

Or,  I  c  -f-  2  a  =. •.  s,  a  representing  area  of  end. 

NOTE.— When  internal  or  convex  surface  alone  is  wanted,  areas  of 
ends  are  omitted. 

EXAMPLE.— Diameter  of  a  cylinder,  b  c,  Fig.  31,  is  30  inches,  and  its 
length,  a  b,  50;  what  is  its  surface? 

30  X  3.1416  =  94.248,  and  94.248  X  50  =  4712.4. 
Then  3o2  X  .7854  —  706.86  =area  of  one  end;  706.86  X  2  =  1413.72 
=  area  of  both  ends,  and  4712.4  + 1413.72  =  6125. 12  square  ins. 

Frisrns. 

DEFINITION.  —  Figures,  sides  of  which  are  parallelograms,  and  ends  equal  and 
parallel. 

NOTE. — When  ends  are  triangles,  they  are  termed  triangular  prisms ;  when  they 
are  square,  square  or  right  prisms  ;  and  when  they  are  a  pentagon,  pentagonal 
prisms,  etc. 

To  Compute  Surface  of  a  Right  3?rism.— Figs.  33  and.  33. 


Fig.  32. 


Fig.  33- 


RULE.  —  Ascertain  areas  of  ends  and  sides,  and 
add  them  together. 

Or,  2  a-}-  n  a'  .•=  s,  a  representing  area  of  ends,  a'  area 
of  sides,  and  n  their  number. 

EXAMPLE.  —Side,  a  b,  Fig.  32,  of  a  square  prism  is  12 
inches,  and  length,  &  c,  30;  what  is  surface? 

12X12  =  144  =  area  of  one  end  ;  144  x  2  =  288  =  area 
of  both  ends  ;  12  X  30  =  360  =  area  of  one  side  ;  360  X  4  = 
1440  =  area  of  four  sides,  and  288  +  1440=  1728  sq.  ins. 

To  Compute  Surface  of  an  Otolique  or  Irregular  iPrism.— 

Fig.  34. 

RULE.—  Multiply  perimeter  of  one  end,  by  perpendic- 
ular height,  a  o.  Or,  multiply  perimeter  as  at  c,  at  a  right 
angle  to  sides  by  actual  length  of  figure,  and  add  area  of  ends. 
EXAMPLE.—  Sides,  a  c,  of  an  oblique  hexagonal  prism,  Fig.  34,  are 
10  inches,  and  perpendicular  height,  a  o,  is  5  feet;  what  is  its  sur- 
face? 

10  X  6  =  60  ins.  =.  length  of  sides. 

60  X  5  X  12  =  3600  square  ins.=:area  of  sides,  and  by  table,  page 
339,  zoo  X  2.  598  08  X  2  =  519.616  square  ins.,  which  added  to  3600  = 
4119.616  square  ins. 


DEFINITION.—  A  wedge  is  a  prolate  triangular  prism,  and  its  surface  is  computed 
by  rule  for  that  of  a  right  prism. 

To   Compute   Surface   of  a  "Wedge.—  Fig.  35. 
Fig.  35.  EXAMPLE.—  Back  of  a  wedge,  abed,  Fig.  35,  is  20  by  2  inches, 

and  its  end,  ef,  20  by  2;  what  is  its  surface? 

i 

2o2  -}-  2  —  j  =  401  =  sum  of  squares  of  half  base,  af,  and 
height,  ef,  of  triangle,  efa. 

1/401  =  20.025  =  square  root  of  above  sum  =.  length  ofea. 
Then  20.025  X  20  X  2  =  801  =  area  of  sides. 

And  20X2  =  40  =  area  of  back;  and  20  X  2-^-2  X2  =  4o  = 
area  of  ends.    Hence  801  -f-  40  -f-  40  =  88  1  square  ins. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       351 


3?rism.oicls. . 

DEFINITION.— Figures  alike  to  a  prism,  having  only  one  pair  of  sides  parallel 
To   Compute    Surface   of  a   IPrismoid.—  Fig.  36. 

RULE.  —  Ascertain  area  of  sides  and  ends  as  by  rules  for 
squares,  triangles,  etc.,  and  add  them  together. 

EXAMPLE.— Ends  of  a  prismoit',  efg  h  and  a  b  c  d,  Fig.  36,  are  10  and 
8  inches  square,  and  its  slant  height,  d  h,  25;  what  is  its  surface? 
10  X  10=  ioo  —  area  of  base  ;  8  X  8=64  =  area  of  top. 

:>-±—  X  25  =:  225,  and  225  X  4  =  900= area  of  sides. 

Then  ioo  -f-  64  +  9<x>  =  1064  =  square  ins. 
To  Compute  Surface  of  an.  Ot>liq.ne  or  Irregular  IPrismoid. 

Proceed  as  directed  for  an  Oblique  or  Irregular  Prism,  page  350. 

TJngulas. 

DEFINITION.— Cylindrical  ungulas  are  the  parts  (including  all  or  part  of  the  base) 
left  by  a  plane  cutting  a  cylinder  through  any  portion  and  at  any  angle. 


TJngxila.— Figs.  37, 
RULE  i.—  Mul- 


Fig-37- 


Fig.  38. 


To    Compute    Curved    Surface   of  an 
38,  39,  and   4O. 

When  Section  is  parallel  to  Axis  of  the  Cylinder,  Fig.  37, 
tiply  length  of  arc  of  one  end  by  height. 

EXAMPLE.— Diameter  of  a  cylinder,  a  c,  from  which  an 
ungula,  Fig.  37,  is  cut,  is  10  inches,  its  length,  6  d,  50,  and 
versed  sine  or  depth  of  ungula  is  5  inches;  what  is  curved 
surface  ? 

10 -f-  2  =  5  =  radius  of  cylinder. 

Hence  radius  and  versed  sine  are  equal;  the  arc,  there- 
fore, of  ungula  is  one  half  circumference  of  the  cylinder, 
which  is  31.416-7-  2  =  15-708,  und  15.708  X  50  =  785.4 
square  ins. 

When  Section  passes  obliquely  through  opposite  Sides  of  Cyl- 
inder, Fig.  ^8.  RULE  2. — Multiply  circumference  of  base  of  cylinder  by 
half  sum  of  greatest  and  least  heights  of  ungula. 

EXAMPLE.— Diameter,  cd,  of  a  cylindrical  ungula,  Fig.  38,  is  10  inches,  and  great- 
er and  less  heights,  b  d  and  a  c,  are  25  and  15  inches;  what  is  its  curved  surface? 

10  diameter  =  31.416  circumference;  25+15  =40,  and  40-^ 2  =  20.  Hence  31.416 
X  20  =  628.32  square  ins. 

When  Section  passes  through  Base  of  Cylinder  and  one  of  its  Sides,  and 
Versed  Sine  does  not  exceed  Sine,  or  Base  is  equal  to  or  less  than  a  Semi- 
circle, Fig.  39.  RULE  3. — Multiply  sine,  a  d,  of  half  arc,  d  g,  of  base,  d  g  f, 
by  diameter,  e  g,  of  cylinder,  and  from  this  product  subtract  product  *  of  arc 
and  cosine,  a  o.  Multiply  difference  thus  found  by  quotient  of  height,  g  c, 
divided  by  versed  sine,  dg. 
NOTE.— The  sine  of  base  is  half  of  the  longest  chord  that  can  be  drawn  in  base. 

EXAMPLE.— Sine,  a  d,  of  half  arc  of  base  of  an  ungula,  Fig.  39,  is  5, 
diameter  of  cylinder,  eg,  is  10,  and  height,  eg,  of  ungula  10  inches; 
what  is  curved  surface  ? 

5  x  10  =  50  =  sine  of  half  arc  by  diameter. 

Length  of  arc,  versed  sine  and  radius  being  equal,  under  Rule,  page 
346  =  15.708,  and  as  versed  sine  and  radius  are  equal,  cosine  is  o. 

Hence,  when  cosine  is  o,  product  is  o.  Therefore  50— 0  =  50  =  dif 
ference  of  product  before  obtained  and  product  of  arc  and  cosine,  and 
50  X  10  -r-  5  =  50  X  2  —  ioo  square  ins. 

*  When  the  cosine  is  o.  this  product  is  o. 


352      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

When  Section  passes  through  Base  of  Cylinder,  and  Versed  Sine,  a  g,  ex- 
ceeds Sine,  or  when  Base  exceeds  a  Semicircle,  Fig.  40.  RULE  4.  —  Multiply 
Fig.  40.  sme  °f  half  the  arc  of  base  by  diameter  of  cylinder,  and  to  thfs 

product  add  product  of  arc  and  the  excess  of  versed  sine  over 
the  sine  of  base.  Multiply  sum  thus  found  by  quotient  of 
height  divided  by  versed  sine. 

EXAMPLE.  —  Sine,  a  d,  of  half  arc  of  an  ungula,  Fig.  40,  is  12  inches; 
versed  sine,  ag,  is  16;  height,  c  g,  16;  and  diameter  of  cylinder,  h  g, 
25  inches;  what  is  curved  surface? 

g      12X25  =  300  •=.  sine  of  half  arc  by  diameter  of  cylinder,  and  length 
of  arc  of  base,  Rule,  page  344  =  arcofd  hf—  circumference  of  base  = 


Then  46.392X16  —  12.5  =  162.372,  and  300-1-162.272  =  462.372;  16-:-  16  =  1,  and 
462.372  X  i  =  462.372  square  ins. 

Fig  41  NOTE.—  When  sine  of  an  arc  is  o,  the  versed  sine  is  equal  to  diameter. 
When  Section  passes  obliquely  through  both  Ends  of  Cylinder, 
Fig.  41.  RULE  5.  —  Conceive  section  to  be  continued  to  m,  till  it 
meets  side  of  cylinder  produced  ;  then,  as  difference  of  versed 
sines,  a  e  and  d  o,  of  arcs  of  two  ends  of  ungula  is  to  versed  sine, 
a  e,  of  arc  of  the  less  end,  so  is  height  of  cylinder,  a  d,  to  the 
part  of  side  produced. 

Ascertain  surface  of  each  of  ungulas  thus  found  by  Rules  3 
and  4,  and  their  difference  will  give  curved  surface. 


DEFINITION.  —  Space  between  intersecting  arcs  of  two  eccentric  circles. 
To   Compute   Area  of  a   Lune.—  T^ig.  43. 
RULE.  —  Ascertain  areas  of  the  two  segments  from  which  lune  is  formed, 
and  their  difference  will  give  area. 

EXAMPLE.—  Length  of  chord  a  c,  Fig.  42,  is  20  inches,  height 
c  d  is  3,  and  e  b  2  ;  what  is  area  of  lune  ? 

By  Rule  2,  page  345,  diameters  of  circles  of  which  lune  is 
formed  are  thus  ascertained: 


Then,  by  Rule  for  Areas  of  Segments  of  a  Circle,  page  267, 
Area  of  a  d  c  is  70.5577  sq.  ins. 

"       aec  "  27.1638     " 
Their  difference  43.3939  *£•  in*« 

Cycloid. 

DEFINITION.—  A  curve  generated  by  revolution  of  a  circle  on  a  plane. 

To    Compute    Area   of*  a    Cycloid.—  Fig.  -4:3. 
i£-  43-  __  *   _^  /X  ,--<•  A        RULE.  —  Multiply  area  of  generating  circle  by  3. 


EXAMPLE.—  Generating  circle  of  a  cycloid,  a  6  c,  Fig.  43, 

oid  ? 


"A 

/   has  an  area  of  115.45  sq.  Inches;  what  is  area  of  cycloid 
•"•'c  1 1 5. 45  X  3  =  346. 35  sgware  in*. 

To   Compute   Length   of  a  Cyoloidal   Curve. 
RULE. — Multiply  diameter  of  generating  circle  by  4. 
EXAMPLE.— Diameter  of  generating  circle  of  a  cycloid,  Fig.  43,  is  8  inches;  what 
is  length  of  curve  d  s  c  ? 

8  X  4  =  32  =  product  of  diameter  and  4  =  ins. 

NOTE. — The  curve  of  a  cycloid  is  line  of  swiftest  descent;  that  is,  a  body  will  fall 
through  arc  of  this  curve,  from  one  point  to  another,  in  less  time  than  through  any 
other  path. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       353 

Circular  Rings. 

DEFINITION.— Space  between  two  concentric  circles. 

To  Compute  Sectional  Area  of  a  Circular  Ring.— Fig.  44. 
RULE. — From  area  of  greater  circle  subtract  that  of  less. 

Cylindrical   Rings. 

DEFINITION.  — A  ring  formed  by  curvature  of  a  cylinder. 

To  Compute   Surface  of  a  Cylindrical  Ring.—  Fig.  4.4.. 
RULE. — To  diameter  of  body  of  the  ring  add  inner  diameter  of  the  ring ; 
multiply  this  sum  by  diameter  of  the  body,  and  product  by  9.8696. 

Fig.  44. Or,  c  X  I  =  surface. 

EXAMPLE. — Diameter  of  body  of  a  cylindrical  ring,  a  6,  Fig.  44, 
is  2  inches,  and  inner  diameter,  b  c,  is  18;  what  is  surface  of  it? 

2  -j-  1 8  =  20  =  thickness  of  ring  added  to  inner  diameter. 
20  x  2  X  9  8696  =  sum  above  obtained  X  thickness  of  ring,  and 
that  product  by  9. 8696  =  394. 784  ins. 

Link. 

DEFINITION.— An  elongated  ring. 

To   Compute    Surface   of  a  H.inlz.— Figs.  45   and  46. 
RULE. — Multiply  length  of  axis  of  link  Tby  circumference  of  a  section  of 
body,  a  b. 

Or,  I X  c  =  surface. 

To    Compute    Length   of  Axis    and    Circumference. 

When  Ring  is  Elongated.  RULE. — To  less  diameter  add  the  diameter  of 
the  body  of  the  link,  and  multiply  sum  by  3.1416;  subtract  less  diameter 
from  greater,  multiply  remainder  by  2,  and  sum  of  these  products  is  length 
Fig.  45.  of  axis.  Fig.  46. 

a  EXAMPLE.— Link  of  a  chain,  Fig.  45,  is  i  inch  in  diameter         a 

of  body,  a  6,  and  its  inner  diameters,  b  c  and  efy  are  12.5 
and  2.5  inches;  what  is  its  circumference? 

2. 5  -}-  i  X  3. 1416  =  10.9956  =  length  of  axis  of  ends. 

1 2. 5  —  2. 5  X  2  =  20  =  length  of  sides  of  body. 
Then  10.9956  -(-  20  =  30.9956  =  length  of  axis  of  link,  and 
30.9956  X  3- 1416  (cir.  of  i  inch)  =  97. 3758  square  ins. 

When  Ring  is  Elliptical,  Fig.  46.  RULE. — Square  diameters  of  axes  of 
ring,  multiply  square  root  of  half  their  sum  by  3.1416,  and  product  is  length 
of  axis. 

Cones. 

DEFINITION.  —  A  figure  described  by  revolution  of  a  right-angled  triangle  about 
one  of  its  legs. 
For  Sections  of  a  Cone,  see  Conic  Sections,  page  379. 

To    Compute    Surface   of  a   Cone.— Fig.  47'. 
RULE. — Multiply  perimeter  or  circumference  of  base  by  slant  height,  or 
side  of  cone ;  divide  product  by  2,  and  add  the  quotient  to  area  of  the  base. 
Fig.  47.  c  Or,  c  X  &  -r-  2  -}-  a'  =  surface,  c  representing  perimeter. 

EXAMPLE.— Diameter,  a  &,  Fig.  47,  of  base  of  a  cone  is  3  feet, 
and  slant  height,  a  c,  15;  what  is  surface  of  cone? 

Circum.  of  3  feet  =  9. 4248,  and  ^^ —  =7o.686  =  swr- 
face  of  side;  area  of  base  3=7.068,  and  70.686+7.068  =  77.754 
square  feet. 

GG* 


354      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

To    Compute    Surface    of  the    Frustum    of  a    Cone.— 

ITig.  48. 

RULE.— Multiply  sum  of  perimeters  of  two  ends  by  slant  height  of  frus- 
tum ;  divide  product  by  2,  and  add  it  to  areas  of  two  ends. 

Or,  ^t-^ \-a  +  a'  =  surface, 

EXAMPLE.— Frustum,  abed,  Fig.  48,  has  a  slant  height,  c  d,  of  26  inches,  and 
„.      s  circumferences  of  its  ends  are  15.71  and  22  inches  respectively; 

^  A  what  is  its  surface? 


JLrf  +  (    j2l6)2  X  .7854  =  58.119  =  areas  of  ends.     Then  490.23 -+• 

58. 119  =  548.349  square  ins. 

Pyramids. 

DEFINITION.— A  figure,  base  of  which  has  three  or  more  sides,  and  sides  of  whicli 
are  plane  triangles. 

To    Compute    Surface   of  a  Pyramid..— Figs.  49  and.  aO. 

RULE.— Multiply  perimeter  of  base  by  slant  height ;  divide  product  by  2, 
and  add  it  to  area  of  base. 
Fig.  49.    c 

EXAMPLE.— Side  of  a  quadrangular  pyramid,  a  b, 
Fig.  49,  is  12  inches,  and  its  slant  height,  o  c,  40; 
what  is  its  surface? 

12X4  —  48=  perimeter  of  base. —  =  960  = 

]b    area  of  sides,  and  1 2  X  'i  2  -j-  960  =  1 104  square  ins. 

To    Compute    Surface    of  Frustum    of  a    Pyramid.— 
Fig.  SI. 

RULE. — Multiply  sum  of  perimeters  of  two  ends  by  slant  height ;  divide 
product  by  2,  and  add  it  to  areas  of  ends. 

Fig.  5I.  Or,  C-±^-*lL  +  a  +  a-  =  surface. 

%- f  EXAMPLE.  —Sides  ab,cd,  Fig.  51,  of  frustum  of  a  quadrangular 

pyramid  are  10  and  9  inches,  and  its  slant  neight  is  20  ;   what 
is  its  surface? 
10  x  4  =  40,  and  9  X  4  =  36 ;  40  +  36  =  76  =  sum  of  perimeters. 

,          76  x  20:=  1520,  and     —  =  760  =  area  of  sides;  10  X  10=  100, 

and  9X9  =  81.    Then  100  -J-  81  -f  760 = 941  =  square  ins. 

When  Pyramid  is  Irregular  sided  or  Oblique.    RULE.  —  The  surfaces  of 
each  of  the  sides  and  ends  must  be  computed  and  added  together. 

Helix   (Screw). 

DEFINITION.— A  line  generated  by  progressive  rotation  of  a  point  around  an  axis 
and  equidistant  from  its  centre. 

To    Compute   Length,    of  a   Helix.— Fig.  52. 
RULE. — To  square  of  circumference  described  by  generating  point,  add 
square  of  distance  advanced  in  one  revolution,  extract  square  root  of  their 
sum,  and  multiply  it  by  number  of  revolutions  of  generating  point. 


MENSURATION  OF  AEEAS,  LINES,  AND  SURFACES.       355 

Fig.  52.  Or,  -\/(P2  -\-l2)n  =  length,  n  representing  number  of  revolutions. 

EXAMPLE.— What  is  length  of  a  helical  line,  Fig.  52,  running  3.5 
times  around  a  cylinder  of  22  inches  in  circumference,  and  advancing 
16  inches  in  each  revolution  ? 

22  2  -J- 162  =  740  =  sum  of  squares  of  circumference  and  of  distance 
advanced.  *  Then  V740  X  3-  5  =  95-  21  ins. 

To    Compete    Length,    of  a    Revolution   of  Thread    of   a 
Screw. 

RULE.— Proceed  as  above  for  length  and  omit  number  of  revolutions. 

Spirals. 

DEFINITION.— Lines  generated  by  the  progressive  rotation  of  a  point  around  a 
fixed  axis. 

A  Plane  Spiral  is  when  the  point  rotates  around  a  central  point. 

A  Conical  Spiral  is  when  the  point  rotates  around  an  axis  at  a  progressing  dis- 
tance from  its  centre,  as  around  a  cone. 

To   Compnte   Length    of   a   Plane    Spiral   Line. —Fig.  543. 

RULE. — Add  together  greater  and  less  diameters ;  divide  their  sum  by  2 ; 
multiply  quotient  by  3.1416,  and  again  by  number  of  revolutions. 

Or,  when  circumferences  are  given,  take  their  mean  length,  and  multiply 
it  by  number  of  revolutions. 

Or,  d  -\-  d'  -f-  2  K  3- 1416  n  =  length  of  line;  Pxn  =  radius,  and 

*  }&  53- p  r2  _i- 1  =pitch.     P  representing  the  pitch. 

EXAMPLE.— Less  and  greater  diameters  of  a  plane  spiral  spring, 
as  a  6,  c  d,  Fig.  53,  are  2  and  20  inches,  and  number  of  revolutions 
\d  10}  what  is  length  of  it? 

sT^flzo  -T-  2  =  ii  =  sum  of  diameters  -r-  2  ;    u  X  3.1416  = 


Then  34.5576  X  10=  345.576  inches. 
NOTE. — Above  rule  is  applicable  to  winding  engines,  see  page  862,  where  it  is  re- 
quired to  ascertain  length  of  a  rope,  its  thickness,  number  of  revolutions,  diameter 
of  drum,  etc. 

To  Comp-ute  Length  of  a  Conical  Spiral  Line.— Fig.  S4r. 

RULE. — Add  together  greater  and  less  diameters;  divide  their  sum  by 
2,  and  multiply  quotient  by  3.1416. 

To  square  of  product  of  this  circumference  and  number  of  revolutions  of 
spiral,  add  square  of  height  of  its  axis,  and  take  square  root  of  the  sum. 


Or,  V(d  -f  d'  -4-  2  X  3. 1416  n  -f  h2)  =  length  of  line. 
EXAMPLE.— Greater  and  less  diameters  of  a  conical  spiral,  Fig.  54,  are 
20  and  2  inches;  its  height,  cd,  10;  and  number  of  revolutions  10;  what 
is  length  of  it? 

20  -f  2  -f-  2  =  ii  x  3.1416  =  34.5576  =  sum  of  diameters  -r-  2,  and  X 
1.1416;  34.5576  X  10  =  345.576. 

I l,i  Then  V345-5762-f-  io2  =  345. 72  inches. 

Spindles. 

DEFINITION.— Figures  generated  by  revolution  of  a  plane  area,  when  the  curve  is 
revolved  about  a  chord  perpendicular  to  its  axis,  or  about  its  double  ordinate,  and 
they  are  designated  by  the  name  of  the  arc  or  curve  from  which  they  are  generated, 
as  Circular,  Elliptic,  Parabolic,  etc. 

*  When  the  spiral  is  other  than  a  line,  measure  diameters  of  it  from  middle  of  body  composing  it. 


356      MENSURATION  OP  AREAS,  LINES,  AND  SURFACES. 


Circular   Spindle. 
To  Compute  Con-vex  Srarface  of  a  Circular  Spindle,  Zone} 

or    Segment   of  it.— figs.  (5S,  £56,  and.   ST. 
RULE. — Multiply  length  by  radius  of  revolving  arc ;  multiply  this  arc  by 
central  distance,  or  distance  between  centre  of  spindle  and  centre  of  revolv- 
ing arc ;  subtract  this  product  from  former,  double  remain- 
der, and  multiply  it  by  3.1416. 


I         /  c  \2 
-(a.^J  r2—  (  —  J  )  2  p  =  surf  ace,  a  representing  length 

I  c  the  spindle  chord. 


Or,  Ir  — 
of  arc,  andcthespi, 

EXAMPLE. — What  is  surface  of  a  circular  spindle,  Fig.  55,  length 
of  it,/c,  being  14.142  inches,  radius  of  its  arc,  o  c,  10,  and  central 
distance,  oe,  7.071? 

14. 142  X  10  =  141.42  =  length  x  radius.    Length  of  arc,  fa  c,  by  Rules,  page  344 
=  15.708. 
15-708  X  7-071  =  111.0713  =  length  of  arc  X  central  distance ;  141.42  — 111.0713  — 


of  products.    Then  30. 3487  X  2  X  3- 1416  =  190.687  square  ins. 

Zone. 

EXAMPLE. — What  is  convex  surface  of  zone  of  a  circular 
spindle,  Fig.  56,  length  of  it,  i  c,  being  7.653  inches,  radius  of 
its  arc,  o  g,  10,  central  distance,  o  e,  7.071,  and  length  of  its 
side  or  arc,  d  b,  7.854  inches? 

7. 653 X 10= 76. $3=length X  radius ;  7. 854  X  7-071  =  55. 5356 
=  length  of 'arc  x  central  distance ;  76. 53  —  55. 5356  =  20.9944 
=  difference  of  products. 
Then  20.9944  X  2  X  3. 1416  =  131.912  square  ins.  d lg'  S7' 

Segment. 

EXAMPLE.— What  is  convex  surface  of  a  segment  of  a  cir- 
cular spindle,  Fig.  57,  length  of  it,  ic,  being  3.2495  inches, 
radius  of  its  arc,  og,  10,  central  distance,  o  e,  7.071,  and  length 
of  its  side,  id,  3.927  inches? 

3.2495  X  10  =  32.495  ==  length  X  radius;  3.927  X  7-0?1  =  27.7678  =  length  of  arc 
X  central  distance  ;  32. 495  —  27. 7678  =  4. 7272  =  difference  of  products. 

Then  4.7272  X  2  X  3.1416  =  29.702  square  ins. 

GENERAL  FORMULA.—  8  =  2  (lr  —  ac)p  =  surface,  I  representing  length  of  spindle, 
segment,  or  zone,  a  length  of  its  revolving  arc,  r  radius  of  generating  circle,  and  c 
central  distance. 

ILLUSTRATION.— Length  of  a  circular  spindle  is  14.142  inches,  length  of  its  revolv- 
ing arc  is  15.708,  radius  of  its  generating  circle  is  10,  and  distance  of  its  centre  from 
centre  of  the  circle  from  which  it  is  generated  is  7.071 ;  what  is  its  surface? 

2  X  (14.142  X  10  —  15.708X7-071)  X  3.1416=  190.687  square  inches. 
NOTE.— Surface  of  a  frustum  of  a  spindle  may  be  obtained  by  division  of  the 
surface  of  a  zone. 

Cycloidal    Spindle. 
To   Compnte  Convex   Snrflaee   of  a   Cycloidal   Spindle.— 

Fie.  58. 

RULE.— Multiply  area  of  generating  circle  by  64,  and  divide  it  by  3. 
Fig.  58.^ 

'     3 

•'c       EXAMPLE.  —Area  of  generating  circle,  a  6  c,  of  a  cycloidal 
spindle,  d  e,  is  32  inches;  what  is  surface  of  spindle? 

32  X  64  =  2048  =  area  of  circle  x  64 ,  and  2048  -4-  3  = 
682. 667  square  ins. 

NOTE.— Area  of  greatest  or  centre  section  of  a  cycloidal  spindle  is  twice  area  of 
the  cycloid. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.       357 


Ellipsoid.,  Paraboloid.,  or   Hyper"boloid   of  Rev- 
olution. 

DEFINITION.  —  Figures  alike  to  a  cone,  generated  by  revolution  of  a  conic  section 
around  its  axis. 

NOTE.  —  These  figures  are  usually  known  as  Conoids. 

When  they  are  generated  by  revolution  of  an  ellipse,  they  are  termed  Ellipsoids, 
and  when  by  a  parabola,  Paraboloids,  etc. 

Revolution  of  an  arc  of  a  conic  section  around  the  axis  of  the  curve  will  give  a 
segment  of  a  conoid. 

Ellipsoid. 
To  Compute   Convex   Surface  of  an.  Ellipsoid.  —Fig.  59. 

RULE.  —  Add  together  square  of  base  and  four  times  square  of  height  ; 
multiply  square  root  of  half  their  sum  by  3.1416,  and  this  product  by  radius 
of  the  base. 


Or' 


3-«4i6r  ^surface. 


EXAMPLE.— Base,  a  6,  of  an  ellipsoid,  Fig.  59,  is  10  inches,  and 
vertical  height,  c  d,  7;  what  is  its  surface? 

io2-f-  72  X  4  =  296  =  sum  of  square  of  base  and  4  times  square 
of  height;  296  -4-  2  =  148,  and  \/I4^  —  I2-  J^55 = square  root  of  half 

above  sum.    Then  12.1655  X  3.1416  X  —  =  191.0957  square  ins. 


To    Compute  Convex   Surfa.ce   of  a   Segment,  Frustum, 
or    Zone    of  an  Ellipsoid..—  Fig.  59. 

See  Rules  for  Convex  Surface  of  a  Segment,  Frustum,  or  Zone  of  a 
Spheroid  or  Ellipsoid,  pages  348-9. 

d  or  d'  X  3-  1416  X  h  =  surface, 


and 


=  h  ;  then  d  X  3.  »4»6  X  »  =  surf**. 


3?ara"boloid. 
To  Compute  Convex  Surface  of  a  IParalDoloid..—  Fig.  6O. 

RULE.  —  From  cube  of  square  root  of  sum  of  four  times  square  of  height, 
and  square  of  radius  of  base,  subtract  cube  of  radius  of  base  ;  multiply  re- 
mainder by  quotient  of  3.1416  times  radius  of  base  divided  by  six  times 
square  of  height. 

Flg-  fe  1  _  E.=  surface, 


Or, 


3_r3  x 


EXAMPLE.—  Axis,  6  d,  of  a  paraboloid,  Fig.  60,  is  40  inches;  ra- 
dius, a  d,  of  its  base  is  18  inches;  what  is  its  convex  surface? 

4o2  X  4  =  6400  =  4  times  square  of  height  ;  6400  +  i82  =  6724  — 
sum  of  above  product  and  square  of  radius  of  base;  (-^6724)3  —  183 
545  536  =  remainder  of  cube  of  radius  of  base  subtracted  from  cube 
of  square  root  of  preceding  sum  ;  3.  1416  X  i8-f-  (6  X  4o2)  =  .005  8905 
•=.  quotient  0/3.1416  times  radius  of  base  -4-6  times  square  of  height. 
Then  545  536  X  .005  890  5  =  3213.48  square  ins. 


Fig.  61.  t  a 


Cylinder    Sections. 
To  Compute  Surface  of  a  Cylinder  Section. 

-Fig.  61. 

RULE.  —  From  entire  surface  of  cylinder  a  o  subtract 
surface  of  the  two  ungulas,  ?•  o,  o  c,  as  per  rule,  page  35  1> 
and  multiply  result  by  4. 


358      MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


Figure   of  Revolution.. 
To  Ascertain  Convex  Surface  of  any  Figure  of  Revolu- 

tion.— Figs.  6£J,  G39  and    64. 

RULB.  —  Multiply  length  of  generating  line  by  circumference  described 
by  its  centre  of  gravity. 

Or,  I  2  r  p  =  surface,  r  representing  radius  of  centre  of  gravity. 

EXAMPLE  i.—  If  generating  line,  a  c,  of  cylinder,  a  cdf,  10  inches 
*  '-     .  in  diameter,  Fig.  62,  is  10,  then  centre  of  gravity  of  it  will  be  in  b, 

radius  of  which  is  6  r  =  5. 


\    /  i  

J-V-nV  Hence  10  X  5  X  2  X  3- 1416  =  314. 16  ins. 

V'\  1  Again,  if  generating  line  is  e  a  c  g,  and  it  is  (e  a  =  5,  a  c— 10, 

' ^        L     and  c  g  =  s)  =  2o,  then  centre  of  gravity,  o,  will  be  in  middle  of 

A. — -j    line  joming  centres  of  gravity  of  triangles  e  a  c  and  a  c  y  =  3. 75 
from  r. 


Hence  20  X  3.75  X  2  X  3.1416  =  471.24  square  ins.=entire  surface. 

w  J  Convex  surface  as  above 314. 16 

N.  {Areaof  each  end,  io*x  .7854  X  2  = .157.08 

471.24  inches. 

Fig.  63.  2. — If  generating  elements  of  a  cone,  Fig.  63,  are  a  d  —  10, 

d  c  =  10,  and  ac,  generating  line,=  14.142,  centre  of  gravity  of 
which  is  in  o,  and  o  r  =  5, 
Then  14.142  XsX2X  3. 1416 =444. 285,  con- 

vex  surface,  and  10  x  2  X  .7854  =  314.16,  area 
of  base.  , 

Hence  444.285  -\-  314. 16  —  758.445,  entire  surface. 
3.— If  generating  elements  of  a  sphere,  Fig.  64,  are  a  c  =  so,  a  &  c 
will  be  15.708,  centre  of  gravity  of  which  is  in  o,  and  by  Rule,  page 
606,  o  r  =  3. 183. 
Hence  15.708  X  3-183  X  2  X  3.1416  =  314.16  square  ins. 

Capillary   TulDe. 
To   Compute   Diameter   of  a   Capillary   Tube. 

RULE. — Weigh  tube  when  empty,  and  again  when  filled  with  mercury ; 
subtract  one  weight  from  the  other ;  reduce  difference  to  grains,  and  divide 
it  by  length  of  tube  in  inches.  Extract  square  root  of  this  quotient,  multi- 
ply it  by  .019  224  5,  and  product  will  give  diameter  of  tube  in  inches. 

Or,  I  —  x  .0192245  =  diameter,  w  representing  difference  in  weights  in  grqins 
and  I  length  of  tube. 

EXAMPLE.— Difference  in  weights  of  a  capillary  tube  when  empty  and  when  filled 
with  mercury  is  90  grains,  and  length  of  tube  is  10  inches;  what  is  diameter  of  it? 

90  -r- 10  —  9  =  weight  of  mercury  -=-  length  of  tube  ;  -\/Q  =  3,  and  3  x  .019  224  5  = 
.057  673  5  —  square  root  of  above  quotient  x  .019  224  5  inches  =  diameter  of  tube. 

PROOF.— Weight  of  a  cube  inch  of  mercury  is  3442.75  grains,  and  diameter  of  a 
circular  inch  of  equal  area  to  a  square  inch  is  1.128  (page  342). 

If,  then,  3442.75  grains  occupy  i  cube  inch,  90  grains  will  require  .026 141 9  cube 
inch,  which,  -4- 10  for  height  of  tube  —  .002  614 19  inch  for  area  of  section  of  tube. 

Then  ^-002  61419  =  .  05 11 29  =  side  of  square  of  a  column  of  mercury  of  this  area. 

Hence  .051 129  x  1.128  (which  is  ratio  between  side  of  a  square  and  diameter  of  a 
circle  of  equal  area)  =  .057  673  5  ins. 

To   Ascertain   A.rea   of  an    Irregular   Figure. 

RULE.— Take  a  uniform  piece  of  board  or  pasteboard,  weigh  it,  cut  out 
figure  of  which  area  is  required,  and  weigh  it ;  then,  as  weight  of  board  or 
pasteboard  is  to  entire  surface,  so  is  weight  of  figure  as  cut  out  to  its  surface, 

Or,  see  rule  page  341,  or  Simpson's  rule,  page  342. 


MENSURATION  OF  AREAS,  LINES,  SURFACES,  ETC.       359 


To   Ascertain  Area  of  any   Plane   Figure, 

RULE.  —  Divide  surfaces  into  squares,  triangles,  prisms,  etc. ;  ascertain 
their  areas  and  add  them  together. 

Reduction  of  an  -Ascending  or  Descending  Line  to  Hor- 
izontal   ^Measurement. 
In  Link  and  Foot. 

Degrees.        Link.  Foot.        Degrees.         Link.  Foot.        Degrees.  I      Link.  Foot. 


.000099 

.000403 

.000904 

.001  61 

.002515 

.003617 


.00015 
.00061 
•  00137 
.00244 
.00381 
.00548 


9 
10 
ii 

12 


.004917 
.006421 
.008  125 
.OIOO25 
.012  124 
.014421 


.00745 
.00973 
.01231 
.015  19 
.01837 
.02285 


13 


3 


016915 
019602 

022  486 
025569 
O28  925 
0323 


02563 

0297 

03407 

03874 

0437 

04894 


O     .003017   .00540      12     .014421   .O2205      10      0323      040 

ILLUSTRATION  i.— In  an  ascending  grade  of  14°,  what  is  reduction  in  500  feet? 

14°  =  500  X  .0297  =  14.85  feet  =  14  feet  10.2  ins. 
2.— What  is  reduction  in  500  links? 

14°  =  500  X  -019  602  =  9. 801  feet  =  9  feet  9.6  ins. 

Reduction  of  Oracle  of  an  Ascending  or  Descending  Line 

to  Degrees. 
Per  100  Links,  Feet,  etc. 


Grade. 

Degrees. 

Grade. 

Degrees. 

Grade. 

Degrees. 

Grade. 

Degrees. 

•25 

/       // 

8  35-2 

1-75 

I     o  10.3 

4-5 

2  34  45-5 

10 

5  44  20.7 

•5 

17  10.3 

2 

i    8  45.5 

5 

2  5i  57-6 

ii 

6  18  55.8 

•75 

25  47-6 

2.5 

i  25  57.6 

6 

3  26  22.7 

12 

6  53  3i 

x-25 

34  22.7 
42  57-9 

3 
3-5 

i  43    8.3 

2      O   2O.7 

7 
8 

4    o  49.6 
4  35  18-6 

13 
14 

7  28  10.3 
8    2  51-7 

i-5 

5i  35-2 

4 

2  17  33-i 

9 

5    9  49-6 

15 

8  37  37-  » 

To   Plot   Angles   -without   a   Protractor. 

On  a  given  line  prick  off  100  with  any  convenient  scale,  and  from  the 
point  so  pricked  off  lay  off  at  right  angle  with  the  same  scale  the  natural 
tangent  due  to  the  angle  (see  table  of  Natural  Tangents  and  Sines) ;  or 
strike  out  a  portion  of  a  circle  with  radius  100  and  lay  off  a  chord  =  2  sin. 
of  half  the  angle  required. 

To   Compute   Chord,  of  an  Angle. 

Double  sine  of  half  angle. 
ILLUSTRATION.— What  is  the  chord  of  21°  30'? 

Sine  of        3°  =  10°  45',  and  sine  of  10°  45'  = .  186  52,  which,  X  2  =  .373 04  chord. 


To  Ascertain  Value  of  a  IPower  of  a  Qu.an.tity. 

RULE. — Multiply  logarithm  of  quantity  by  fractional  exponent,  and  prod- 
uct is  logarithm  of  required  number. 

EXAMPLE.— What  is  the  value  of  16%  ? 

X  X  log.  16  =  &  x  x.  204  la  = . 903  09.    Number  for  which  =  & 


360 


MENSURATION    OF   VOLUMES. 


MENSURATION   OF  VOLUMES. 
Cubes   and   IParallelopipedons. 

Cube. 

DEFINITION.— A  volume  contained  by  six  equal  square  sides. 
Fig.  i.     h  To   Compute   Volume   of  a   Cube.—  T^ig.  1. 

RULE.— Multiply  a  side  of  cube  by  itself,  and  that  product 
again  by  a  side. 

Or,  «3  —  V,  s  representing  length  of  a  side,  and  V  volume. 

EXAMPLE.—  Side,  a  b,  Fig.  i,  is  12  inches;  what  is  volume  of  it? 

12  X  12  X  12  —  1728  cube  ins. 

IParallelopipedon. 

DEFINITION.— A  volume  contained  by  six  quadrilateral  sides,  every  opposite  two 
of  which  are  equal  and  parallel. 

To  Compute  "Volume  of  a  IParallelopipedon. 

—Fig.  2. 

RULE. — Multiply  length  by  breadth,  and  that  product 
again  by  depth. 

Or,  lbd  =  V. 

IPrisms,  IPrismoids,  and  "Wedges. 
Prisms. 

DEFINITION.— Volumes,  ends  of  which  are  equal,  similar,  and  parallel  planes,  and 
sides  of  which  are  parallelograms. 

NOTE.—  When  ends  of  a  prism  or  prismoid  are  triangles,  it  is  termed  a  triangular 
prism  or  prismoid;  when  rhomboids,  a  rhomboidal  prism,  and  when  squares,  a 
square  prism,  etc. 


Fig  3 


Fig  5- 


To    Compute    Volume    of    a    Prism.— 

Figs.  3    and.   4. 

RULE.— Multiply  area  oPbase  by  height. 
Or,  a  h  =  V. 

EXAMPLE.— A  triangular  prism,  a  b  c,  Fig.  4,  has  sides 
of  2. 5  feet,  and  a  length,  c  b,  of  10;  what  is  its  volume  ? 

By  Rule,  page  339,  2.52  x  .433  =  2.70625  =  area  of 
end  a  6vand  2.706  25  X  10  =  27.0625  cube  feet. 


When  a  Prism  is  Oblique  or  Irregular. 
RULE.  —  Multiply  area  of  an  end  by  height,  as  a  o ;  or, 
multiply  area  taken  at  a  right  angle  to  sides,  as  at  c,  by 
actual  length. 


To    Compute  Volume    of  any  ITrustxxm    of  a 
IPrism,  whether    Right    or    Oblique.—  Figs. 


Fig.  6. 


Fig.  7. 


RULE.— Multiply  area  of  base  by  perpendicular 
distances  between  it  and  centre  of  gravity  of  upper 
or  other  end. 

EXAMPLE.— Area  of  base,  a  o,  of  frustum  of  a  rectan- 
gular or  cylindrical  prism,  Fig.  6,  is  15  inches,  and 
height  to  centre  of  gravity,  c,  is  12 ;  what  is  its  vol- 
ume? 

10  X  12  =  120  cube  ins. 


MENSURATION    OF    VOLUMES. 


IPrismoids.  * 

To   Compute  "Volume   of*  a    I»rismoid.— Fig.  8. 
RULE. — To  sum  of  areas  of  the  two  ends  add  four  times  area  of  middle 
section,  parallel  to  them,  and  multiply  this  sum  by  one  sixth  of  perpendicu- 
lar height. 

NOTE.— This  is  the  general  rule,  and  known  as  the  Prismoidal  Formula,  and  it 
applies  equally  to  all  figures  of  proportionate  or  dissimilar  ends. 


Fig.  8. 


Or,  a  +  a'  -f  4  ra  x  h  -r-  6  =  V,  a  and  a'  representing  areas  of  ends, 
and  m  area  of  middle  section. 

EXAMPLE.  —  What  is  volume  of  a  rectangular  prismoid,  Fig.  8, 
lengths  and  breadths,  e  g  and  g  h,  a  b  and  b  d,  of  two  ends  being 
7X6  and  3X2  inches,  and  height  15  feet  ? 

7X6+3X2  =  42 -f6  =  48  =  mm  of  areas  of  two  ends ;  7  -f  3  -r- 
2  =  5  =  length  of  middle  section  ;  6  +  2-^2  =  4  =  breadth  of  middle 
section  ;  5X4X4  =  80  =four  times  area  of  middle  section. 


Then  48  + 80  X 


15  X  12 


=  128  X  30  =  3840  cube  ins. 


NOTE  i.— Length  and  breadth  of  middle  section  are  respectively  equal  to  half 
sum  of  lengths  and  breadths  of  the  two  ends. 

2.— Prismoids,  alike  to  prisms,  derive  their  designation  from  figure  of  their  ends, 
as  triangular,  square,  rectangular,  pentagonal,  etc. 

When  it  is  Irregular  or  Oblique  and  their  ends  are  united  by  plane  or 
curved  surfaces,  through  which  and  every  point  of  them,  a  right  line  may  be 
drawn  from  one  of  the  ends  or  parallel  faces  to  the  other. — Figs.  9,  io,and  n. 

Fig.  10.  Fig.  ii. 

Fig.  9. 


EXAMPLE.— Areas  of  ends,  a  c  and  o  r  s,  Fig.  10,  a  b  c  d,  and  i  m  n  w,  Fig.  n,  and 
abce  and  v  x  w  z,  Fig.  9,  are  each  10  and  30  inches,  that  of  their  middle  section 
20,  and  their  perpendicular  heights  18;  what  is  their  volume? 

10  +  30+ 20  x  4  =  120  =  sum  of  areas  of  ends -\- 4  times  middle  section.     And 

120  X  -T-  =  360  cube  ins. 
6 

TVedge. 

To    Compnte   "Volvitne   of  a  "Wedge.— Fig.  12. 
RULE. — To  length  of  edge  add  twice  length  of  back ;  multiply  this  sum 
by  perpendicular  height,  and  then  by  breadth  of  back,  and  take  one  sixth 
of  product. 
Fig.  12.  Or,  (I  -f-  V  X  2  X  h  b)  -i-  6  =  V. 

EXAMPLE.— Length  of  edge  of  a  wedge,  eg,  is  20  inches,  back, 
abed,  is  20  by  2,  and  its  height,  e/,  20;  what  is  its  volume? 

20  +  20  X  2  =  60  —  length  of  edge  added  to  twice  length  of 
back  ;  60  X  20  X  2  =  2400  —  above  sum  multiplied  by  height,  and 
that  product  by  breadth  of  back. 

Then  2400  -r-  6  =  400  cube  ins. 

NOTE.  — When  a  wedge  is  a  true  prism,  as  represented  by 
2,  volume  of  it  is  equal  to  area  of  an  end  multiplied  by  its  length. 


*  An  excavation  or  embankment  of  a  road,  when  terminated  by  parallel  cross  sections,  is  a  rectan- 
gular prismoid. 

H  H 


362 


MENSURATION    OF    VOLUMES. 


To    Compute    Frustum   of  a.   "Wedge.— Fig.  13. 

RULE. — To  sum  of  areas  of  both  ends,  add  4  times  area 
of  section  parallel  to  and  equally  distant  from  both  ends, 
and  multiply  sum  by  one  sixth  of  length. 


Or,A-J-a-f-4a/XT-  =  V. 
o 

EXAMPLE.  — Lengths  of  edge  and  back  of  a  frustum  of  a  wedge 
a  b  and  c  d  are  20  X  i  and  20  X  2  ins. ,  and  height  o  r  is  20  ins. ; 
what  is  its  volume  ? 

X  2  +  4  X  (20  X  — fe^  X  ^  =  60  +  120  X  —  =  600  cube  ins. 
2  \  2    /       6  6 

NOTE.— When  frustum  is  a  true  prism,  as  represented  Fig.  13,  volume  of  it  is  equal 
to  mean  area  of  ends  multiplied  by  its  length. 

Regular  Bodies   (3Polyh.ed.rons). 

DEFINITION. — A  regular  body  is  a  solid  contained  under  a  certain  number  of  simi- 
lar and  equal  plane  faces,*  all  of  which  are  equal  regular  polygons. 

NOTE  i.— Whole  number  of  regular  bodies  which  can  possibly  be  formed  is  five. 

2. — A  sphere  may  always  be  inscribed  within,  and  may  always  be  circumscribed 
about  a  regular  body  or  polyhedron,  which  will  have  a  common  centre. 


Fig.  14. 


Fig.  16. 


Fig.  17- 


1.  Tetrahedron,  or  Pyramid,  Fig.  14,  which  has  four  triangular  faces. 

2.  Hexahedron,  or  Cube,  Fig.  i,  which  has  six  square  faces. 

3.  Octahedron,  Fig.  15,  which  has  eight  triangular  faces. 

4.  Dodecahedron,  Fig.  16,  which  has  twelve  pentagonal  faces. 

5.  Icosahedron,  Fig.  17,  which  has  twenty  triangular  faces. 

Xo  Compute  Klements  of  any  Regular  Body.— Figs.  14:, 
15,  16,  and  IT. 

To  Compute  Radius  of  a  Sphere  that  will  Circumscribe  a  given  Regular 
Body,  or  that  may  be  Inscribed  within  it. 

When  Linear  Edge  is  given.  RULE. — Multiply  it  by  multiplier  opposite 
to  body  in  columns  A  and  B  in  following  Table,  under  head  of  element  re- 
quired. 

EXAMPLE.— Linear  edge  of  a  hexahedron  or  cube,  Fig.  i,  is  2  inches;  what  are 
radii  of  circumscribing'and  inscribed  spheres? 

2  x  .  866  02  =  i .  732  04  inches  =  radius  of  circumscribing  sphere  ;  2  X  .  5  =  i  inch  = 
radius  of  inscribed  sphere. 

When  Surface  is  given.  RULE. — Multiply  square  root  of  it  by  multiplier 
opposite  to  body  in  columns  C  and  D  in  following  Table,  under  head  of 
element  required. 

When  Volume  is  given.  RULE.  —  Multiply  cube  root  of  it  by  multiplier 
opposite  to  body  in  columns  E  and  F  in  following  Table,  under  head  of  ele- 
ment required. 

*  Angle  of  adjacent  facei  of  a  polygon  is  termed  diedral  angle. 


MENSURATION    OF    VOLUMES.  363 

When  one  of  the  Radii  of  Circumscribing  or  Inscribed  Sphere  alone  is  re-< 
quired,  the  other  being  given.  RULE. — Multiply  given  radius  by  multiplier 
opposite  to  body  in  columns  G  and  H  in  Table,  page  364,  under  head  ef 
other  radius. 

To   Compute   Linear   Edge. 

When  Radius  of  Circumscribing  or  Inscribed  Sphere  is  given.  RULE. — 
Multiply  radius  given  by  multiplier  opposite  to  body  in  columns  I  and  K  in 
Table,  page  364. 

When  Surface  is  given.  RULE.— Multiply  square  root  of  it  by  multiplier 
opposite  to  body  in  column  L  in  Table,  page  364. 

When  Volume  is  given.  RULE.  —  Multiply  cube  root  of  it  by  multiplier 
opposite  to  body  hi  column  M  in  Table,  page  364. 

To    Compute    Surface. 

When  Radius  of  Circumscribing  Sphere  is  given.  RULE. — Multiply  square 
of  radius  by  multiplier  opposite  to  body  hi  column  N  hi  Table,  page  364. 

When  Radius  of  Inscribed  Sphere  is  given.  RULE. — Multiply  square  of 
radius  by  multiplier  opposite  to  body  in  column  O  in  Table,  page  364. 

When  Linear  Edge  is  given.  RULE. — Multiply  square  of  edge  by  multi- 
plier opposite  to  body  hi  column  P  in  Table,  page  364. 

When  Volume  is  given.  RULE. — Extract  cube  root  of  volume,  and  multi- 
ply square  of  root  by  multiplier  opposite  to  body  in  column  Q  hi  Table, 
page  364. 

To   Compute   "Volume. 

When  Linear  Edge  is  given.  RULE. — Cube  linear  edge,  and  multiply  it 
by  multiplier  opposite  to  body  in  column  R  in  Table,  page  364. 

When  Radius  of  Circumscribing  Sphere  is  given.  RULE. — Multiply  cube 
of  radius  given  by  multiplier  opposite  to  body  hi  column  S  hi  Table, 
page  364. 

When  Radius  of  Inscribed  Sphere  is  given.  RULE.  —  Multiply  cube  of 
radius  given  by  multiplier  opposite  to  body  in  column  T  in  Table,  page  364. 

When  Surface  is  given.  RULE. — Cube  surface  given,  extract  square  root, 
and  multiply  the  root  by  multiplier  opposite  to  body  hi  column  U  in  Table, 
page  364. 

Fig.  18.  Cylinder. 

To    Compute  Volume    of  a   Solid    Cylinder.— 

Fig.  18. 

RULE.— Multiply  area  of  base  by  height. 
EXAMPLE.— Diameter  of  a  cylinder,  b  c,  is  3  feet,  and  its  length,  a  6, 
7  feet;  what  is  its  volume? 

Area  of  3  feet  =  7.068.    Then  7.068  X  7  =  49.476  cube  feet. 

To   Compute  "Volume   of*  a   Hollow  Cylinder. 
RULE. — Subtract  volume  of  internal  cylinder  from  that  of  cylinder. 

Fig.  ig.  a  Cone. 

To  Compute  "Volume  of  a  Cone.— Fig.  19. 

RULE. — Multiply  area  of  base  by  perpendicular  height, 
and  take  one  third  of  product. 

EXAMPLE.— Diameter,  a  &,  of  base  of  a  cone  is  15  inches,  and 
height,  c  e,  32. 5  inches ;  what  is  its  volume  f 

Are*  of  15  inches  =  176.7146.    Then  176'7I5X32'5  =  1914.4125  cube  int. 


364 


MENSURATION    OF   VOLUMES. 


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MENSURATION    OF   VOLUMES. 


Fig.  20. 


To   Compute  "Volume   of  Frustum  of  a   Cone.— Fig.  2O. 

RULE. — Add  together  squares  of  the  diameters  or  circumferences  of  greater 
and  lesser  ends  and  product  of  the  two  diameters  or  circumferences ;  mul- 
tiply  their  sum  respectively  by  .7854  or  .07958,  and  this  product  by  height; 
then  divide  this  last  product  by  3. 

Or,  d2  +  a"'2  -f  dxtf  X  .  7854  h  -r-  3  =  V. 
Or,  c2  +  c/2  +  c  X  c'  X  -079  58  h-r-  3  =  V. 
EXAMPLE.— What  is  volume  of  frustum  of  a  cone,  diameters 
of  greater  and  lesser  ends,  bd,ac,  being  5  and  3  feet,  and  height, 
eo,  9? 

52 -f-  32H-5X3  =  49 ;  and  49  X  .7854  =  38-4846  =  above  sum 
by  .7854;  and  38'4^6  X  9  =  115.4538  cube  feet. 

3Pyrain.icU 

NOTE.— Volume  of  a  pyramid  is  equal  to  one  third  of  that  of  a  prism  having  equal 
bases  and  altitude. 


Fig.  si. 


Fig.  22. 


To    Compute    "Volume    of  a    Pyramid.-—  ITig.  SI. 

RULE.  —  Multiply  area  of  base  by  perpendicular  height,  and 
take  one  third  of  product. 

EXAMPLE.—  What  is  the  volume  of  a  hexagonal  pyramid,  Fig.  21, 
a  side,  a  b,  being  40  feet,  and  its  height,  e  c,  60? 

40*  X  2.5981  (tabular  multiplier,  page  339)  =4156.96  =  area  of  base. 

4156  96  X  *>  =  83  x39.2  ate  fed. 

To  Compute  Volume  of  Frustum  of  a  Pyramid.—  Fig.  22. 

RULE.  —  Add  together  squares  of  sides  of  greater  and  lesser  ends,  and 

product  of  these  two  sides  ;  multiply  sum  by  tabular  multiplier  for  areas  in 

Table,  page  339»  and  this  product  by  height  ;  then  divide  last  product  by  3. 

Or,  *2  +  s'2  -fsxs'  X  tab.  mult,  x  h  -r-  3  =  V. 

When  A  reas  of  Ends  are  known,  or  can  be  obtained  without  reference  to 
a  tabular  multiplier,  use  following. 

Or,  a  +  a'-f-  Vax  a'X  A-r-s  =  V. 

EXAMPLE.  —  What  is  the  volume  of  the  frustum  of  a  hexagonal 
pyramid,  Fig.  22,  the  lengths  of  the  sides  of  the  greater  and  lesser 
ends,  ab,cd,  being  respectively  3.75  and  2.5  feet,  and  its  perpen- 
dicular height,  e  o,  7.5? 

3.752-r-2.52  =  2o.3i25=$Mw  of  squares  of  sides  of  greater  and 
lesser  ends;  20.3125+  3.75  x  2.  5  =  29.  6875  =  above  sum  added  to 
product  of  the  two  sides  ;  29.6875  X  2.5981  X  7-5  —  578.48  x  tab. 
mult.,  and  again  by  the  height,  which,  -7-3—192.83  cube  feet. 

When  Ends  of  a  Pyramid  are  not  those  of  a  Regular  Polygon,  or  when 
Areas  of  Ends  are  given 

RULE.  —  Add  together  areas  of  the  two  ends  and  square  root  of  their  prod- 
uct ;  multiply  sum  by  height,  and  take  one  third  of  product. 

Or,  a  -\-  a'  +  Vaa'  X  h  -r-  3  =  V. 

EXAMPLE.  —What  is  the  volume  of  an  irregular  sided  frustum  of  a  pyramid,  the 
areas  of  the  two  ends  being  22  and  88  inches,  and  the  length  20? 

22-j-88=no  =  5wm  of  areas  of  ends;  22  X  88  =  1936,  and  ^1936  =  44  =  square 
root  of  product  of  areas.    Then  —  — 

3HH* 


=  1026.66  cube  int. 


366 


MENSURATION    OF    VOLUMES. 


Spherical   3?yramid. 

A  Spherical  Pyramid  is  that  part  of  a  sphere  included  within  three  or  more  ad- 
joining plane  surfaces  meeting  at  centre  of  sphere.  The  spherical  polygon  defined 
by  these  plane  surfaces  of  pyramid  is  termed  the  base,  and  the  lateral  faces  are 
sectors  of  circles. 

NOTE.— To  compute  the  Elements  of  Spherical  Pyramids,  see  Docharty  and  Hack- 
ley's  Geometry. 

Cylindrical   TJngnlas. 

DEFINITION.— Cylindrical  Ungulas  are  frusta  of  cylinders.  Conical  Ungulas  arc 
frusta  of  cones. 

To  Coiripxite  "Volume  of  a  Cylindrical  TJngula.— Fig.  23C 
i.  When  Section  is  parallel  to  Axis  of  Cylinder.    RULE. — Multiply  area 
Fig.  23.        of  base  by  height  of  the  cylinder. 

Or,  a  h  =  V. 

EXAMPLE.— Area  of  base,  d  ef,  Fig.  23,  of  a  cylindrical  ungula  is  15.5 
inches,  and  its  height,  a  e,  20;  what  is  its  volume? 
15. 5  X  20  =  310  cube  ins. 

2.  When  Section  passes  Obliquely  through  opposite  sides  of 
Cylinder,  Fig.  24.  RULE.— Multiply  area  of  base  of  cylinder  by 
half  sum  of  greatest  and  least  lengths  of  ungula.  Fig.  24. 

Or,  ax  J-f  r-5-2  =  V. 

EXAMPLE.— Area  of  base,  c  d,  of  a  cylindrical  ungula,  Fig.  24,  is  25      iV^^'ft 
inches,  and  the  greater  and  less  heights  of  it,  ac,bd,  are  15  and  17: 
what  is  its  volume? 

25  X  Ig          =  400  cube  ins. 

3.  When  Section  passes  through  Base  of  Cylinder  and  one  of  its  Sides, 
and  Versed  Sine  does  not  exceed  Sine,  or  the  Base  is  equal  to  or  less  than  a 
Semicircle,  Fig.  25.     RULE. — From  two  thirds  of  cube  of  sine  of  half  arc 
of  base  subtract  product  of  area  of  base  and  cosine  *  of  half  arc.    Multiply 
difference  thus  found  by  quotient  arising  from  height  divided  by  versed  sine. 

Fiff.  25.  ^2  sin.3      —          h 

Or, a  c  X : —  =  v ,  v.  sin.  representing  versed  sine. 

3  v.  sin. 

EXAMPLE.— Sine,  a  d,  of  half  arc,  d  ef,  of  base  of  an  ungula,  Fig.  25, 
is  5  inches,  diameter  of  cylinder  is  10,  and  height,  e  g,  of  ungula  10; 
what  is  its  volume  ? 

Two  thirds  of  53  —  83.333  =  ^0  thirds  of  cube  of  sine.  As  versed 
sine  and  radius  of  base  are  equal,  cosiae  is  o.  Henoe,  area  or  base  x 
cosine  =  o,  and  83. 333  —  o  x  10  -r-  5  =  166.666  cube  ins. 

4.  When  Section  passes  through  Base  of  Cylinder,  and  Versed  Sine  exceeds 
Radius,  «r  when  the  Base  exceeds  a  Semicircle,  Fig.  26.     RULE. — To  two 
thirds  of  cube  of  sine  of  half  arc  of  base  add  product  of  area  of  base  and 
cosine.    Multiply  sum  thus  found  by  quotient  arising  from  height,  divided 
by  versed  sine. 

2_wn.3      —     _A__V 

3  v.  sin. 

EXAMPLE.— Sine,  a  d,  of  half  arc  of  an  ungula,  Fig.  26,  is  12  inches, 
versed  sine,  a  g,  is  16,  height,  g  c,  10,  and  diameter  of  cylinder  25 ; 
c  what  is  its  volume? 

Two  thirds  of  i23  =  n52  —  two  thirds  of  cube  of  sine  of  half  arc  of 
base.  Area  of  base  =331. 78;  1152  +  331.78  x  16  —  12.5  =  2313.23  = 
sum  of  two  thirds  of  cube  of  sine  of  half  the  arc  of  base,  and  product  of 
area  of  base  and  cosine.  Then  2313. 23  x  20-^-16  =  2891. 5375  cube  ins. 

*  Wh«n  ih«  «ouu«  is  ot  the  product  is  o. 


MENSURATION  OF  VOLUMES.  367 

5.  When  Section  passes  Obliquely  through  both  Ends  of  Cylinder,  Fig.  27. 
RULE.  —  Conceive  section  to  be  continued  till  it  meets  side  of  cylinder 
produced ;  then,  as  the  difference  of  versed  sines  of  the  arcs  of  the  two  ends 
of  ungula  is  to  the  versed  sine  of  arc  of  less  end,  so  is  the  height  of  cylinder 
to  the  part  of  side  produced. 

Ascertain  volume  of  each  of  the  ungulas  by  Rules  3  and  4,  and  take  their 
difference. 

Or, '- — : — ;  =  h',  v.  sin.  and  v.  sin. '  representing  versed  sines 

v.  sin.  —  v.  sin. 

of  arcs  of  the  two  ends,  h  height  of  cylinder,  and  h'  height  of  part  pro- 
duced. 

EXAMPLE. — Versed  sines,  ae,do,  and  sines,  e  and  o,  of  arcs  of  two 
ends  of  an  ungula,  Fig.  27,  are  assumed  to  be  respectively  8.5  and  25, 
and  11.5  and  o  inches,  length  of  ungula,  bo,  within  cylinder,  cut  from 
one  having  25  inches  diameter,  d  o,  is  20  inches;  what  is  height  of  un- 
gula produced  beyond  cylinder,  and  what  is  volume  of  it? 

25  <-o  8.5  :  8.5  ::  20  :  10.303  =  height  of  ungula  produced  beyond  cyl- 
inder. 

Greater  ungula,  sine  o  being  o,  versed  sine  =  the  diameter.  Base  of  ungula  being 
a  circle  of  25  inches  diameter,  area  =  490.875.  Versed  sine  and  diameter  of  base 

being  equal  (25),  sine  =  o.     490.875  X  25  <x>  —  =  6135.9375  =product  of  area  of  base 

and  cosine,  or  excess  of  versed  sine  over  sine  of  base.     30. 303  -4-25  =  1.21212  =  quo- 
tient of  height  -r-  versed  sine. 

Then  6135.9375  x  1.212  12  =  7437.4926  cube  inches;  and  by  Rules  3  and  4,  volumes 
of  less  and  greater  ungulas  =  515.444,  and  6922.0486  =  7437.4926  cube  inches. 

Sphere. 

DEFINITION.— A  solid,  surface  of  which  is  at  a  uniform  distance  from  the  centre. 
Fig'  2i--5=^  To  Compute  Volume  of  a  Sphere.— Fig.  28. 

RULE.— Multiply  cube  of  diameter  by  .5236. 
a  Or,  <P  x  •  5236  =  V,  d  representing  diameter. 

EXAMPLE.— What  is  volume  of  a  sphere,  Fig.  28,  its  diameter, 
a  b,  being  10  inches  ? 

io3  =  1000,  and  looo  X  .  5236  =  523.6  cube  ins. 

To   Compute  Volume   of  a   Hollow    Sphere. 

RULE.— Subtract  volume  of  internal  space  from  that  of  sphere. 
Or,  V  —  v  —  volume. 

Segment   of  a   Sphere. 

DEFINITION.— A  section  of  a  sphere. 
To  Compute  Volume  of  a  Segment  of  a  Sphere.— Fig.  29. 

RULE  i.  —  To  three  times  square  of  radius  of  its  base  add  square  of  its 
height ;  multiply  this  sum  by  height,  and  product  by  .5236. 
Fig.  29.  or,! 


2. — From  three  times  diameter  of  sphere  subtract  twice 
height  of  segment ;  multiply  this  remainder  by  square  of 
height,  and  product  by  .5236. 

Or,  3  d  —  2  h  h2  X  .  5236  •=.  V. 

EXAMPLE.— Segment  of  a  sphere,  Fig.  29,  has  a  radius,  a  o,  of  7 
inches  for  its  base,  and  a  height,  6  o,  of  4 ;  what  is  its  volume  ? 

7  2  x  3  -}-  42=  163  =  the  sum  of  three  times  square  of  radius  and 
square  of  height;  163  X  4  X  .5236  =  341.3872  cube  ins. 


368 


MENSUKATION    OP   VOLUMES. 


Spherical    Zone   (or   JEnrnstum   of  a   Sphere). 

DEFINITION. — Part  of  a  sphere  included  between  two  parallel  chords. 

To    Coxnpnte   "Volume    of  a    Spherical    Zone.— Fig.  3O. 

DEFINITION. — Part  of  a  sphere  included  between  two  parallel  planes. 

RULE. — To  sum  of  squares  of  the  radii  of  the  two  ends  add  one  third  of 
square  of  height  of  zone ;  multiply  this  sum  by  height,  and  again  by  1.5708. 
Fig.30.  _ 

EXAMPLE.— What  is  the  volume  of  a  spherical  zone,  Fig.  30, 
greater  and  less  diameters, /&  and  de,  being  20  and  15  inches, 
and  distance  between  them,  or  height  of  zone,  eg,  being  10  ins.? 

I02  _|_  7. 52  _  j  56. 25  =  sum  of  squares  of  radii  of  the  two  ends ; 
156.25  -}-  io2  -r-  3  =  189.58  =  above  sum  added  to  one  third  of 
square  of  the  height. 

Then  189.58  x  io  x  1.5708  =  2977.9226  cube  ins. 

Cylindrical  Ring. 

DEFINITION.— A  ring  formed  by  the  curvature  of  a  cylinder. 
To    Compute   "Voltime    of*  a    Cylindrical    Ring.— Fig.  31. 

RULE. — To  diameter  of  body  of  ring  add  inner  diameter  of  ring ;  multi- 
ply sum  by  square  of  diameter  of  body,  arid  product  by  2.4674. 

Fig.  31.  _  Or,  d  +  d'  dy  2. 4674  =  V. 

Or,  a  I  =  V,  a  representing  area  of  section  of  body,  and  I  length 
of  axis  of  body.. 

EXAMPLE. — What  is  volume  of  an  anchor  ring,  Fig.  31,  diameter 
of  metal,  a  b,  being  3  inches,  and  inner  diameter  of  ring,  b  c,  8? 

3_j_8X32  =  99=  product  of  sum  of  diameters  and  square  of  di- 
ameter of  body  of  ring. 
Then  99  X  2.4674  =  244.2726  cube  ins. 

Spheroids   (Ellipsoids). 

DEFINITION. — Solids  generated  by  the  revolution  of  a  semi  ellipse  about  one  of  its 
diameters.  When  the  revolution  is  about  the  transverse  diameter  they  are  termed 
Prolate,  and  when  about  the  conjugate  they  are  Oblate. 

To    Compute   "Volume    of  a    Spheroid..  —Fig.  32. 
RULE.— Multiply  square  of  revolving  axis  by  fixed  axis,  and  this  product 
by  .5236. 

Or,  a2  a'  X  •  5236  =  V,  a  and  a'  representing  revolving  and 
fixed  axes. 
Or,  4-^3X3- 1416  r2  r'= V,  r  and  r'  representing  semi-axes. 

EXAMPLE.— In  a  prolate  spheroid,  Fig.  32,  fixed  axis,  a  &,  is 
14  inches,  and  revolving  axis,  cd,  io;  what  is  its  volume? 

io2  X  14  =  1400=  product  of  square  of  revolving  axis  and 
fixed  axis.     Then  1400  X  5236  =  733.04  cube  ins. 
NOTE. Volume  of  a  spheroid  is  equal  to  %  of  a  cylinder  that  will  circumscribe  it. 

Segments   of  Splieroids. 

To  Compn te  "Volx\me  of  Segment  of  a  Spheroid.— Fig.  33. 
When  Base,  ef,  is  Circular,  or  parallel  to  revolving  Axis,  as  c  d,  Fig.  33, 
or  as  efto  Axis  a  6,  Fig.  34.  RULE.— Multiply  fixed  axis  by  3,  height  of 
segment  by  2,  and  subtract  one  product  from  the  other ;  multiply  remainder 
by  square  of  height  of  segment,  and  product  by  .5236.  Then,  as  square  of 
fixed  axis  is  to  square  of  revolving  axis,  so  is  last  product  to  volume  of 
segment. 


Fig.  32. 


MENSUEATION   OF   VOLUMES. 


369 


Fig-  33- 


Or, 


3  a~  2 


a/2 


_y 

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EXAMPLE.— In  a  prolate  spheroid,  Fig.  33,  fixed  or  trans- 

l--j — .jfc  verse  axis,  a  6,  is  100  inches,  revolving  or  conjugate,  c  d,  60, 

'     and  height  of  segment,  ao,  10;  what  is  its  volume? 


^ — i +r  ioo  x  3  — 10  x  2  =  280=  twice  the  height  of  segment  sub- 

d  tracted  from  three  times  Jixed  axis  ;    280  X  10 2  X  •  5236  = 

14660.8  inches  =  product  of  above  remainder,  square  of  height,  and  .5236.    Then 
ioo2  :  6o2  ::  14669.8  :  5277.888  cube  ins. 

When  Base,  ef,  is  Elliptical,  or  perpendicular  to  revolving  Axis,  a  b,  Fig. 
33,  or  as  ef  to  Axis  c  d,  Fig.  34.  RULE.  —  Multiply  fixed  axis  by  3, 
and  height  of  segment  by  2,  and  subtract  one  from  the  other ;  multiply  re- 
mainder by  square  of  height  of  segment,  and  product  by  .5236.  Tnen,  as 
fixed  axis  is  to  revolving  axis,  so  is  last  product  to  volume  of  segment. 

^  34-  3a'--***X.5*36Xa  =  y 

«U*??=S5=^/ 

^v         EXAMPLE.— Diameters  of  an  oblate  spheroid,  Fig.  34,  are 
— \b   ioo  and  60  inches,  and  height  of  a  segment  thereof  is  12; 
what  is  its  volume? 


ioo  X  3  — 12  X  2  =  276  =  twice  the  height  of  the  segment  sub- 
tracted from  three  times  the  revolving  axis  ;  276  X  12 2  X  -5236 
=  20  809. 9584=  product  of  above  remainder,  the  square  of  height,  and  .5236. 
Then  ioo  :  60  ::  20809.9584  :  12485.975  cube  ins. 


Frnsta   of  Spheroids. 

To  Compute  "Volume  of  Middle  Frustum  of  a  Spheroid.— 
Fig.  35. 

When  Ends,  ef  and  g  h,  are  Circular,  or  parallel  to  revolving  Axis,  as  c  d, 
Fig.  35,  or  a  b,  Fig.  36.  RULE.— To  twice  square  of  revolving  axis  add 
square  of  diameter  of  either  end ;  multiply  this  sum  by  length  of  frustum, 
and  product  by  .2618. 

Or,  2  a'2+d2  X  ^.2618  =  V. 

EXAMPLE. —  Middle  frustum  of  a  prolate  spheroid,  i  o, 
Fig.  35,  is  36  inches  in  length,  diameter  of  it  being,  in 
middle,  c  d,  50,  and  at  its  ends,  ef  and  g  h,  40;  what  is  its 
volume  ? 

5o2  X  2  -f  4o2  =  6600  =  sum  of  twice  square  of  middle  di- 
ameter added  to  square  of  diameter  of  ends.  Then  6600  X 
36  X  2618  =  62  203.68  cube  ins. 

When  Ends,  efand  g  h,  are  Elliptical,  or  perpendicular  to  revolving  Axis, 
a  b,  Fig.  35,  or  ef  and  g  h  to  Axis,  c  d,  Fig.  36.  RULE. — To  twice  product 
of  transverse  and  conjugate  diameters  of  middle  section,  add  product  of 
transverse  and  conjugate  of  either  end;  multiply  this  sum  by  length  of 
frustum,  and  product  by  .2618. 

Fig.  36.  Or,  d  <T  X  2  +  ~ddf  I X  .2618  =  V. 

EXAMPLE. — In  middle  frustum  of  a  prolate  spheroid,  Fig. 
36,  diameters  of  its  middle  section  are  50  and  30  inches,  its 
ends  40  and  24,  and  its  length,  o  i,  18;  what  is  its  volume? 

50  X  30  X  2  =  3000  =.  twice  product  of  transverse  and  con- 
jugate diameters  ;   3000  -j-  40  X  24  =  3960  =.  sum  of  above 
product  and  product  of  transverse  and  conjugate  diameters 
of  ends. 
Then  3960  x  18  X  .2618  =  18661.104  cube  ins. 


370 


MENSURATION    OF    VOLUMES. 


Links. 

DEFINITION. — Elongated  or  Elliptical  rings. 

Elongated,   or   Elliptical   Links. 

To  Compute  "Volume  of  an  Elongated,  or  Elliptical  Link. 
—Figs.  37    and.    38. 

RULE. — Multiply  area  of  a  section  of  the  body  of  link  by  its  length,  or 
circumference  of  its  axis. 

Or,  a  I  or  c  =  V. 

NOTE.— By  Rule,  page  353,  Circumference  or  length  of  axis  of  an  Elongated  link 
=  the  sum  of  3.1416  times  sum  of  less  diameter  added  to  thickness  of  ring,  and 
product  of  twice  remainder  of  less  diameter  subtracted  from  greater. 

Also,  Circumference  or  length  of  axis  of  an  Elliptical  ring  =  square  root  of  half 
Bum  of  diameters  added  to  thickness  of  ring  or  axes  squared  x  3. 1416. 
Fig.  37.  EXAMPLE.— Elongated  link  of  a  chain,  Fig.  37,  is  i  inch  in  diameter 

of  body,  a  6,  and  its  inner  diameters,  b  c  and  ef,  are  10  and  2.5  inches; 
what  is  its  volume  ? 

Area  of  i  inch  =1.7854;  2. 5  -}- 1  X  3- 1416=  10.9956  =13. 1416  times  sum 
of  less  diameter  and  thickness  of  ring  =.  length  of  axis  of  ends;  10  —  2.5 
X  2  =  15  =  twice  remainder  of  the  less  diameter  subtracted  from  greater 
=  length  of  sides  of  body.  F.  g 

Then  10.9956  -f- 15  =  25.9956  =  length  of  axis  of  length. 
Hence  .7854  X  25.9956  =  20.417  cube  ins. 

2.— Elliptical  link  of  a  chain,  Fig.  38,  is  of  the  same  dimensions  as 
preceding;  what  is  its  volume? 


2. 5  + 1  -f- 10  -|- 1  =  133. 25  =  diameter  of  axes  squared  ;    /I33'25  3. 1416 

=  25. 643  =  square  root  of  half  sum  of  diameters  squared  X  3. 1416  =  cir- 
cumference of  axis  of  ring.    Area  of  i  inch  = .  7854. 
Then  25.643  X  .7854  =  20.14  cube  ins. 

Spherical   Sector. 

DEFINITION.— A  figure  generated  by  the  revolution  of  a  sector  of  a  circle  about  a 
straight  line  through  the  vertex  of  the  sector  as  an  axis. 

NOTE.— Arc  of  sector  generates  surface  of  a  zone,  termed  base  of  sector  of  a 
sphere,  and  the  radii  generate  surfaces  of  two  cones,  having  a  vertex  in  common 
with  the  sector  at  the  centre  of  the  sphere. 

To  Compute   Volume   of*  a   Spherical    Sector.— Fig.  39. 

RULE.— Multiply  external  surface  of  zone,  which  is  base  of  sector,  by  on« 
third  of  the  radius  of  sphere. 

Or,  a  r  -r-  3  =  V,  a  representing  area  of  base. 

NOTE.— Surface  of  a  spherical  sector  =  sum  of  surface  of  zone  and  surfaces  of  the 
two  cones. 

Fig.  39.  EXAMPLE.— What  is  volume  of  a  spherical  sector,  Fig. 

39,  generated  by  sector,  c  a  h,  height  of  zone,  a  b  c  d,  be 
ing  a  o,  12  inches,  and  radius,  g  A-,  of  sphere  15  ? 

12  X  94.248  =  1130.976  =r  height  of  zone  X  circumference 
of  sphere  =  external  surface  of  zone  (see  page  350). 

1130  976  X  15  -r-  3  =  surface  X  one   third   of  radius  = 
5654  88  cube  ins. 

Spindles. 

DEFINITION.  —Figures  generated  by  revolution  of  a  plane  area  bounded  by  a  curve, 
when  the  curve  is  revolved  about  a  chord  perpendicular  to  its  axis  or  about  its 
double  ordinate,  and  they  are  designated  by  the  name  of  arc  from  which  they  are 
generated,  as  Circular,  Elliptic,  Parabolic,  etc. 


MENSURATION  OF  VOLUMES. 


371 


Circular   Spindle. 
To   Compute   "Volume    of  a    Circular    Spindle.— Fig.  4O. 

RULE. — Multiply  central  distance  by  half  area  of  revolving  segment-, 
subtract  product  from  one  third  of  cube  of  half  length,  and  multiply  re- 
mainder by  12.5664. 

Or, (  ~2) (c  X  |J  X  12.5664  =  V,  a  representing  area  of  revolving  segment. 

EXAMPLE.— What  is  volume  of  a  circular  spindle,  Fig.  40,  when 
central  distance,  oe,  is  7.071067  inches,  length, /c,  14.142  13,  and 
radius,  oc,  10? 

NOTE.— Area  of  revolving  segment;  fe,  being  =  side  of  square 
that  can  be  inscribed  in  a  circle  of  20,  is  2o2  x  .  7854  — 14. 142 13* 
•4-  4  =  28. 54  area. 

7.071 067X28. 54 -^-2=100. 9041  =  central  distance  X  half  area  of 

revolving  segment ;  — 100. 9041  =  16. 947  =  remainder  of 

above  product  and  one  third  of  cube  of  half  length. 

Then  16.497  X  12.5664  =  212.9628  cube  ins. 

Frastum   or   Zone   of  a   Circular   Spindle.* 

To  Compute  "Volume  of  a  Frustum  or  Zone  of  a  Circular 

Spindle. — F"ig.  41. 

RULE. — From  square  of  half  length  of  whole  spindle  take  one  third  of 
square  of  half  length  of  frustum,  and  multiply  remainder  by  said  half  length 
of  frustum ;  multiply  central  distance  by  revolving  area  which  generates 
the  frustum ;  subtract  this  product  from  former,  and  multiply  remainder  by 
6.2832. 

Or,  £-=-2 -X (c  X  a)  X  6.2832  =  V,  I  and  V  representing  lengths  of 

spindle  and  of  frustum,  and  a  area  of  revolving  section  of  frustum. 

NOTE.  — Revolving  area  of  frustum  can  be  obtained  by  dividing  its  plane  into  a 
segment  of  a  circle  and  a  parallelogram. 

EXAMPLE.— Length  of  middle  frustum  of  a  circular  spindle, 
tc,  Fig.  41,  is  6  inches;  length  of  spindle,/^,  is  8;  central  dis- 
tance, o  e,  is  3 ;  and  area  of  revolving  or  generating  segment 
is  10 ;  what  is  volume  of  frustum  ? 

(8  -T-  2)2  —  *  ~       =  13,  and  13  X  3  —  39  =  product  of  — 

length  of  frustum,  and  remainder  of  one  third  square  of  half 
length  of  frustum  subtracted  from  square  of  half  length  of 

spindle  ;  39  —  3  x  10 = 9  =  product  of  central  distance  and  area  of  segment  subtracted 

from  preceding  product. 

Then  9X6. 2832  =  56. 5488  cube  ins. 

Segment   of  a   Circular    Spindle. 

To    Compute    "Volxixne    of   a    Segment    of  a    Circular 
Spindle.— Fig.  4:2. 

RULE. — Subtract  length  of  segment  from  half  length  of  spindle ;  double 
remainder,  and  ascertain  volume  of  a  middle  frustum  of  this  length.  Sub- 
tract result  from  volume  of  whole  spindle,  and  halve  remainder.! 

Or,  C  —  c^-  2  =  V,  C  and  c  representing  volume  of  spindle  and  middle  frustwn. 


*  Middle  frustum  of  a  Circular  Spindle  U  ene  of  the  various^  forms  of  casks. 
•  Mm*  first  ob'tiin* 


t  This  rule  if  applicable  to  segment  of  any  Spindle  or  any  Conoid,  yolume  of  the  figure  and  fru»tu* 

•     i*d. 


3/2  MENSURATION    OF   VOLUMES. 

Fig.  42.  EXAMPLE.  —  Length  of  a  circular  spindle,  i  a,  Fig.  42,  IP 

^  14.14213  inches;  central  distance,  o  e,  is  7.07107;  radius  of 

jf. ^>,0  arc,  o  a,  is  10;  and  length  of  segment,  i  c,  is  3.535  53;  what  is 

.-;/  its  volume  ? 
_J..— •*'*/ 

j   /  —^  —  3. 535  53  X  2  =  7.071 07  =  double  remainder  of 

**%  length  of  segment  subtracted  from  half  length  of  spindle  — 

length  of  middle  frustum. 

NOTK. — Area  of  revolving  or  generating  segment  of  whole  spindle  is  28.54  inches, 
and  that  of  middle  frustum  is  19.25. 

The  volume  of  whole  spindle  is 212.9628  cube  ins. 

"         "     middle  frustum  is 162.8982    "     " 

Hence 50.0646 H-  2  =  25.0323  cube  ins. 

Cycloidal    Spindle.* 
To   Compute  Volume  of  a  Cycloidal   Spindle.— Fig.  4:3. 

RULE. — Multiply  product  of  square  of  twice  diameter  of  generating  circle 
and  3.927  by  its  circumference,  and  divide  this  product  by  8. 

_. or, .9..4.   = 

v\     'c 
d  (- — ^==^-'"'       °f  c*VcZe,  or  half  width  of  spindle. 

EXAMPLE. — Diameter  of  generating  circle,  a  6  c,  of  a  cy- 
cloid, Fig.  43,  is  10  inches;  what  is  volume  of  spindle,  d  e? 

2 

10  X  2  X  3-927  =  1570.8  =1  product  of  twice  diameter  squared  and  3.927. 
Then  1570.8  x  iox  3.1416-7-8  =  6168.5316  cube  ins. 

Elliptic    Spindle. 
To    Compute    Volume    of  an    Elliptic    Spindle.— Fig.  44. 

RULE.  —  To  square  of  its  diameter  add  square  of  twice  diameter  at  one 
fourth  of  its  length ;  multiply  sum  by  length,  and  product  by  .1309-! 

. 2 

Or,  d2  -f-  2  d'  1. 1309  =  V,  d  and  d'  representing  diameters  as  above. 
Fig.  44.  EXAMPLE.  —  Length  of  an  elliptic  spindle,  a  6,  Fig.  44,  \& 

75  inches,  its  diameter,  cd,  35,  and  diameter,  e/,  at  .25  of  its 
length,  25 ;  what  is  its  volume  ? 


352  -f-  25  X  2  =  3725  =  sum  of  squares  of  diameter  of 
spindle  and  of  twice  its  diameter  at  one  fourth  of  its  length; 
3725  X  75  —  279  375  =  above  sum  X  length  of  spindle. 
Then  279  375  x  .  1309  =  36  570. 1875  cube  ins. 

NOTK.— For  all  such  solid  bodies  this  rule  is  exact  when  body  is  formed  by  a 
conic  section,  or  a  part  of  it,  revolving  about  axis  of  section,  and  will  always  be 
rery  near  when  figure  revolves  about  another  line. 

To    Compute   "Volume    of*  Middle    FmstxTm    or    Zone    of 
an    Elliptic    Spindle.— Fig.  45. 

RULE. — Add  together  squares  of  greatest  and  least  diameters,  and  square 
of  double  diameter  in  middle  between  the  two ;  multiply  the  sum  by  length, 
and  product  by  .13094 

2 

Or,  d2  -f-  d'2  +  2  d"  1. 1309  =  V,  d,  d',  and  d"  representing  different  diameters. 


*  Volume  of  a  Cycloidal  Spindle  is  equal  to  .625  of  ite  circumscribing  cylinder, 
t  See  preceding  Note.  J  See  Note  abore. 


MENSURATION    OF    VOLUMES. 


373 


Fig.  45- 


qa 


EXAMPLE.— Greatest  and  least  diameters,  a  b  and  cd,  of 
the  frustum  of  an  elliptic  spindle,  Fig.  45,  are  68  and  50 
inches,  its  middle  diameter,  g  h,  60,  and  its  length,  g/,  75; 
what  is  its  volume? 


682  -\-  so2  -|-  60  X  2  =  21  524  =  sum  of  squares  of  greatest 
and  least  diameters  and  of  double  middle  diameter. 
Then  21  524  X  75  X  .1309  =  211  311.87  cube  ins. 

To  Compute  "Volume  of*  a  Segment  of  an.  Elliptic  Spin- 
dle.— Fig.  46. 


RULE. — Add  together  square  of  diameter  of  base  of  segment  and  square 
of  double  diameter  in  middle  between  base  and  vertex ;  multiply  sum  by 
length  of  segment,  and  product  by  .1309.* 

2 

Or,  d2+  2  d"  I X  .  1309  =  V,  d  and  d"  representing  diameters. 

EXAMPLE.— Diameters,  c  d  and  g  h,  of  the  segment  of  an 
elliptic  spindle,  Fig.  46,  are  20  and  12  inches,  and  length, 
*"\%     o  ey  is  16 ;  what  is  its  volume  ? 

""/      2o2  -f  12  x  2  =  976  =  sum  of  squares  of  diameter  at  base 
^**'      and  in  middle. 
>J — "*  Then  976  x  16  X  .  1309  =  2044. 134  cube  ins. 

3?ara"bolic   Spindle. 

To  Compute  "Volume    of  a   ^Parabolic    Spindle.— Fig.  47'. 

RULE  i.  —  Multiply  square  of  diameter  by  length,  and  the  product  by 

.41888.1 


RULE  2.  —  To  square  of  its  diameter  add  square  of  twice  diameter  at  one 
fourth  of  its  length  ;  multiply  sum  by  length,  and  product  by  .13094 

Or,  d2  +  ~zdf  I  X  -1309  —  V. 

EXAMPLE.—  Diameter  of  a  parabolic  spindle,  a  bt  Fig. 

47,  is  40  ins.,  and  its  length,  cd,  10;  what  is  its  volume? 

402  X  10  =  16  ooo  =  square  of  diameter  X  length. 

Then  16000  X  .418  88  =  6702.08  cube  ins. 
Again,  If  middle  diam.  at  .25  of  its  length  is  30,  Then, 

by  Rule  2,  4o2  +  30  X  2  X  40  X  .  1309  =  6806.8  cube  ins. 

To  Compute  "Volume  of  ^Middle  Frustum  of  a  3?ara"bolio 

Spindle.—  Fig.  48. 

RULE  i.  —  Add  together  8  times  square  of  greatest  diameter,  3  times 
square  of  least  diameter,  and  4  times  product  of  these  two  diameters  ;  mul- 
tiply sum  by  length,  and  product  by  .052  36. 


RULE  2.  —  Add  together  squares  of  greatest  and  least  diameters  and 
square  of  double  diameter  in  middle  between  the  two ;  multiply  the  sum 
by  length,  and  product  by  .1309. 

Or,  d2  -f-  d'2  -f-  2  d"2  I  X  •  1309  =  V,  d"  representing  diameter  between  the  two. 
pjff    o  EXAMPLE. — Middle  frustum  of  a  parabolic  spindle,  Fig. 

48,  has  diameters,  a  b  and  ef,  of  40  and  30  inches,  and  its 
Ill>y         length,  cd,  is  10;  what  is  its  volume? 


4o2  X  8  +  30-'  X  3  +  40  X  30  X  4  =  20  300  =  sum  of  8 
times  square  of  greatest  diameter,  3  times  squeire  of  least 
diameter,  and  4  times  product  of  these. 

Then  20  300  x  10  X  -052  36  =  10629.08  cube  ins. 


*  See  Note,  page  372. 


t  8-15  of  .7854. 


!-i5  of .; 
Il 


J  See  Note,  page  372. 


374 


MENSURATION   OF   VOLUMES. 


To    Compute    "Volume    of  a    Segment    of  a    IParatoolic 
Spindle.—  Fig.  49. 

RULE.  —  Add  together  square  of  diameter  of  base  of  segment  and  square 
of  double  diameter  in  middle  between  base  and  vertex  ;  multiply  sum  by 
height  of  segment,  and  product  by  .1309. 


EXAMPLE.—  Segment  of  a  parabolic  spindle,  Fig.  49,  has 
diameters,  e/and  g  h,  of  15  and  8.75  inches,  and  height, 
c  d,  is  2.5  ;  what  is  its  volume  ? 

152  -f-  8.  75  x  2  =  531.25  =  sum  of  square  of  base  and  of 
double  diameter  in  middle  of  segment.  Then  531.25  X  2.5 
X  .1309  =  173.852  cube  ins. 


Hyperbolic   Spindle. 
To  Compute  Volume  of*  a  Hypertoolie  Spindle.—  Fig.  SO. 

RULE.  —  To  square  of  diameter  add  square  of  double  diameter  at  one 
fourth  of  its  length  ;  multiply  sum  by  length,  and  product  by  .1309.* 

Fig.  50.     a  Or,  d2  +l^o?l  x  .  1309  =  V. 

EXAMPLE  —  Length,  aft,  Fig.  50,  of  a  hyperbolic  spindle 
is  ioo  inches,  and  its  diameters,  cd  and  ef,  are  150  and 
no;  what  is  its  volume? 


I5°2H~  II0  X  2  X  109  =  7090000  =  product  of  sum  of 
squares  of  greatest  diameter  and  of  twice  diameter  at  one 
fourth  of  length  of  spindle  and  length.  Then  709ooooX 
.  1309  =  928  08  1  cube  inches. 

To  Compute  Volume   of  Middle  Frustum   of  a  Hyper- 
toolie    Spindle.—  Fig.  SI. 

RULE.—  Add  together  squares  of  greatest  and  least  diameters  and  square 
of  double  diameter  in  middle  between  the  two  ;  multiply  this  sum  by  length, 
and  product  by  .1309^ 


Fi 


Or,  d2  +  d'2  -f  (2  d")2  I  X  .  1309  =  V. 
EXAMPLE.—  Diameters,  a  b  and  c  d",  of  middle  frustum  of  a 
hyperbolic  spindle,  Fig.  51,  are  150  and  no  inches;  diam- 
eter, g  h,  140;  and  length,  eft  50;  what  is  its  volume? 

i5o2+  iio2-|-  140  X  2  =  113  000  =  sum  of  squares  ofgreat- 
egt  and  feast  diameters  and  of  double  middle  diameter.  Then 
113000  X  5°  X  •  1309  =  739  585  cube  ins. 

To  Compute  Volume  of  a  Segment  of  a  Hyper"bolic  Spin- 
dle.— Fig.  S3. 

RULE.  —  Add  together  square  of  diameter  of  base  of  segment  and  square 
of  double  diameter  in  middle  between  base  and  vertex  ;  multiply  sum  by 
length  of  segment,  and  product  by  .1309. 

Or,  d2  +  d"2lX.i3og  =  V. 

EXAMPLE.  —  Segment  of  a  hyperbolic  spindle,  Fig.  52,  has 
diameters,  e/and  g  h,  of  no  and  65  inches,  and  its  length,  a  6, 
25;  what  is  its  volume? 


i  io2  -f-  65  x  2  =  29  ooo  =  sum  of  squares  of  diameter  of  base 
and  of  double  middle  diameter. 

Then  29  ooo  X  25  x  •  1309  =  94  902.5  cube  ins. 


*  See  Note,  page  372. 


MENSURATION  OF  VOLUMES. 


375 


Ellipsoid,  I?ara"boloid,  and.   Hypertooloid   of  Revo- 
lution*  (Conoids). 

DEFINITION.  —Figures  like  to  a  cone,  described  by  revolution  of  a  conic  section 
around  and  at  a  right  angle  to  plane  of  their  fixed  axes. 

Ellipsoid   of*  Revolution   (Spheroid). 

DEFINITION.  —  An  ellipsoid  of  revolution  is  a  semi-spheroid.    (See  page  368.) 

IParaboloid   of*  Revolntion.t 

To   Compute  Volume   of  a   Paraboloid.   of  Revolution.— 

Fig.  S3. 

RULE.  —  Multiply  area  of  base  by  half  height 
Fig.  53.    0  Or,afc-i-2  =  V. 

NOTE.  —  This  rule  will  hold  for  any  segment  of  paraboloid, 
whether  base  be  perpendicular  or  oblique  to  axis  of  solid. 

EXAMPLE.—  Diameter,  a  6,  of  base  of  a  paraboloid  of  revolution, 
Fig.  53,  is  20  inches,  and  its  height,  d  c,  20;  what  is  its  volume? 

Area  of  20  inches  diameter  of  base  :=  314.16.    Then  314.  16  X 
20  -i-  2  —  3141.6  cube  ins. 

of  a  I?ara"boloid   of  Revolution. 

To  Compute  "Volume  of  a  Frustum  of  a  IParatooloid.  of 
Revolvition.—  Fig.  £54. 

Fig.  54.  RULE.  —  Multiply  sum  of  squares  of  diameters  by 

height  of  frustum,  and  this  product  by  .3927. 

Or,(d2  +  <r2)Ax.3927=V. 

EXAMPLE.  —  Diameters,  a  b  and  d  c,  of  the  base  and  vertex 
of  frustum  of  a  paraboloid  of  revolution,  Fig.  54,  are  20  and 
11.5  inches,  and  its  height,  ef,  12.6;  what  is  its  volume? 

2o2  -f  1  1.  s2  =  532.  25  =  sum  of  squares  of  diameters.  Then 
532.25  X  12.6  X  .3927  =  2633.5837  cube  ins. 

Segment   of  a  3?ara"boloid   of  Revolution. 

To  Compute  Volume  of  Segment  of  a  JParatooloid  of  Revo- 
lution.— Fig.  5£5. 


55-  / 


RULE.  —  Multiply  area  of  base  by  half  height. 


Or,  a  x  ft-f-2  =  V. 

NOTE.—  This  rule  will  hold  for  any  segment  of  paraboloid, 
whether  base  be  perpendicular  or  oblique  to  axis  of  solid. 

EXAMPLE.  —  Diameter,  a  6,  of  the  base  of  a  segment  of  a  para- 
boloid of  revolution,  Fig.  55,  is  115  inches,  and  its  height,  «/,  is 
7.4;  what  is  its  volume? 

Area  of  11.5  inches  diameter  of  base  =  103.869.    Then  103.869 
X  7.4  -f-  2  =  384.  315  cube  ins. 

Hyperboloid   of  Revolution. 

To   Compute  "Volume   of  a  HypertDoloid.  of  Revolution. 
—Fig.  SB. 

HULE.  —  To  square  of  radius  of  base  add  square  of  middle  diameter  ;  mul- 
tiply this  sum  by  height,  and  product  by  .5236. 

*  These  figures  have  been  known  aa  Conoid*.    For  the  definition  of  a  Conoid,  see  HatvelVi  Men 
turai'on,  page  233. 
t  Volume  of  a  Paraboloid  of  Revolution  is  =  .5  of  its  circumference. 


376 


MENSURATION    OP   VOLUMES. 


Fig.  56. 


Or,  r*-\-d2  ftX-5236=V,  d  representing  middle  diameter 
EXAMPLE.  —  Base,  a  6,  of  a  hyperboloid  of  revolution, 

Fig.  56,  is  80  inches;  middle  diameter,  c  d,  66;  and  height, 

ef,  60;  what  is  its  volume? 

*        80  -r-  2  -j-  662  =  5956  =  sum  of  square  of  radius  of  base  and 
0    middle  diam.    Then  5956  X  60  X  •  5236  =  87 113.7  cuoe  ins- 

Segment   of  a   Hyper"boloid.   of  Revolution. 
To   Compute  "Volxime   of  Segment   of  a    Hyperboloid    of 

Revolution.,  as    Fig.  £56. 

KUT.E. — To  square  of  radius  of  base  add  square  of  middle  diameter ;  mul- 
tiply this  sum  by  height,  and  product  by  .5236. 

Or,  r2  -f-  d"2  h  X  .5236  =  V,  r  representing  radius  of  base. 

EXAMPLE.— Radius,  a  e,  of  base  of  a  segment  of  a  hyperboloid  of  revolution,  as 
Fig  56,  is  21  inches;  its  middle  diameter,  c  d,  is  30;  and  its  height,  efy  15;  what  is 
its  volume? 

21 2  -}-  3o2  X  1 5  =  20 1 15  =  product  of  sum  of  squares  of  radius  of  base  and  middle 
diameter  multiplied  by  height.  Then  20 115  X  .  5236  =  10  532. 214  cube  ins. 

"Frustum,   of  a   Hyperboloid    of  Revolution. 
To   Compute  Volume  of  Frustum    of  a   Hyperboloid.   of 

devolution.— Fig.  S7". 

RULE.— Add  together  squares  of  greatest  and  least  semi-diameters  and 
square  of  diameter  in  middle  of  the  two ;  multiply  this  sum  by  height,  and 
product  by  .5236. 

Or,  (-)  +  (— )  +  d"2  A  X  •  5236  =  V,  d,  d',  and  d"  representing  several  diameters. 


Fig-  57 


EXAMPLE.— Frustum  of  a  hyperboloid  of  revolution,  Fig. 
57,  is  in  height,  e  i,  50  inches;  diameters  of  greater  and 
lesser  ends,  a  b  and  c  d,  are  no  and  42 ;  and  that  of  middle 
diameter,  g  h,  is  80;  what  is  volume? 

110-^-2  —  55,  and  42-4-9  =  21.  Hence  ss2-)-  2i2-f8o2 
j  =9866  =  sum  of  squares  of  semi-diame^jrs  of  ends  and  of 
0  middle  diam.  Then  9866  X  50  X  •  5236  =  258  291. 88  cube  ins. 

.A.ny   Figure    of  Revolution. 
To    Compute   Volume   of  any   Figxire   of  Revolution.— 

Fig.  SS. 

RULE. — Multiply  area  of  generating  surface  by  circumference  described 
by  its  centre  of  gravity. 

Or,  a  2rp  =  V,  r  representing  radius  of  centre  of  gravity. 

Fig-  58-  ILLUSTRATION  i.  —  If  generating  surface,  ab  c  d,  of  cylinder, 

e  b  e  df,  Fig.  58,  is  5  inches  in  width  and  io  in  height,  then  will 
a  b  =  5  and  6  d  =  io,  and  centre  of  gravity  will  be  in  o,  the  ra- 


, 

dius  of  which  is  r  o  =.  5  -4-  2  =  2.  5. 
of  generating  surface. 


, 
Hence  10  X  5  =  50  =  area 


Then  50  x  2. 5  x  2  X  3- 1416  =  785. 4  =  area 
"~-~lf  of  generating  surface  X  circumference  of  its 
.---'' J  centre  of  gravity  =  volume  of  cylinder. 


PROOF.—  Volume  of  a  cylinder  10  inches  in  diameter  and  10 
inches  in  height.     io2  X  .7854  =  78.  54,  and  78.54  x  10  =  785.4. 

2.  _lf  generating  surface  of  a  cone,  Fig.  59,  is  a  e  =  io,  d  e  — 
5,  then  will  aci—  11.18.  and  area  of  triangle  =  io  X  5-7-2  =  25, 
centre  of  gravity  of  which  is  in  o,  and  o  r,  by  Rule,  page  607,   d 
=  1.666. 


Hence,  25  x  1.666  X  2  x  3.1416  —  261.8  =  area  of  generating  surface  X  circum- 
ference of  its  centre  of  gravity  =  volume  of  cone. 


MENSURATION   OF  VOLUMES.  377 

3.— If  generating  surface  of  a  sphere,  Fig.  60,  is  abc,  and  ac 

=  10,  a  b  c  will  be  f  — '-?—^\  —  39.27,  centre  of  gravity  of 

which  is  in  o,  and  by  Rule,  page  607,  o  r  =  2. 122. 

/       Hence,  39.27  X  2. 122  X  2  X  3- 1416  =  523.6  =  area  of  general- 
s'   ing  surface  X  circumference  of  its  centre  of  gravity  =  volume  oj 
sphere. 

Irregnlar  Bodies. 
To    Compute   "Volume    of  an    Irregular    Body. 

RULE. — Weigh  it  both  in  and  out  of  fresh  water,  and  note  difference  in 
Ibs. ;  then,  as  62.5*  is  to  this  difference,  so  is  1728!  to  number  of  cube  inches 
in  body. 

Or,  divide  difference  in  Ibs.  by  62.5,  and  quotient  will  give  volume  in 
cube  feet. 

NOTE.  —If  salt  water  is  to  be  used,  ascertained  weight  of  a  cube  foot  of  it,  or  64,  ia 
to  be  used  for  62. 5. 

EXAMPLE.— An  irregular-shaped  body  weighs  15  Ibs.  in  water,  and  30  out;  what 
is  its  volume  in  cube  inches  ? 

30  — 15  —  15  =  difference  of  weights  in  and  out  of  water. 

62.5  :  15  ::  1728  :  414.72  =  volume  in  cube  ins. 

Or,  1 5 ---62.5  =  .24,  and  .24  x  1728  =  414. 72?=  volume  in  cube  ins. 


CASK   GAUGING. 

Varieties   of  Casks. 

To    Compute   "Volume   of  a  Cask. 

ist  Variety.    Ordinary  form  of  middle  frustum  of  a  Prolate  Spheroid. 
This  class  comprises  all  casks  having  a  spherical  outline  of  staves,  as  Rum 
puncheons,  Whiskey  barrels,  etc. 

RULE. — To  twice  square  of  bung  diameter  add  square  of  head  diameter; 
multiply  this  sum  by  length  of  the  cask,  and  product  by  .2618,  and  it  will 
give  volume  in  cube  inches,  which,  being  divided  by  231,  will  give  result  in 
gallons. 

2d  Variety.    Middle  frustum  of  a  Parabolic  Spindle. 
This  class  comprises  all  casks  in  which  curve  of  staves  quickens  at  the  chime, 
as  Brandy  casks  and  Provision  barrels. 

RULE. — To  square  of  a  head  diameter  add  double  square  of  bung  diam- 
eter, and  from  sum  subtract  .4  of  square  of  difference  of  diameters ;  multiply 
remainder  by  length,  and  product  by  .2618,  which,  being  divided  by  231, 
will  give  volume  in  gallons. 

3^  Variety.    Middle  frustum  of  a  Paraboloid. 

This  class  comprises  all  casks  in  which  curve  of  staves  quickens  slightly  at 
bilge,  as  Wine  casks. 

RULE. — To  square  of  bung  diameter  add  square  of  head  diameter ;  mul- 
tiply sum  by  length,  and  product  by  .3927,  which,  being  divided  by  231, 
will  give  volume  in  gallons. 

^th  Variety.    Two  equal  frustums  of  Cones. 

This  class  comprises  all  casks  in  which  curve  of  staves  quickens  sharply  at 
bilge,  as  Gin  pipes. 

RULE. — Add  square  of  difference  of  diameters  to  three  times  square  of 
their  sum  ;  multiply  sum  by  length,  and  product  by  .065  66,  and  it  will  give 
Tolume  in  cube  inches,  which,  being  divided  by  231,  will  give  result  in 
gallons. 

*  Weight  of  a  cube  foot  of  fresh  water.  f  Number  of  inches  in  a  cube  foot. 


378  MENSURATION   OF   VOLUMES. 

EXAMPLE. — Bung  and  head  diameters  of  a  cask  are  24  and  16  inches,  and  length 
36;  what  is  its  volume  in  gallons? 


24  — i6  +  (24  +  i6)2X3  =  4864,  which  X  36  =  175  i°4>  and  '75  104  X  .06566  = 
ii  497.329,  which-:- 231  =  49.77  gallons. 

Generally. 

Dd-f-M2  .061 692  L  =  U.  S.  gallons,  and  .001 416 2  =  Imperial  gallons. 
D,  d,  and  M  representing  interior,  head  and  bung  diameters,  and  L  length  of  cask 
in  inches. 

To   Ascertain.    Mean   Diameter   of  a   Cask. 

RULE.— Subtract  head  diameter  from  bung  diameter  in  inches,  and  mul- 
tiply difference  by  following  units  for  the  four  varieties ;  add  product  to 
heacl  diameter,  and  sum  will  give  mean  diameter  of  varieties  required. 

ist  Variety 7       I      sd   Variety 56 

2d  Variety 68     |      4th  Variety 52 

EXAMPLE. — Bung  and1  head  diameters  of  a  cask  of  ist  variety  are  24  and  20  inch- 
es; what  is  its  mean  diameter? 

24  —  20  =  4,  and  4  x  .7  =  2.8,  which,  added  to  20,  =  22.8  ins. 

ULLAGE  CASKS. 
To   Compute   Volume   of*  Ullage   Casks. 

When  a  cask  is  only  partly  filled,  it  is  termed  an  ullage  cask,  and  is  com 
sidered  in  two  positions,  viz.,  as  lying  on  its  side,  when  it  is  termed  a  /Seg- 
ment Lying,  or  as  standing  on  its  end,  when  it  is  termed  a  Segment  Standing. 

To    Ullage    a    Lying    Cask. 

RULE. — Divide  wet  inches  (depth  of  liquid)  by  bung  diameter ;  find  quo- 
tient in  column  of  versed  sines  in  table  of  circular  segments,  page  267,  and 
take  its  corresponding  segment ;  multiply  this  segment  by  capacity  of  cask 
in  gallons,  and  product  by  1.25  for  ullage  required. 

EXAMPLE.— Capacity  of  a  cask  is  90  gallons,  bung  diameter  being  32  inches;  what 
is  its  volume  at  8  inches  depth? 

8-:-  32  =  .25,  tab.seg.  of  which  is.  153  55,  which  x  90  =  13.8195,  and  again  x  1.25  = 
i7.2744.ya/Zons. 

To   Ullage    a   Standing    Cask. 

RULE. — Add  together  square  of  diameter  at  surface  of  liquor,  square  of 
head  diameter,  and  square  of  double  diameter  taken  in  middle  between  the 
two;  multiply  sum  by  wet  inches,  and  product  by  .1309,  and  divide  by  231 
for  result  in  gallons. 

To  Compute  "Volume   of  a    Cask  "by    Four  Dimensions. 

RULE. — Add  together  squares  of  bung  and  head  diameters,  and  square  of 
double  diameter  taken  in  middle  between  bung  and  head ;  multiply  the  sum 
by  length  of  cask,  and  product  by  .1309,  and  divide  this  product  by  231  for 
result  in  gallons. 

To  Compute  Volume  of  any  Cask  from  Tliree  Dimen- 
sions  only. 

RULE. — Add  into  one  sum  39  times  square  of  bung  diameter,  25  times 
square  of  head  diameter,  and  26  times  product  of  the  two  diameters ;  mul- 
tiply sum  by  length,  and  product  by  .008  726 ;  and  divide  quotient  by  231 
for  result  in  gallons. 

For  Rules  in  Gauging  in  all  its  conditions  and  for  description  and  use  of 
instruments,  see  HaswelVs  Mensuration,  pages  307-23. 


CONIC    SECTIONS. 


379 


Fig.  i. 


CONIC  SECTIONS. 

A  Cone  is  a  figure  described  by  revolution  of  a  right-angled  triangle 
about  one  of  its  legs,  or  it  is  a  solid  having  a  circle  for  its  base,  and 
terminated  in  a  vertex. 

Conic  Sections  are  figures  made  by  a  plane  cutting  a  cone. 

If  a  cone  is  cut  by  a  plane  through  vertex  and  base,  section  will  be  a  triangle, 
and  if  cut  by  a  plane  parallel  to  its  base,  section  will  be  a  circle. 

Axis  is  line  about  which  triangle  revolves.  Base  is  circle  which  is  described  by 
revolving  base  of  triangle. 

An  Ellipse  is  a  figure  generated  by  an  oblique  plane  cut- 
ting a  cone  above  its  base. 

Transverse  axis  or  diameter  is  longest  right  line  that  can  be 
drawn  in  it,  as  a  fc,  Fig.  i. 

Conjugate  axis  or  diameter  is  a  line  drawn 
through  centre  of  ellipse  perpendicular  to  trans- 
verse axis,  as  c  d. 

A  Parabola  is  a  figure  generated  by  a 
plane  cutting  a  cone  parallel  to  its  side,  as  a  o  c,  Fig.  2. 
Axis  is  a  right  line  drawn  from  vertex  to  middle  of  base,  as  bo. 
NOTE. — A  parabola  has  not  a  conjugate  diameter. 

A  Hyperbola  is  a  figure  generated  by  a  plane 
cutting  a  cone  at  any  angle  with  base  greater  than  that  of 
side  of  cone,  as  a  b  c,  Fig.  3. 

Transverse  axis  or  diameter,  o  6,  is  that  part  of  axis,  e  b,  which, 
if  continued,  as  at  o,  would  join  an  opposite  cone,  ofr. 

Conjugate  axis  or  diameter  is  a  right  line  drawn  through  centre, 
g,  of  transverse  axis,  and  perpendicular  to  it. 

Straight  line  through  foci  is  indefinite  transverse  axis;  that  part 
of  it  between  vertices  of  curves,  as  o  b,  is  definite  transverse  axis. 
Its  middle  point,  g,  is  centre  of  curve. 
Eccentricity  of  a  hyperbola  is  ratio  obtained  by  dividing  distance  from  centre  to 
either  focus  by  semi-transverse  axis. 

Parameter  is  cord  of  curve  drawn  through  focus  at  right  angles  to  axis. 
Asymptotes  of  a  hyperbola  are  two  right  lines  to  which  the  curve  continually  ap- 
proaches, touches  at  an  infinite  distance  but  does  not  pass;  they  are  prolongations 
of  diagonals  of  rectangle  constructed  on  extremes  of  the  axes. 

Two  hyperbolas  are  conjugate  when  transverse  axis  of  one  is  conjugate  of  the 
other,  and  contrariwise. 

<3-eneral   Definitions. 

An  Ordinate  is  a  right  line  from  any  point  of  a  curve  to  either  of  diameters,  as 
a  e  and  do,  Fig.  4,  and  a  b  and  d/,  are  double  ordinates;  cb}  Fig.  5,  is  an  ordinate, 
and  a  6  an  abscissa. 

Fig.  4  c  d  An  Abscissa  is  that  part  of  diameter  which  is  contained  between 

vertex  and  an  ordinate,  as  ce.  go,  Fig. 4,  and  a  b.  Wia 

Fig.  5- 

Parameter  of  any  diameter  is  equal  to  four  times 
°j  distance  from  focus  to  vertex  of  curve;  parameter 
*  of  axis  is  least  possible,  and  is  termed  parameter 
of  curve. 
Parameter  of  curve  of  a  conic  section  is  equal 

_^          to  chord  of  curve  drawn  through  focus  perpendic-  ^ — 

/  ular  to  axis.  & 

Parameter  of  transverse  axis  is  least,  and  is  termed  parameter  of  curve. 
Parameter  of  a  conic  section  and  foci  are  sufficient  elements  for  construction 
of  curve. 


CONIC    SECTIONS. 


d 


_\  s 


A  Focus  is  a  point  on  principal  axis  where  double  ordinate  to  axis,  through  point, 
is  equal  to  parameter,  as  ef,  Fig.  5. 

It  may  be  determined  arithmetically  thus:  Divide  square  of  ordinate  by  four 
times  abscissa,  and  quotient  will  give  focal  distances,  as  and  s,  in  preceding  figures. 
Fig.  6.  Directrix  of  a  conic  section  is  a  right  line  at  right  angles  to 

-f*  major  axis,  and  it  is  in  such  a  position  that 

f:g:\u:  o. 

Here  a  d,  Fig.  6,  is  directrix,  and  o  is  offset  to  directrix. 
Latus  Rectum,  or  principal  parameter,  passes  through  a  focus; 
it  is  a  double  ordinate,  which  is  a  third  proportion  to  the  axis. 

Or,  A  :  a  :  :  a  :  L. 

A  and  a  representing  major  and  minor  axes.     (See  HasweWs 
Mensuration,  page  232.  ) 

A  Conoid  is  a  warped  surface  generated  by  a  right 
line  being  moved  in  such  a  manner  that  it  will  touch 
a  straight  line  and  curve,  and  continue  parallel  to  a 
given  plane.  Straight  line  and  curve  are  called  di- 
rectrices, plane  a  plane  directrix,  and  moving  line  the 
generatrix. 

Thus,  let  a  b  a',  Fig.  7,  be  a  circle  in  a  horizontal  plane, 
.  and  d  d'  projection  of  right  lines  perpendicular  to  a  ver- 
tical  plane,  r'  b  e  ;  if  right  lines,  d  a,  r  s,  r'  b,  r"  s,  and  d'  a, 
be  moved  so  as  to  touch  circle  and  right  line  d  d'  and  be 
constantly  parallel  to  plane  r'  b  e,  it  will  generate  conoid 
dab  a'  d'. 

Radii  vectores  are  lines  drawn  from  the  foci  to  any  point  in  the  curve;  hence  a 
radius  vector  is  one  of  these  lines. 

Traced  angle  is  angle  formed  by  the  radii  vectores  and  the  transverse  diameter. 
Ellipsoid,  Paraboloid,  and  Hyperboloid  of  Revolution—  Figures  generated 
by  the  revolution  of  an  ellipse,  parabola,  etc.,  around  their  axes.     (See  Men- 
suration of  Surf  aces  and  Solids,  pages  357-75.) 

NOTE  i.  —  All  figures  which  can  possibly  be  formed  by  cutting  of  a  cone  are  men- 
tioned in  these  definitions,  and  are  five  following—  viz.,  a  Triangle,  a  Circle,  an  El- 
lipse, a  Parabola,  and  a  Hyperbola  ;  but  last  three  only  are  termed  Conic  Sections. 

2.  —  In  Parabola  parameter  of  any  diameter  is  a  third  proportional  to  abscissa 
and  ordinate  of  any  point  of  curve,  abscissa  and  ordinate  being  referred  to  that 
diameter  and  tangent  at  its  vertex. 

3.  —  In  Ellipse  and  Hyperbola  parameter  of  any  diameter  is  a  third  proportional 
to  diameter  and  its  conjugate. 

To  Determine  Parameter  of  an  Ellipse  or  Hyperbola. 

Fig.  8.  RULE.  —  Divide  product   of  conjugate  a   Fig.  9. 

c  diameter,  multiplied  by  itself,  by  trans- 

verse,  and   quotient    is    equal    to   para- 
meter. 
rts  —  '     \°      In  annexed  Figs.  8  and  9,  of  an  Ellipse 

and  Hyperbola,  transverse  and  conjugate  c~ 
diameters,  ab,  cd,  are  each  30  and  20. 

Then  30  :  20  ::  20  :  13.  333=  parameter. 
Parameter  of  curve  =  ef,  a  double  ordinate  passing  through 
focus,  s.  o 

Ellipse. 

To   Describe   Ellipses.     (See  Geometry,  page  226.) 
To    Compute   Terms    of  an    Ellipse. 

When  any  three  of  four  Terms  of  an  Ellipse  are  given,  viz.,  Transverse 
and  Conjugate  Diameters,  an  Ordinate,  and  its  Abscissa,  to  ascertain  remain- 
ing Terms. 


CONIC    SECTIONS.  381 

To    Compute    Ordinate. 

Transverse  and  Conjugate  Diameters  and  Abscissa  being  given.  RULE.— As  trans- 
verse diameter  is  to  conjugate,  so  is  square  root  of  product  of  abscissae  to  ordinate 
which  divides  them. 

Fig.  10.  EXAMPLE.  —  Transverse  diameter,  a  b,  of  an  ellipse,  Fig. 

10,  is  25;  conjugate,  c  d,  16;  and  abscissa,  ai,  7;  what  is 
length  of  ordinate,  t  e? 

25  —  7  =  18  less  abscissa  ;  Vy  X  18  =  1 1. 225. 
Hence  25  :  16  ::  11.225  :  7.184  ordinate. 

Or,  A/c2  —  I ^ J  =  any  ordinate,  c  and  t  representing 

semi-conjugate  and  transverse  diameters,  and  x  distance  of  ordinate  from  centre  of 
figure. 

To    Compute   .AJbscissae. 

Transverse  and  Conjugate,  Diameters  and  Ordinate  being  given.  RULE.  — As  conju- 
gate diameter  is  to  transverse,  so  is  square  root  of  difference  of  squares  of  ordinate 
and  semi -conjugate  to  distance  between  ordinate  and  centre;  and  this  distance  be- 
ing added  to,  or  subtracted  from,  semi-transverse,  will  give  abscissae  required. 

EXAMPLE. — Transverse  diameter,  a  b,  of  an  ellipse,  Fig.  10,  is  25;  conjugate,  c  d, 
16;  and  ordinate,  ie,  7.184;  what  is  abscissa,  t6? 

V82  —  7.  i842  =  3-519  943-    Hence,  as  16  :  25  ::  3.52  :  5.5. 
Then  25  -4-  2  =  12. 5,  and  12. 5  +  5. 5  =  18  =  6  t, )  abscissce 
25-7-2  =  12.5,  and  12.5  — 5.5=  7  =  at,)"° 

To    Compute   Transverse   Diameter. 

Conjugate,  Ordinate,  and  Abscissa  being  given.  RULE.— To  or  from  semi-conju. 
gate,  according  as  great  or  less  abscissa  is  used,  add  or  subtract  square  root  of  dif- 
ference of  squares  of  ordinate  and  semi-conjugate.  Then,  as  this  sum  or  difference 
is  to  abscissa,  so  is  conjugate  to  transverse. 

EXAMPLE.  —  Conjugate  diameter,  c  d,  of  an  ellipse,  Fig.  10,  is  16;  ordinate,  ie, 
7.184;  and  abscissae,  6  i,  i  a,  18  and  7 ;  what  is  length  of  transverse  diameter? 

V(i6-=-2)2-7.i842  =  3-  52. 
16 -i- 2 -f- 3. 52  :  18  ::  16  :  25;  16-7-2 — 3.52  :  7  ::  16  :  25  transverse  diameter. 

To    Compute    Conjugate   Diameter. 

Transverse,  Ordinate,  and  Abscissa  being  given.  RULE.— As  square  root  of  prod- 
uct of  abscissae  is  to  ordinate,  so  is  transverse  diameter  to  conjugate. 

EXAMPLE.— Transverse  diameter,  a  6,  of  an  ellipse,  Fig.  10,  is  25;  ordinate,  i  t, 
7. 184 ;  and  abscissae,  b  i  and  i  a,  18  and  7 ;  what  is  length  of  conjugate  diameter  ? 

Vi8  X  7  =  ii  225.     Hence  11.225  '•  7- 184  ::  25  :  16  conjugate  diameter. 

To    Compute    Circumference    of  an    Ellipse. 
RULE.  —Multiply  square  root  of  half  sum  of  the  squares  of  two  diameters  by 
3- 1416- 

EXAMPLE.— Transverse  and  conjugate  diameters,  a  b  and  cd,  of  an  ellipse,  Fig.  10, 
are  24  and  20;  what  is  its  circumference? 

'—  =  488,  and  x/488  =  22.09.   Hence  22.09  X  3. 1416  =  69. 398  circumference. 

To    Compute    Area   of  an    Ellipse. 

RULE. — Multiply  the  diameters  together,  and  the  product  by  .7854.  Or,  multiply 
one  diameter  by  .7854,  and  the  product  by  the  other. 

EXAMPLE.— The  transverse  diameter  of  an  ellipse,  a  b,  Fig.  10,  is  12,  and  its  con 
jugate,  c  d,  9 ;  what  is  its  area  ? 

12  x  9  X  .7854  =  84.8232  area. 

NOTE.  —Area  of  an  ellipse  is  a  mean  proportional  between  areas  of  two  circles, 
diameter  of  one  being  major  axis  and  of  the  other  minor  axis. 

ILLUSTRATION.  —  Area  of  circle  of  40  =  1256.64;  area  of  ellipse  40  X  20  =  628.32; 
area  of  circle  of  20  =  314.16,  mean  of  the  two  circles  1256.644-314.16  =  785.4. 
Therefore  the  conjugate  diameter  of  an  ellipse  of  an  area  of  785.4  sq.  ins.,  its  trans 
verse  being  40,  is  25  feet,  as  40  X  25  x  .7854  =  785.4  sq.  ins. 


382 


CONIC   SECTIONS. 


Segment   of  an   Ellipse. 

To   Compute   Area   of  a   Segment   of  an   Ellipse. 
When  its  Base  is  parallel  to  either  Axis,  as  e  if.     RULE.— Divide  height  of  seg- 
ment, b  t,  by  diameter  or  axis,  a  &,  of  which  it  is  a  part,  and  find  in  Table  of  Areas 
of  Segments  of  a  Circle,  page  267,  a  segment  having  same  versed  sine  as  this  quo- 
tient; then  multiply  area  of  segment  thus  found  and  the 
axes  of  ellipse  together. 

EXAMPLE.—  Height,  b  t,  Fig.  n,  is  5,  and  axes  of  ellipse  are 
30  and  20;  what  is  area  of  segment? 

5  -r-  30  =  .  1666  tabular  versed  sine,  the  area  of  which 
(page  267)  is  .085  54. 

Hence  .085  54  X  30  X  20  =  51.324  area. 

To  Ascertain  Length  of  an  Elliptic  Curve  which  is  less 

than,   half  of  entire    Figure. 

Fig.  ja.          ^jjfc  Let  curve  of  which  length  is  required  be  A  6  C, 

Fig.  12. 
Extend  versed  sine  6  d  to  meet  centre  of  curve  in  e. 

A/,      __  <f{ ..._A}Cvn      Draw  line  e  C,  and  from  e,  with  distance  e  &,  describe 

j                         ""^"AM    b  h't  b'sect  h  C  in  t,  and  from  e,  with  radius  e  i,  de- 
L^ Ui-.Il' -J    scribe  k  i,  and  it  is  equal  to  half  arc  A  6  C. 

To  Ascertain  Length  -when  Curve   is  greater  than  half 
entire   Figure. 

Ascertain  by  above  problem  curve  of  less  portion  of  figure ;  subtract  it  from  cir- 
cumference of  ellipse,  and  remainder  will  be  length  of  curve  required. 


To  Describe  a 


I?ara~bola. 
3?arat>ola.    (See  Geometry,  page  229.) 


To  Compute  either  Ordinate  or  Abscissa  of  a  Parabola. 

When  the  other  Ordinate  and  Abscissa,  or  other  Abscissa  and  Ordinates  are 
given.  RULE.  —  As  either  abscissa  is  to  square  of  its  ordinate,  so  is  other  abscissa  to 
square  of  its  ordinate. 

Or,  as  square  of  any  ordinate  is  to  its  abscissa,  so  is  square  of  other  ordinate  to 
its  abscissa. 

EXAMPLE  i.  —  Abscissa,  a  6,  of  parabola,  Fig.  13,  is  9;  its  ordi- 
nate, b  c,  6;  what  is  ordinate,  d  e,  abscissa  of  which,  a  d,  is  16  ? 

Hence  9  :  62  ::  16  :  64,  and  1/64  =  8  length. 
2.—  Abscissae  of  a  parabola  are  9  and  16,  and  their  correspond- 
ing ordinates  6  and  8;  any  three  of  these  being  taken,  it  is  re- 
quired to  compute  the  fourth. 


x.    _        =  g  ordinate. 

9 

I6X62 
3.  — — —  =  9  less  abscissa. 


- 

ID 


=  6  ordinate. 


=  rf  abscissa. 


!Para"bolic    Curve. 

To  Compute   Length  of  Curve  of  a  IParatoola  out  off  t>y 
a   Double    Ordinate.— Fig.  13. 

RULE.— To  square  of  ordinate  add  —  of  square  of  abscissa,  and  square  root  of 
this  sum,  multiplied  by  two,  will  give  length  of  curve  nearly. 

EXAMPLE.— Ordinate,  d  e,  Fig.  13,  is  8,  and  its  abscissa,  a  d,  16;  what  is  length  of 
curve,  fa  e  ? 

82  +  4Xl62  =  405-  333,  and  ^405. 333X2  =  40. 267  length. 


CONIC   SECTIONS.  383 

jrjg  x .  To   Compute  Area  of  a  JParabola. 

RULE.— Multiply  base  by  height,  and  take  two  thirds  of  product 
Corollary. — A  parabola  is  two  thirds  of  its  circumscribing  par- 
allelogram. 

EXAMPLE.— What  is  area  of  parabola,  a  b  c,  Fig.  14,  height,  &«, 
being  16,  and  base,  or  double  ordinate,  a  c,  16  ? 

16  x  16  =  256,  and  —  0/256  =  170.667  area. 

To    Compute   Area   of  a    Segment  of  a   Farabola. 

RULE. — Multiply  difference  of  cubes  of  two  ends  of  segment,  a  c,  df,  by  twice  Hi 
height,  e  o,  and  divide  product  by  three  times  difference  of  squares  of  ends. 

EXAMPLE. — Ends  of  a  segment  of  a  parabola,  a  c  and  df,  Fig.  14,  are  10  and  6,  and 
height,  e  o,  is  10;  what  is  its  area? 

io3a.63  X  10  X  2  =  15680,  and  •+•  io2  ^  62  X  3  =  81.667  area. 

NOTE.— Any  parabolic  segment  is  equal  to  a  parabola  of  the  same  height,  the  base 
of  which  is  equal  to  base  of  segment,  increased  by  a  third  proportional  to  sum  of 
the  two  ends  and  lesser  end. 

Hyperbola. 
To  Describe   a  Hyperbola.    (See  Geometry,  page  230.) 

To    Compute    Ordinate    of  a  Hyperbola, 

Transverse  and  Conjugate  Diameters  and  Abscissa  being  given.  RULE.— As  trans- 
verse diameter  is  to  conjugate,  so  is  square  root  of  product  of  abscissae  to  ordinate 
required. 

Fig.  15.  5        EXAMPLE.  —Hyperbola,  a  be,  Fig.  15,  has  a  transverse 

diameter,  a  t,  of  120;  a  conjugate,  df,  of  72 ;  and  abscissa, 
ae,  40;  what  is  the  length  of  ordinate,  ec? 

40-}- 120  =  160  greater  abscissa,  and 
120  172::  V(4°  X  160)  :  48  ordinate. 
NOTE  i.— In  hyperbolas  lesser  abscissa,  added  to  axis 
(the  transverse  diameter),  gives  greater. 

2. — Difference  of  two  lines  drawn  from  foci  of  any  hyperbola  to  any  point  in  curve 
is  equal  to  its  transverse  diameter. 

To    Compute   Abscissae, 

Transverse  and  Conjugate  Diameters  and  Ordinate  being  given.  RULE. — As  con- 
Jugate  diameter  is  to  transverse,  so  is  square  root  of  sum  of  squares  of  ordinate  and 
semi  conjugate  to  distance  between  ordinate  and  centre,  or  half  sum  of  abscissae. 
Then  the  sum  of  this  distance  and  semi-transverse  will  give  greater  abscissa,  and 
their  difference  the  lesser  abscissa. 

EXAMPLE.— Transverse  diameter,  a  t,  of  a  hyperbola,  Fig.  15,  is  120;  conjugate,  d/, 
72 ;  and  ordinate,  e  c,  48 ;  what  are  lengths  of  abscissae,  t  e  and  ae? 

72  :  120  ::  -\/482  +  (72  ~-  2)2  =  60  :  100  half  sum  of  abscissa,  and  ioo-f-(i20-r-2)  = 
160  greater  abscissa,  and  100  —  (120-?-  2)  =  40  lesser  abscissa. 

To   Compute   Conjugate   Diameter, 

Transverse  Diameter,  Abscissa,  and  Ordinate  being  given.  RULE.— As  square  root 
of  product  of  abscissae  is  to  ordinate,  so  is  transverse  diameter  to  conjugate. 

EXAMPLE. — Transverse  diameter,  at,  of  a.  hyperbola,  Fig.  15,  is  120;  ordinate, ec, 
48;  and  abscissae,  te  and  ae,  160  and  40;  what  is  length  of  conjugate,  df? 

1/40  X  160  =  80  :  48  ::  120  :  72  conjugate. 


384  CONIC   SECTIONS. 

To    Compute   Transverse   Diameter, 

Conjugate,  Ordinate,  and  an  Abscissa  being  given.  RULE. — Add  square  of  ordinate 
to  square  of  semi  conjugate,  and  extract  square  root  of  their  sum. 

Take  sum  or  difference  of  semi-conjugate  and  this  root,  according  as  greater  or 
lesser  abscissa  is  used.  Then,  as  square  of  ordinate  is  to  product  of  abscissa  and 
conjugate,  so  is  sum  or  difference  above  ascertained  to  transverse  diameter  required. 

NOTE.  —  When  the  greater  abscissa  is  used,  the  difference  is  taken,  and  con- 
trariwise. 

EXAMPLE.— Conjugate  diameter,  df,  of  a  hyperbola,  Fig.  15,  is  72;  ordinate,  e  c, 
48 ;  and  lesser  abscissa,  a  e,  40 ;  what  is  length  of  transverse  diameter,  at? 

•v/482  -p-  (72  -r-  2)2  =  60,  and  60  -f-  72  -r-  2  =  96  lesser  abscissa,  and  40  X  72  =  2880. 
Hence,  482  :  2880  ::  96  :  120  transverse  diameter. 

To  Compute   Length  of  any  Arc  of  a  HyperTaola,  com- 
mencing  at   Vertex. 

RULE.— To  19  times  transverse  diameter  add  21  times  parameter  of  axis. 

To  9  times  transverse  diameter  add  21  times  parameter,  and  multiply  each  of 
these  sums  respectively  by  quotient  of  lesser  abscissa  divided  by  transverse  di- 
ameter. 

To  each  of  products  thus  ascertained  add  15  times  parameter,  and  divide  former 
by  latter;  then  this  quotient,  multiplied  by  ordinate, will  give  length  of  arc,  nearly. 

NOTE.— To  Compute  Parameter,  divide  square  of  conjugate  by  transverse  diam- 
eter. 

Fig.  16.  5  EXAMPLE.— In  hyperbola,  a  be,  Fig.  16,  transverse  diameter  is  120, 
conjugate,  72,  ordinate,  e  c,  48,  and  lesser  abscissa,  a  e,  40;  what  is 
length  of  arc,  a  b  ? 

—  =  43. 2  parameter.    120  X  19  +  43.2X21  X  —  =  1062. 4. 


40 


120  X  9  +  43. 2  X  21  X  —  =662.4.     Then  1062.4  +  43.2X15-^662.4 
+  43. 2  X  15  =  1.305,  which  x  48  =  62.64  length. 

NOTE.— As  transverse  diameter  is  to  conjugate,  so  is  conjugate  to  parameter. 
(See  Rule,  page  380.) 

To    Compete   A.rea  of  a   Hypert>ola, 

Transverse,  Conjugate,  and  Lesser  Abscissa  being  given.  RULE.— To  product  of 
transverse  diameter  and  lesser  abscissa  add  five  sevenths  of  square  of  this  abscissa, 
and  multiply  square  root  of  sum  by  21. 

Add  4  times  square  root  of  product  of  transverse  diameter  and  lesser  abscissa  to 
product  last  ascertained,  and  divide  sum  by  75. 

Divide  4  times  product  of  conjugate  diameter  and  lesser  abscissa  by  transverse 
diameter,  and  this  last  quotient,  multiplied  by  former,  will  give  area,  nearly. 

EXAMPLE.  —Transverse  diameter  of  a  hyperbola,  Fig.  16,  is  60,  conjugate  36,  and 
lesser  abscissa  or  height,  a  e,  20;  what  is  area  of  figure  ? 

60  X  20  H of  2o2  =  1485. 7143,  and  1/1485. 7143  X  21  =  809. 43,  and  V6o  X  20  X 

4  +  809.43 =901.02,  which  -r-  75  =  12.0136  and  3   X^°X  4  X  12.0136  =  576.653  area. 
NbM.— For  ordinates  of  a  parabola  in  divisions  of  eighths  and  tenths,  see  page  229. 
Delta   Metal. 

Delta  Metal  is  an  improved  composition  of  Aluminium  and  its  alloys ;  it  is 
non-corrosive,  capable  of  being  cast,  forged,  and  hot  rolled. 

Tensile  Strength  per  Sq.  Inch. 

Cast  in  green  sand 48  380  Ibs.    I   Rolled,  annealed 60  920  lb& 

Rolled,  hard 75260    u     |   Wire,  No.  22  WG 140000   " 


PLANE   TBIGONOMETRY.  385 

PLANE  TRIGONOMETRY. 

By  Plane  Trigonometry  is  ascertained  how  to  compute  or  determine 
four  of  the  seven  elements  of  a  plane  or  rectilinear  triangle  from  the 
other  three,  for  when  any  three  of  them  are  given,  one  of  which  being 
a  side  or  the  area,  the  remaining  elements  may  be  determined ;  and 
this  operation  is  termed  Solving  the  Triangle. 

The  determination  of  the  mutual  relation  of  the  Sines,  Tangents,  Secants, 
etc.,  of  the  sums,  differences,  multiples,  etc.,  of  arcs  or  angles  is  also  classed 
under  this  head. 

For  Diagram  and  Explanation  of  Terms,  see  Geometry,  pp.  219-21. 
Right-angled.  Triangles. 

For  Solution  by  Lines  and  Areas,  see  Mensuration  of  Areas,  Lines, 
and  Surf  aces,  pp.  335-39. 

To   Compute  a   Side. 

When  a  Side  and  its  Opposite  Angle  is  given.  RULE.— As  sine  of  angle 
opposite  given  side  is  to  sine  of  angle  opposite  required  side,  so  is  given  side 
to  required  side. 

To   Compute   an   Angle. 

RULE.— As  side  opposite  to  given  angle  is  to  side  opposite  to  required 
angle,  so  is  sine  of  given  angle  to  sine  of  required  angle. 

To  Compute  Base   or  Perpendicular  in   a  Right-angled 
Triangle. 

When  A  ngles  and  One  Side  next  Right  A  ngle  are  given.  RULE. — As  ra- 
dius is  to  tangent  of  angle  adjacent  to  given  side,  so  is  this  side  to  other  side. 

To   Compute   the   other   Side. 

When  Two  Sides  and  Included  Angle  are  given.  RULE.— As  sum  of  two 
given  sides  is  to  their  difference,  so  is  tangent  of  half  sum  of  their  opposite 
angles  to  tangent  of  half  their  difference ;  add  this  half  difference  to  half 
sum,  to  ascertain  greater  angle ;  and  subtract  half  difference  from  half  sum, 
to  ascertain  less  angle.  The  other  side  may  then  be  ascertained  by  Rule 
above. 

To   Compute  Angles. 

When  Sides  are  given.  RULE. — As  one  side  is  to  other  side,  so  is  radius 
to  tangent  of  angle  adjacent  to  first  side. 

To    Compute   an   Angle. 

When  Three  Sides  are  given.  RULE  i. — Subtract  sum  of  logarithms  of 
sides  which  contain  required  angle,  from  20 ;  to  remainder  add  logarithm 
of  half  sum  of  three  sides,  and  that  of  difference  between  this  half  sum  and 
side  opposite  to  required  angle.  Half  the  sum  of  these  three  logarithms  is 
logarithmic  cosine  of  half  required  angle.  The  other  angles  may  be  ascer- 
tained by  Rule  above. 

2.  — Subtract  sum  of  logarithms  of  two  sides  which  contain  required 
angle,  from  20,  and  to  remainder  add  logarithms  of  differences  between 
these  two  sides  and  half  sum  of  the  three  sides.  Half  result  is  logarithmic 
sine  of  half  required  angle. 

NOTE. — In  all  ordinary  cases  either  of  these  rules  will  give  sufficiently  accurate 
results.  Rule  i  should  be  used  when  required  angle  exceeds  90° ;  and  Rule  2  when 
it  ts  less  than  00°. 

Ks 


386 


PLANE   TRIGONOMETRY. 


EXAMPLE.  —The  sides  of  a  triangle  are  3,  4,  and  5 ;  what  are  the  angles  of  the 
hypothenuse  ? 

20  —  (Log.  4  =  .602 06  +  Log.  5  —  .698 97)  =  18.698 97 ;  Log.  3  +  4  +  5-^2  —  4  = 
.301 03;  and  Log.  3 -}•-  4  +  5  -r-  2  — -  5:=  o. 

Then  18.698  97  + .  301 03  =  19,  which  -f-  2  =  9. 5  =  log.  sin.  of  half  angle  =  18°  26', 
which  X  2  =  36°  52'  angle. 

Hence  90°  —  36°  52'  =  53°  8'  remaining  angle. 

In  following  figures,  i  and  2 : 

A  =  9o°,  B  =  45°,  0  =  45°,  Radius  =  i,  Secant  =  1.4142,  Cosine  =  .7071,  Sin.  45° 
—  .7071,  Tangent  =  i,  Area  =  .25. 

By  Sin.,  Tan.,  Sec.,  etc.,  A  B,  etc.,  is  expressed  Sine,  Tangent,  Secant,  etc.,  of 
angles,  A,  B,  etc. 

To   Compute    Sides   A  C   arid.   B  C.— Figs.  1   and.   3. 
When  Hyp.,  Side  B  A,  and  Angles  B  and  C  are  given. 


Fig.  i. 


Sin.  B  X  B  A  _ 

Sin.  C 

B  A  X  Cot.  C  =  A  C. 
Hyp.  X  Cos.  C  =  A  C. 
Hyp.  X  Sin.  B  =  A  C. 
BA 


Fig.  2. 


I 


O      Cosine.      A  Vers? 


Sin.  C 

AC 

Sin.  B 


=  BC. 


To   Compute   Side   A  C   and   Angles. 
When  Hyp.  and  Side  B  A  are  given. — Fig.  i  and  2. 


Hyp. 


=  Sin.  B. 


BA  BAXSJD.B 

HTpT  sln7c 


B  C  X  Sin.  B  =  A  C. 


To  Compute    Side   B  C  and   Hyp.  or   Angles. 
When  both  Sides  are  given. — Fig.  2. 


Sin.  C 
BA 
BC 


=  BC. 

=  Sin.  C. 


AC 


=  Tan.  C. 


Fig.  3- 


To   Compute    Sides.— Figs.  3   and 
When  a  Side  and  an  Angle  are 


given. 


B  C  x  Cos.  B  =  B  A. 
B  C  X  Sin.  B  =  A  C. 
A  B  X  Sec.  B  =  B  C.' 


=  BA. 


ACxSin.C 
Sin.  B 


=  BA. 


Rad. 


=  BC.  -= 

Sin.  B 


Tangent. 


In  BAG,  Fig.  5,  a  right-angled  triangle,  C  A,  is  assumed  to  be  radius ; 
B  A  tangent  of  C,  and  B  C  secant  to  that  radius ;  Or,  dividing  each  of  these 
by  base,  there  is  obtained  the  tangent  and  secant  of  C  respectively  to  radius  i. 


PLANE    TRIGONOMETRY. 


387 


Fig.  5- 


Radius       C  A  = 

Secant  C  B  = 
Tangent  A  B  = 
Co-secant  CB  = 
Co -tangent  e  B  = 

VAC2H-BA2  =  hyp.  B  C. 
A  C  -r-  Cos.  C  =  hyp.  B  C. 


4142 
4142 


O    Radius,  ff 


B  C  X  Cos.  C  =  Rad. 
B  A  X  Tan.  B  =  Rad. 
BC-^-BA  =  Sec.  B. 
B  C  X  Cos.  B  =  B  A. 


Cos.  G 
Sin.  C 


Sine  dg—  .7071 

Cosine  Cgorod=  .7071 
Versed  sine  g  A  =  .  2929 
Co- versed  sine  o  e  =  .2929 
Angle  C  A  B  =  90° 

BA-:-Sin.  C  =  hyp.  BC. 
i  -f-  Tan.  C  =  Cot.  C. 
B  C2  X  Sin.  2  C 


=  CotC. 


-  =Area, 


B  A  x  Sec.  B  =  B  C. 

B  AX  Cot.  C  =  Rad.  B  C  X  Sin.  B  =  Rad. 

B  C  X  Sin.  C  =  B  A.  A  C  X  Tan  C  =  B  A. 

i  -i-  Sin.  C  =  Cosec.  C.  i  —  Sin.  C  =  Co-ver.  sin. 

Cos.  C-=-Sin.  C^Cot.  C.          C  B  x  Sin.  B  =  AC. 


Trigonometrical    Equivalents. 

Perp.  -r-  hyp.  =  Sin.  C.         Hyp.  -r-  base  =  Sec.  C.         Perp.  -r-  base  =  Tan.  C. 
Base  -r-  perp.  =  Tan.  B.        Hyp.  -=-  perp.  =  Sec.  B. 
Perp.  -4-  hyp.  =  Cos.  B.        Hyp.  -r-  perp.  —  Cosec.  C. 
Hyp.  —  Base  =  Versin.     Hyp.  —  Perp.  =  Co-ver.  sin.  Ci 
Tan.  -4-  sin.     =  Sec. 
Tan.  -4-  sec.    =  Sin. 
Tan.  X  cot.     =  Rad. 
V(i— cos.2)  =  Sin. 
i       -f-  cot.     =  Tan. 
i      -4-  sin.     =  Cosec. 
ILLUSTRATIONS.— Assume  side  A  B  of  a  right-angled  triangle  is  100,  and  angle  C 
53°  8';  what  are  its  elements? 

Fig.  6.  B  Oblique-angled.   Triangles. 

To   Compute    Sides    B  A   and    B  C. 

When  Side  A  C  and  Angles  are  given. — Fig.  6. 


Base  -4-  hyp.  =  Cos.  C. 

Base  -4-  hyp.  =  Sin.  B. 

Base  -4-  perp.  =  Cotan.  C. 

V  (i  — sin.2)  =  Cos. 
Sin.  -4-  tan.  =  Cos. 
Sin.  x  cot.  =  Cos. 
Sin.  -4- cos.  =Tan. 
Cos.  -T-  cot  =  Sin. 
Cos.  -4-  sin.  =  Cot. 


-f-  cos.  =  Sec. 
-f-  cosec.  =  Sin. 
-r-  sec.  =  Cos. 

—  cos.     =  Versiu. 

—  sin.     =  Co-ver.  sin. 
-r-  tan.     =  Cotan. 


Sin.  C  X  A  C 
Sin.B 


=  B  A. 

Sin.  Ax  AC 
Sin  B      : 


Sin.  C  X  B  C 
Sin.  A       = 


To   Compute   Angles   and    Side   A  C. 
When  Sides  A  B,  B  C,  and  one  of  the  Angles  are  given. — Fig.  6. 


B  C  X  Sin.  B 
AJC 


Fig.  7. 


=  Sin.  A. 


Sin.  C  X  A  C 


B  A 
Sin.  BxBC 


=  Sin.  B. 


A  B  x  Sin.  B 
~ACT 


=  Sin.  C. 


=  AC. 


Sin.  A 

To  Compute  Sides  B  A  and  B  C. 
When  Side  A  C  and  A  ngles  are  given. — Fig.  7. 
Sin.  CXBC  Sin.  AxAC_ 

Sin.  A  Sin.  B 

When  Side  B  C  and  Angles  are  given. — Fig.  7. 
B  C  x  Sin.  C  Sin.  C  X  A  C 


Sin.  A 


Sin.  B 


NOTE.  — Sine  and  Cosine  of  an  arc  are  each  equal  to  sine  and  cosine  of  their  sup- 
plements. 

Spherical  Triangles,  Right-angled  and  Oblique.    For  full  formulas  See 
Molesworth,  Lond.,  1878,  pp.  435-6. 


388 


PLANE    TRIGONOMETBY. 


To    Compute   Angles   and    Side   AC. 
When  Sides  A  B,  B  C,  and  A  ngle  B  are  given. — Fig.  7. 
Sin.  BxBC  BCxSin.  B 


Sin.  A 

AC  X  Sin.  A 

BC 


=  Sin.  B. 


AC 
B  A  X  Sin.  A 
BC 


=  Sin.  A. 


=  Sin.  C. 


To    Compute   all    the    Angles. 

When  all  the  Sides  are  given,  Figs.  6  and  7.  RULE.  —  Let  fall  a  perpen- 
dicular, B  d,  opposite  to  required  angle.  Then,  as  A  C  :  sum  of  A  B,  B  C  :  : 
their  difference  :  twice  d  g,  the  distance  of  perpendicular,  B  d,  from  middle 
of  the  base. 

Hence  A  d,  C  g  are  known,  and  triangle,  A  B  C,  is  divided  into  two  right- 
angled  triangles,  B  C  d,  B  A  d  ;  then,  by  rules  for  right-angled  triangles, 
ascertain  angle  A  or  C. 

OPERATION.—  AC,  Fig.  6,  .5014  :  AB-j-BC,  1.1174+  1.4142  =  2.5316:^  B  GO  BC, 
1.4142  —  1.  1174  =  .  2968  :  a  x  d  g  =  i  4986. 


Consequently,  triangle  B  d  C,  Fig.  6,  is  divided  into  two  triangles,  BAG  and  B  d  A. 

To   Compute    Side   A  B   and   Angles. 

When  Two  Sides  and  One  Angle,  or  One  Side  and  Two  Angles,  are  given.  — 
Fig.  6. 


A  C  x  Sin.  C 


=  Sin.  B. 


=  AB. 


B  C  X  Sin.  B 


AC 
ABxSin.  B 


=  Sin.  A. 


=  Sin.  C. 


A  C  X  Sin.  A 
AB— (ACxCos.  A)" 
AC  X  Sin.  C 


=  Tan.  B 


2  Area 


AC  BC— (ACXCOS.C)" 

To   Compute  Area  of  a  Triangle. —Fig.  8. 
BAxBCxSin.  B    ACxBCxSin.  C    BAx  AC X  Sin.  A 

222 

Sin.  2  C,  B  C2    A  C2,  Tan.  C        .  B  A2,  Cot.  C 

, ,  and =  Area. 

42  2 

NOTE.— For  other  rules,  see  Mensuration  of  Areas,  Lines,  and 
_  Surfaces,  page  335. 

To   Compute    Sides. 

When  Areas  and  Angles  are  given. — Figs.  6  and  7. 
2  Area 


:  =  AC. 


-  =  BA. 


B  C,  Sin.  C  ~"  AC,  Sin.  A 

To  Ascertain  Distance  of  Inacces- 
si"ble  Otjects  on  a  Level  IPlane.— 
Figs.  9  and  1O. 


/     2  Area,  Sin.  A      _ 
V  Sin.  C,Sin.  (A  +  C)~ 


Fig.  ia 


Fig.  9. 


OPERATION.— Lay  off  perpendic- 
ulars to  line  A  B,  Fig.  9,  as  B  c,  d  e, 
on  line  A  d,  terminating  on  line 
e  A. 

Then  ed  —  cB:cB::Bd:  BA. 

When  there  are  Tioo  Inacces- 
sible Objects,  as  Fig.  10. 

OPERATION.  —  Measure  a  base 
line,  A  B,  Fig.  10,  and  angles  c  A  B, 
dBA,  dAB,  cBA,eto.  Then  pro- 
ceed by  formulas,  page  387,  to  deduce  cd. 

NOTE.— If  course  of  cd  is  required,  take  difference  of  angles 
d  c  A  and  c  d  JB  from  course  A  B. 


PLANE   TKIGONOMETBT. 


389 


When  the  Objects  can  be  aligned. — 
Fig.  ii. 

OPERATION. — Align  c  B,  Fig.  n,  at  A, 
measure  a  base  line  at  any  angle  there- 
to, as  A  o,  and  angles  o  A  c,  c  o  A,  and 
B  o  A.  Then  proceed  as  per  formula, 
page  386,  to  deduce  c  B. 

To  Compute  Distance  from, 
a  Griveii  feint  to  an  In- 
accessible Object.  —  Fig. 
12. 


Fig.  13. 


OPERATION.— Measure  a  level  line,  A  c,  Fig.  12,  and  ascertain  angles,  B  A  c,  B  c  A, 
Hence,  having  side,  A  c,  and  two  angles,  proceed  as  per  formula,  page  386,  to  de- 
termine A  B. 


To   Compute   Height  of  an.   Elevated   Point.— Fig.  13. 

Fig.  13.  OPERATION.  —  Measure     B\  Fig.  14. 

distance  on  a  horizontal 
line,  A  c,  Fig.  13 ;  ascertain 
Angle  B  A  c.  Then  pro- 
ceed as  per  formulas,  pp. 
386-8,  to  ascertain  B  c. 

When  a  Horizontal 
Base  is  not  Attainable. 
—Fig.  14. 

OPERATION.— Measure  or 
compute  distance  A  c,  Fig. 

14;  ascertain  angle  of  depression   cAo  and  of  elevation 
B  A  c.    Then  proceed  as  per  formula,  page  386,  to  ascertain  B  c. 


Fig.  15- 


When  a  Full  Base  Line  is  not  Attain* 
able. — Fig.  15. 

OPERATION.  —  Measure  a  base 
line,  A  c,  Fig.  15,  and  ascertain 
angles  A  c  B,  c  A  B. 

Then  proceed  as  per  for- 
mula, page  386,  to  ascer- 
tain d  B. 


Fig.  16. 


Without  Use  of  an  Instrument. 
—Fig.  16. 

OPERATION. —Lay  off  any  suitable  and  level  distance,  d  d,  set  up  a  staff  at  each  ex- 
tremity at  like  elevation  from  base  line  d  d,  and  note  distances  y  and  z,  at  which 
the  lines  of  sight  of  object  range  with  tops  of  the  staffs;  deduct  height  of  eye  from 
length  of  staffs,  and  ascertain  heights  h. 

D  h 
Then 1-  h  -\-  s  =  height.    $  representing  height  of  line  of  sight  from  base  d  d, 

and  D  length  of  lined  d. 

• 


390 


NATURAL    SINES   AND    COSINES. 
Natural   Sines   and.   Cosines. 


QO 

1° 

20 

30 

• 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

o 

.00000 

I 

01745 

•99985 

0349 

•99939 

•05234 

.99863 

I 

.00029 

I 

.01774 

•99984 

•03519 

.99938 

.05263 

.99861 

2 

.00058 

X 

01803 

.99984 

•03548 

•99Q37 

.05292 

.9986 

3 

.00087 

I 

.01832 

.99983 

•03577 

.99936 

•05321 

.99858 

4 

.00116 

I 

.01862 

.99983 

03606 

•99935 

•0535 

•99857 

5 

.00145 

I 

.01891 

.99982 

03635 

•99934 

•05379 

•99855 

6 

.00175 

X 

.0192 

.99982 

.03664 

•99933 

.05408 

.99854 

7 

.00204 

I 

.01949 

99981 

03693 

•99932 

•05437 

.99852 

8 
9 

.00262 

I 
I 

.01978 
.02007 

.9998 
.9998 

•03723 
•03752 

•99931 
•9993 

.05466 
•05495 

.99851 
.99849 

10 

.00291 

I 

.02036 

•99979 

03781 

.99929 

•05524 

•99847 

ii 

12 

.0032 
.00349 

.99999 
.99999 

02065 
.02094 

•99979 
.99978 

.0381 
.03839 

.99927 
.99926 

•05553 
.05582 

.99846 
.99844 

'3 

.00378 

.99999 

.02123 

•99977 

03868 

.99925 

.05611 

.99842 

X4 

.00407 

•99999 

.02152 

•99977 

.03897 

.99924 

.0564 

.99841 

»5 

.00436 

•99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

16 

.00465 

•99999 

.O22II 

.99976 

•03955 

.99922 

.05698 

.99838 

'7 

.00495 

•99999 

.0224 

•99975 

.03984 

.99921 

.05727 

.99836 

18 

.00524 

•99999 

.  02269 

•99974 

.04013 

.99919 

•05756 

.99834 

*9 

•00553 

.99998 

.02298 

•99974 

.04042 

.99918 

•05785 

•99833 

20 

.00582 

•9999s 

.02327 

•99973 

.04071 

.99917 

.05814 

.99831 

21 

.00611 

.99998 

.02356 

.99972 

.041 

.99916 

.05844 

.99829 

22 

.0064 

.99998 

.02385 

•99972 

.04129 

99915 

•05873 

.99827 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

•999X3 

.05902 

.99826 

24 

.00698 

.99998 

.02443 

•9997 

.04188 

.99912 

•05931 

.99824 

25 

.00727 

•99997 

.02472 

.99969 

.04217 

.99911 

.0596 

.99822 

26 

.00756 

•99997 

.02501 

.99969 

.04246 

.9991 

.05989 

.99821 

27 

.00785 

•99997 

•0253 

.99968 

•04275 

.99909 

.06018 

.99819 

28 

.00814 

•99997 

.0256 

.99967 

.04304 

.99907 

.06047 

.99817 

29 

.00844 

.99996 

.02589 

.99966 

•04333 

.99906 

.06076 

.99815 

3o 

.00873 

.99996 

.026l8 

.99966 

.04362 

.99905 

.06105 

.99813 

3i 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

32 

.00931 

.99996 

.02676 

.99964 

.0442 

.99902 

.06163 

.9981 

33 

.0096 

•99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

34 

.00989 

•99995 

.02734 

.99963 

.04478 

•999 

.06221 

.99806 

35 

01018 

•99995 

•02763 

.99962 

.04507 

.99898 

.0625 

.99804 

36 

.01047 

•99995 

.02792 

.99961 

•04536 

.99897 

.06279 

.99803 

37 

.01076 

•99994 

.02821 

.9996 

•04565 

.99896 

.06308 

.99801 

38 

.01105 

•99994 

0285 

•99959 

.04594 

•99894 

•06337 

•99799 

39 

.01134 

.99994 

.02879 

•99959 

.04623 

.99893 

.06366 

•99797 

40 

.01164 

•99993 

.02908 

.99958 

.04653 

.99892 

•06395 

•99795 

4i 

.01193 

•99993 

.02938 

•99957 

.04682 

%o6424 

•99793 

42 

.01222 

•99993 

.02967 

.99956 

.O47n 

.99889 

06453 

.99792 

43 

.01251 

.99992 

02996 

•99955 

.0474 

.99888 

.06482 

9979 

44 

.0128 

.99992 

.03025 

•99954 

.04769 

.99886 

.06511 

.99788 

45 

01309 

.99991 

.03054 

•99953 

.04798 

.99885 

.0654 

.99786 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

47 

.01367 

.99991 

.63112 

.99952 

.04856 

.99882 

06598 

.99782 

48 

.01396 

•9999 

.03141 

•99951 

04885 

.99881 

06627 

•9978 

49 

.01425 

.9999 

•0317 

•9995 

04914 

99879 

.06656 

.99778 

So 

.01454 

.99989 

.03199 

•99949 

04943 

.99878 

06685 

.99776 

5i 

.01483 

•99989 

.03228 

.99948 

.04972 

.99876 

.06714 

•99774 

52 

•OI5I3 

.99989 

.03257 

•99947 

.05001 

•99875 

.06743 

•99772 

53 

.01542 

.99988 

.03286 

.99946 

•0503 

.99873 

.06773 

•9977 

54 

.01571 

.99988 

.03316 

•99945 

05059 

.99872 

.06802 

.99768 

5I 

.Ol6 

.99987 

•03345 

•99944 

.05088 

•9987 

.06831 

•99766 

56 

.01629 

.99987 

•03374 

•99943 

.05117 

.99869 

.0686 

.99764 

11 

.01658 
.01687 

.99986 
.99986 

•03403 
•03432 

•99942 
.99941 

,05146 
05175 

.99867 
.99866 

.06889 
.06918 

.99762 
•9976 

f9 

.01716 

.99985 

.03461 

•9994 

.05205 

.99864 

.06947 

•99758 

60 

•01745 

.99985 

•0349 

•99939 

•05234 

.99863 

.06976 

•99756 

N.  cos. 

N.  sine. 

N.  cos.   N.  sine.   N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

89° 

88°     '     87°    [ 

86° 

NATUBAL   SINES   AND   COSINES. 


4 

0 

5 

o 

C 

o 

1 

ro 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

0 

.06976 

•99756 

^>87i6 

.99619 

•  0453 

•99452 

.12187 

•99255 

I 

.07005 

•99754 

.08745 

.99617 

.  0482 

•99449 

.12216 

.99251 

2 

.07034 

•99752 

.08774 

.99614 

.  0511 

.99446 

.12245 

.99248 

3 

.07063 

•9975 

.08803 

.99612 

•  054 

-99443 

.  12274 

•99244 

4 

.07092 

.99748 

.08831 

.99609 

.  0569 

•9944 

.  12302 

•9924 

5 

.07121 

.99746 

.0886 

.99607 

•  0597 

•99437 

•12331 

•99237 

6 

•0715 

•99744 

.08889 

.99604 

.  0626 

-99434 

.1236 

.99233 

7 

.07179 

.99742 

.08918 

.99602 

•  0655 

•99431 

•12389 

•9923 

8 
9 

.07208 
.07237 

•9974 
•99738 

^08976 

•99599 
.99596 

.  0684 
.  0713 

.99428 
•99424 

.12418 
.12447 

.99226 
.99222 

10 

.07266 

•99736 

.09005 

-99594 

.  0742 

.99421 

.12476 

.99219 

ii 

.07295 

•99734 

•09034 

•99591 

.  0771 

.99418 

.12504 

.99215 

12 

.07324 

•99731 

.09063 

.99588 

.  08 

•99415 

•12533 

.99211 

13 

•07353 

.99729 

.09092 

.99586 

.  0829 

.99412 

.12562 

.99208 

*4 

.07382 

.99727 

.09121 

•99583 

.  0858 

.99409 

.12591 

.99204 

J5 

.07411 

•99725 

.0915 

•9958 

.  0887 

.99406 

.1262 

•992 

16 

.0744 

•99723 

.09179 

•99578 

.  0916 

.99402 

.12649 

.99197 

'7 

.07469 

.99721 

.09208 

•99575 

0945 

•99399 

.12678 

•99193 

18 

.07498 

.99719 

.09237 

•99572 

«  0973 

•99396 

.12706 

.99189 

X9 

.07527 

.95716 

.09266 

•9957 

.  IOO2 

•99393 

•12735 

.99186 

20 

•07556 

.99714 

.09295 

•99567 

1031 

•9939 

.12764 

.99182 

21 

•07585 

.99712 

.09324 

•99564 

.  106 

.99386 

•12793 

.99178 

22 

.07614 

.9971 

.09353 

.99562 

.  1089 

•99383 

.12822 

•99*75 

23 

.07643 

.99708 

.09382 

•99559 

.  1118 

.9938 

.12851 

.99171 

24 

.07672 

•99705 

.09411 

•99556 

•  "47 

•99377 

.1288 

.99167 

25 

.07701 

•99703 

.0944 

•99553 

1176 

•99374 

.  12908 

.99163 

26 

•0773 

.99701 

.09469 

•99551 

.  1205 

•9937 

•12937 

.9916 

2Z 

•07759 

.99699 

.09498 

.99548 

1234 

.99367 

.12966 

.99156 

28 

.07788 

.99696 

.09527 

•99545 

•  1263 

•99364 

.12995 

.99152 

29 

.07817 

.99694 

•09556 

•99542 

.  1291 

•9936 

.13024 

.99148 

3o 

.07846 

.99692 

•09585 

•9954 

.  132 

•99357 

13053 

.99144 

3i 

.07875 

.99689 

.09614 

•99537 

«  1349 

•99354 

13081 

.99141 

32 

.07904 

.99687 

.09642 

•99534 

.  1378 

•99351 

.13" 

•99'37 

33 

•07933 

.99685 

.09671 

•99531 

.  1407 

•99347 

.13139 

•99133 

34 
35 

.07962 
.07991 

.99683 
.9968 

.097 
.09729 

.99528 
.99526 

.  1436 
.  1465 

•99344 
•99341 

.13168 
•13197 

.99129 
.99125 

36 

.0802 

.99678 

.09758 

.99523 

.  1494 

-99337 

.13226 

.99122 

37 

.08049 

.99676 

.09787 

•9952 

•  1523 

•99334 

•13254 

.99118 

38 

.08078 

•99673 

.09816 

•99517 

•  1552 

•99331 

•13283 

.99114 

39 

.08107 

.99671 

.09845 

.99514 

•99327 

.13312 

.9911 

40 

.08136 

.99668 

.09874 

•  995" 

.  1609 

.99324 

.13341 

.99106 

4i 

.08165 

.99666 

.09903 

.99508 

•  1638 

•9932 

•1337 

.99102 

42 

.08194 

.99664 

•09932 

.99506 

.  1667 

•99317 

•13399 

.99098 

43 

.08223 

.99661 

.09961 

.99503 

.  1696 

•99314 

13427 

.99094 

44 

.08252 

•99659 

•°999 

•995 

•  1725 

•9931 

•13456 

45 

.08281 

•99657 

.10019 

•99497 

•  1754 

•99307 

•13485 

.99087 

46 

.0831 

.99654 

.10048 

•99494 

1783 

•99303 

•I35I4 

.99083 

47 

•08339 

.99652 

10077 

.99491 

.  1812 

•993 

•13543 

.99079 

48 

.08368 

•99649 

.10106 

.99488 

.  184 

.99297 

.13572 

.99075 

49 

.08397 

.99647 

•10135 

.99485 

.  1869 

.99293 

•  136 

.99071 

50 

.08426 

.99644 

.10164 

.99482 

.  1898 

.9929 

.13629 

.99067 

5i 

•08455 

.99642 

.  10192 

.99479 

•  i927 

99286 

.13658 

.99063 

52 

.08484 

•99639 

.IO22I 

•99476 

•  1956 

.99283 

.13687 

.99059 

53 

.08513 

•99637 

.1025 

•99473 

.  1985 

.99279 

.13716 

•99°55 

54 

.08542 

•99635 

.  10279 

•9947 

.  2014 

.99276 

•  13744 

.99051 

55 

.08571 

•99632 

.10308 

•99467 

.  2043 

.99272 

•13773 

•99047 

56 

.086 

'9963 

•10337 

.99464 

.  2071 

.99269 

.13802 

.99043 

57 

.08629 

.99627 

.10366 

.99461 

.  21 

.99265 

•13831 

.99039 

58 

.08658 

.99625 

•  10395 

.99458 

,  2129 

.99262 

•^386 

•99°35 

59 

.08687 

.99622 

.  10424 

•99455 

.  2158 

.99258 

.13889 

99031 

60 

.08716 

.99619 

•10453 

.99452 

.  2187 

•99255 

•i39I7 

.99027 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.C08. 

N.  sine. 

N.  cos. 

N.  sine. 

a 

0  -    i 

# 

1° 

s: 

JO 

82 

!° 

392 


NATURAL    SINBS   AND    COSINES. 


8° 

90 

100 

11° 

' 

N.sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

0 

•  139*7 

.99027 

•15643 

.98769 

.17365 

.9848* 

.  19081 

.98163 

I 

.13946 

.99023 

.15672 

.98764 

•17393 

.98476 

.19109 

2 

•13975 

.99019 

.15701 

.9876 

.17422 

.98471 

.19138 

•98152 

3 

.14004 

.99015 

•1573 

•98755 

•I745I 

.98466 

.19167 

.98146 

4 

•  H033 

.99011 

•15758 

•98751 

•17479 

.98461 

.9814 

5 

.14061 

.99006 

•15787 

.98746 

.17508 

•98455 

.19224 

6 

.1409 

.15816 

.98741 

•17537 

•9845 

.19252 

.98129 

7 

.14119 

.  98998 

.15845 

•98737 

•17565 

.98445 

.19281 

.98124 

8 

.14148 

•98994 

•15873 

.98732 

•17594 

•9844 

.19309 

.98118 

9 

.14177 

.9899 

.15902 

.98728 

.17623 

•98435 

.19338 

.98112 

10 

.14205 

.98986 

•*593» 

.98723 

.17651 

•9843 

.19366 

.98107 

ii 

.14234 

.98982 

•*S959 

.98718 

.1768 

•98425 

.98101 

12 

.14263 

.98978 

.15988 

.98714 

.17708 

.9842 

.19423 

.98096 

13 

.  14292 

•98973 

.16017 

.98709 

•17737 

.98414 

.19452 

.9809 

14 

.1432 

.98969 

.  16046 

.98704 

.17766 

.98409 

.19481 

.98084 

15 

•i  4349 

•98965 

.16074 

•987 

•17794 

.98404 

•  19509 

.98079 

16 

•14378 

.98961 

.16103 

.98695 

.17823 

.98399 

•19538 

.98073 

17 

•  I44°7 

•98957 

.16132 

.9869 

•17852 

.98394 

.  19566 

.98067 

18 

.14436 

•98953 

.1616 

.98686 

.1788 

•98389 

.98061 

19 

.14464 

.98948 

.16189 

.98681 

.17909 

•98383 

•  19623 

.98056 

20 

•  H493 

.98944 

.16218 

.98676 

•17937 

•98378 

.19652 

•9805 

21 

.14522 

.9894 

.  16246 

.98671 

.  i  7966 

•98373 

.1968 

.98044 

22 

•I455I 

.98936 

.16275 

.98667 

•17995 

.98368 

.19709 

.98039 

23 

.1458 

.98931 

.  16304 

.98662 

.18023 

.98362 

•19737 

•98033 

24 

.  14608 

.98927 

•16333 

.98657 

.18052 

.98357 

.19766 

.98027 

25 

•14637 

.98923 

.16361 

.98652 

.18081 

•98352 

.19794 

.  9802  i 

26 

.  14666 

.98919 

.1639 

.98648 

.18109 

•98347 

.19823 

.98016 

27 

.  14695 

.98914 

.16419 

•98643 

.18138 

.98341 

.19851 

.9801 

28 

•14723 

.9891 

.16447 

.98638 

.18166 

•98336 

.1988 

.98004 

29 

•14752 

.98906 

.16476 

•98633 

.18195 

•98331 

.19908 

.97988 

3° 

.14781 

.  98902 

•  16505 

.98629 

.18224 

•98325 

.97992 

31 

.1481 

.98897 

•16533 

.98624 

.18252 

.9832 

.19965 

.97987 

S2 

.14838 

.98893 

.16562 

.98619 

.18281 

•98315 

•  19994 

.97981 

33 

.14867 

.98889 

.16591 

.98614 

.18309 

.9831 

.  20022 

•97975 

34 

.14896 

.98884 

.1662 

.98609 

•18338 

•98304 

.2OO5I 

.97969 

35 

.14925 

.9888 

.16648 

.98604 

.18367 

.98299 

.20079 

.97963 

36 

•14954 

.98876 

.16677 

.986 

•i8395 

.98294 

-2OIO8 

•97958 

37 

.14982 

.98871 

.16706 

•98595 

.18424 

.98288 

.20136 

•97952 

38 

.15011 

.98867 

•16734 

•9859 

•18452 

.98283 

.20l65 

.97946 

39 

.1504 

.98863 

•16763 

•98585 

.18481 

.98277 

.20193 

•9794 

40 

.15069 

.98858 

.16792 

•  9858 

.18509 

.98272 

.20222 

•97934 

41 

•15097 

.98854 

.1682 

•98575 

•18538 

.98267 

.2O25 

.97928 

42 

.15126 

.98849 

.16849 

•9857 

.18567 

.98261 

.20279 

.97922 

43 

•I5I55 

•98845 

.16878 

•98565 

•18595 

.98256 

.20307 

.97916 

44 

.15184 

.98841 

.16906 

.98561 

.18624 

.9825 

.20336 

.9791 

45 

.15212 

.98836 

.16935 

•98556 

.18652 

.98245 

.  20364 

.  97905 

46 

.15241 

•98832 

.16964 

•98551 

.18681 

.9824 

•20393 

.97899 

47 

•1527 

.98827 

.16992 

.98546 

.1871 

.98234 

.2O42I 

•97893 

48 

.15299 

.98823 

.17021 

.98541 

.18738 

.98229 

.2045 

.97887 

49 

.15327 

.98818 

•1705 

•98536 

.18767 

.98223 

.20478 

.97881 

50 

•15356 

.98814 

17078 

•98531 

.98218 

.20507 

•97875 

51 

•15385 

.98809 

.17107 

.98526 

.18824 

.98212 

•20535 

.97869 

52 

•I54H 

.98805 

.17136 

.98521 

.18852 

.98207 

•20563 

.97863 

53 

.15442 

.988 

.17164 

•98516 

.18881 

.98201 

.  20592 

•97857 

54 

.98796 

•17193 

•98511 

.1891 

.98196 

.2062 

•97851 

55 

.155 

.98791 

.17222 

.98506 

.18938 

.9819 

.20649 

•97845 

56 

•15529 

.98787 

•1725 

.98501 

.18967 

.98185 

.20677 

•97839 

57 

•15557 

.98782 

.17279 

.98496 

.18995 

.98179 

.  20706 

•97833 

58 

.15586 

.98778 

•17308 

.98491 

.19024 

.98174 

•20734 

.97827 

59 

•15615 

•98773 

•17336 

.98486 

•  19052 

.98168 

.20763 

.97821 

60 

•15643 

.98769 

•17365 

.98481 

.19081 

.98163 

.20791 

•97815 

N.  cos. 

N.  sine. 

N.  cos.  I  N.sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

81° 

80° 

790 

78° 

NATURAL   SINES   AND    COSINES. 


393 


II 

12° 

130 

14° 

15° 

27 

1 

N.  sine. 

N.  cos 

N.  sine. 

N.  cos. 

N.  sine 

N.  cos 

N.  sine 

N.  cos. 

o 

o  \  .20791 

.97815 

.22495 

•97437 

.24192 

•9703 

.25882 

•96593 

0 

I   .  2082 

.97809 

.22523 

•9743 

.2422 

.97023 

.2591 

•96585 

X 

2 

.  20848 

.97803 

•22552 

•97424 

.24249 

•97015 

.25938 

.96578 

I 

3 

.20877 

•97797 

.2258 

•97417 

.24277 

.97008 

.25966 

•9657 

2 

4 

.20905 

.97791 

.22608 

.97411 

•24305 

.97001 

.25994 

.96562 

2 

5 

•20933 

.97784 

.22637 

.97404 

•24333 

.26022 

•96555 

3 

6 

.20962 

•97778 

.22665 

•9739s 

.24362 

.  9698- 

.2605 

•96547 

3 

7 

.2099 

.97772 

.22693 

•  9*739! 

•2439 

.9698 

.26079 

•9654 

4 

8 

.21019 

.97766 

.22722 

•97384 

.24418 

•96973 

.  26107 

•96532 

4 

9 

.21047 

•9776 

.2275 

•97378 

.24446 

.96966 

•26135 

.96524 

5 

o 

.21076 

•97754 

.22778 

•97371 

•  24474 

.96959 

.2616; 

.96517 

5 

i 

.21104 

.97748 

.22807 

•97365 

•24503 

.96952 

.26191 

.96509 

2 

.21132 

.97742 

•22835 

•97358 

•24531 

•96945 

.26219 

.96502 

6 

3 

.21161 

•97735 

.22863 

•97351 

•24559 

•96937 

.  26247 

•96494 

61  4 

.21189 

.97729 

.22892 

•97345 

.24587 

•9693 

•26275 

.96486 

7   5 

.21218 

•97723 

.2292 

•97338 

.24615 

•96923 

.26303 

.96479 

7   6 

.21246 

.97717 

.22948 

•97331 

.24644 

.96916 

•26331 

.96471 

8   7 

.21275 

.97711 

.22977 

•9732; 

.24672 

.96909 

•26359 

.96463 

8   8 

.21303 

•97705 

.23005 

.247 

.26387 

•96456 

9   9 

•21331 

.97698 

•23033 

•973" 

.24728 

.  96894 

•26415 

.96448 

9  j  o 

•2136 

.97692 

.23062 

•97304 

•24756 

.9688; 

.26443 

.9644 

9   ' 

.21388 

.97686 

.2309 

.97298 

.24784 

.9688 

.26471 

•96433 

10    2 

.21417 

.9768 

.23118 

.97291 

.24813 

.96873 

•265 

.96425 

10  j  3  |  .21445 

•97673 

•23146 

.97284 

.24841 

.96866 

.26528 

.96417 

ii  |  4 

.21474 

.97667 

•23175 

.97278 

.24869 

.96858 

.26556 

.9641 

11  I  5 

.21502 

.97661 

•23203 

.97271 

.24897 

•96851 

.26584 

.96402 

12   26 

•2153 

•97655 

•23231 

.97264 

•24925 

.96844 

.26612 

.96394 

12   27 

13  i  28 

•21559 
•21587 

.97648 
•97642 

.2326 
.23288 

•97257 
•97251 

•24954 
.24982 

.96837 
.96829 

.2664 
.26668 

.96386 
•96379 

13  ;  29   .2l6l6 

.97636 

.23316 

.97244 

•  2501 

-96822 

.26696 

•96371 

14   30   .21644 

•9763 

•23345 

97237 

.25038 

•96815 

.26724 

•96363 

X4  31 

.21672 

•97623 

•23373 

•9723 

.25066 

.96807 

26752 

•96355 

14  32 

.21701 

.97617 

•23401 

.97223 

.25094 

.968 

.2678 

•96347 

»S  33 

.21729 

.97611 

.23429 

.97217 

.25122 

•96793 

.26808 

•9634 

15  34 

.21758  .97604 

•23458 

.9721 

•25151 

.96786 

.26836 

.96332 

16  35 

.21786 

•97598 

.23486 

.97203 

•25179 

.96778 

.26864 

.96324 

16  36  .21814 

•97592 

•23514 

.97196 

.25207 

.96771 

.26892 

.96316 

17  37  |  .21843 

•97585 

•23542 

.97189 

•25235 

.96764 

.2692 

.96308 

17  38  .21871 

•97579 

•23571 

.97182 

.25263 

.96756 

.26948 

.96301 

18  39  ;  .21899 

•97573 

•23599 

.97176 

•25291 

•96749 

.26976 

96293 

18  40  !  .21928 

.97566 

.23627 

.97169 

•2532 

.96742 

.27004 

96285 

18 

41  .21956 

•9756 

.23656 

97162 

•25348 

•96734 

.27032 

96277 

19 

42  21985 

•97553 

.23684 

97155 

•25376 

.96727 

.2706 

96269 

19 

43  -22013 

•97547 

.23712 

97148 

.25404 

96719 

.27088 

96261 

20 

44 

.22041 

•97541 

•2374 

97141 

254J2 

96712 

.27116 

96253 

20 

45 

.2207 

•97534 

•23769 

97134 

2546 

96705 

27144 

96246 

21 

46 

.22098 

•97528 

23797 

97127 

25488 

96697 

27172 

96238 

21 

47  j  .22126 

•97521 

23825 

9712 

272 

9623 

22 

48 

•22155 

•97515 

23853 

97"3 

25545 

96682 

27228 

96222 

22 

49 

.22183 

•975o8 

23882 

97106 

25573 

96675 

27256 

96214 

23 

50 

.22212 

.97502 

2391 

971 

25601 

96667 

27284 

96206 

23 

51 

.2224 

•97496 

23938 

97093 

25629 

9666 

27312 

96198 

23 

52 

.  22268 

97489 

23966 

97086 

25657 

96653 

2734 

9619 

24 

53 

22297 

97483 

23995 

97079 

25685 

96645 

27368 

96182 

24 

54 

.22325 

97476 

24023 

97072 

25713 

96638 

27396 

96174 

25 

55 

•22353 

9747 

24051 

97065 

25741 

9663 

27424 

96166 

25 

56 

.22382 

•97463 

24079 

.97058 

25769 

96623 

27452 

96158 

26 

57 

.2241 

97457 

24108 

.97051 

25798 

2748 

96i5 

26 

58 

.22438   9745 

24136 

.97044 

25826 

96608 

27508 

96142 

27 

59 

.22467 

•97444 

24164 

•97037 

25854 

966 

27536 

96i34 

27 

60 

.22495 

•97437 

24192 

•9703 

25882 

96593 

27564 

96126 

N.  cos. 

N.  sine. 

N.  cos.  |  N.  sine. 

N.  cos.  J 

N.  sine. 

N.  cos. 

N.  sine. 

770 

76° 

75° 

74°     1 

394 


NATURAL   SINES   AND    COSINES. 


16° 

170 

18° 

igo 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  COB. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

o 

.27564 

.96126 

•29237 

•9563 

.30902 

.95106 

•32557 

•94552 

I 

.27592 

.96118 

.  29265 

.95622 

.30929 

•95097 

•32584 

•94542 

2 

.2762 

.9611 

•29293 

•95613 

•30957 

.95088 

.32612 

•94533 

3 

.27648 

.96102 

•29321 

•95605 

•30985 

•95079 

.32639 

•94523 

4 

.27676 

.96094 

29348 

•95596 

.31012 

•9507 

.  32667 

•945H 

5 

.27704 

.96086 

.29376 

.95588 

.3104 

.95061 

.32694 

.94504 

6 

•27731 

.96078 

.29404 

•95579 

.31068 

.95052 

.32722 

•94495 

7 

•27759 

.9607 

.29432 

•95571 

•31095 

•95043 

.32749 

.94485 

8 

.27787 

.96062 

.2946 

.95562 

.3"23 

•95033 

•32777 

.94476 

9 

.27815 

.96054 

.29487 

•95554 

•3IJ5i 

.95024 

.32804 

.94466 

10 

.27843 

.  96046 

29515 

•95545 

.31178 

•95015 

.32832 

•94457 

ii 

.27871 

.96037 

29543 

.95536 

.31206 

.95006 

•32859 

•94447 

12 

.27899 

.96029 

29571 

•95528 

•31233 

•94997 

•32887 

•94438 

13 

.27927 

.96021 

.29599 

•95519 

.31261 

.94988 

.32914 

.94428 

14 

•27955 

.96013 

.  29626 

•955" 

.31289 

•94979 

.32942 

.94418 

'5 

27983 

.96005 

29654 

•95502 

•31316 

•9497 

.32969 

.94409 

16 

.28011 

•95997 

29682 

95493 

•31344 

.94961 

•32997 

•94399 

17 

.  28039 

.95989 

.2971 

•95485 

31372 

•94952 

•33024 

•9439 

18 

28067 

9598i 

29737 

•95476 

•31399 

•94943 

.33051 

•9438 

J9 

.28095 

95972 

29765 

95467 

•3H27 

•94933 

•33079 

•9437 

20 

.28123 

.95964 

29793 

•95459 

•3H54 

.94924 

.33106 

.94361 

21 

.2815 

•95956 

29821 

•9545 

.31482 

•949J5 

•33134 

94351 

22 

.28178 

.95948 

29849 

•95441 

3i5i 

.94906 

•33161 

.94342 

23 

.  28206 

•9594 

.29876 

95433 

•31537 

.94897 

•33i89 

•94332 

24 

.28234 

•95931 

29904 

95424 

•31565 

.94888 

.33216 

•94322 

25 

.28262 

•95923 

29932 

•95415 

31593 

.94878 

.33244 

•94313 

26 

.2829 

•959'S 

2996 

•95407 

.3162 

.94869 

•33271 

•94303 

27 

.28318 

•959°7 

.29987 

95398 

.31648 

.9486 

•33298 

•94293 

28 

.28346 

.30015  .95389 

•31675 

.94851 

.33326 

.94284 

29 

.28374 

•9589 

.30043  |  .9538 

•3i703 

.94842 

•33353 

.94274 

30 

.  28402 

.95882 

.30071  .95372 

•3i73 

•94832 

•3338i 

.94264 

31 

.  28429 

•95874 

.30098  .95363 

•31758 

.94823 

•33408 

.94254 

32 

.28457 

•95865 

.30126  |  .95354 

.31786 

.94814 

•33436 

•94245 

33 

.28485 

•95857 

.30154  -95345 

•31813 

•94805 

•33463 

•94235 

34 

•28513 

.95849 

.30182 

•95337 

.31841 

•94795 

•3349 

•94225 

35 

.28541 

.95841 

30209 

.95328 

.31868 

.94786 

•33518 

.94215 

36 

.28569 

•95832 

•30237 

•95319 

.31896 

•94777 

•33545 

.  94206 

37 

.28597 

.95824 

.30265 

•9531 

•3i923 

.94768 

•33573 

.94196 

38 

.28625 

95816 

.30292 

•95301 

•3i95i 

•94758 

•336 

.94186 

39 
40 

.28652 
.2868 

.95807 
•95799 

•3032 
•30348 

•95293 
.95284 

•31979 
.32006 

•94749 
•9474 

•33627 
•33655 

.94176 
.94167 

4i 

.28708 

•95791 

30376 

•95275 

•32034 

•9473 

.33682 

•94157 

42 

.28736 

•95782 

•30403 

.95266 

.32061 

•94721 

•3371 

.94147 

43 

.28764 

•95774 

•30431 

•95257 

.32089 

.94712 

•33737 

•94137 

44 

.28792 

.95766 

•  30459 

.95248 

.32116 

.94702 

•33764 

.94127 

45 

.2882 

•95757 

.  30486 

.9524 

•32144 

.94693 

•33792 

.94118 

46 

.28847 

•95749 

.30514  .95231 

.32171 

94684 

•33819 

.94108 

47 

•28875 

•9574 

30542 

.95222 

.32199 

•94674 

•33846 

.94098 

48 

.28903 

•95732 

•3057 

•95213 

.32227 

.94665 

•33874 

.94088 

49 

.28931 

•95724 

•30597 

.95204 

•32254 

.94656 

•33901 

.94078 

50 

•28959 

•95715 

.30625 

•95195 

.32282 

.94646 

•33929 

.94068 

5i 

.28987 

•95707 

30653 

.95186 

•32309 

•94637 

•33956 

.94058 

52 

•29015 

.95698 

.3068 

.95177 

•32337 

•94627 

•33983 

.94049 

53 

.29042 

•9569 

.30708 

.95168 

•32364 

.94618 

.34011 

.94039 

54 

.2907 

.95681 

•30736 

•95159 

•32392 

.94609 

•34038 

.94029 

55 

.29098 

.95673 

30763 

•95i5 

.32419 

•94599 

•34065 

.94019 

56 

.29126 

.95664 

.30791 

.95142 

•32447 

•9459 

•34093 

.94009 

57 

.29154 

•95656 

.30819 

.95133 

•32474 

•9458 

.3412 

58 

.29182 

•95647 

.30846 

.95124 

•32502 

•94571 

•34H7 

.  93989 

59 

.29209 

•95639 

•  30874 

•95"5 

•32529 

.94561 

•34175 

•93979 

60 

.29237 

•9563 

.30902 

.95106 

•32557 

•94552 

.34202 

•93969 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

73° 

72° 

710 

70° 

NATURAL    SINES    AND    COSINES. 


395 


if 

2( 

)° 

21° 

21 

jo 

23° 

27 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  eofl. 

N.  sine. 

N.  cos. 

0 

O 

.34202 

.93969 

•35837 

•93358 

>374oo 

.92718 

•39°73 

.9205 

o 

I 

.34229 

•93959 

•35864 

•9334s 

•37488 

.92707 

•39* 

.92039 

I 

2 

•34257 

•93949 

•3589* 

•93337 

•375*5 

.92697 

•39*27 

.92028 

I 

3 

•34284 

93939 

•359l8 

•93327 

•37542 

.92686 

•39*53 

.92016 

2 

4 

•343" 

•93929 

•35945 

•37569 

•92675 

•39*8 

.92005 

2 

5 

•34339 

•939*9 

•35973 

.93306 

•37595 

.92664 

.39207 

•9*994 

3 

6 

•34366 

•36 

•93295 

•37622 

•92653 

•39234 

.91982 

3 

7 

•34393 

.93899 

•36027 

•93285 

•37649 

.  92642 

.3926 

.91971 

4 

8 

•34421 

.93889 

•36054 

•93274 

•37676 

.92631 

•39287 

•9*959 

4 

9 

.34448 

93879 

.36081 

.93264 

•37703 

.9262 

•393*4 

.91948 

5 

10 

•34475 

•93869 

.36108 

•93253 

•3773 

.92609 

•3934* 

.91936 

5 

ii 

•34503 

•93859 

36135 

•93243 

•37757 

•92598 

•39367 

•9*925 

5 

12 

•3453 

.93849 

.36162 

•93232 

•37784 

.92587 

•39394 

•9*9*4 

6 

13 

•34557 

•93839 

.3619 

.93222 

.37811 

•92576 

.39421 

.91902 

6 

14 

•34584 

.93829 

•36217 

.93211 

•37838 

•92565 

.39448 

.91891 

7 

15 

34612 

93819 

•36244 

.93201 

•37865 

•92554 

•39474 

.91879 

7 

1  6  34639 

93809 

36271 

•93*9 

.37892 

•92543 

•39501 

.91868 

8 

17  :  .34666 

•93799 

.36298 

•93*8 

•379*9 

•92532 

•39528 

.91856 

8 

i  8  .34694 

•93789 

•36325 

•93*69 

•37946 

•92521 

•39555 

.91845 

9 

'9 

•34721 

•93779 

•36352 

•93*59 

•37973 

.9251 

•3958i 

•9l833 

9 

20 

•34748 

93769 

•36379 

.93148 

•92499 

.39608 

.91822 

9 

21  -34775 

•93759 

.36406 

•93*37 

.  38026 

.92488 

.39635 

.9181 

10 

22 

•34803 

•93748 

•36434 

•93*27 

•38053 

.92477 

.39661 

•9*799 

10 

23 

•3483 

•93738 

.36461 

•93"6 

.3808 

.92466 

.39688 

•9*787 

ii 

24 

34857 

•93728 

.36488 

.93106 

.38107 

•92455 

•397*5 

•9*775 

ii 

12 

3 

.34884 
•349*2 

.93708 

•365*5 
.  36542 

•93095 
.93084 

.38161 

•92444 
.92432 

•3974* 
.39768 

•9*764 
•9*752 

12 

27 

•34939 

•  36569 

•93074 

.38188 

.92421 

39795 

•9*74* 

13 

28 

•34966 

.93608 

•  36596 

.93063 

•38215 

.9241 

.39822 

.91729 

*3 

29 

•34993 

•93677 

•36623 

•93052 

.38241 

.92399 

.39848 

.91718 

30 

•35021 

•93667 

•3665 

93042 

.38268 

.92388 

•39875 

.91706 

*4 

3* 

•35048 

•93657 

•36677 

•9303* 

•38295 

•92377 

.39902 

.91694 

*4 

32 

•35075 

•93647 

•36704 

•9302 

.38322 

.92366 

.39928 

•9*683 

15 

33 

•35*02 

•93637 

•3673* 

,9301 

•38349 

•92355 

•39955 

.91671 

15 

34 

•35*3 

.93626 

•36758 

.92999 

•38376 

•92343 

.39982 

.9166 

16 

35 

•35*57 

.93616 

•36785 

.  92988 

•38403 

.92332 

.40008 

.91648 

16 

36 

35*84 

.93606 

.36812 

.92978 

•3843 

•92321 

.40035  \  .91636 

*7 

37 

•352" 

•93596 

•  36839 

.92967 

•38456 

.9231 

.40062  \  .91625 

38 

•35239 

•93585 

.  36867 

.92956 

•38483 

.92299 

.40088  .91613 

18  39 

•35266 

•93575 

•36894 

•92945 

•385* 

.92287 

.40115  .91601 

18 

18 

40 
4* 

•35293 
•3532 

•93565 
•93555 

.36921 
.36948 

•92935 
.92924 

•38564 

.92276 
.92265 

.40141 
.40168 

•9*59 
•9*578 

*9 

42 

•35347 

•93544 

•36975 

•929*3 

•3859* 

.92254 

•40195 

.91566 

*9 

43 

•35375 

•93534 

•37002 

•38617 

.92243 

.40221 

•9*555 

20 

44 

.35402 

•93524 

.92892 

.38644 

.92231 

.40248 

•9*543 

20 

4I 

•  35429 

•935*4 

.  37056 

.92881 

.38671 

.9222 

.40275 

•9*53* 

41 

46 

•  35456 

•93503 

.  37083 

.9287 

.38698 

.92209 

.40301 

•9*5*9 

£1 

47 

•35484 

•93493 

•37" 

.92859 

•38725 

.92198 

.  40328 

.91508 

22 

48 

•355" 

•93483 

•37*37 

.92849 

•38752 

.92186 

•40355 

•9*496 

22 

49 

•35538 

•93472 

•37*64 

.92838 

.38778 

•92*75 

.40381 

.91484 

23 

50 

•35565 

.93462 

•37*9* 

.92827 

•38805 

.92164 

.40408 

•9*472 

23 

5* 

•35592 

•93452 

•37218 

.92816 

•38832 

.92152 

.40434 

.91461 

23 

52 

•35619 

•9344* 

•37245 

92805 

.38859 

.92141 

.40461 

•9*449 

24 

24 

53 
54 

•35647 
•35674 

•9343* 
•9342 

•37272 
•37299 

.92794 
•92784 

.38886 
.38912 

•92*3 
.92119 

.40488 
.40514 

•9*437 
•9*425 

25 

55 

•  357°* 

•934* 

•37326 

•92773 

•38939 

.92107 

.40541 

•9*4*4 

25 

56 

•35728 

•934 

•37353 

.92762 

.38966 

.92096 

•40567 

.91402 

26 

57 

•35755 

•93389 

•3738 

•9275* 

•38993 

.92085 

.40594 

•9*39 

26 

58 

•35782 

•93379 

•37407 

•9274 

.3902 

•92073 

.40621 

•9*378 

27 

59 

•93368 

•37434 

.92729 

•39046 

.92062 

.40647 

•9*366 

27 

60 

•35837 

•93358 

•3746i 

.92718 

•39°73 

.9205 

.40674 

•9*355 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  COB. 

N.  Bine. 

690 

680 

67° 

66° 

396 


NATUKAL    SINKS   AND   COSINES. 


ti 

24° 

25° 

26° 

27° 

X|  0. 

26 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

o 

o 

.40674 

•9*355 

.42262 

.90631 

•43837 

.89879 

•45399 

.89101 

0 

i 

.407 

•9I343 

.42288 

.90618 

.43863 

.89867 

•45425 

.89087 

I 

2 

.40727 

•9*33* 

•42315 

.90606 

.43889 

.89854 

•45451 

.89074 

I 

3 

•40753 

•9*3*9 

.42341 

•9°594 

.43916 

.89841 

•45477 

.89061 

a 

4 

.4078 

-913<>7 

.42367 

.90582 

.43942 

.89828 

•45503 

.89048 

2 

5 

.40806 

.91295 

.42394 

.90569 

.43968 

.89816 

45529 

•89035 

3 

6 

•40833 

.91283 

.4242 

•90557 

•43994 

.89803 

•45554 

.89021 

3 
3 

i 

.4086 
.40886 

.91272 
.9126 

.42446 
•42473 

•9°545 
•90532 

.4402 
.44046 

,8979 
.89777 

45f 
.45606 

;fg*| 

4 

9 

.40913 

.91248 

•42499 

.9052 

.44072 

.89764 

•45632 

.8898i 

4 

10 

.40939 

.91236 

•42525 

.90507 

.44098 

.89752 

•45658 

.88968 

5 

ii 

.40966 

.91224 

•42552 

.90495 

.44124 

•89739 

.45684 

•88955 

12 

.40992 

.91212 

.42578 

.90483 

•44i5i 

.89726 

•4571 

.88942 

6 

13 

.41019 

.912 

.42604 

•9°47 

•44177 

•89713 

•45736 

.88928 

6 

H 

.41045 

.91188 

.42631 

.90458 

.44203 

•897 

•45762 

.88915 

7 

IS 

.41072 

.91176 

•42657 

.90446 

.44229 

.89687 

•45787 

.88902 

7 

16 

.41098 

.91164 

.42683 

•9°433 

•44255 

.89674 

•458i3 

.88888 

7 

17 

.41125 

.91152 

.42709 

.90421 

.4428! 

.89662 

•45839 

.88875 

8 

18 

.41151 

.9114 

•42736 

.90408 

•44307 

.89649 

•45865 

.88862 

8 

i9 

.41178 

.91128 

.42762 

.90396 

•44333 

.89636 

.45891 

.88848 

9 

20 

.41204 

.91116 

.42788 

•90383 

•44359 

.89623 

•459*7 

•88835 

9 

21 

.41231 

.91104 

.42815 

•9037i 

.44385 

.8961 

.45942 

.88822 

10 

22 

•41257 

.91092 

.42841 

•90358 

.44411 

•89597 

45968 

.88808 

10 

23 

.41284 

.9108 

.42867 

.90346 

•44437 

i  89584 

•45994 

.88795 

10 

24 

•4i3i 

.91068 

.42894 

•90334 

•44464 

89571 

.4602 

.88782 

II 
II 

11 

•41337 
•41363 

.91056 
.91044 

.4292 
.42946 

.90321 
•90309 

•4449 
.44516 

.89558 
•89545 

.46046 
.46072 

.88768 
•88755 

12 

12 

11 

•4139 
.41416 

.91032 
.9102 

.42972 
.42999 

.90296 
.90284 

•44542 
.44568 

•89532 
.89519 

.46097 
.46123 

.88741 
.88728 

13 

29 

•41443 

.91008 

.43025 

90271 

•44594 

.89506 

.46149 

.88715 

13 

30 

.41469 

.90996 

•43051 

.90259 

.4462 

•89493 

•46175 

.88701 

13 

31 

.41496 

.90984 

•43077 

.90246 

.44646 

.8948 

46201 

.88688 

H 

32 

.41522 

.90972 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

14 
15 

33 
34 

•41549 
•41575 

9096 
.90948 

4313 
•43156 

.90221 
.90208 

.44698 
.44724 

.89454 
.89441 

•46252 
.46278 

.88661 
.88647 

15 

35 

.41602 

.90936 

•43182 

.90196 

•4475 

.89428 

.46304 

.88634 

16 

36 

.41628 

.90924 

.43209 

.90183 

.44776 

.89415 

•4633 

.8862 

16 

37 

•41655 

•43235 

.90171 

.44802 

.89402 

•46355 

.88607 

16 

38 

.41681 

.  90899 

•43261 

90158 

.44828 

.89389 

.46381 

•88593 

17 

39 

.41707 

.  90887 

.43287 

.90146 

.44854 

.89376 

.46407 

.8858 

17 

40 

•41734 

•  9°875 

•43313 

9OI33 

.4488 

•89363 

•46433 

.88566 

18 

4i 

.4176 

.90863 

•4334 

.9012 

.44906 

•8935 

.46458 

•88553 

18 

42 

.41787 

.90851 

•43366 

.90108 

•44932 

•89337 

.46484 

•88539 

J9 

43 

.41813 

.90839 

•43392 

•90095 

.44958 

.89324 

.4651 

.88526 

^9 
20 

44 
45 

.4184 
.41866 

.  90826 
.90814 

.43418 
•43445 

90082 
.9007 

.44984 
.4501 

•89311 
.89298 

.46536 
.46561 

.88512 
.88499 

20 

46 

.41892 

.90802 

•43471 

.90057 

•45036 

.89285 

.46587 

.88485 

2O 

47 

.41919 

.9079 

•43497 

.90045 

.45062 

.89272 

.46613 

.88472 

21 

48 

•41945 

.90778 

•43523 

.90032 

.45088 

.89259 

.88458 

21 

49 

.41972 

.90766 

•43549 

.90019 

•45"4 

.89245 

.  46664 

.88445 

22 

50 

.41998 

•9°753 

•43575 

4514 

.89232 

.4669 

.88431 

22 

5i 

.42024 

.90741 

.43602 

89994 

.45166 

.89219 

.46716 

.88417 

23 

52 

•42051 

.90729 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

23 

53 

.42077 

.90717 

•43654 

.89968 

452i8 

.89193 

•46767 

.8839 

23  54 

.42104 

.90704 

4368 

.89956 

•45243 

.8918 

•46793 

•88377 

24 

55 

•4213 

.90692 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

24 

56 

.42156 

.9068 

•43733 

•8993 

•45295 

•89153 

.46844 

.88349 

25 

57 

.42183 

.90668 

•43759 

.89918 

45321 

.8914 

.4687 

.88336 

25 

58 

.42209 

•90655 

.43785  .89905 

•45347 

.89127 

.46896 

.88322 

26 

26 

59 
60 

•42235 
.42262 

.90643 
.90631 

.43811 
•43837 

i  .89892 
.89879 

•45373 
•45399 

.89114 
.89101 

46921 
•46947 

88308 
88295 

N.  cot. 

N.  sine. 

N.  cot. 

N.  sine. 

N.  cos. 

N.  tine. 

N.  cos. 

N.  tine. 

65° 

64° 

630 

62° 

NATURAL    SINES   AND    COSINES. 


397 


2 

BO 

2 

50 

3 

30 

3] 

LO 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  COB. 

o 

.46947 

.88295 

.48481 

.87462 

•  5 

.86603 

•51504 

.85717 

I 

•46973 

.88281 

.48506 

.87448 

.50025 

.86588 

•51529 

.85702 

2 

.46999 

.88267 

•48532 

•87434 

•5005 

•86573 

•51554 

.85687 

3 

.47024 

.88254 

.48557 

.8742 

.50076 

•86559 

•51579 

.85672 

4 

•4705 

.8824 

.48583 

.87406 

.50101 

.86544 

.51604 

•85657 

5 

.47076 

.88226 

.48608 

•87391 

.50126 

•8653 

.51628 

.85642 

6 

.47101 

.88213 

.48634 

.87377 

•50151 

.86515 

•51653 

.85627 

7 

.47127 

.88199 

•48659 

•87363 

•50176 

86501 

.51678 

.85612 

8 

•47I53 

.88185 

.48684 

•87349 

.  50201 

.86486 

•51703 

.85597 

9 

.47178 

.88172 

.4871 

•87335 

.50227 

.86471 

.51728 

.85582 

10 

.47204 

.88158 

.48735 

•87321 

.  50252 

.86457 

•51753 

.85567 

ii 

.47229 

.88144 

.48761 

.87306 

.50277 

.86442 

.51778 

.85551 

12 

•47255 

.8813 

.48786 

.87292 

.50302 

.86427 

.51803 

.85536 

13 

.47281 

.88117 

.48811 

.87278 

•  50327 

.86413 

.51828 

.85521 

J4 

.88103 

.48837 

.87264 

•50352 

.86398 

•51852 

.85506 

15 

•47332 

.88089 

.48862 

.8725 

•50377 

.86384 

•51877 

.85491 

16 

•47358 

.88075 

.48888 

•87235 

.50403 

.86369 

•51902 

.85476 

17 

•47383 

.88062 

.48913 

.87221 

.  50428 

•86354 

•51927 

.85461 

18 

.47409 

.88048 

.48938 

.87207 

•50453 

•  8634 

•51952 

.85446 

'9 

•47434 

.88034 

.48964 

•87193 

•50478 

•86325 

•51977 

.85431 

20 

•4746 

.8802 

.48989 

.87178 

•50503 

•  863I 

.52002 

.85416 

21 

.47486 

.88006 

.49014 

.87164 

.50528 

•86295 

.  52026 

.85401 

22 

•475H 

•87993 

.4904 

•8715 

.50553 

.8628! 

•  52051 

.85385 

23 

•47537 

.87979 

.49065 

.87136 

•  50578 

.86266 

•52076 

•8537 

24 

•47562 

.87965 

•49°9 

.87121 

.86251 

.52101 

.85355 

25 

.47588 

•87951 

.49116 

.87107 

.50628 

•86237 

.52126 

•8534 

26 

.47614 

•87937 

.49141 

87093 

•50654 

.86222 

•52I5I 

•85325 

11 

•47639 
.47665 

.87923 
.87909 

.49166 
.49192 

.87079 
.87064 

•50679 
.50704 

.86207 
.86192 

•52175 
•  522 

•8531 
.85294 

29 

.4769 

.87896 

.49217 

.8705 

•  50729 

.86178 

•52225 

•85279 

30 

.47716 
•47741 

.87882 
.87868 

.  49268 

.87036 
.87021 

•50754 
•50779 

.86163 
.86148 

•5225 
•52275 

.85264 
.85249 

32 

•47767 

•87854 

.49293 

.87007 

.50804 

.86133 

•52299 

•85234 

33 

•47793 

.8784 

.49318 

.86993 

.50829 

.86119 

•52324 

.85218 

34 

.47818 

.87826 

•49344 

.86978 

•50854 

.86104 

.52349 

.85203 

35 

.47844 

.87812 

•49369 

.86964 

.50879 

.86089 

.52374 

.85188 

36 

.47869 

.87798 

•49394 

.86949 

.50904 

.86074 

.52399 

•85173 

37 

•47895 

.87784 

.49419 

•86935 

.50929 

.86059 

•52423 

•85157 

38 

.4792 

.8777 

•49445 

.86921 

.50954 

.86045 

.52448 

.85142 

39 

•47946 

•87756 

•4947 

.86906 

•50979 

.8603 

•52473 

.85127 

40 

•47971 

•87743 

•49495 

.86892 

.51004 

.86015 

.52498 

.85112 

41 

•47997 

.87729 

.49521 

.86878 

.51029 

.86 

.52522 

.85096 

42 

.48022 

•87715 

•49546 

.86863 

•51054 

•85985 

•52547 

.85081 

43 

.48048 

.87701 

•49571 

.86849 

•51079 

•8597 

.85066 

44 

.48073 

.87687 

.49596 

.86834 

.51104 

.85956 

.52597 

.85051 

45 

.48099 

.87673 

.49622 

.8682 

.51129 

.85941 

.52621 

•85035 

46 

.48124 

.87659 

•49647 

.86805 

.85926 

.52646 

.8502 

47 

.4815 

.87645 

.49672 

.86791 

•51179 

.85911 

•52671 

.85005 

48 

•48i75 

•87631 

.49697 

.86777 

.51204 

.85896 

.52696 

.84989 

49 

48201 

.876I7 

•49723 

.86762 

.51229 

.85881 

•5272 

.84974 

So 

.48226 

.87603 

.49748 

.86748 

•51254 

.85866 

•52745 

•  84959 

51 

.48252 

.87589 

•49773 

•86733 

.51279 

.85851 

•5277 

•84943 

52 

.48277 

•87575 

.49798 

.86719 

•51304 

•85836 

•52794 

.84928 

53 

•48303 

.87561 

.49824 

.86704 

•51329 

.85821 

.52819 

.84913 

54 

.48328 

•87546 

.49849 

.8669 

.51354 

.85806 

•52844 

.84897 

55 

•48354 

•87532 

.49874 

•86675 

•51379 

.85792 

.52869 

.84882 

56 

.48379 

•87518 

.49899 

.86661 

.51404 

•85777 

•52893 

.84866 

u 

.48405 
.4843^ 

.87504 
.8749 

.49924 
•4995 

.86646 
,86632 

.51429 
•51454 

•85762 
•85747 

•52918 
•  52943 

.84851 
.84836 

59 

.48456 

.87476 

•49975 

.86617 

•5H79 

•85732 

•52967 

.8482 

60 

.48481 

.87462 

•5 

.86603 

•51504 

•85717 

•  52992 

.84805 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  Bin*. 

6 

1° 

6 

0° 

5 

JO 

51 

P 

L 

L 

398 


NATURAL    SINES    AND    COSINES. 


35 

,o 

33 

o 

34 

t° 

35 

o 

' 

N.  sine. 

N.  cos. 

N.sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

o 

•  52992 

.84805 

.54464 

.83867 

•559J9 

.82904 

.57358 

.81915 

I 

•53017 

.84789 

.54488 

.83851 

.  82887 

.57381 

.81899 

2 

•84774 

•54513 

•83835 

.  559°8 

.82871 

•  57405 

.81882 

3 

^  53066 

•84759 

•54537 

.83819 

•55992 

•82855 

•57429 

.81865 

4 

•53091 

•84743 

•5456i 

.83804 

.56016 

.82839 

•57453 

.81848 

5 

•53"5 

.84728 

•54586 

•83788 

.5604 

.82822 

•57477 

.81832 

6 

•5314 

.847I2 

.5461 

.83772 

.56064 

.82806 

•57501 

.81815 

7 

•53164 

.84697 

•54635 

.83756 

.56088 

.8279 

•57524 

.81798 

8 

•53189 

.84681 

•54659 

•8374 

.56112 

.82773 

•57548 

.81782 

9 

10 

•53214 
•53238 

.84666 
•  8465 

.54683 
.54708 

.83724 
.83708 

.56136 
.5616 

.82757 
.82741 

•57572 
•57596 

•81765 
.81748 

ii 

12 

•53263 
.53288 

•84635 
.84619 

•54732 
•54756 

.83692 
.83676 

•56184 
.56208 

.82724 
.82708 

-57619 
.57643 

.81731 
.81714 

13 

•53312 

.84604 

•5478i 

.8366 

•56232 

.82692 

•57667 

.81698 

14 

•53337 

.84588 

.54805 

•83645 

.56256 

•82675 

•57691 

.81681 

15 

•5336i 

.84573 

.54829 

.83629 

.5628 

.82659 

•57715 

.81664 

16 

•53386 

•84557 

.54854 

.83613 

•56305 

.82643 

.57738 

.81647 

!7 

•534" 

•84542 

.54878 

.83597 

•56329 

.82626 

.57762 

.81631 

18 

•53435 

.84526 

.54902 

.83581 

•56353 

.8261 

.57786 

.81614 

19 

•5346 

•84511 

.54927 

.83565 

•56377 

•82593 

.5781 

•8l597 

20 

•53484 

•84495 

.54951 

.83549 

.56401 

.82577 

•57833 

.8158 

21 

•53509 

.8448 

•54975 

.83533 

•56425 

.82561 

.57857 

•81563 

22 

•53534 

.84464 

•54999 

.83517 

•56449 

.82544 

•57881 

.81546 

23 

•53558 

.84448 

.55024 

.83501 

•56473 

.82528 

•  57904 

.8i53 

24 

•53583 

.84433 

.55048 

•83485 

•56497 

.82511 

.57928 

•81513 

25 

•53607 

.84417 

•55072 

.83469 

•56521 

.82495 

•57952 

.81496 

26 

•53632 

.84402 

.55097 

.83453 

•56545 

.82478 

•57976 

.81479 

27 

•53656 

.84386 

•55121 

•83437 

•56569 

.82462 

•  57999 

.81462 

28 

•53681 

.8437 

•55145 

.83421 

•56593 

.82446 

.58023 

•8i445 

29 

•53705 

•84355 

•55169 

•83405 

.56617 

.82429 

.58047 

.81428 

30 

•5373 

•84339 

•55194 

•83389 

.56641 

•82413 

•5807 

.81412 

31 

•53754 

.84324 

•55218 

.83373 

.  56665 

.82396 

.58094 

.81395 

32 

•53779 

.84308 

•55242 

.83356 

.56689 

.8238 

.58118 

•81378 

33 

•53804 

.84292 

•55266 

.8334 

•56713 

.82363 

.58141 

.81361 

34 

•53828 

.84277 

•55291 

.83324 

•56736 

.82347 

.58165 

.81344 

35 

•53853 

.84261 

.83308 

.0676 

•8233 

.58189 

.81327 

36 

.53877 

.84245 

•55339 

.83292 

.56784 

.82314 

.58212 

.8131 

37 

•  53902 

.8423 

•55363 

•83276 

.56808 

.82297 

•58236 

.81293 

38 

•53926 

.842I4 

•55388 

.8326 

.56832 

.82281 

.5826 

.81276 

39 

•53951 

.84198 

•55412 

.83244 

.56856 

.82264 

.58283 

.81259 

40 

•53975 

.84182 

.55436 

.83228 

.5688 

.82248 

.58307 

.81242 

•54 

.84167 

.5546 

.83212 

.56904 

.82231 

•5833 

.81225 

42 

•54024 

.84151 

•55484 

.83195 

.56928 

.82214 

.58354 

.81208 

43 

.54049 

•84135 

•55509 

.83179 

•56952 

.82198 

•58378 

.81191 

44 

•54073 

.8412 

•55533 

.83163 

•56976 

.82181 

.58401 

.81174 

45 

•54097 

.84104 

•55557 

.83147 

•57 

.82165 

•58425 

.81157 

46 

.54122 

.84088 

•5558i 

•83131 

•57024 

.82148 

.  58449 

.8114 

47 

.54146 

.84072 

•55605 

•83115 

•57047 

.82132 

.58472 

.81123 

48 

•54I7I 

.84057 

.5563 

.83098 

-57071 

.82115 

.58496 

.81106 

49 

•54I95 

.84041 

•55654 

.83082 

•57095 

.82098 

•58519 

.81089 

50 

.5422 

.84025 

•55678 

.83066 

•57"9 

.82082 

.58543 

.81072 

Si 

.54244 

.84009 

•55702 

•8305 

.57143 

.82065 

•58567 

•81055 

52 

.  54269 

.83994 

•55726 

•83034 

•57167 

.82048 

•5859 

.81038 

53 

.54293 

.83978 

•5575 

•83017 

•57191 

.82032 

•586,4 

.81021 

54 

•54317 

83962 

•55775 

.83001 

•57215 

.82015 

•  58637 

.81004 

55 

•54342 

.83946 

•55799 

.82985 

•57238 

.81999 

.58661 

.80987 

56 

•54366 

•8393 

•55823 

.  82969 

•  57262 

.81982 

.58684 

.8097 

57 

•54391 

•83915 

•55847 

•82953 

.57286 

•81965 

.58708 

•80953 

58 

•544I5 

.83899 

•55871 

.82936 

•5731 

.81949 

•58731 

.80936 

59 

•5444 

•83883 

•55895 

.8292 

•57334 

•81932 

•58755 

.80919 

60 

•54464 

.83867 

•559*9 

.82904 

•57358 

.81915 

•58779 

.80902 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N  cos 

N.  sine. 

N.  cos. 

N  sine 

5 

70 

Si 

5° 

5 

5° 

5< 

1° 

NATURAL   SINES   AND   COSINES. 


399 


360 

370 

380 

390 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

0 

•  58779 

.80902 

.60182 

.79864 

.61566 

.78801 

-.62932 

•77715 

j  I 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

•62955 

•  77696 

2 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

3 

.58849 

.8085 

.60251 

.79811 

•61635 

.78747 

.63 

.7766 

4 

•58873 

.80833 

.60274 

•79793 

.61658 

.78729 

.63022 

.77641 

5 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

•63045 

•77623 

6 

.5892 

.80799 

.60321 

•79758 

.61704 

.78694 

.63068 

.77605 

7 

•58943 

.80782 

•60344 

•79741 

.61726 

.78676 

.6309 

•77586 

8 

•58967 

.80765 

.60367 

•79723 

.61749 

.78658 

.63113 

•77568 

X9 

•5899 

.80748 

.6039 

•797o6 

.61772 

.7864 

•63135 

•7755 

10 

.59014 

•8073 

.60414 

.79688 

•6i795 

.78622 

.63158 

•77531 

ii 

•59°37 

•80713 

•60437 

.79671 

.61818 

.78604 

.6318 

•77513 

12 

.59061 

.80696 

.6046 

•79653 

.61841 

•78586 

.63203 

•77494 

13 

.59084 

.80679 

.60483 

•79635 

.61864 

.78568 

•63225 

.77476 

.59108 

.80662 

.60506 

.79618 

.61887 

•7855 

.63248 

•77458 

15 

.80644 

•60529 

.796 

.61909 

•78532 

.63271 

•77439 

16 

•59I54 

.80627 

•60553 

79583 

61932 

•78514 

.63293 

.77421 

'7 

.59178 

.8061 

.60576 

•79565 

•61955 

.78496 

.63316 

.77402 

18 

.59201 

•80593 

.60599 

•79547 

.61978 

.78478 

.63338 

•77384 

19 

.59225 

•80576 

.60622 

•7953 

62001 

.7846 

.63361 

•77366 

20 

.  59248 

•80558 

.60645 

•79512 

.62024 

•78442 

.63383 

•77347 

21 

.59272 

•80541 

.60668 

•79494 

.62046 

.78424 

.63406 

.77329 

22 

.59295 

.80524 

.60691 

•79477 

.62069 

.78405 

•63428 

-773I 

23 

.80507 

.60714 

•79459 

.62092 

.78387 

•63451 

.77292 

24 

•59342 

.  80489 

•60738 

.79441 

.62115 

.78369 

•63473 

.77273 

25 

•59365 

.80472 

.60761 

.79424 

.62138 

•78351 

63496 

.77255 

26 

•59389 

•80455 

.60784 

.79406 

.6216 

•78333 

.63518 

•77236 

3 

.59412 
•59436 

.80438 
.8042 

.60807 
.6083 

.79388 
•79371 

.62183 
.62206 

•78315 
.78297 

•6354 
•63563 

.77218 
.77199 

29 

•59459 

.80403 

•60853 

•79353 

.62229 

.78279 

•63585 

.77181 

30 

.59482 

.80386 

.60876 

•79335 

.62251 

.78261 

.63608 

.77162 

•595o6 

.80368 

.60899 

.62274 

.78243 

•6363 

•77*44 

32 

•59529 

•80351 

.60922 

•793 

.62297 

.78225 

•63653 

•77125 

33 

•59552 

•80334 

•60945 

.79282 

.6232 

.78206 

•63675 

.77107 

34 

•59576 

.80316 

60968 

.79264 

62342 

.78188 

.63698 

.77088 

35 

•59599 

.80299 

.60991 

.79247 

.62365 

.7817 

•6372 

.7707 

36 

.59622 

.80282 

.61015 

.79229 

.62388 

78152 

•63742 

•77051 

37 

.59646 

.80264 

.61038 

.79211 

.62411 

•78134 

•63765 

.77033 

38 

.59669 

.80247 

.61061 

•79J93 

•62433 

78116 

63787 

.77014 

39 

•59693 

.8023 

.61084 

.79176 

.62456 

.78098 

6381 

.76996 

40 

•597i6 

.80212 

.61107 

•79158 

.62479 

.78079 

•63832 

•76977 

41 

•59739 

.80195 

.6113 

.7914 

.62502 

.78061 

•63854 

•76959 

42 

.59763 

.80178 

•6u53 

.79122 

.62524 

.78043 

.63877 

•7694 

43 

.59786 

.8016 

.61176 

•79I05 

•62547 

.78025 

.63899 

.76921 

44 

.59809 

.80143 

.61199 

.79087 

.6257 

.78007 

.63922 

•76903 

45 

•59832 

.80125 

.61222 

.79069 

62592 

.77988 

•63944 

.76884 

46 

•59856 

.80108 

.61245 

•79051 

.62615 

•7797 

.63966 

.76866 

47 

•59879 

.80091 

.61268 

•79033 

.62638 

•77952 

.63989 

.76847 

48 

.59902  .80073 

.61291 

.79016 

.6266 

•77934 

.64011 

.76828 

49 
50 

.59926  .80056 

.59949  !  .80038 

•61314 
•61337 

.78998 
.7898 

.62683 
.62706 

.77916 
.77897 

•64033 
-64056 

.768x 
76791 

Si 

.59972  :  .80021 

•6136 

.78962 

.62728 

.77879 

.64078 

.76772 

52 

59995  .80003 

•61383 

•78944 

.62751 

.77861 

.641 

•76754 

53 

.60019  -79986 

61406 

.78926 

.62774 

•77843 

.64123 

•76735 

54 

.60042  .79968 

.61429 

.78908 

.62796 

.77824 

64145  1  .76717 

55 

•60065  .79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

56 

•60089  .79934 

.61474 

.78873 

62842 

.77788 

.6419 

.76679 

57 

.60112  .79916 

.61497 

•78855 

.62864 

.77769 

64212 

.76661 

58 

.60135  .79899 

.6152 

.78837 

.62887  .77751 

64234 

.76642 

59 

.60158  .79881 

•6i543 

78819 

.62909  .77733 

•64256 

.76623 

60 

.60182 

.79864 

.61566 

.78801 

.62932 

•777*5 

.64279 

.76604 

N.  cos. 

N.  sine. 

N.  cos.  N.sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

53°    I 

52°     I 

61° 

60°    i 

400 


NATURAL    SINES   AND    COSINES. 


si 

4 

3° 

4 

L° 

4 

go 

4 

3° 

22 

• 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

0 

O 

.64279 

.76604 

.65606 

•75471 

•669x3 

•743*4 

.682 

•73135 

O 

I 

.64301 

•76586 

.65628 

•75452 

.66935 

•74295 

.68221 

.73116 

2 

•64323 

.76567 

•6565 

•75433 

.66956 

•74276 

.68242 

.73096 

3 

•76548 

.65672 

•75414 

.66978 

74256 

.68264 

.73076 

4 

.  64368 

•7653 

•65694 

•75395 

.66999 

•74237 

.68285 

•73056 

5 

•6439 

.76511 

•65716 

•75375 

.67021 

.74217 

.68306 

•73036 

6 

.64412 

'.76492 

•65738 

75356 

.67043 

74198 

•68327 

.73016 

7 

•64435 

•76473 

•65759 

•75337 

.67064 

.74178 

•68349 

.72996 

3 

8 

•64457 

76455 

65781 

•753i8 

.67086 

•74J59 

•6837 

.72976 

3 

9 

•64479 

•76436 

•65803 

•75299 

.67107 

•74139 

.68391 

•72957 

4 

10 

.64501 

.76417 

.65825 

.7528 

.67129 

.7412 

.68412 

•72937 

4 

ii 

.64524 

.76398 

.65847 

.75261 

.67151 

.741 

.68434 

.72917 

4 
5 

12 
13 

.64546 
.64568 

.7638 
.76361 

.65869 
.65891 

75241 
•75222 

.67172 
.67194 

.7408 
.74061 

•68455 
.68476 

.72897 
.72877 

5 

14 

•6459 

•76342 

.75203 

.67215 

.74041 

.68497 

.72857 

6 

15 

.64612 

•76323 

•65935 

•75184 

.67237 

.  74022 

.68518 

.72837 

6 

16 

.64635 

.76304 

65956 

•75165 

.67258 

.74002 

•68539 

.72817 

6 

17 

.64657 

.76286 

.65978 

•75146 

.6728 

•73983 

.68561 

•72797 

7 

18 

.64679 

.76267 

.66 

.75126 

•67301 

•  73963 

.68582 

.72777 

7 

19 

.64701 

.76248 

.66022 

•75107 

•67323 

•73944 

.68603 

•72757 

7 

20 

.64723 

.76229 

.66044 

.75088 

•67344 

•73924 

.68624 

•72737 

8 

21 

.64746 

.7621 

.66066 

.75069 

.67366 

•739°4 

.68645 

.72717 

8 

22 

.64768 

.76192 

.66088 

•7505 

.67387 

•73885 

.68666 

.  72697 

8 

23 

•6479 

•76173 

.66109 

•7503 

.67409 

•73865 

.68688 

72677 

9 

24 

.64812 

.66131 

.75011 

•6743 

•73846 

.68709 

.72657 

9 

25 

.64834 

.76135 

•66153 

.74992 

.67452 

.73826 

•6873 

.72637 

10 

26 

.64856 

.76116 

.66175 

•74973 

•67473 

.73806 

.68751 

.72617 

10 

27 

.64878 

.76097 

.66197 

•74953 

•67495 

•73787 

.68772 

•72597 

xo 

28 

.64901 

.76078 

.66218 

•  74934 

•67516 

•73767 

•68793 

•  72577 

II 

29 

•64923 

.76059 

.6624 

•67538 

•73747 

.68814 

•72557 

II 

30 

.64945 

.  76041 

.66262 

.74896 

•67559 

•73728 

•68835 

•72537 

II 

31 

.64967 

.  76022 

.66284 

.74876 

.6758 

.73708 

•68857 

•72517 

12 

32 

.64989 

.76003 

.66306 

•74857 

.67602 

.73688 

.68878 

72497 

12 

33 

.65011 

•75984 

•66327 

.74838 

•67623 

•73669 

.68899 

•72477 

12 

34 

•65033 

•75965 

.66349 

.74818 

.67645 

•  73649 

.6892 

•72457 

13 

35 

•65055 

•75946 

.66371 

•74799 

.67666 

.73629 

.68941 

72437 

13 

36 

.65077 

•75927 

•66393 

.7478 

.67688 

.68962 

.72417 

14 

37 

.651 

.664x4 

•7476 

.67709 

•7359 

.68983 

72397 

14 

38 

.65122 

.75889 

.66436 

7474i 

•6773 

•7357 

.69004 

•72377 

14 

39 

.65144 

•7587 

.66458 

•74722 

.67752 

•73551 

.69025 

•72357 

IS 

40 

.65166 

•75851 

.6648 

•747°3 

•67773 

•73531 

.69046 

•72337 

15 

41 

.65188 

75832 

.66501 

•74683 

•67795 

•735" 

.  69067 

•72317 

15 

42 

•  6521 

.66523 

.74664 

.67816 

•73491 

.  69088 

.72297 

16 

43 

•65232 

•75794 

•66545 

74644 

.67837 

•73472 

.69109 

.72277 

16 

44 

•65254 

•75775 

.66566 

.74625 

•67859 

•73452 

.6913 

•72257 

17 

45 

.65276 

•75756 

.66588 

.74606 

.6788 

•73432 

•69151 

.72236 

17 

46 

•65298 

•75738 

.6661 

.74586 

.67901 

•73413 

.69172 

.72216 

17 

47 

•6532 

•75719 

.66632 

•74567 

.67923 

•73393 

.69193 

.72196 

18 

48 

.65342 

•757 

.66653 

•74548 

•67944 

•73373 

.69214 

.72176 

18 

49 

•65364 

.7568 

•66675 

.74528 

•67965 

•73353 

•69235 

72156 

18 

50 

.65386 

•75661 

.66697 

•74509 

.67987 

•73333 

.69256 

.72136 

19 

51 

.65408 

75642 

.66718 

.74489 

.68008 

•73314 

.69277 

.72116 

19 

52 

•6543 

•75623 

.6674 

7447 

.68029 

•73294 

.69298 

72095 

19 

53 

•65452 

.75604 

.66762 

•74451 

.68051 

•73274 

.69319 

72075 

20 

54 

•65474 

•75585 

•66783 

•74431 

.68072 

•73254 

•6934 

•72055 

2O 

55 

•65496 

•  75566 

.66805 

.74412 

.68093 

•7323'. 

.69361 

•72035 

21 

56 

•65518 

•75547 

.66827 

•74392 

.68115 

•73215 

.69382 

.72015 

21 

57 

•6554 

•75528 

.66848 

•74373 

.68136 

•73195 

69403 

•71995 

21 

58 

.65562 

•75509 

.6687 

•74353 

68157 

•73175 

.69424 

•71974 

22 

59 

•65584 

•7549 

.66891 

•74334 

68179 

•73155 

.  69445 

•71954 

22 

60 

.65606 

•75471 

•66913 

•74314 

.682 

•73135 

.69466 

•71934 

N.  cos. 

N.sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sin*. 

4< 

>° 

4i 

ti 

4' 

1° 

4C 

>° 

NATUKAL    SINES    AND    COSINES. 


401 


n 

ii 

"  , 

44 

N.  sine. 

1° 
N.  cog. 

£3 
&l 

19 

it 

22 

, 

44 

N.  sine. 

1° 

N.  cos. 

if 

9 

o 

o 

.69466 

.71934 

60 

19 

II 

31 

.70112 

•71305 

29 

9 

o 

I 

I 

2 

.69487 
.69508 

.71914 
.71894 

59 

58 

19 
18 

12 
12 

32 

33 

.70132 
•70153 

.71284 
.71264 

28 
27 

9 
9 

I 

3 

.69529 

•71873 

57 

18 

12 

34 

.70174 

•71243 

26 

8 

I 

4 

.69549 

•71853 

56 

18 

13 

35 

.70195 

.71223 

25 

8 

2 

5 

•6957 

•71833 

55 

17 

13 

36 

.70215 

.71203 

24 

8 

2 

6 

.69591 

.71813 

54 

17 

14 

37 

.70236 

.71182 

23 

7 

3 

7 

.69612 

.71792 

53 

i7 

14 

38 

.70257 

.71162 

22 

7 

3 

8 

.69633 

.71772 

52 

16 

14 

39 

.70277 

.71141 

21 

7 

3 

9 

-69654 

•71752 

51 

16 

15 

40 

.70298 

.71121 

20 

6 

4 

10 

•69675 

•71732 

So 

16 

15 

4i 

•70319 

.711 

19 

6 

4 

ii 

.69696 

.71711 

49 

16 

15 

42 

•70339 

.7108 

18 

6 

4 

12 

.69717 

.71691 

48 

15 

16 

43 

•7036 

-71059 

17 

5' 

5 

13 

•69737 

.71671 

47 

15 

16 

44 

.70381 

•7I039 

16 

5 

5 

14 

.69758 

•7165 

46 

15 

17 

45 

.70401 

.71019 

IS 

5 

6 

15 

.69779 

•7163 

45 

14 

17 

46 

.70422 

.70998 

14 

4 

6 

16 

.698 

.7161 

4* 

*4 

'7 

47 

•70443 

.70978 

13 

4 

6 

7 

;? 

.69821 
.69842 

•7159 
•71569 

43 
42 

14 
13 

18 
18 

48 
49 

•70463 
.70484 

•70957 
•70937 

12 
II 

4 
3 

7 

19 

.69862 

•7I549 

4i 

13 

18 

So 

•70505 

.70916 

10 

3 

7 

20 

.69883 

•71529 

4° 

13 

J9 

Si 

.70525 

.70896 

9 

3 

8 

21 

.69904 

.71508 

39 

12 

J9 

52 

.70546 

.70875 

8 

3 

8 

22 

.69925 

.71488 

38 

12 

X9 

53 

.70567 

•70855 

7 

2 

8 

23 

.69946 

.71468 

37 

12 

20 

54 

.70587 

.70834 

6 

2 

9 

24 

.69966 

•7H47 

36 

II 

20 

55 

.70608 

.70813 

5 

2 

9 

25 

.69987 

.71427 

35 

II 

21 

56 

.70628 

•70793 

4 

I 

o 

26 

.70008 

.71407 

34 

II 

21 

57 

.70649 

.70772 

3 

I 

o 

27 

.70029 

•71386 

33 

10 

21 

58 

•7067 

.70752 

2 

I 

o 

28 

.70049 

.71366 

32 

10 

22 

59 

.7069 

•70731 

I 

0 

I 

29 

.7007 

•7I345 

3i 

IO 

22 

60 

•70711 

.70711 

O 

O 

I 

30 

.70091 

•71325 

30 

10 

N.  cos. 

N.  sine. 

' 

N.C08. 

N.  sine. 

~~' 

4i 

>° 

4 

5° 

Preceding  Table  contains  Natural  Sine  and  Cosine  for  every  minute 
of  the  Quadrant  to  Radius  i. 

If  Degrees  are  taken  at  head  of  columns,  Minutes,  Sine,  and  Cosine  must 
be  taken  from  head  also ;  and  if  they  are  taken  at  foot  of  column,  Minutes, 
etc.,  must  be  taken  from  foot  also. 

ILLUSTRATION— .3173  is  sine  of  18°  30',  and  cosine  of  71°  30'. 

To    Compute    Sine   or   Cosine   for   Seconds. 

When  Angle  is  less  than  45°.  RULE. — Ascertain  sine  or  cosine  of  angle 
for  degrees  and  minutes  from  Table;  take  difference  between  it  and  sine 
or  cosine  cf  angle  next  below  it.  Look  for  this  difference  or  remainder,* 
if  Sine  is  required,  at  head  of  column  of  Proportional  Parts,  on  left  side ; 
and  if  Cosine  is  required,  at  head  of  column  on  right  side;  and  in  these 
respective  columns,  opposite  to  number  of  seconds  of  angle  in  column,  is 
number  or  correction  in  seconds  to  be  added  to  Sine,  or  subtracted  from 
Cosine  of  angle. 

ILLUSTRATION  i.— What  is  sine  of  8°  9'  10"? 

Sine  of  8°   9',  per  Table  =  .141 77;)  0  ,.~   ^ 

Sine  of  80  ,o',        «        =.14205!}  -00028  *ff*™<*> 

In  left  side  column  of  proportional  parts,  under  28,  and  opposite  to  10',  is  5,  cor- 
rection  for  10',  which,  being  added  to  .141 77  =  .141 82  Sine. 

*  Tke  table  in  some  instances  will  give  a  unit  too  much,  but  this,  in  general,  is  of  little  importance, 
L  L* 


4O2  NATURAL    SINES   AND    COSINES. 

2.  —What  is  cosine  of  8°  9'  10"? 


In  right-side  column  of  proportional  parts,  under  4,  and  opposite  to  10',  is  i,  tho 
correction  for  10',  which,  being  subtracted  from  .98990—98989  cosine. 

When  Angle  exceeds  45°.  RULE.  —  Ascertain  sine  or  cosine  for  angle  in 
degrees  and  minutes  from  Table,  taking  degrees  at  the  foot  of  it  ;  then  take 
difference  between  it  and  sine  or  cosine  of  angle  next  above  it.  Look  for  re- 
mainder, if  Sine  is  required,  at  head  of  column  of  Proportional  Parts,  on  right 
side  ;  and  if  Cosine  is  required,  at  head  of  column  on  left  side  ;  and  in  these 
respective  columns,  opposite  to  seconds  of  angle,  is  number  or  correction  in 
seconds  to  be  added  to  Sine,  or  subtracted  from  Cosine  of  angle. 
ILLUSTRATION.—  What  is  the  Sine  and  Cosine  of  81°  50'  50"? 

1SS  SIS  £;  ""P"8::^'}  **  *&*«. 

In  right-side  column  of  proportional  parts,  and  opposite  to  50',  is  3,  which,  added 
to  .  989  86  =  .  989  89  Sine. 


In  left-side  column  of  proportional  parts,  and  opposite  to  50',  is  24,  which,  sub- 
tracted from  .  142  05  =  .  141  81  Cosine. 

To  Ascertain,  or  Compute  3S"um"ber  of  Degrees,  Minutes, 
and  Seconds  of*  a  given  Sine  or  Cosine. 

When  Sine  is  given.  RULE.  —  If  given  sine  is  in  Table,  the  degrees  of  it 
will  be  at  top  or  bottom  of  page,  and  minutes  in  marginal  column,  at  left  or 
right  side,  according  as  sine  corresponds  to  an  angle  less  or  greater  than  45°. 

If  given  sine  is  not  in  Table,  take  sine  in  Table  which  is  next  less  than  the 
one  for  which  degrees,  etc.,  are  required,  and  note  degrees,  etc.,  for  it.  Sub- 
tract this  sine  from  next  greater  tabular  sine,  and  also  from  given  sine. 

Then,  as  tabular  difference  is  to  difference  between  given  sine  and  tabu- 
lar sine,  so  is  60  seconds  to  seconds  for  sine  given. 

EXAMPLE.—  What  are  the  degrees,  minutes,  and  seconds  for  sine  of  .75? 

Next  less  sine  is  .74992,  arc  for  which  is  48°  35'.  Next  greater  sine  is  .75011, 
difference  between  which  and  next  less  is  .75011  —  .749  92  =  .00019.  Difference  be- 
tween less  tabular  sine  and  one  given  is  .  75  —  .  749  92  =  8. 

Then  19  :  8  ::  60  :  25+,  which,  added  to  48°  35'  =  48°  35'  25". 

When  Cosine  is  given.  RULE.  —  If  given  cosine  is  found  in  Table,  degrees 
of  it  will  be  found  as  in  manner  specified  when  sine  is  given. 

If  given  cosine  is  not  in  Table,  take  cosine  in  Table  which  is  next  greater 
than  one  for  which  degrees,  etc.,  are  required,  and  note  degrees,  etc.,  for  it. 
Subtract  this  cosine  from  next  less  tabular  cosine,  and  also  from  given  cosine. 

Then,  as  tabular  difference  is  to  difference  between  given  cosine  and  tabu- 
lar cosine,  so  is  60  seconds  to  seconds  for  cosine  given. 

EXAMPLE.  —  What  are  the  degrees,  minutes,  and  seconds  for  cosine  of  .75? 

Next  greater  cosine  is  .750  u,  arc  for  which  is  41°  24'.  Next  less  cosine  is  .749  92, 
difference  between  which  and  next  greater  is  .750  u  —  -74992  =  .000  19.  Difference 
between  greater  tabular  cosine  and  one  given  is  .750  n  —.75000=  u. 

Then  19  :  n  :  :  60  :  35  —  ,  which,  added  to  41°  24'  =  41°  24'  35". 

To    Compnte  Versed    Sine    of  an   Angle. 

Subtract  cosine  of  angle  from  i. 

ILLUSTRATION.—  What  is  the  versed  sine  of  21°  30'? 

Cosine  of  21°  30'  is  .930  42,  which,  —  i  =  .069  58  versed  sine. 

To   Compnte   Co-versed   Sine   of  an  Angle. 
Subtract  sine  of  angle  from  i. 
ILLUSTRATION.—  What  is  the  co-versed  sine  of  21°  30'? 

The  sine  of  21°  30'  is  .3665,  which,  —  i  =  .6335  co-versed  sine. 


NATUKAL   SECANTS   AND   CO-SECANTS. 


403 


N"atnral   Secants    and   Co  -secants. 

0< 

>  .. 

lc 

2C 

11             V 

> 
CO-SKC'T. 

0 

Infinite. 

1.  0001 

57-299 

i.  0006 

28.654  i 

1.0014 

19.107 

I 

i 

3437-7 

.0001 

6-359 

.0006 

8.417   i 

.0014 

9.002 

2 

j 

1718.9 

.0002 

5-45 

.0606 

8.184 

.0014 

8.897 

3 

i 

145-9 

.0002 

4-57 

.0006 

7-955 

.0014 

8.794 

4 

859.44 

.0002 

3.718 

.0006 

7-73 

.0014 

8.692 

5 

687-55 

1.0002 

52.891 

1.0007 

27-508 

1.0014 

18.591 

6 

572-96 

.OOO2 

2.09 

.0007 

7.29 

.0015 

8.491 

7 

491.11 

.0002 

1.313 

.0007 

7-075 

.0015 

8-393 

s 

29.72 

.0002 

0.558 

.0007 

6.864 

.0015 

8.295 

9. 

.OOO2 

49.826 

.0007 

6.655 

.0015 

8.198 

10 

343-77 

1.0002 

49.114 

1.0007 

26.45 

1.0015 

18.103 

rl 

12.52 

.OOO2 

8.422 

.0007 

6.249 

.0015 

8.008 

12 

13 

286.48 
64.44 
45-55 

.0002 
.0002 
0002 

7-75 
7.096 
6.46 

.0007 
.0007 
.0008 

6.05 

5-854 
5.661 

.0016 
.0016 
.0016 

7.914 
7.821 
7-73 

j  - 

229.18 

1.0002 

45.84 

1.0008 

25.471 

1.0016 

17-639 

ii 

14.86 

.OOO2 

5-237 

.0008 

5-284 

.0016 

7-549 

17 

02.22 

.0002 

4-65 

.0008 

.0016 

7-46 

18 

190.99 

.OOO2 

4.077 

.0008 

4.'9i8 

.0017 

7-372 

J9 

80.73 

.0003 

3-52 

.0008 

4-739 

.0017 

7.285 

20 

171.89 

1.0003 

42.976 

1.0008 

24.562 

1.0017 

17.198 

21 

63-7 

.0003 

2-445 

.0008 

4.358 

.0017 

7-"3 

22 

56.26 

.0003 

1.928 

.0008 

4.216 

.0017 

7.028 

23 

49-47 

.OOO3 

1.423 

.0009 

4.047 

.0017 

6-944 

24 

43-24 

.0003 

40-93 

.0009 

3.88 

.0018 

6.861 

25 

I37-5I 

1.0003 

40.448 

1.0009 

23.716 

1.0018 

16.779 

26 

32.22 

.0003 

39.978 

.0009 

3-553 

.0018 

6.698 

27 
28 

27.32 
22.78 

.0003 
.OOO3 

9.518 
9.069 

.0009 
.0009 

3-393 
3-235 

.0018 
.0018 

6.617 
6.538 

29 

18.54 

.0003 

8.631 

.0009 

3-079 

.0018 

6-459 

30 

"4-59 

1.0003 

38.201 

1.0009 

22.925 

1.0019 

16.38 

31 

10.9 

.0003 

7.782 

.001 

2.774 

.0019 

6.303 

32 

07-43 

.0003 

7-371 

.001 

2.624 

.0019 

6.226 

33 

04.17 

.0004 

6.969 

.001 

2.476 

.0019 

6.15 

34 

OI.II 

.0004 

6.576 

.001 

233 

.0019 

6-075 

35 

98.223 

1.0004 

36-  191 

I.OOI 

22.186 

1.0019 

16 

36 

5-495 

.0004 

5.814 

.001 

2.044 

.002 

5.926 

37 

2.914 

.0004 

5-445 

.001 

1.904 

.002 

5-853 

38 

.0001 

2.469 

.OOO4 

5-084 

.001 

1-765 

.OO2 

578 

39 

.0001 

88.149 

.0004 

4-729 

.OOII 

1.629 

.002 

5-708 

40 

1.  0001 

85.946 

1.0004 

I.OOII 

21.494 

I.OO2 

15.637 

41 

.0001 

3.849 

.0004 

4.042 

.OOII 

1.36 

.0021 

5.566 

42 

.0001 

1-853 

.0004 

3.708 

.OOII 

1.228 

.0021 

5-496 

43 

.0001 

7995 

.0004 

3-38i 

.OOII 

1.098 

.OO2I 

5-427 

44 

.0001 

8.133 

.0004 

3-o6 

.OOII 

20.97 

.0021 

5.358 

45 

1.  0001 

76-396 

I.OOO5 

32-745 

I.OOII 

20.843 

I.OO2I 

15.29 

46 

.0001 

4-736 

.0005 

2-437 

.0012 

0.717 

.0022 

5-222 

.0001 

3.146 

.0005 

2.134 

[       .0012 

0.593 

.OO22 

5-155 

48 

.0001 

1.622 

.0005 

1.836 

.0012 

0.471 

.0022 

5-089 

49 

.0001 

1.16 

.0005 

1-544 

.0012 

o.35 

.0022 

5-023 

1.  0001 

68.757 

I.OOO5 

3I-257 

I.OOI2 

20.23 

1.0022 

14.958 

51 

.0001 

7-409 

.0005 

30.  976 

.0012 

0.  112 

.0023 

4-893 

52 

0001 

6.113 

.OOO5 

0.699 

.OOI2 

19.995 

.0023 

4.829 

53 

.0001 

4.866 

.0005 

0.428 

.0013 

9-88 

.0023 

4.765 

54 

.0001 

3.664 

OOO5 

o.  161 

.0013 

9.766 

.0023 

4.702 

55 

1.  0001 

62.507 

1.0005 

29.899 

I.OOI3 

1.0023 

14.64 

56 

.0001 

I-391 

.0006 

9.641 

.0013 

9-541 

.0024 

4578 

57 

.0001 

.OOO6 

9.388 

.OOI3 

943' 

.OO24 

4-5I7 

58 

.0001 

59-  274 

.0006 

9-r39 

.0013 

9.322 

.0024 

59 

.0001 

8.27 

.0006 

8.894 

.0013 

9.214 

.OO24 

4-395 

60 

1.  0001 

57-299 

I.  OOO6 

28.654 

I.OOI4 

19.107 

1.0024 

14-335 

/ 

CO-SKC'T 

SECANT. 

CO-SEC  'T 

SECANT 

CO-SKC'T 

SECANT. 

CO-SEC'T 

SECANT. 

890 

88° 

87° 

86° 

404 


NATURAL   SECANTS   AND   CO-SECANTS. 


4°     1 

50 

6° 

70 

SECANT. 

CO-SIC'T. 

SKCANT. 

CO-SKC'T. 

SKCANT. 

CO-SKC'T. 

SXCANT.  |  CO-SKC'T. 

1.0024 
.0025 

14-335 
4.276 

1.0038 
.0038 

i  .474 
•436 

1.0055 
•0055 

9.5668 
•5404 

1.0075 
.0075 

8.2055 
.1861 

.0025 

4.217 

.0039 

.398 

.0056 

•SHI 

.0076 

.1668 

.0025 

4.159 

.0039 

•  36 

.0056 

.488 

.0076 

.1476 

.0025 

4.101 

.0039 

•323 

.0056 

.462 

.0076 

.1285 

1.0025 

14.043 

1.0039 

i  .286 

1.0057 

9.4362 

1.0077 

8.1094 

.0026 

3-986 

.004 

.249 

.0057 

.4105 

.0077 

.0905 

.0026 

3-93 

.004 

.213 

•0057 

.385 

.0078 

.0717 

.0026 

3-874 

.004 

.176 

.0057 

•3596 

.0078 

.0529 

.0026 

3.818 

.004 

.14 

.0058 

•3343 

.0078 

.0342 

1.0026 

13-763 

1.0041 

i  .104 

1.0058 

9.3092 

1.0079 

8.0156 

.0027 

3.708 

.0041 

.069 

.0058 

.2842 

.0079 

7.9971 

.0027 

3'654 

.0041 

•033 

.0059 

•2593 

.0079 

.9787 

.0027 

3.6 

.0041 

0.988 

•  0059 

•2346 

.008 

.9604 

.0027 

3-547 

.0042 

0.963 

.0059 

.21 

.008 

.9421 

1.0027 

13-494 

1.0042 

10.929 

1.006 

9-I855 

i.  008 

7.924 

.0028 

3-441 

.0042 

0.894 

.006 

.l6l2 

-0081 

•9059 

.0028 

3-389 

.0043 

0.86 

.006 

•137 

.0081 

.8879 

.0028 

3-337 

.0043 

0.826 

.0061 

.1129 

.0082 

.87 

.0028 

3.286 

.0043 

0.792 

.0061 

.089 

.0082 

.8522 

1.0029 

13-235 

1.0043 

10.758 

1.0061 

9.0651 

1.0082 

7-8344 

.0029 

3.184 

.0044 

0.725 

.0062 

.0414 

.0083 

.8168 

.0029 

3-*34 

.0044 

0.692 

.0062 

.0179 

.0083 

.7992 

.0029 
•0029 

3-084 
3-034 

.0044 
.0044 

0-659 
0.626 

.0062 
.0063 

8.9944 
-97II 

.0084 
.0084 

.7817 
.7642 

1.003 

12.985 

1.0045 

10.593 

1.0063 

8-9479 

1.0084 

7.7469 

.003 

2-937 

.0045 

0.561 

.0063 

.9248 

.0085 

.7296 

.003 

2.888 

.0045 

0.529 

.0064 

.9018 

.0085 

.7124 

.003 

2.84 

.0046 

0.497 

.0064 

.879 

.0085 

•6953 

.0031 

z-793 

.0046 

0.465 

.0064 

.8563 

.0086 

.6783 

1.0031 

12.745 

1.0046 

10-433 

1.0065 

8-8337 

1.0086 

7.6613 

.0031 

2.698 

.0046 

0.402 

.0065 

.8112 

.0087 

.6444 

.0031 

2.652 

.0047 

0.371 

.0065 

.7888 

.0087 

.6276 

.0032 

2.606 

.0047 

o-34 

.0066 

.7665 

.0087 

.6108 

.0032 

2.56 

.0047 

0.309 

.0066 

•7444 

.0088 

•5942 

1.0032 

12.514 

1.0048 

10.278 

1.0066 

8.7223 

1.0088 

7-5776 

.0032 

2.469 

.0048 

0.248 

.0067 

.7004 

.0089 

.5611 

.0032 

2.424 

.0048 

0.217 

.0067 

.6786 

.0089 

•5446 

•  0033 

2-379 

.0048 

0.187 

.0067 

.6569 

.0089 

.5282 

.0033 

2-335 

.0049 

o.i57 

.0068 

•6353 

.009 

•5"9 

1.0033 

12.291 

1.0049 

10.127 

i.  0068 

8.6138 

1.009 

7-4957 

•0033 

2.248 

.0049 

0.098 

.0068 

•5924 

.009 

-4795 

.0034 

2.204 

.005 

0.068 

.0069 

•57" 

.0091 

•4634 

.0034 

2.161 

.005 

0.039 

.0069 

•5499 

.0091 

•4474 

.0034 

2.118 

.005 

0.01 

.0069 

.5289 

.0092 

•43*5 

1.0034 

12.076 

1.005 

9.9812 

1.007 

8.5079 

1.0092 

7-4156 

•0035 

2.034 

•  0051 

•9525 

.007 

.4871 

.0092 

.3998 

•0035 

1.992 

.0051 

.9239 

.007 

•  4663 

.0093 

.384 

.0035 

i-95 

.0051 

•8955 

.0071 

•4457 

•  0093 

•3683 

.0035 

1.909 

.0052 

.8672 

.0071 

.4251 

.0094 

•3527 

1.0036 

n.868 

1.0052 

9.8391 

1.0071 

8.4046 

1.0094 

7-3372 

.0036 

1.828 

.0052 

.8112 

.0072 

•3843 

.0094 

•3217 

.0036 

1.787 

•0053 

-7834 

.0072 

.3640 

.0095 

.3063 

.0036 

1-747 

•0053 

.7558 

.0073 

•3439 

.0095 

.2909 

.0037 

1.707 

•0053 

.7283 

.0073 

•3238 

.0096 

•2757 

1-0037 

n.668 

1-0053 

9.701 

1.0073 

8-3039 

1.0096 

7.2604 

.0037 

1.628 

•  0054 

•6739 

.0074 

.2840 

.0097 

•2453 

.0037 

1.589 

.0054 

.6469 

.0074 

.2642 

.0097 

.2302 

.0038 

'•55 

•  0054 

.62 

.0074 

.2446 

.0097 

.2152 

.0038 

1.512 

•0055 

-5933 

.0075 

.225 

.0098 

.2002 

1.0038 

11.474 

1.0055 

9.5668 

1.0075 

8.2055 

.0098 

7-1853 

CO-8KC'T. 

SKCANT. 

CO-BKC'T. 

SKCANT. 

CO-SKC'T. 

SKCANT. 

CO-BEC'T. 

SECANT. 

85° 

84° 

83° 

82° 

NATURAL   SECANTS  AND    CO-SECANTS. 


405 


8 

0 

9 

c 

II 

)0 

13 

L° 

SKCANT. 

CO-SKC'T. 

SKCANT. 

CO-SKC'T. 

SKCANT. 

CO-SKC'T. 

SKCANT. 

Co-SBC'T. 

' 

1.0098 

7-^853 

1.0125 

6.3924 

1.0154 

5.7588 

1.0187 

5.2408 

60 

.0099 
.0099 

.1704 
.1557 

.0125 
.0125 

.3807 
•369 

•0155 
•0155 

•7493 
•7398 

.0188 
.0188 

•233 
.2252 

ti 

.0099 
.01 

.1409 
•  1263 

.0126 
.0126 

•3574 
•3458 

.0156 
.0156 

•7304 
.721 

.0189 
.0189 

.2174 
.2097 

57 
56 

1.  01 

7.1117 

1.0127 

6-3343 

1.0157 

5>7"7 

1.019 

5.2019 

55 

.0101 

.0972 

.0127 

.3228 

•0157 

•7023 

.0191 

.1942 

54 

.0101 

.0827 

.0128 

.3113 

.0158 

•693 

.0191 

.1865 

53 

.0102 

.0683 

.0128 

.2999 

.0158 

.6838 

.0192 

.1788 

52 

.0102 

•0539 

.0129 

.2885 

.0159 

•6745 

.0192 

.1712 

51 

I.OIO2 

7.0396 

1.0129 

6.  2772 

1-0159 

5-6653 

1.0193 

5.1636 

50 

.OIO3 
.0103 

.0254 

.0112 

•  013 
.013 

.2659 
•2546 

.016 
.016 

.6561 
.647 

.0193 
.0194 

.156 
.1484 

49 
43 

.OIO4 

6.9971 

.0131 

•2434 

.0161 

•6379 

.0195 

.1409 

47 

.0104 

•983 

.0131 

.2322 

.0162 

.6288 

.0195 

•'333 

46 

I.OI04 

6.969 

1.0132 

6.22II 

1.0162 

5.6197 

1.0196 

5-1258 

45 

•0105 

•955 

.0132 

.21 

.0163 

.6107 

.0196 

.1183 

44 

.0105 

•94" 

•0133 

.199 

.0163 

.6017 

.0197 

.1109 

43 

.OIO6 

•9273 

•0133 

.188 

.0164 

.5928 

.0198 

.1034 

42 

.OI06 

•9135 

.0134 

.177 

.0164 

•5838 

.0198 

.096 

I.OIO7 

6.8998 

1.0134 

6.1661 

1.0165 

5-5749 

1.0199 

5.0886 

40 

.OIO7 

.8861 

•0135 

•1552 

.0165 

.566 

.0199 

.0812 

39 

.0107 

.8725 

•  0135 

•1443 

.0166 

•5572 

.02 

-0739 

38 

.OI08 

•8589 

.0136 

•1335 

.0166 

•5484 

.0201 

.0666 

37 

.OI08 

•8454 

.0136 

.1227 

.0167 

•5396 

.O2OI 

•0593 

36 

I.OI09 

6.832 

1.0136 

6.  1  12 

1.0167 

5-53o8 

1.0202 

5-052 

35 

.0109 

.8185 

•0137 

.1013 

.0168 

.5221 

.O2O2 

.0447 

34 

.Oil 

.8052 

•0137 

.0906 

.0169 

•5134 

.0203 

•0375 

33 

.Oil 

.7919 

.0138 

.08 

.0169 

•5047 

.0204 

.0302 

32 

.0111 

.7787 

.0138 

.0694 

.017 

.496 

.O2O4 

.023 

I.OIII 

6-7655 

1.0139 

6.0588 

1.017 

5-4874 

1.0205 

5-0158 

30 

.0111 

•7523 

.0139 

•0483 

.0171 

.4788 

.0205 

.0087 

29 

.0112 

•7392 

.014 

•0379 

.0171 

.4702 

.0206 

.0015 

28 

.OII3 

•7*32 

.0141 

.017 

.0172 

•4532 

.0207 

•9873 

26 

I.OII3 

6.7003 

1.0141 

6.0066 

1.0173 

5-4447 

I.  O2O8 

4.9802 

25 

.OII4 

.6874 

.0142 

5*9963 

.0174 

.4362 

.0208 

•9732 

24 

.0114 

•6745 

.0142 

.986 

.0174 

.4278 

.0209 

.9661 

23 

.OII5 

.6617 

.0143 

•9758 

•0175 

.4194 

.021 

•9591 

22 

.0115 

•649 

•0143 

•9655 

•0175 

.411 

.021 

.9521 

21 

I.OII5 

6.6363 

1.0144 

5-9554 

1.0176 

5.4026 

Z.02II 

4.9452 

20 

.OIl6 

•6237 

.0144 

•9452 

.0176 

•3943 

.O2II 

•  9382 

'9 

.OIl6 

.6m 

.0145 

•9351 

.0177 

.386 

.0212 

•9313 

18 

.0117 

•5985 

.0145 

•925 

.0177 

•3777 

.O2I3 

•9243 

17 

.0117 

•  586 

.0146 

•9*5 

.0178 

•3695 

.0213 

•9*75 

16 

1.0118 

6.5736 

1.0146 

5.9049 

1.0179 

5-3612 

I.O2I4 

4.9106 

15 

.0118 

.5612 

.0147 

•895 

.0179 

•353 

.0215 

•9°37 

14 

.0119 

•  5488 

.0147 

.885 

.018 

•3449 

.O2I5 

.8969 

13 

.0119 

•5365 

.0148 

•8751 

.018 

•3367 

.02l6 

.8901 

12 

.0119 

•5243 

.0148 

.8652 

.0181 

.3286 

.O2l6 

•8833 

II 

1.  012 

6.5121 

1.0149 

5-8554 

1.0181 

5-3205 

I.02I7 

4-8765 

IO 

.012 

•4999 

.015 

.8456 

.0182 

.3124 

.02l8 

.8697 

9 

.0121 

.4878 

.015 

•8358 

.0182 

.02l8 

.863 

8 

.0121 

•4757 

.0151 

.8261 

.0183 

•  2963 

.0219 

•8563 

7 

.OI22 

•4637 

.0151 

.8163 

.0184 

.2883 

.022 

.8496 

6 

1.  0122 

6.4517 

1.0152 

5-8067 

1.0184 

5-2803 

1.022 

4.8429 

5 

.0123 

.4398 

.0152 

•797 

.0185 

.2724 

.0221 

.8362 

4 

•0123 

4279 

•0153 

.7874 

.0185 

.2645 

.0221 

.8296 

3 

.0124 

.416 

•0153 

•7778 

.0186 

.2566 

.0222 

.8229 

2 

.0124 

.4042 

.0154 

.7683 

.0186 

.2487 

.0223 

,8163 

I 

I.OI25 

6.3924 

1.0154 

5-7588 

1.0187 

5.2408 

1.0223 

4.8097 

O 

CCH»EC'T. 

SECANT. 

CO-SEC'T. 

SECANT,  j 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

'  ' 

8] 

0 

80 

0 

7S 

>o 

7« 

o 

406 


NATURAL    SECANTS    AND    CO-SECANTS. 


u 

SECANT. 

H 

CO-BBC  'T. 

1 

SECANT. 

3° 
CO-SKC'T. 

1' 

SECANT. 

t° 
CO-SEC'T. 

150 

SECANT.  |  CO-SEC'T. 

1.0223 

4.8097 

1.0263 

4-4454 

1.0306 

4.I336 

1-0353 

3-8637 

.0224 

.8032 

.0264 

.4398 

.0307 

.1287 

•0353 

•8595 

.0225 

.7966 

.0264 

•4342 

.0308 

.1239 

•0354 

.8553 

.0225 

.7901 

.0265 

.4287 

.0308 

.1191 

•0355 

.8512 

,0226 

•7835 

.0266 

.4231 

.0309 

.1144 

•OSS6 

.847 

1.0226 

4-777 

1.0266 

4.4176 

1.031 

4.  1096 

1.0357 

3.8428 

.0227 

.7706 

.0267 

.4121 

.0311 

.1048 

•0358 

•8387 

.0228 

.7641 

.0268 

.4065 

.0311 

.1001 

-0358    .8346 

.0228 

•7576 

.0268 

.4011 

.0312 

•0953 

•0359 

.8304 

.0229 

-7512 

.0269 

•3956 

•0313 

.0906 

.036 

.8263 

1.023 

4.7448 

1.027 

4.3901 

1.0314 

4.0859 

1.0361 

3.8222 

.023 

'7384 

.0271 

•3847 

.0314 

.0812 

.0362    .8181 

.0231 

•732 

.0271 

•3792 

•0315 

.0765 

.0362    .814 

.0232 

•7257 

.0272 

.3738 

.0316 

.0718 

.0363  \   .81 

.0232 

•7*93 

.0273 

.3684 

•0317 

.0672 

.0364    .8059 

1.0233 

4-7!3 

1.0273 

4-363 

1.0317 

4.0625 

1.0365   3.8018 

•0234 

.7067 

.0274 

•3576 

.0318 

•0579 

.0366  |   .7978 

•0234 

.7004 

.0275 

•3522 

.0319 

•0532 

•0367  j   .7937 

•0235 

.6942 

.0276 

•3469 

.032 

.0486 

.0367  j  .7897 

•0235 

.6879 

.0276 

.3415 

.032 

.044 

.0368 

•7857 

1.0236 

4.6817 

1.0277 

4-3362 

1.0321 

4.0394 

1.0369 

3-7816 

.0237 

•6754 

.0278 

•3309 

.0322 

.0348 

•037 

.7776 

.0237 

.6692 

.0278 

•3256 

^0323 

.0302 

•0371 

•7736 

.0238 

.6631 

.0279 

•  3203 

.0323 

.0256 

•Q371 

.7697 

.0239 

.6569 

.028 

•315 

.0324 

.021  1 

.0372 

•7657 

1.0239 

4.6507 

1.028 

4.3098 

1.0325 

4.0165 

1.0373 

3-76i7 

.024 

.6446 

.0281 

•3045 

.0326 

.012 

•0374 

•7577 

.0241 

.6385 

.0282 

•2993 

.0327 

.0074 

•0375 

•7538 

.0241 

.6324 

.0283 

.2941 

.0327 

.0029 

.0376 

.7498 

.0242 

.6263 

.0283 

.2888 

.0328 

3.9984 

.0376 

•7459 

1.0243 

4.  6202 

1.0284 

4-2836 

1-0329 

3-9939 

1-0377- 

3-742 

.0243 

6142 

0285 

.2785 

•033 

.9894 

.0378 

.738 

.0244 

6081 

.0285 

•2733 

•033 

•985 

•0379 

-734I 

.0245 

.6021 

.0286 

.2681 

•0331 

.9805 

.038 

.7302 

.0245 

.5961 

.0287 

.263 

•0332 

.976 

.0381 

.7263 

1.0246 

4-  5901 

1.0288 

4-2579 

1-0333 

3-97i6 

1.0382 

3.7224 

.0247 

.5841 

.0288 

.2527 

•0334 

.9672 

.0382 

.7186 

.0247 

.5782 

.0289 

.2476 

•0334 

.9627 

•0383 

•7H7 

.0348 

.5722 

.029 

.2425 

•0335 

•9583 

.0384 

.7108 

.0249 

•5663 

.0291 

•2375 

•0336 

•9539 

.0385 

.707 

1.0249 

4-  5604 

1.0291 

4.2324 

1-0337 

3-9495 

1.0386 

3-703i 

.025 

•5545 

.0292 

.2273 

•0338 

•9451 

.0387 

•6993 

,0251 

.5486 

.0293 

.2223 

•0338 

.9408 

.0387 

•6955 

.0251 

•5428 

.0293 

•2173 

•0339 

•9364 

.0388 

.6917 

.0252 

•5369 

.0294 

.2122 

•034 

•932 

.0389 

.6878 

1-0253 

4-53" 

1.0295 

4.2072 

1.0341 

3-9277 

1.039 

3-684 

•0253 

•5253 

.0296 

.2022 

.0341 

•9234 

.0391 

.6802 

.0254 

•5*95 

.0296 

.1972 

.0342 

.919 

.0392 

.6765 

•0255 

•5137 

.0297 

.1923 

•0343 

.9147 

•0393 

.6727 

•0255 

•5079 

.0298 

•l873 

•0344 

.9104 

•0393 

.6689 

1.0256 

4.5021 

1.0299 

4.1824 

1-0345 

3.9061 

1.0394 

3-6651 

.0257 

4964 

.0299 

•1774 

•0345 

.9018 

•0395 

.6614 

.0257 

.4907 

•03 

•1725 

.0346 

.8976 

.0396 

.6576 

.0258 

485 

.0301 

.1676 

•0347 

•8933 

•0397 

•6539 

.0259 

•4793 

0302 

.  1627 

.0348 

.899 

•0398 

.6502 

1.026 

4-4736 

1.0302 

4-I578 

1.0349 

3.8848 

1.0399 

3-6464 

.026 

.4679 

•0303 

.1529 

•0349 

.8805 

•0399 

.6427 

.0261 

.4623 

.0304 

.1481 

•035 

.8763 

.04 

•639 

.0262 

.4566 

•0305 

.1432 

•0351 

.8721 

.0401 

.6353 

.0262 

•451 

•0305 

.1384 

•0352 

.8679 

.0402 

.6316 

1.0263 

4-4454 

1.0306 

4-I336 

1-0353 

3-8637 

1.0403 

3.6279 

CO-SEC  'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

770 

76° 

75° 

74° 

NATURAL   SECANTS   AND   CO-SECANTS. 


407 


|     160 

17° 

180 

19° 

'  |.  SECANT. 

CO-SEC'T. 

SKCANT.  CO-SEC'T. 

SECANT,  j  COHBEC'T. 

SECANT. 

CO-SBC'T. 

o 

1.0403 

3.6279 

1-0457 

3-4203 

1-0515 

3-2361 

1.0576 

3-0715 

(>l 

.0404 

.0458 

.417 

.0516 

•  2332 

•0577 

.069 

2 

.0405 

.6206 

•0459 

.4138 

•0517 

.2303 

.0578 

.0664 

3 

.0406 

.6169 

.046 

.4106 

.0518 

.2274 

•°579 

.0638 

4 

.0406 

•6i33 

.0461 

•4073 

.0519 

.2245 

.058 

.0612 

5 

1.0407 

3.6096 

1.0461 

3.4041 

1.052 

3.2216 

1.0581 

3-0586 

6 

.0408 

.606 

.0462 

.4009 

.0521 

.2188 

.0582 

.0561 

7 

.0409 

.6024 

.0463 

•3977 

.0522 

.2159 

.0584 

•0535 

8 

.041 

.5987 

.0464 

•3945 

.0523 

.2131 

•0585 

.0509 

9 

.0411 

•5951 

.0465 

•39*3 

.0524 

.2102 

.0586 

.0484 

10 

1.0412 

3-59*5 

1.0466 

3-3881 

1.0525 

3.2074 

1.0587 

3-0458 

ii 

.0413 

•5879 

.0467 

•3849 

.0526 

.2045 

.0588 

•0433 

12 

.0413 

•5843 

.0468 

•3817 

•0527 

.2017 

•0589 

.0407 

13 

.0414 

•5807 

.0469 

•3785 

.0528 

.1989 

•059 

.0382 

14 

.0415 

•5772 

.047 

•3754 

.0529 

.196 

.0591 

•0357 

15 

1.0416 

3.5736 

1.0471 

3-3722 

1-053 

3.1932 

1.0592 

3-033I 

16 

.0417 

•57 

.0472 

•369 

•0531 

.1904 

•0593 

.0306 

;i 

.0418 
.0419 

-5665 
.5629 

•0473 
•0474 

•3659 
•3627 

•0532 
•°533 

.1876 
.1848 

•0594 
•0595 

.0281 
.6256 

IQ 

.042 

•5594 

•°475 

•3596 

•°534 

.182 

.0596 

.0231 

20 

1.042 

3-5559 

1.0476 

1-0535 

3-I792 

1.0598 

3.0206 

21 

.0421 

•5523 

.0477 

•3534 

•0536 

.1764 

•0599 

.0181 

22 

.0422 

.5488 

.0478 

•3502 

•0537 

•1736 

.06 

.0156 

23 

.0423 

•5453 

•0478 

•347* 

•0538 

.1708 

.0601 

.0131 

24 

.0424 

.5418 

•0479 

•344 

•0539 

.1681 

.0602 

.0106 

25 

1.0425 

3.5383 

1.048 

3-3409 

1.054 

3-1653 

1.0603 

3.0081 

26 

.0426 

•5348 

.0481 

•3378 

.0541 

1625 

.0604 

.0056 

27 

.0427 

•5313 

.0482 

•3347 

.0542 

.1598 

.0605 

0031 

28 

.0428 

•5279 

•0483 

•0543 

•*57 

.0606    .0007 

29 

.0428 

•5244 

.0484 

!3286 

•0544 

•1543 

.0607   2.9982 

30 

1.0429 

3.5209 

1.0485 

3-3255 

1-0545 

1.0608  i  29957 

31 

•043 

•5175 

.0486 

•3224 

.0546 

.1488 

•  0609   -9933 

32 

.0431 

.0487 

•3*94 

•°547 

.1461 

.0611    -9908 

33 
34 

.0432 
•0433 

.5106 
•5072 

.0488 
.0489 

•3163 

•  0548 
•0549 

.1406 

.0612 

.9884 
•9859 

35 

1.0434 

3-5037 

1.049 

3-3102 

1-055 

3-1379 

1.0614 

2.9835 

36 

•0435 

.5003 

.0491 

.3072 

•0551 

•1352 

.0615 

.981 

37 

.0436 

.4969 

.0492 

•  3042 

•0552 

•1325 

.0616 

.9786 

38 

•0437 

•4935 

•0493 

.3011 

•0553 

.1298 

.0617 

.9762 

39 

.0438 

.4901 

.0494 

.2981 

•0554 

.1271 

.0618 

•9738 

40 

1.0438 

3-4867 

1.0495 

3-2951 

1-0555 

3.1244 

1.0619 

2.9713 

4i 

•0439 

•4833 

.0496 

.2921 

•05fi6 

.1217 

.062 

.9689 

42 

•044 

•4799 

.0497 

.2891 

•0557 

.119 

.0622 

.9665 

43 

.0441 

.4766 

.0498 

.2861 

•0558 

.1163 

.0623 

.9641 

44 

.0442 

•4732 

•0499 

.2831 

•0559 

H37 

.0624 

.9617 

45 

1.0443 

3.  4698 

1.05 

3.2801 

1.056 

1.0625 

2-9593 

46 

•0444 

.4665 

.0501 

.2772 

.0561 

.1083 

.0626 

•9569 

47 

-0445 

.4632 

.0502 

.2742 

.0562 

•1057 

.0627 

•9545 

48 

.0446 

•4598 

.0503 

.2712 

.0563 

.103 

.0628 

.9521 

49 

•0447 

-4565 

•0504 

.2683 

•0565 

.1004 

.0629 

•9497 

50 

1.0448 

3-4532 

1.0505 

3-2653 

1.0566 

3-0977 

1.063 

2-9474 

51 

.0448 

.4498 

.0506 

.2624 

.0567 

.0951 

.0632 

•945^ 

52 

.0449 

•4465 

.0507 

•2594 

.0568 

.0925 

.0633 

.9426 

53 

•045 

•4432 

.0508 

•2565 

.0569 

.0898 

.0634 

.9402 

54 

.0451 

.0509 

•2535 

•057 

.0872 

•0635 

•9379 

55 

1.0452 

3-4366 

1.051 

3.2506 

1.0571 

3.0846 

1.0636 

2-9355 

56 

•0453 

•4334 

.0511 

.2477 

•0572 

.082 

.0637 

•9332 

57 

•0454 

.4301 

.0512 

.2448 

•0573 

•0793 

.0638 

.9308 

58 

•0455 

.4268 

•0513 

.2419 

•0574 

.0767 

•0639 

.9285 

59 

.0456 

4236 

.0514 

•239 

•0575 

.0741 

.0641 

.9261 

60 

1-°457 

3.4203 

1-0515 

3.2361 

1.0576 

3-0715 

1.0642 

2.9238 

' 

CO-SEC'T. 

SECANT. 

CO-SBC'T.  SECANT. 

CO-SBC'T. 

SECANT.  |  CO-SBC'T. 

SECANT. 

73° 

720 

710     |i     70o 

408 


NATURAL   SECANTS   AND   CO-SECANTS. 


20° 

21° 

22° 

23° 

SECANT.  CO-SEC'T 

SECANT. 

CO-SEC'T 

SECANT. 

CO-SEC'T 

SECANT.  CO-SEC'T. 

1.0642 

2.9238 

1.0711 

2.7904 

1.0785 

2.6695 

1.0864 

2-5593 

.0643 
.0644 

.9215 
.9191 

•0713 
.0714 

.7883 
.7862 

.0787 
.0788 

.6675 
.6656 

.0865 
.0866 

•5575 
•5558 

.0645 

.9168 

•0715 

•7841 

.0789 

.6637 

.0868 

•554 

.0646 

•9J45 

.0716 

.782 

.079 

.6618 

.0869 

•5523 

1.0647 

2.9122 

1.0717 

2-7799 

1.0792 

2.6599 

1.087 

2.  5506 

.0648 

.9098 

.0719 

.7778 

•0793 

.658 

.0872 

-5488 

.065 

•9°75 

.072 

•7757 

.0794 

.6561 

.0873 

•547i 

.0651 

.9052 

.0721 

•7736 

•0795 

.6542 

.0874 

5453 

.0652 

.9029 

.0722 

•7715 

.0797 

•6523 

.0876 

•5436 

1-0653 

2.9006 

1.0723 

2.7694 

1.0798 

2.6504 

1.0877 

2.5419 

.0654 

.8983 

.0725 

.7674 

.0799 

6485 

.0878 

.5402 

•0655 

.896 

.0726 

•7653 

.0801 

.6466 

.088 

•5384 

.0656 

•8937 

.0727 

.7632 

.0802 

.6447 

.0881 

•5367 

.0658 

.8915 

.0728 

.7611 

.0803 

.6428 

.0882 

•535 

1.0659 

2.8892 

1.0729 

2-7591 

1.0804 

2.641 

1.0884 

2-5333 

.066 

.8869 

•0731 

•757 

.0806 

6391 

.0885 

•53i6 

.0661 

.8846 

.0732 

•755 

.0807 

6372 

.0886 

•5299 

.0662 

.8824 

•0733 

•7529 

.0808 

•6353 

.0888 

.5281 

.0663 

.8801 

.0734 

•7509 

.081 

-6335 

.0889 

.5264 

1.0664 

2.8778 

1.0736 

2.7488 

1.0811 

2.6316 

1.0891 

2-5247 

.0666 

.8756 

•0737 

.7468 

.0812 

.6297 

.0892 

•523 

.0667 

•8733 

.0738 

•7447 

.0813 

.6279 

.0893 

•5213 

.0668 

.87n 

•0739 

.7427 

.0815 

.626 

.0895 

.5196 

.0669 

.8688 

.074 

.7406 

.0816 

.6242 

.0896 

•5179 

1.067 

2.8666 

1.0742 

2.7386 

1.0817 

2.6223 

1.0897 

2.5163 

.0671 

.8644 

•0743 

.7366 

.0819 

.6205 

.0899 

.5146 

.0673 

.8621 

.0744 

.7346 

.082 

.6186 

.09 

.5129 

.0674 

•8599 

•0745 

•7325 

.0821 

.6168 

.0902 

.5112 

•°^7i 
1.0676 

.8577 
2-8554 

.0747 
1.0748 

•7305 
2-7285 

.0823 
1.0824 

.615 
2.6131 

.0903 
1.0904 

•5095 
2.5078 

.0677 

•8532 

.0749 

.7265 

.0825 

.61x3 

.0906 

.5062 

.0678 

.851 

•075 

•7245 

.0826 

.6095 

.0907 

5045 

.0679 

.8488 

0751 

.7225 

.0828 

.6076 

.0908 

.5028 

.0681 

.8466 

•°753 

•7205 

.0829 

.6058 

.091 

.5011 

1.0682 

2.8444 

i-°754 

2-7185 

1.083 

2.604 

1.0911 

2.4995 

.0683 

.8422 

•0755 

•7165 

.0832 

.6022 

.0913 

.4978 

.0684 

•84 

.0756 

•7145 

•0833 

.6003 

.0914 

.4961  \ 

.0685 

.8378 

.0758 

•7125 

•  0834 

•5985 

.0915 

•4945  ! 

.0686 

•8356 

•0759 

•7I05 

.0836 

•  5967 

.0917 

.4928 

1.0688 

2-8334 

1.076 

2.7085 

1.0837 

2-5949 

1.0918 

2.4912 

.0689 

.8312 

.0761 

.7065 

.0838 

•5931 

.092 

•4895 

.069 

.829 

.0763 

•7045 

.084 

•59*3 

.0921 

.4879 

.0691 

.8269 

.0764 

.7026 

.0841 

•5895 

.0922 

.4862 

.0692 

.8247 

.0765 

.7006 

.0842 

•5877 

.0924 

.4846 

1.0694 

2.8225 

1.0766 

2.6986 

1.0844 

2-5859 

1.0925 

2.4829 

.0695 

.8204 

.0768 

.6967 

.0845 

.5841 

.0927 

.4813 

.0696 

.8182 

.0769 

.6947 

.0846 

.5823 

.0928 

•4797 

.0697 

.816 

.077 

.6927 

.0847 

•5805 

.0929 

•478 

.0698 

•8139 

.0771 

.6908 

.0849 

•5787 

.0931 

.4764 

1.0699 

2.8117 

I-Q773 

2.6888 

1.085 

2-577 

1.0932 

2.4748 

.0701 

.8096 

.0774 

.6869 

.0851 

•5752 

•0934 

•4731 

.0702 

.8074 

•0775 

.6849 

•0853 

•5734 

•0935 

•47i5 

.0703 

•8053 

.0776 

.683 

.0854 

•57i6 

.0936 

.4699 

.0704 

-8032 

.0778 

.681 

•0855 

•5699 

.0938 

-4683 

1.0705 

2.801 

1.0779 

2.6791 

1.0857 

2.5681 

1.0939 

2.4666 

.0707 

.7989 

.078 

.6772 

.0858 

•5663 

.0941 

•465 

.0708 

.7968 

.0781 

.6752 

.0859 

.5646 

.0942 

•4634 

.0709 

•7947 

.0783 

•6733 

.0861 

.5628 

•0943 

.4618 

.071 

•7925 

.0784 

.6714 

.0862 

.561 

•0945 

.4602 

1.0711 

2.7904 

1.0785 

2.6695 

1.0864 

2-  5593 

.0946 

2.4586 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT,  j 

69° 

68° 

6?o 

660     j 

NATUBAL  SECANTS  AND    CO-SECANTS. 


409 


•:    24° 

25° 

260 

27° 

' 

SECANT. 

CO-SKC'T 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SKC'T. 

SECANT.  |  CC-BKC'T. 

• 

o 

1.0946 

2.4586 

1-1034 

2.3662 

1.1126 

2.2812 

1.1223 

2.  2027 

i  60 

I 

.0948 

•457 

•1035 

•3647 

.1127 

.2798 

.1225 

.2014   59 

2 

.0949 

•4554 

.1037 

•  3632 

.1129 

.2784 

.1226 

.  2OO2  i  58 

3 

.0951 

.4538 

.1038 

.3618 

.1131 

.2771 

.1228 

.1989   57 

4 

.0952 

•4522 

.104 

•  3603 

•  "32 

•2757 

.123 

.1977   56 

5 

1-0953 

2.4506 

1.1041 

2.3588 

1.1134 

2.2744 

1.1231 

2.1964   55 

6 

•0955 

•449 

.1043 

•3574 

•"35 

-273 

•1233 

.1952   54 

7 

.0956 

•4474 

.1044 

•3559 

•"37 

.2717 

-1235 

•1939   53 

8 

•0958 

•4458 

.1046 

•3544 

•"39 

.2703 

•1237 

.1927   52 

9 

.0959 

•4442 

.1047 

•353 

.114 

.269 

.1238 

.1914   5I 

10 

1.0961 

2.4426 

1.  1049 

2.3515 

1.1142 

2.2676 

1.124 

2.  1902   50 

ii 

.0962 

•44" 

.105 

•3501 

•"43 

.2663 

.1242 

.1889   49 

12 

.0963 

•4395 

.1052 

.3486 

•"45 

.265 

.1243 

.1877   48 

'3 

.0965 

•4379 

•1053 

•3472 

."47 

.2636 

.1245 

.1865   47 

14 

.0066 

•4363 

•1055 

•3457 

.1148 

.2623 

.1247 

-1852   46 

15  !  1.0968 

a-  4347 

1.  1056 

2-3443 

1.115 

2.261 

1.1248  |  2.184  i  45 

16 

.0969 

•4332 

.1058 

•  3428 

•  1151 

.2596 

.125 

.1828   44 

17 

.0971 

.4316 

.1059 

•34H 

•"53 

.2583 

.1252 

.1815  \  43 

18 

.0972 
•0973 

.4285 

.1061 
.1062 

•3399 
•3385 

."55 
."56 

•  257 
•  2556 

•1253 
•1255 

.1803   42 
.1791   41 

20  |  1.0975 

2.4269 

1.1064 

2-3371 

1.1158 

2-2543 

1.1257 

2.1778   40 

21 

.0976 

•4254 

.1065 

•3356 

•"59 

•253 

.1258 

.1766  j  39 

22 

.0978 

.4238 

.1067 

•3342 

.1161 

.2517 

.126 

•1754   38 

23 

.0979 

.4222 

.1068 

•3328 

.1163 

-2503 

.1262 

•1742   37 

24 

.0981 

.4207 

.107 

.1164 

.249 

.1264 

•173   36 

25 

1.0982 

2.4191 

1.1072 

2.3299 

1.1166 

2.2477 

1.1265 

2.1717 

35 

26 

.0984 

.4176 

•1073 

•3285 

.1167 

.2464 

.1267 

.1705 

34 

27 

.0985 

.416 

•1075 

•3271 

.1169 

.2451 

.1269 

.1693 

33 

28 

.0986 

•4145 

.1076 

•3256 

.1171 

•2438 

.127 

.1681 

32 

29 

.0988 

•413 

.1078 

•3242 

.1172 

.2425 

.1272 

.1669 

30 

1.0989 

2.4114 

1.1079 

2.3228 

1.1174 

2.2411 

1.1274 

2.1657 

30 

31 

.0991 

.4099 

.1081 

.3214 

.1176 

.2398 

•1275 

.1645 

29 

32 

.0992 

•  4083 

.1082 

.32 

•"77 

-2385 

.1277 

•1633 

28 

33 

.0994 

.4068 

.1084 

•  3186 

•"79 

-2372 

.1279 

.162 

27 

34 

•0995 

-4053 

.1085 

•3172 

.118 

•2359 

.1281 

.1608 

26 

35 

I  0997 

2-4037 

1.1087 

2.3158 

1.1182 

2.2346 

1.1282 

2.1596 

25 

36 

.0998 

.4022 

.1088 

•3143 

.1184 

•2333 

.1284 

.1584 

24 

37 

*x 

.4007 

.109 

.3129 

•  1185 

•  232 

.1286 

•1572 

23 

38 

.1001 

•3992 

.1092 

.1187 

.2307 

.1287 

•  156 

22 

39 

.1003 

•3976 

.1093 

.3101 

.1189 

.2294 

.1289 

.1548 

21 

40 

1*1004 

2.3961 

1.1095 

2.3087 

1.119 

2.2282 

1.1291 

2.1536 

20 

4* 

.1005 

•3946 

.1096 

•3073 

.1192 

.2269 

.1293 

•1525 

19 

42 

.1007 

•3931 

.1098 

-3°59 

•"93 

.2256 

,1294 

•1513 

18 

43 

.1008 

.3916 

.1099 

•3046 

•"95 

•2243 

.1296 

.1501 

'7 

44 

.101 

.3901 

.1101 

•3032 

•"97 

•  223 

.1298 

.1489 

16 

45 

I.IOII 

2.3886 

1.  1  102 

2.3018 

1.1198 

2.2217 

1.1299 

2.1477 

15 

46 

.1013 

•3871 

.IIO4 

.3004 

.12 

.2204 

.1301 

.1465 

14 

47 

.1014   .3856 

.II06 

•299 

.I2O2 

.2192 

•1303 

•1453 

13 

48 

.I0l6       .3841   : 

.1107 

.2976 

.1203 

•2179 

•1305 

.1441 

12 

49 

.1017   .3826 

.1109 

.2962 

•  I2O5 

.2166 

.1306 

.143 

II 

50 

1.1019 

2.3811 

I.  Ill 

2.2949 

I.I207 

2.2153 

1.1308 

2.1418 

10 

51 

52 

.102 
.1022 

•3796 
•378i  ! 

.1112 
.1113 

.2935 
.2921 

.1208 
.121 

.2141 
.2128 

ll3I2 

•  1406 
•1394 

I 

53 

.IO23 

.3766 

.1115 

.2907 

.1212 

•  2115 

•I3I3 

.1382 

7 

54 

•I025  I   .3751 

.1116 

.2894 

.1213 

.2103 

•I3I5 

•I37I 

6 

55 

I.I026  |  2.3736 

1.1118 

2.288 

I.I2I5 

2.209 

1-1317 

2-1359 

5 

56 

.1028  j 

•3721 

•  112 

.2866 

.1217 

.2077 

•1319 

•1347 

4 

57 

.1029 

.1121 

.2853 

.I2l8 

.2065 

.132 

•1335 

3 

58 

.1031 

.3691 

•"23 

•  2839 

.122 

.2052 

.1322 

.1324 

2 

59 

.1032 

•3677 

•"24 

.2825 

.1222 

•2039 

•1324 

•  1312 

I 

00 

I.I034 

2.3662 

I.II26 

2.2812 

I.I223 

2.2027 

1.1326 

2.13 

0 

T~ 

CO-SBC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

9 

65° 

64° 

63° 

620 

MM 

4io 


NATURAL    SECANTS   AND    CO-SECANTS. 


m 

1° 

2< 

)° 

3( 

)° 

3] 

L° 

SECANT. 

CO-BEC'T. 

SECANT. 

CO-SBC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

1.1326 

2-  3 

I-I433 

2.0627 

I-I547 

2 

1.  1666 

1.9416 

•1327 

.  289 

•1435 

.0616 

•1549 

1.999 

.1668 

9407 

.1329 

•  277 

•1437 

.0605 

.998 

.167 

•9397 

.  266 

•1439 

•0594 

.1553 

•997 

.1672 

.9388 

»i333 

•  254 

.1441 

•0583 

•'555 

•996 

.1674 

•9378 

I-I334 

2.  242 

I-I443 

2-0573 

I-I557 

1-995 

i  1676 

i  9369 

.1336 

.  231 

•M45 

.0562 

•1559 

•994 

1678 

•936 

•1338 

.  219 

.1446 

•0551 

.1561 

•993 

.1681 

•935 

•134 

.  208 

.1448 

•054 

.1562 

•992 

.1683 

•9341 

•I34I 

.  I96 

•145 

•053 

.1564 

.991 

•1685 

•9332 

I-I343 

2.  185 

1.1452 

2.0519 

1.1566 

1.99 

1.1687 

1.9322 

•  1345 

•  173 

•1454 

.0508 

.1568 

•989 

.1689 

.9313 

•1347 

.  162 

.1456 

.0498 

.157 

.988 

1691 

•93°4 

•'349 

•  15 

.1458 

.0487 

.1572 

•987 

.1693 

•9295 

•135 

•  139 

.1459 

.0476 

•1574 

.986 

.1695 

•9285 

I-I352 

2.  127 

1.1461 

2.0466 

1.1576 

1.985 

1.1697 

1.9276 

•1354 

.  116 

.1463 

•0455 

•1578 

-984 

.1699 

.9267 

.1356 

.  104 

.1465 

.0444 

.158 

.983 

.1701 

.9258 

•1357 

.  093 

.1467 

•0434 

.1582 

.982 

•1703 

.9248 

•1359 

.  082 

.1469 

.0423 

.1584 

.9811 

•1705 

•9239 

1.1361 

2.  07 

1.1471 

2.0413 

1.1586 

1.9801 

1.1707 

1.923 

•1363 

•  059 

•1473 

.0402 

•  1588 

.9791 

.1709 

.9221 

•1365 

.  048 

.1474 

•0392 

•'59 

.9781 

.1712 

.9212 

.1366 

•  036 

.1476 

.0381 

.1592 

.9771 

.1714 

.9203 

•  1368 

025 

.1478 

•037 

•1594 

.9761 

.1716 

•9J93 

I-I37 

2.  014 

1.148 

2.036 

1.1596 

1-9752 

1.1718 

1.9184 

•1372 

.  OO2 

.1482 

•0349 

.1598 

.9742 

.172 

•9*75 

•'373 

.0991 

.1484 

•0339 

.16 

•9732 

.1722 

.9166 

•'375 

.098 

.1486 

.0329 

.1602 

.9722 

.1724 

•9I57 

•'377 

.0969 

.1488 

.0318 

.1604 

.1726 

.9148 

2.0957 

1.1489 

2.0308 

1.  1606 

I-9703 

1.1728 

.1381 

.0946 

.1491 

•0297 

.1608 

•9693 

•173 

•9'3 

.1382 

-0935 

•M93 

.0287 

.161 

.9683 

•1732 

.9121 

.1384 

.0924 

.1495 

.0276 

.1612 

.9674 

•1734 

.9112 

.1386 

.0912 

.1497 

.0266 

.1614 

.9664 

•1737 

.9102 

1.1388 

2.0901 

1.1499 

2.0256 

1.  1616 

1.9654 

I-I739 

1.9093 

•139 

.089 

.1501 

.0245 

.1618 

•9645 

.1741 

.9084 

.0879 

•1503 

•0235 

.162 

•9635 

•1743 

•9°75 

•1393 

.0868 

•1505 

.0224 

.1622 

.9625 

•1745 

.9066 

•1395 

•0857 

•1507 

.0214 

.1624 

.9616 

•1747 

•9°57 

I-I397 

2.0846 

1.1508 

2.0204 

1.1626 

1.9606 

1.1749 

1.9048 

•'399 

•0835 

•IS' 

•  0194 

.1628 

.9596 

.1751 

•9°39 

.1401 

.0824 

.1512 

.0183 

.163 

•9587 

-1753 

•9°3 

.1402 

.O8l2 

•1514 

.0173 

.1632 

•9577 

•1756 

.9021 

.1404 

.O8oi 

.1516 

.0163 

.1634 

.9568 

.1758 

.9013 

1.1406 

2.079 

1.1518 

2.0152 

1.1636 

I.9558 

1.176 

.1408 

.0779 

.152 

.0142 

.1638 

•9549 

.1762 

.141 

.0768 

.1522 

.0132 

.164 

•9539 

.1764 

. 

.1411 

.0757 

.1524 

.0122 

.1642 

•953 

.1766 

.8977 

.0746 

.1526 

.OIII 

.1644 

•952 

.1768 

.8968 

1.1415 

2.0735 

1.1528 

2.0IOI 

1.1646 

1.177 

1.8959 

.1417 

.0725 

•153 

.0091 

.1648 

.9501 

.1772 

.895 

.1419 

.0714 

.1531 

.0081 

.165 

.9491 

•1775 

.8941 

.1421 

.0703 

•1533 

.0071 

.1652 

.9482 

.1777 

.8932 

.1422 

.0692 

•1535 

.Oo6l 

.1654 

•9473 

.1779 

.8924 

1.1424 

2.0681 

I-I537 

2.005 

1.1656 

1-9463 

1.1781 

1.8915 

.1426 

.067 

•1539 

.004 

.1658 

•9454 

-1783 

.8906 

.!428 

•0659 

.1541 

.003 

.166 

•9444 

•1785 

.8897 

•143 

.0648 

•1543 

.002 

.1662 

•9435 

.1787 

.8888 

.1432 

.0637 

•1545 

.OOI 

.1664 

-9425 

•179 

.8879 

2.0627 

i-  1547 

2 

1.  1666 

1.9416 

1.1792 

1.8871 

CO-BEC'T. 

SECANT. 

CO-BEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

6 

L° 

6( 

)° 

5< 

61 

1? 

NATURAL    SECANTS   AND   CO-SECANTS. 


411 


32° 

33° 

34° 

35° 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SBC'T. 

SECANT. 

CO-SEC'T. 

1.1792 

1.8871 

1.1924 

1.8361 

1.2062 

1.7883 

1.2208 

1-7434 

.1794 
.1796 

.8862 
•8853 

.1926 
.1928 

•8352 
.8344 

.2064 
.2067 

•7875 
.7867 

.221 
.2213 

.7427 
.742 

.1798 

.8844 

•193 

8336 

.2069 

.786 

.2215 

•7413 

.18 

.8836 

-1933 

.8328 

.2072 

.7852 

.22l8 

•74°5 

1.  1802 

1.8827 

I-I935 

1.832 

1.2074 

1.7844 

1.222 

I-7398 

.1805 

.8818 

•1937 

.8311 

2076 

-7837 

.2223 

-7391 

.1807 
.1809 

.8809 
.8801 

.1939 
.1942 

-8303 
.8295 

.2079 
.2081 

.7829 
.7821 

.2225 
.2228 

-7384 
•7377 

.1811 

.8792 

.1944 

.8287 

.2083 

.7814 

223 

•7369 

1.1813 

1.8783 

1.1946 

1.8279 

1.2086 

1.7806 

1.2233 

1-7362 

.1815 

.8785 

.1948 

.8271 

.2088 

.7798 

-2235 

•7355 

.1818 

.8766 

.8263 

.209! 

.7791 

.2238 

•7348 

.182 

-8757 

•1953 

-8255 

.2093 

.7783 

.224 

•7341 

.1822 

.8749 

•1955 

.8246 

.2095 

.7776 

.2243 

-7334 

1.1824 
.1826 

1.874 
•8731 

1.1958 
.196 

1.8238 
•823 

•  21 

1.7768 
.776 

1-2245 
.2248 

1-7327 

.1828 

.8723 

.1962 

.8222 

.2103 

•7753 

.225 

-7312 

.1831 

.8714 

.1964 

.8214 

•2105 

-7745 

•2253 

•7305 

•1833 

.8706 

.1967 

•  8206 

.2IO7 

•7738 

•2255 

.7298 

1-1835 

1.8697 

1.1969 

1.8198 

1.  211 

1-773 

1.2258 

1.7291 

-1837 

.8688 

.1971 

.819 

.2112 

•7723 

.226 

.7284 

.1839 

.868 

.1974 

.8182 

.2115 

•7715 

.2263 

.7277 

.1841 
.1844 

.8671 
.8663 

.1976 
.1978 

-8174 
.8166 

.2II7 
.2119 

.7708 
•77 

.2265 
.2268 

.727 
.7263 

1.1846 

1.8654 

i.I98 

1.8158 

I.  2122 

1-7693 

1.227 

1.7256 

.1848 

.8646 

.1983 

.815 

.2124 

.7685 

.2273 

•7249 

.185 

.8637 

.1985 

.8142 

.2127 

.7678 

.2276 

.7242 

.1852 

.8629 

.1987 

.8134 

.2129 

.767 

.2278 

•7234 

•1855 

.862 

.199 

,8126 

.2132 

i'7f3 

.228l 

.7227 

1.1857 

1.8611 

1.1992 

1.8118 

I.2I34 

1.2283 

1.722 

.1859 

.8603 

.1994 

.811 

.2136 

.7648 

.2286 

•7213 

.1861 

.8595 

.1997 

.8102 

.2139 

-764 

.2288 

.7206 

•  1863 

.8586 

.1999 

.8094 

.2141 

•7633 

.2291 

.7199 

.1866 

.8578 

.2001 

.8086 

.2144 

.7625 

.2293 

.7192 

1.1868 

1.8569 

I.2OO4 

1.8078 

1.2146 

1.7618 

1.2296 

1.7185 

.187 

.8561 

.2006 

.807 

.2149 

.761 

.2298 

.7178 

.1872 

.8552 

.2008 

.8062 

.2151 

•7603 

.2301 

.7171 

.1874 

.8544 

.201 

.8054 

•2153 

•7596 

.2304 

.7164 

.1877 

.8535 

.2OI3 

.8047 

•2156 

.7588 

.2306 

.7157 

1.1879 

1.8527 

I.20I5 

1.8039 

I.2I58 

1.7581 

1.2309 

.1881 

.8519 

.2017 

.8031 

.2l6l 

-7573 

.2311 

.7144 

.1883 

.851 

.202 

.8023 

.2163 

.7566 

.2314 

.1886 

.8502 

.2022 

.8015 

.2166 

•7559 

.2316 

•7'3 

.1888 

•8493 

.2024 

.8007 

.2168 

•7551 

.2319 

•7123 

1.189 

1.8485 

I.2O27 

1.7999 

I.2I7I 

1-7544 

J.2322 

1.7116 

.1892 

.8477 

.2029 

.7992 

•2173 

•7537 

•2324 

.7109 

.1894 

.8468 

.2031 

.7984 

•2175 

•7529 

•2327 

.7102 

.1897 

.846 

.2034 

•7976 

.2178 

.7522 

.2329 

•7095 

.1899 

.8452 

.2036 

.7968 

.218 

•7514 

.2332 

.7088 

1.1901 

1.8443 

1.2039 

1.796 

I.2I83 

I-7507 

1-2335 

1.7081 

.1903 

'  -8435 

.2041 

•7953 

.2185 

•75 

•2337 

•7°75 

.1906 

.8427 

.2043 

•7945 

.2188 

•7493 

•234 

.7068 

.1908 

.8418 

.2046 

•7937 

.219 

7485 

•2342 

.7061 

.191 

.841 

.2048 

.7929 

.2193 

.7478 

•2345 

•7054 

1.1912 

1.0402 

1.205 

1.7921 

I.2I95 

1.7471 

1.2348 

1.7047 

.1915 

•8394 

•2053 

.7914 

.2198 

•7463 

•235 

.704 

.1917 

•8385 

•2055 

.22 

•7456 

•2353 

•7033 

.1919 

•8377 

.2057 

.7898 

.2203 

•7449 

2355 

.7027 

•  1921 

.8369 

.206 

.7891 

•2205 

•7442 

•2358 

.702 

1.1922 

1.8361 

1.2062 

1-7883 

1.  2208 

1-7434 

1.2361 

1.7013 

CO-SBC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

570 

56° 

55° 

54° 

412 


NATUKAL   SECANTS   AND    CO-SECANTS. 


3* 
SECANT. 

>° 
CO-SEC'T. 

& 
SECANT. 

ro 
CO-SEC'T. 

3E 

SECANT. 

1° 
CO-SEC'T. 

35 

SECANT. 

>o 
CO-SEC'T. 

1.2361 

1.7013 

1.2521 

i.  6616 

1.269 

1.6243 

1.2867 

1.589 

•  2363 

.7006 

•2524 

.661 

.2693 

•6237 

.2871 

.5884 

.2366 

.6999 

•2527 

.6603 

.2696 

.6231 

.2874 

•5879 

.2368 

•6993 

•253 

•6597 

.2699 

.6224 

.2877 

-5873 

•2371 

.6986 

.2532 

.6591 

.2702 

.6218 

.288 

.5867 

1-2374 

1.6979 

1-2535 

1.6584 

1.2705 

1.6212 

1.2883 

1.5862 

.2376 

.6972 

.2538 

.6578 

.2707 

.6206 

.2886 

.5856 

•2379 

.6965 

.2541 

.6572 

.271 

.62 

.2889 

-585 

.2382 
.2384 

•6959 
.6952 

•2543 
.2546 

•6565 
•6559 

•2713 
.2716 

.6194 
.6188 

.2892 
.2895 

-5845 
•5839 

1.2387 

1.6945 

1.2549 

1.6552 

1.2719 

1.6182 

1.2898 

I-5833 

.2389 

.6938 

•2552 

.6546 

.2722 

.6176 

.2901 

.5828 

•  2392 

.6932 

•2554 

•654 

.2725 

.617 

.2904 

.5822 

•2395 

.6925 

•2557 

•6533 

.2728 

.6164 

.2907  j   .5816  : 

•  2397 

.6918 

.256 

•6527 

•2731 

.6159 

.291 

.5811  ; 

1.24 

1.6912 

1.2563 

1.6521 

1-2734 

1.6153 

1.2913 

1.5805 

.2403 

.6905 

.2565 

.6514 

•2737 

.6147 

.2916     -5799 

•  2405 

.6898 

.2568 

.6508 

•2739 

.6141 

.2919    .5794 

.2408 

.6891 

•2571 

.6502 

.2742 

•6i35 

.2922  |  .5788 

.2411 

.6885 

•2574 

.6496 

•2745 

.6129 

.2926    .5783 

1.2413 

1.6878 

1-2577 

1.6489 

1.2748 

1.6123 

1.2929 

1-5777 

.2416 

.6871 

•2579 

.6483 

•2751 

.6117 

.2932 

•5771 

.2419 

.6865 

.2582 

•6477 

•2754 

.6111 

•2935 

.5766 

.2421 

.6858 

•2585 

.647 

•2757 

.6105 

.2938 

•576 

.2424 

•  6851 

.2588 

.6464 

.276 

.6099 

.2941 

•5755 

1.2427 

1.6845 

1.2591 

1.6458 

1.2763 

1.6093 

1.2944 

1-5749 

.2429 
.2432 

.6838 
.6831 

•2593 
.2596 

.6452 
•6445 

.2766 
.2769 

.6087 
.6081 

•2947 
•295 

•5743 
.5738 

•2435 

•  6825 

•2599 

•6439 

.2772 

.6077 

•2953 

•5732 

•2437 

.6818 

.2602 

•6433 

•2775 

.607 

.2956 

•5727 

1.244 

i.  6812 

1.2605 

1.6427 

1.2778 

1.6064 

1.296 

1.5721 

•2443 

.6805 

.2607 

.642 

.2781 

.6058 

.2963 

.5716 

•2445 

.6798 

.261 

.6414 

.2784 

.6052 

.2966 

•571 

.2448 

.6792 

•  2613 

.6408 

.2787 

.6046 

.2969 

•5705 

.2451 

.6785 

.2616 

.6402 

.279 

.604 

.2972 

-5699 

1-2453 

1.6779 

1.2619 

1.6396 

1-2793 

1.6034 

1-2975 

1.5694 

.2456 

.6772 

.2622 

.6389 

•2795 

.6029 

.2978 

.5688 

•2459 

.6766 

.2624 

•6383 

.2798 

.6023 

.2981 

-5683 

.2461 

•6759 

.2627 

•6377 

.2801 

.6017 

.2985 

•5677 

.2464 

.6752 

.263 

.6371 

.2804 

.6011 

.2988 

.5672 

1.2467 

1.6746 

1.2633 

1.6365 

i.  2807 

1.6005 

1.2991 

1.5666 

.247 

•6739 

.2636 

•6359 

.281 

.6 

•2994 

.5661 

.2472 

•2639 

•6352 

.2813 

•5994 

•2997 

•5655 

•2475 

.6726 

.2641 

.6346 

.2816 

.5988 

•  3 

-565 

.2478 

.672 

.2644 

-634 

.2819 

.5982 

•  3003 

•5644 

1.248 

1.6713 

1.2647 

I-6334 

1.2822 

I-5976 

1.3006 

I-5639 

.2483 

.6707 

.265 

.6328 

.2825 

•5971 

.301 

•5633 

.2486 

.67 

•2653 

.6322 

.2828 

•5965 

•3013 

.5628 

.2488 

.6694 

.2656 

.6316 

.2831 

•5959 

.3016 

.5622 

.249 

.6687 

•2659 

•  6309 

•2834 

•5953 

.3019 

•5617 

1.2494   I-  6681 

1.2661 

1.6303 

1-2837 

1-5947 

1.3022 

1.5611 

.2497    .6674 
.2499   .6668 

.2664 
.2667 

.6297 
.6291 

.284 
.2843 

•5942 
•5936 

•3025 
.3029 

.5606 
•56 

.2502   .6661 

.267 

.6285 

.2846 

•593 

.3032 

•5595 

•2505 

•6655 

•2673 

.6279 

.2849 

•5924 

•3035 

•559 

1.2508 

1.6648 

1.2676 

1.6273 

1.2852 

1-5584 

.251 

.6642 

.2679 

.6267 

-2855 

•59*3 

.3041 

•5579 

•2513 

.6636 

.2681 

.6261 

.2858 

•59°7 

•3044 

•5573 

•  2516 

.6629 

.2684 

•6255 

.2861 

.3048 

.5568 

.2519 

.6623 

.2687 

.6249 

.2864 

.5896 

•5563 

1.2521 

i.  6616 

1.269 

1.6243 

1.2867 

1.589 

1-3054 

1-5557 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

53° 

520 

510 

50° 

NATURAL   SECANTS   AND   CO-SECANTS. 


413 


4C 

1°     ! 

41 

0 

42 

jo 

42 

1° 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SBC'T. 

I-3054 

1-5557 

I-325 

1-5242 

I-3456 

1-4945 

I-3673 

1-4663 

•3057 

•5552 

•3253 

•5237 

.346 

•494 

•3677 

.4658 

-306 

•5546 

•3257 

•5232 

•3463 

•4935 

-3681 

•4654 

.3064 

•5541 

.326 

•5227 

•3467 

•493 

.3684 

•4649 

.3067 

•5536 

•3263 

.5222 

•347 

•4925 

.3688 

.4644 

1-307 

1-553 

1.3267 

1.5217 

1-3474 

1.4921 

1.3692 

1.464 

3073 

•5525 

•327 

.5212 

•3477 

.4916 

•3695 

•4635 

3076 

•552 

•3274 

.5207 

.3481 

•49" 

•3699 

.4631 

.308 

•5514 

.3277 

.5202 

.3485 

.4906 

•3703 

.4626 

-3083 

•5509 

•328 

•5197 

.3488 

.4901 

•3707 

.4622 

1.3086 

I-5503 

1.3284 

1.5192 

1.3492 

1.4897 

1.4617 

-3089 

•5498 

.3287 

•5187 

•3495 

.4892 

•3714 

.4613 

.3092 

•5493 

•329 

.5182 

-3499 

.4887 

.4608 

.3096 

•5487 

•3294 

•5177 

•3502 

.4882 

•3722 

.4604 

•3099 

-5482 

•3297 

.5171 

•35o6 

.4877 

•3725 

•4599 

1.3102 

1-5477 

I-330I 

1.5166 

I-3509 

1.4873 

I-3729 

1-4595 

•3105 

•5473 

•3304 

.5161 

•3513 

.4868 

•3733 

•459 

.3109 

.5466 

•3307 

•5156 

•3517 

.4863 

•3737 

.4586 

.3112 

.5461 

•33" 

•5I51 

•352 

.4858 

•374 

.4581 

•3"5 

•5456 

•3314 

.5146 

.3524 

•4854 

•3744 

•4577 

1.3118 

1-545 

1.5141 

1-3527 

1.4849 

I-3748 

1-4572 

.3121 

•5445 

•3321 

•5136 

•3531 

•4844 

•3752 

.4568 

•3125 

•544 

•3324 

•5131 

•3534 

-4839 

•3756 

•4563 

.3128 

•5434 

•3328 

.5126 

•3538 

•4835 

•3759 

•4559 

•3I3I 

•5429 

•3331 

.5121 

•3542 

•483 

•3763 

•4554 

1.3134 

1.5424 

1-3335 

1-5116 

1-3545 

1-4825 

1-3767 

1-455 

•3138 

•3338 

.5111 

•3549 

.4821 

•4545 

•5413 

•3342 

.5106 

•3552 

.4816 

•3774 

-3144 

.5408 

•3345 

.5101 

•3556 

.4811 

•3778 

•4536 

.3148 

•5403 
1.5398 

•3348 
1-3352 

.5096 
1-5092 

•  356 
1-3563 

.4806 
1.4802 

.3782 
1-3786 

4532 
M527 

•3154 

•5392 

•3355 

•5087 

•3567 

•4797 

•379 

•4523 

•5387 

•3359 

.5082 

•3571 

•4792 

•3794 

.4518 

.3161 

•5382 

•  3362 

•5077 

•3574 

.4788 

•3797 

•4514 

.3164 

•5377 

-3366 

.5072 

•3578 

•4783 

.3801 

•451 

1.3167 

.5366 

•3372 

.5062 

•3585 

•4774 

1-3805 
•  3809 

^SOS 
.4501 

-3174 

•3376 

•5057 

•3589 

.4769 

•3813 

.4496 

•5177 
.318 

•5356 
•5351 

•3379 
-3383 

•5052 
•5047 

•3592 
•359° 

•4764 
•476 

.3816 
-382 

•4492 
-4487 

1-3184 

1-5345 

1-3386 

1-5042 

1-36 

1-4755 

1-3824 

1.4483 

.3187 

•534 

•339 

•5037 

-3603 

•475 

•  3828 

•4479 

•5335 

•3393 

•5032 

.3607 

•4746 

•3832 

•4474 

•3193 

•533 

•3397 

.5027 

.3611 

.4741 

•3836 

•447 

•3197 

•5325 

•34 

.5022 

.3614 

•4736 

•3839 

•4465 

1.32 

I-53I9 

1.3404 

1.5018 

1.3618 

1-4732 

1-3843 

1.4461 

-3203 

•53H 

•3407 

•5013 

.3622 

•4727 

•3847 

•4457 

.3207 

•5309 

•34" 

.5008 

•3625 

•4723 

•3851 

•4452 

.321 

•53°4 

•34H 

.5003 

-3629 

.4718 

•3855 

.4448 

•3213 

•5299 

.3418 

•4998 

•3633 

•3859 

•4443 

1.3217 

1.5294 

1.3421 

1-4993 

1.3636 

1.4709 

1.3863 

1-4439 

•  322 

-5289 

•3425 

.4988 

•364 

.4704 

•3867 

•4435 

•  3223 

-5283 

•  3428 

•4983 

•3644 

•4699 

•387 

•443 

.3227 

•5278 

•3432 

•4979 

•3647 

•4695 

•3874 

.4426 

•323 

•5273 

•3435 

•4974 

•3651 

.469 

.3878 

.4422 

1-3233 

1.5268 

1-3439 

1.4969 

1.4686 

1.3882 

1.4417 

•3237 

•5263 

•3442 

.4964 

''3658 

.4681 

.3886 

•44*3 

•324 

-5258 

•3446 

•4959 

.3662 

.4676 

•389 

.4408 

•3243 

•5253 

•3449 

•4954 

.3666 

.4672 

•3894 

.4404 

•3247 

-5248 

•3453 

•4949 

-3669 

.3667 

.3898 

•44 

1-325 

1.5242 

I-3456 

1-4945 

1-3673 

1-4663 

1.3902 

1-4395 

CO-BHC'T. 

SKCANT. 

CO-SKC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SKCANT. 

4< 

)° 

41 

30 

4' 

7° 

« 

>° 

M 

M* 

NATURAL    SECANTS   AND    CO-SECANTS. 


44 

t<> 

44 

t° 

44 

0 

' 

SECANT. 

CO-SEC'T. 

' 

' 

SECANT. 

CO-SEC'T. 

r 

1 

SECANT. 

CO-SKC'T. 

'.- 

0 

1.3902 

1-4395 

60 

21 

1.3984 

i-43°5 

39 

41 

1.4065 

1.4221 

^0 

I 

•39°5 

•439* 

59 

22 

•  3988 

.4301 

38 

42 

.4069 

.4217 

18 

2 

•39°9 

•4387 

58 

23 

3992 

.4297 

37 

43 

•4°73 

.4212 

ll 

3 

•39*3 

.4382 

56 

24 

•3996 

.4292 
1.4288 

36 

44 

.4077 

.4208 

16 

4 

5 

•3917 
1.3921 

•4378 
1-4374 

5U 
55 

11 

.4004 

.4284 

35 
34 

45 
46 

.4085 

.42 

H 

6 

•3925 

•437 

54 

27 

.4008 

.428 

33 

47 

.4089 

.4196 

13 

7 

3929 

•4365 

53 

28 

.4012 

.4276 

32 

48 

•4°93 

•4I9,o 

12 

8 

•3933 

.4361 

52 

29 

.4016 

.4271 

3i 

49 

.4097 

.4188 

11 

9 

•3937 

•4357 

Si 

3° 

1.402 

1.4267 

30 

50 

1.4101 

1.4183 

10 

0 

i-394i 

1-4352 

5° 

3i 

.4024 

.4263 

29 

5i 

.4105 

.4179 

9 

i 

•3945 

•4348 

49> 

32 

.4028 

'•4259 

28 

52 

.4109 

•4175 

8 

2 

•3949 

•4344 

48 

33 

.4032 

4254 

27 

53 

•4"3 

.4171 

7 

3 

•3953 

•4339 

47 

34 

.4036 

•425 

26 

54 

.4117 

.4167 

6 

4 

•3957 

•4335 

46 

35 

1.404 

1.4246 

25 

55 

1.4122 

1.4163 

5 

1.396 
•3964 

I-433I 
•4327 

45 
44 

36 
37 

.4044 
.4048 

.4242 
•  4238 

24 
23 

5* 
57 

.4126 
.413 

•4159 

•4!54 

••'  4 
3 

7 

.3968 

.4322 

43 

38 

.4052 

•4333 

22 

58 

•4i34 

•415 

2 

8 

•3972 

.4318 

42 

39 

.4056 

.4229 

21 

39 

.4138 

.4146 

^  * 

9 

•3976 

•4314 

4i 

40 

1.406 

1.4225 

20 

60 

1.4142 

1.4142 

0 

20 

1.398 

J-43I 

40 

/ 

CO-SEC'T. 

SECANT. 

i 

/ 

CO-SEC'T. 

SECANT. 

1 

i 

CO-SEC'T. 

SBCANT. 

I 

4i 

5° 

4 

5° 

4, 

50 

Preceding  Table  contains  Natural  Secants  and  Co-secants  for  every 
minute  of  the  Quadrant  to  Radius  i. 

If  Degrees  are  taken  at  head  of  column,  Minutes,  Secant,  and  Co-secant 
must  be  taken  from  head  also;  and  if  they  are  taken  at  foot  of  column, 
Minutes,  etc.,  must  be  taken  from  foot  also. 

ILLUSTRATION.— 1.05  is  secant  of  17°  45'  and  co-secant  of  72°  15'. 

To    Compute   Secant  or   Co-secant  of  any  Angle. 
RULE.— Divide  i  by  Cosine  of  angle  for  Secant,  and  by  Sine  for  Co-secant. 
EXAMPLE  i.— What  is  secant  of  25°  25'? 

Cosine  of  angle  =  .903  21.    Then  i  -f-  .903  21  =  1.1072,  Secant. 
2.— What  is  co  secant  of  64°  35'? 

Sine  of  angle  =  .903  21.    Then  i  -r-  .903  21  =  1. 1072,  Co-secant. 

To  Compute  Degrees,  IVTiiiTites,  and  Seconds  of*  a  Secant 
or    Co-secant. 

When  Secant  is  given, 

Proceed  as  by  Rule,  page  402,  for  Sines,  substituting  Secants  for  Sines. 
EXAMPLE. — What  is  secant  for  1.1607? 
The  next  less  secant  is  1.1606,  arc  for  which  =  30°  30'. 

Next  greater  secant  is  1.1608,  difference  between  which  and  next  less  is  1.1608  — 
1. 1606  =  .0002. 

Difference  between  less  tab.  secant  and  one  given  is  1. 1607  —  1. 1606  =  .0001. 
Then  .0002  :  .0001  ::  60  :  30,  which,  added  to  30°  30'  =  30°  30'  30". 

When  Co-secant  is  given, 
Proceed  as  by  Rule,  page  402,  Substituting  Co-secants  for  Cosines. 


NATURAL   TANGENTS   AND   CO-TANGENTS. 


Natural   Tangents   and.    Co-tangents. 

0°                ||             1° 

2° 

3° 

TANO. 

CO-TANG.    1  1    TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG.    | 

CO-TANG. 

.00000 

Infinite. 

.01746 

57-29 

.03492 

28.6363 

•  05241 

19.0811 

.00029 

3437-75 

•01775 

6.3506 

.03521 

8.3994 

.0527 

8-9755 

.00058 

1718.87 

.01804 

5-44I5 

•0355 

8.  1664 

•05299 

8.8711 

.00087 

145.92 

01833 

4-56i3 

•035  79 

7-9372 

•05328 

8.7678 

.001  16 

859-436 

01862 

3-7086 

036  09 

7.7117 

•05357 

8.6656 

.00145 

687.549 

01891 

52.8821 

.03638 

27.  4899 

•05387 

18.5645 

.00175 
.00204 

572-957 
491.106 

0192 
01949 

2.0807 
1.3032. 

.03667 
.03696 

7-27*l 
7.0566 

.05416 
•05445 

8.4645 
8-3655 

.00233 

29.718 

01978 

0-5485 

037  25 

6-845 

•05474 

8.2677 

.00262 

381.971 

02007 

49.8157 

•037  54 

6.6367 

•05503 

8.1708 

.00291 

343-774 

02036 

49-  1039 

•037  83 

26.4316 

•05533 

18.075 

.OO3  2 

12.521 

02066 

8.4121 

.038  12 

6.2296 

•  05562 

7.9802 

.00349 

286.478 

.02095 

7-7395 

.03842 

6.0307 

•05591 

7.8863 

•00378 

64.441 

.021  24 

7-0853 

.03871 

5-8348 

.0562 

7-7934 

.00407 

45-552 

.021  53 

6.4489 

•039 

5-6418 

•  05649 

7-70I5 

.00436 

229.  182 

.021  82 

45.8294 

.039  29 

25-45I7 

.05678 

17.6106 

.00465 

14-858 

.02211 

5.2261 

•03958 

5-2644 

•  05708 

7-5205 

.00495 

02.219 

.O224 

4.6386 

.03987 

5.0798 

•05737 

7-43I4 

.00524 

190.984 

.02269 

4.0661 

.040  16 

4.8978 

•  05766 

7-3432 

•00553 
.OO5  82 

80.932 
171-885 

.02298 
.02328 

3-5081 
42.9641 

.04046 
.040  75 

4-7i85 
24.5418 

•05795 
.05824 

7-2558 
17.1693 

.006  II 

63-7 

•023  57 

2-4335 

.041  04 

4-3675 

•05854 

7.0837 

.0064 

56-  259 

.023  86 

1.9158 

•041  33 

4.1957 

.05883 

6.999 

,00669 

49.465 

•02415 

1.4106 

.041  62 

4.0263 

.05912 

6.915 

.00698 

43-237 

.024  44 

0.9174 

04191 

3-8593 

.05941 

6.8319 

.00727 

I37-507 

.02473 

40.4358 

.0422 

23-6945 

•0597 

16.7496 

.00756 

32.219 

.025  02 

39-9655 

0425 

3-5321 

•05999 

6.  668  1 

.00785 

27.321 

.02531 

9-  5059 

.04279 

3-37I8 

.06029 

6.5874 

.00814 

22-774 

.0256 

9.0568 

.04308 

3-2137 

.06058 

6-5075 

.00844 

18.54 

.025  89 

8.6177 

•04337 

3-0577 

.06087 

6.4283 

.00873 

114-589 

.026  19 

38.1885 

.043  66 

22.9038 

.06116 

16.3499 

.00902 

10.  892 

.02648 

7.7686 

•043  95 

2.7519 

•  06145 

6.2722 

.00931 

07.426 

.026  77 

7-3579 

.044  24 

2.602 

•06175 

6.1952 

.0096 

04.171 

.02706 

6.956 

.04454 

2-4541 

.06204 

6.119 

.00989 

01.107 

•02735 

6.5627 

.04483 

2.  3081 

•06233 

6-0435 

.OIOlS 

98.2179 

.02764 

36.  1776 

.04512 

22.  164 

.06262 

15-9687 

.01047 

5-4895 

•02793 

5.8006 

.04541 

2.0217 

.06291 

5-8945 

.01076 

2.9085 

.02822 

5-43I3 

.0457 

I.88I3 

.06321 

5.8211 

.01105 

0-4633 

.02851 

5-0695 

.04599 

1.7426 

•0635 

5-7483 

•01135 

88.1436 

.02881 

4-7x5i 

.04628 

1.6056 

.06379 

5.6762 

.on  64 

85.9398 

.029  i 

34.3678 

.046  58 

21.4704 

.06408 

15.6048 

.01193 

3-8435 

.02939 

4-0273 

.04687 

L3369 

.06437 

5-534 

.OI2  22 

1.847 

.02968 

3-6935 

.047  16 

1.2049 

.06467 

5.4638 

.01251 

79-9434 

.02997 

3.3662 

•04745 

1.0747 

.06496 

5-3943 

.0128 

8.1263 

.03026 

3-0452 

•04774 

0.946 

.06525 

5-3254 

.01309 

76-39 

•03055 

32-7303 

.04803 

20.8l88 

•065  54 

15-2571 

•01338 

4.7292 

.03084 

2.4213 

.04832 

0.6932 

.06584 

5-1893 

.01367 

3-J39 

.031  14 

2.1181 

.048  62 

0.5691 

.066  I3 

5.1222 

.01396 

1.6151 

•031  43 

1.8205 

.04891 

0.4465 

.06642 

5-0557 

.01425 

0.1533 

•  031  72 

1.5284 

.0492 

0.3253 

.06671 

4.9898 

•01455 

68.7501 

.03201 

31.2416 

•04949 

20.  2056 

.067 

14.9244 

.01484 

7.4019 

•0323 

0-9599 

.049  78 

0.0872 

.0673 

4-8596 

•OI5I3 

6.  1055 

•032  59 

0.6833 

.05007 

19.9702 

.067  59 

4-7954 

.01542 

4-858 

.03288 

0.4116 

•05037 

9.8546 

.06788 

4-73I7 

.01571 

3-6567 

•03317 

o.  1446 

.05066 

9-7403 

.06817 

4.6685 

.Ol6 

62.4992 

•03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

.01629 

1.3829 

•033  76 

9.6245    .05124 

9-5I56 

.06876 

4-5438 

.01658 

o.  3058 

•03405 

9-37" 

•05153 

9.4051 

.06905 

4-4823 

.01687 

59-2659 

•034  34 

9.122 

.051  82 

9.2959 

.06934 

4.4212 

;.OI7l6 

8.2612 

•03463 

8.8771 

.052  12 

9.1879 

.06963 

4.3607 

.017  46 

57-29 

.03492 

28.6363 

•05241 

19.0811 

•  06993 

14.3007 

CO-TANG 

TANG. 

CO-TANG 

TANG. 

CO-TANG 

TANG. 

CO-TANG 

TANS.  • 

89° 

88° 

87° 

86° 

416 


NATURAL   TANGENTS   AND    CO-TANGENTS. 


4 

i° 

5 

0 

C 

3 

•3 

0 

TANS. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANS. 

.06993 

14.3007 

.08749 

i  -430I 

.105  i 

9.5H36 

.12278 

8.M435 

.07022 

4.2411 

.08778 

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•  105  4 

.48781 

.12308 

.12481 

.07051 

4.1821 

.08807 

•354 

•  105  69 

.46141 

•  12338 

•  105  36 

.0708 

4-1235 

•08837 

•3163 

•  105  99 

•43515 

.12367 

.086 

.071  I 

4-°655 

.08866 

.2789 

.  106  28 

.40904 

•12397 

.06674 

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14.0079 

.08895 

i  .2417 

.10657 

9-38307 

.  124  26 

8.047  S6 

.071  68 

3-9507 

.089  25 

.2048 

.  106  87 

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.12456 

.02848 

.07197 

3-894 

•08954 

.1681 

.107  16 

•33I54 

.12485 

8.00948 

.07227 

3-8378 

.08983 

.1316 

.10746 

•30599 

•12515 

7.99058 

.07256 

3.7821 

.09013 

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.28058 

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.97176 

.072  85 

13.7267 

.09042 

i  .0594 

.  108  05 

9-2553 

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7-95302 

•073  14 

3-67i9 

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1.0237 

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.23016 

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.07344 

3-6i74 

.09101 

0.9882 

.10863 

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•  12633 

.91582 

.073  73 

3.5634 

.0913 

0.9529 

.10893 

.18028 

.12662 

•89734 

.074  02 

3.  5098 

•09159 

0.9178 

.10922 

•15554 

.12692 

.87895  j 

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13.4566 

.091  89 

10.  882  9 

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7.86064  ; 

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3-4039 

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0.8483 

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3.35I5 

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0.8139 

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.08211 

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3.2996 

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0.7797 

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.1281 

.  806  22 

.07548 

3-248 

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0-7457 

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.1284 

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13-1969 

•093  35 

10.711  9 

.11099 

9.00983 

.12869 

7-77035 

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3.1461 

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0.6783 

.11128 

8.98598 

.12899 

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3-0958 

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0.645 

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0.6118 

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0.5789 

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12.9469 

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10.  546  2 

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8.89185 

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7.68208 

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2.8981 

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0.5136 

.11276 

.86862 

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2.8496 

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0.4813 

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.13076 

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2.8014 

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0.4491 

•"335 

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.63005 

.07841 

2.7536 

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0.4172 

.11364 

.79964 

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12.7062 

.09629 

10.  385  4 

•"394 

8.77689 

•13165 

7-59575 

.07899 

2.6591 

.09658 

0-3538 

.11423 

•75425 

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.57872 

.07929 

2.6124 

.09688 

0.3224 

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.13224 

•  56176 

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2.566 

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0.2913 

.11482 

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2.5199 

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.11511 

.68701 

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12.4742 

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10.2294 

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8.66482 

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7.51132 

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2.4288 

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0.1988 

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2.3838 

.09834 

o.  168  3 

.116 

.62078 

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.47806 

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2-339 

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0.1381 

.11629 

•59893 

.13402 

.46154 

.081  34 

2.  2946 

.09893 

0.108 

.11659 

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.44509 

.081  63 

12.2505 

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10.078 

.11688 

8-55555 

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7.42871 

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2.  2067 

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0.048  3 

.11718 

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2.1632 

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0.018  7 

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9.9893 

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2.0772 

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.11806 

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12.0346 

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9.93101 

.11836 

8.44896 

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7.34786 

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1.9923 

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1.9504 

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1.9087 

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11.8262 

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9-788x7 

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8.34496 

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7.26873 

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1.7448 

•  102  75 

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.13846 

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1.6645 

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11.6248 

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9-64935 

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7.19125 

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•  103  93 

.62205 

.    21  6 

.22344 

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.08661 

I.546I 

.  104  22 

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.20352 

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1.5072 

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1.4685 

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.  140  24 

.13042 

.08749 

11.4301 

.1051 

9-5I436 

.  122  78 

8.14435 

.14054 

7-"537 

CO-TANS. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

8, 

5° 

8 

40 

8 

JO 

8 

2° 

NATUBAL   TANGENTS  AND   CO-TANGENTS. 


417 


h            8 

I 

!         s 

0 

1( 

30 

1 

LO 

* 

TANG. 

CO-TANG. 

(    TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

•   o 

-  140  54 

7-"537 

.15838 

6.31375 

•17633 

5.67128 

.19438 

5-I4455 

I 

.14084 

.10038 

.15868 

.30189 

.17663 

.66165 

.19468 

.13658 

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.14113 

.08546 

.15898 

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.65205 

.19498 

.12862 

3 

•I4M3 

.07059 

.15928 

.27829 

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.64248 

•19529 

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4 

•I4I73 

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•15958 

.26655 

•'7753 

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5 

.14202 

7.041  05 

.15988 

6.25486 

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5-62344 

.19589 

5-  104  9 

6 

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7 

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8 

.14291 

6.997  1  8 

.16077 

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9 

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6.96823 

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6.  197  03 

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5-57638 

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5.06584 

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.  162  83 

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15 

.14499 

6.  896  88 

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6.14023 

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5-53007 

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5.02734 

16 

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17 

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•  163  46 

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•  18143 

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18 

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•  163  76 

.10664 

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.14618 

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4.99695 

20 

.14648 

6.82694 

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6.08444 

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5-48451 

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4.9894 

21 

.14678 

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1  -20073 

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22 

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1  .20103 

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6.75838 

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6.02962 

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5-43966 

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4-95201 

26 

.  148  26 

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27 

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28 

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29 

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6.691  16 

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4.91516 

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32 

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37 

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38 

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40 

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6.56055 

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5.8708 

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5.30928 

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•  18925 

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6.43484 

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5-76937 

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4.77286 

5i 
52 

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156 

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5.671  28 

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4-  7°4  63 

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CO-TANG. 

TAMO. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG.   * 

8 

1» 

8 

0° 

7 

90 

7 

8° 

418 


NATURAL    TANGENTS    AND    CO-TANGENTS. 


1 

TANG. 

20 

3 

L3° 

I  »J 

i~ 

CO-TANG. 

] 

TANG. 

L50 
CO-TANG. 

.21256 

4.70463 

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4-33148 

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4.01078 

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3-73205 

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.69791 

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•  26857 

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3-99S92 

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4.67121 

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4.30291 

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3.98607 

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3.71046 

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4.63825 

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4.27471 

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3.961  65 

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3.68909 

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•  234  24 

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1  .21651 
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j   .217  12 

4.60572 

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4.3604 

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4.01078 

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TANG. 

CO-TANG. 

TANG. 

CO^TANG. 

TANG. 

CO-TANG. 

TANG. 

7 

7° 

7 

50               1 

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1° 

NATURAL   TANGENTS   AND   CO-TANGENTS. 


419 


f  ;         16° 

170 

18° 

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CO-TANG. 

TANG. 

CO-TANG. 

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0 

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3.48741 

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3.27085 

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3.07768 

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CO-TANG. 

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730 

720 

71° 

70° 

420 


NATURAL    TANGENTS    AND    CO-TANGENTS. 


« 

2 

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DO 

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2 

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LO 
CO-TANG. 

2 

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20 
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2 

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2.  605  09 

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CO-TANG. 

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CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

690 

68° 

67° 

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NATURAL   TANGENTS   AND   CO-TANGENTS. 


421 


24° 

25° 

260 

27° 

TANG'.     '  CO-TANK. 

TANG. 

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CO-TANG. 

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•  48163 

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•50331 

•'98684 

•52538 

•9°337    ! 

.46065 

.17083 

.48198 

.07476 

•  50368 

•9854 

.52575 

.90203    j 

.461  01 

2.169  17 

.48234 

2.07321 

.50404 

1-98396 

.52613 

1.90069 

.461  36 

.167  51 

.4827 

.07167 

.50441 

•98253 

•5265 

.89935 

.46171 

•  16585 

.48306 

.07014 

•50477 

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.52687 

.89801 

.46206 

.1642 

•48342 

.0686 

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.52724 

.89667 

.46242 

-16255 

.48378 

.06706 

•5055 

.97823 

•  52761 

•89533 

.46277 

2.1609 

.48414 

2.06553 

•50587 

1.9768 

.52798 

1.894 

;  -46312 

•15925 

•4845 

.064 

•50623 

•9753s 

.52836 

.89266 

.46348 

.1576 

.48486 

.06247 

.5066 

•97395 

•52873 

•89133 

1  -46383 

•I5596 

.48521 

.06094 

.50696 

•97253 

.5291 

.89 

.46418 

•15432 

•48557 

.05942 

•50733 

.97111 

•52947 

.88867 

!  -46454 

2.15268 

•48593 

2-0579 

.  507  69 

1.96969 

.52984 

1.88734 

!  .46489 

.15104 

.48629 

•05637 

.  508  06 

.96827 

.53022 

.88602 

•46525 

.1494 

.48665 

•05485 

•50843 

.96685 

•53059 

.88469 

.4656 

•14777 

.48701 

•05333 

.50879 

•96544 

•53096 

•88337 

•46595 

.14614 

•48737 

.051  82 

.50916 

.96402 

•53134 

.  882  05 

.46631 

2.14451 

•48773 

2.0503 

•50953 

1.96261 

•53I7I 

1.88073 

CO-TANG,  i     TANG. 

CO-TANG. 

TANG. 

CO-TANG 

TANG. 

CO-TANG 

TANG. 

65° 

64° 

63° 

62° 

422 


NATURAL   TANGENTS   AND   CO-TANGENTS. 


28° 

29° 

30° 

31°               \ 

TAWO. 

CO-TANO. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

•53I7I 

1.88073 

•55431 

1.80405 

•57735 

1.73205 

.60086 

1.66428 

•  53208 

.87941 

•55469 

.80281 

•57774 

.73089 

.601  26 

.66318 

.87809 

•55507 

.  801  58 

•57813 

•72973 

.60165 

.66209 

'•53283 

.87677 

•55545 

.80034 

•57851 

.72857 

.60205 

.66099 

.5332 

.87546 

•55583 

.79911 

•5789 

.72741 

.60245 

•6599 

•53358 

1.87415 

•55621 

1.79788 

•  579  29 

1.72625 

.60284 

1.65881 

•53395 

.87283 

•55659 

.79665 

.57968 

•72509 

60324 

.65772 

•53432 

•55697 

•79542 

.58007 

•72393 

.60364 

•65663 

•5347 

.87021 

•55736 

.79419 

.58046 

.72278 

.60403 

•65554 

53507 

.8689! 

•55774 

.79296 

.58085 

.72163 

.60443 

•65445 

53545 

1.8676 

•558i2 

1.79174 

•  58124 

1.72047 

.60483 

I-65337 

•53582 

.8663 

•5585 

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.58162 

.71932 

.60522 

.65228 

•5362 
•53657 

.86499 
.86369 

55888 
•  559  26 

.78929 
.78807 

.58201 
•5824 

.71817 
.71702 

.60562 
.60602 

.651  2 
.65011 

•53694 

.86239 

•  55964 

•  78685 

.58279 

.71588 

.60642 

.64903 

•53732 

1.86109 

.56003 

1.78563 

•  58318 

I-7I473 

.60681 

1.64795 

•53769 

.85979 

.56041 

.78441 

•58357 

.71358 

.60721 

.64687 

•53807 

•8585 

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.78319 

•58396 

.71244 

.607  61 

•64579 

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561  17 

.78198 

•58435 

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.60801 

•  64471 

•  53882 

•85591 

.56156 

.78077 

•58474 

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.60841 

.64363 

.5392 

1.85462 

56194 

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1.70901 

.60881 

1.64256 

•53957 

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.64148 

•53995 

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•  58631 

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.5867 

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1.70332 

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1.63719 

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•54183 

•84561 

.56462 

7711 

.58787 

.70106 

.6116 

•63505 

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•84433 

565 

•7699 

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.6124 

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•  54296 

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•56577 

1.76749 

.58904 

1.69766 

.6128 

1.63185 

•54333 

.84049 

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.7663 

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.69653 

6132 

.63079 

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56654 

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•  6136 

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.54409 
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1.69203 

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62336 

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1.82906 

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75675 
I-75556 

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.68754 
1.68643 

.61641 
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6223 
I.62I  25 

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.8278 

-57 

•75437 

•  593  36 

.68531 

.61721 

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•54786 

.82528 

.57078 

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59415 

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61801 

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.54824 

.82402 

.57116 

.75082 

•  594  54 

.68196 

61842 

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•  54862 

1.82276 

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1.74964 

59494 

1.68085 

61882 

1.61598 

•549 

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.74846 

59533 

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•  59612 

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,  620  03 

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.57309 

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•  62043 

.61179 

•55051 

1.81649 

•57348 

'•74375 

.59691 

1-6753 

.62083 

1.61074 

.55089 

.81524 

•57386 

•74257 

•5973 

.67419 

.621  24 

.6097 

•55127 

.81399 

•57425 

.7414 

•5977 

.67309 

621  64 

.60865 

•55165 

.8I274 

•57464 

.74022 

.59809 

.67198 

.62204 

.60761 

•55203 

•  8115 

•57503 

•73905 

•  598  49 

.67088 

.62245 

•60657 

.55241 

1.81025 

•57541 

1.73788 

.59888 

1.66978 

62285 

1-60553 

.55279 

.80901 

•5758 

•73671 

.  599  28 

.66867 

.62325 

.60449 

.55317 

.80777 

•57619 

-73555 

•  599  67 

•66757 

.62366 

.60345 

•55355 

•  80653 

•57657 

.73438 

.60007 

.66647 

.  624  06 

.60241 

•5539? 

.80529 

.57696 

.73321 

.60046 

.66538 

.62446 

.60137 

•55431 

1.80405 

•57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033   ' 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG  . 

TANG. 

610 

60°   _ 

59° 

58° 

NATURAL   TANGENTS   AND   CO-TANGENTS. 


423 


32° 

33° 

340                | 

35°               ] 

TAKO. 

CO-TANO. 

TANG. 

CO-TANO. 

TANG. 

CO-TANO. 

TANO. 

CO-TANO. 

.62487 
.62527 

1.60033 

.649  82 

1.53986 
•  53888 

•6745* 
•67493 

1.48256 

.48163 

.70021 
.70064 

1.42815 
.42726 

.62568 

.  598  26 

.65023 

•5379* 

•67536 

.4807 

.70107 

.42638 

.62608 

•59723 

•  65065 

•53693 

.67578 

•47977 

.70151 

.4255 

.62649 

.5962 

.65106 

•53595 

.6762 

.47885 

.70194 

.42462 

.62689 

1-595*7 

.65148 

1-53497 

•  67663 

1.47792 

.70238 

1.42374 

.6273 

-594*4 

.65189 

•534 

.67705 

.47699 

.70281 

.42286 

.6277 
.62811 

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.59208 

.65231 
.65272 

•53302 
•53205 

•67748 
.6779 

•47607 
•475*4 

70325 
•70368 

.42198 
.4211 

.62852 

•59*05 

•653*4 

•53*07 

.67832 

•47422 

.70412 

.42022 

.62892 

1.59002 

•65355 

1.5301 

•67875 

1-4733 

•70455 

1.41934 

•62933 

•589 

.65397 

•529*3 

•679*7 

•47238 

.70499 

•4*847 

.62973 

•58797 

.65438 

.52816 

.6796 

•47*46 

•70542 

•4*759 

.63014 

.58695 

.6548 

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.68002 

•47°53 

.70586 

.41672 

•63055 

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•  65521 

.526  22 

.68045 

.  469  62 

.70629 

•4*584 

•63095 

1.5849 

•65563 

I.52525 

.68088 

1.4687 

.70673 

1-4*497  i 

•  63136 

•  58388 

.65604 

.52429 

•6813 

.46778 

.70717 

•4*4°9  : 

•63177 

.58286 

.65646 

•52332 

.68!  73 

.46686 

.7076 

.41322 

.632  17 

.58184  ||  .65688 

•52235 

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•46595 

.70804 

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.63258 

.58083 

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1.57981 

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1.52043 

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1.46411 

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1.41061 

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•  63421 

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1-57474 

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1.51562 

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1.40627 

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•  45682 
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.40367 
.40281 

•63707 

1.56969 

.66189 

1.51084 

.68728 

I-4550I 

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1.40195 

•63748 

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.68771 

•4541 

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•  567  67 

.66272 

•  508  93 

.688  14 

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.40022 

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•  56566 

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•  63912 

1.56466 

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1.50607 

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1.45049 

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I-39764 

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•  505*2 

.68985 

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.56265 

.66482 

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.69028 

.44868 

•7*637 

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•64035 

.56165 

.66524 

•  50322 

.69071 

•44778 

71681 

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.64076 

.56065 

.66566 

.  502  28 

.69114 

.44688 

•7*725 

.39421 

.641  17 

1.55966 

.66608 

i-5oi33 

69*57 

1.44598 

•7*769 

I-39336 

•  641  58 

.55866 

.6665 

.50038 

.692 

.445o8 

•7*8*3 

.3925 

.64199 

.55766 

.66692 

•  499  44 

•69243 

.44418 

•7*857 

.39*65 

.6424 

.55666 

.66734 

•49849 

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•44329 

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.39079 

.64281 

.55567 

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.69329 

•44239 

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.38994 

•  64322 

1-55467 

.66818 

1.49661 

•69372 

1.44149 

•7*99 

1.38009 

.64363 

.55368   1  .6686 

.49566 

.69416 

.4406 

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.55269  1   .66902 

.49472 

69459 

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.66944 

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.69502 

.43881 

.72122 

.38653 

.  644  87 

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.66986 

.492  84 

•69545 

•43792 

.72166 

.38568 

.64528 

1.54972 

.67028 

1.4919 

.69588 

1-43703 

.722  ii 

1.38484 

•  64569 

•54873 

1  .67071 

•49°97 

.69631 

.43614 

•72255 

38399 

.6461 

•54774      -671*3 

.49003 

•69675 

•43525 

.72299 

•383*4 

.646  52 

•54675 

•67*55 

.48909 

.69718 

•43436 

•72344 

.  382  29 

.646  93 

•54576 

•67*97 

.488  16 

.69761 

•43347 

.72388 

•38*45 

•64734 

1.54478 

•67239 

1.48722 

.69804 

1.43258 

•72432 

1.3806 

•64775 

•54379 

.67282 

.48629 

.69847 

•43*69 

.72477 

•37976 

.64817 
.64858 

.54281 
•54*83 

.67324 
-67366 

•48536 
.48442 

.69891 
.69934 

.4308 
.42992 

.72521 
•72565 

.3789* 
.37807 

.64899 

•54085 

.67409 

.48349 

.69977 

.42903 

.7261 

.37722 

.64941 

1.53986 

•6745* 

1.48256 

.70021 

1.42815 

.72654 

*-37638 

CO-TANG. 

'IANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANO. 

CO-TANG. 

TANG. 

57° 

56° 

550 

54° 

424 


NATURAL    TANGENTS    AND    CO-TANGENTS. 


S6° 

37°            |1             38° 

39°            J 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

1    TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

.72654 

1-37638 

•75355 

1.32704 

.78129 

1.27994 

.80978 

1.2349 

.72699 

•37554 

•75401 

.326  24 

.27917 

.81027 

.23416 

•72743 

•3747 

•75447 

•32544 

.782  22 

.27841 

.81075 

•23343 

.72788 

•37386 

•75492 

.32464 

.78269 

.27764 

.81123 

.2327 

.72832 

•37302 

•75538 

•32384 

.78316 

.27688 

.81171 

.23196 

.72877 

1.37218 

•75584 

1.32304 

•78363 

1.276  ii 

.8122 

1.231  23 

.72921 

•37134 

.75629 

.32224 

.7841 

•27535 

.81268 

•2305 

.72966 

•3705 

•75675 

.32144 

.78457 

.27458 

.81316 

.22977 

•7301 

•75721 

.32064 

.78504 

•  27382 

.81364 

.22904 

•73055 

.  368  83 

•75767 

.31984 

•78551 

•  27306 

•81413 

.22831 

•731 

1.368 

.75812 

1.31904 

.78598 

1.2723 

.81461 

1.22758 

•731  44 

.367  16 

•75858 

.31825 

.78645 

•27153 

.8151 

.22685 

.36633 

•75904 

•31745 

.78692 

.27077 

.81558 

.226  12 

•73234 

•36549 

•7595 

.31666 

•78739 

.27001 

.81606 

•22539 

•73278 

1.36466 

•75996 

•31586 

.78786 

.269  25 

.816  55  |    .22467 

•73323 

.36383 

.76042 

1-31507 

.78834 

1.26849 

.81703 

1.22394 

•73368 

.363 

.76088 

•3I427 

.78881 

.26774 

•81752 

.223  21 

•73413 

•  36217 

•76134 

•31348 

.78928 

.26698 

.818 

.22249 

•73457 

•36133 

.7618 

.31269 

.78975 

.2662-2 

.81849 

.221  76 

•73502 

•  36051 

.76226 

.79022 

.26546 

.81898 

.22104 

•73547 

1.35968 

.76272 

1.311  i 

•79°7 

1.26471 

.81946 

I.2203I 

•73592 

.35885 

.76318 

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.79117 

•26395 

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•73637 

•  35802 

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.79164 

.26319 

.82044 

.21886 

.73681 

.35719 

.7641 

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.79212 

.26244 

.82092 

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.76456 

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.82141 

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1-35554 

•  76502 

1.30716 

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1.26093 

.8219 

1.2167 

.73816 

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1-76548 

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.21598 

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.79401 

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.82287 

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•35307 

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.3048 

-79449 

.25867 

.82336 

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.76686 

.30401 

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.25792 

.82385 

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•73996 

I-35I42 

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1.30323 

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1.25717 

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I.2I3I 

74041 

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.76779 

•  30244 

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.25642 

.82483 

.212  38 

.74086 

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.76825 

.  301  66 

•79639 

•25567 

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.741  31 

.34896 

.76871 

.30087 

.79686 

.25492 

.8258 

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.74176 

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.30009 

-79734 

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.82629 

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i  .74221 

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.76964 

1.29931 

.79781 

1-25343 

.82678 

1.20951 

-742  67 

•3465 

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.298  53 

.79829 

.25268 

.82727 

.20879 

•74312 

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•77057 

•29775 

•79877 

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.82776 

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•74357 

•34487 

•77103 

.  296  96 

•  799  24 

.25118 

.82825 

.207  36 

.74402 

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.29618 

.79972 

.25044 

.82874 

.20665 

•74447 

I-34323 

.77196 

1.29541 

.8002 

1.24969 

.82923 

1.20593 

.74492 

•34242 

.77242 

•  29463 

.80067 

.24895 

.82972 

.20522 

•74538 

.3416 

.77289 

•29385 

.80115 

.2482 

.83022 

•20451 

•74583 

•34079 

'77335 

•  293  07 

.80163 

.24746 

.83071 

.20379 

.74628 

•33998 

•77382 

.292  29 

.80211 

.24672 

.8312 

.  2O3  08 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1-24597 

.83169 

1.20237 

•74719 

•33835 

•77475 

.29074 

.  803  06 

•24523 

.83218 

.  201  66 

.74764 

•33754 

•77521 

.28997 

•80354 

•24449 

.83268 

.20095 

•748i 

.33673 

•77568 

.28919 

.  804  02 

•24375 

•83317 

.  200  24 

•74855 

.33592 

•77615 

.28842 

.8045 

•24301 

•83366 

.19953 

•749 

1-335" 

.77661 

1.28764 

.80498 

1.24227 

•83415 

1.19882 

•74946 

•3343 

.77708 

.28687 

.80546 

-24153 

•83465 

.19811 

.74991 

•33349 

•77754 

.2861 

.80594 

.24079 

•83514 

.1974 

•75037 

.33268 

.77801 

-28533 

.80642 

.24005 

•83564 

.I9669 

.75082 

•33187 

.77848 

•28456 

.8069 

•23931 

.83613 

•19599 

.75128 

1-33107 

•77895 

1.28379 

.807  38 

1.23858 

.83662 

I.I95  28 

•75173 

.33026 

•77941 

.28302 

.  807  86 

.23784 

.83712 

•19457 

•75219 

.32946 

.77988 

.28225 

.80834 

•2371 

•83761 

.19387 

.75264 

.32865 

•78035 

.28148 

.80882 

•23637 

.83811 

.19316 

•7531 

•32785 

.78082 

.28071 

.8093 

•23563 

.8386 

.19246 

•75355 

1.32704 

.78129 

1.27994 

.80978 

1.2349 

.8391 

i-  19*  75 

CO-TAKG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG, 

CO-TANG. 

TANG. 

53° 

52° 

510 

50° 

NATURAL   TANGENTS    AND   CO-TANGENTS, 


425 


40° 

410 

42° 

43° 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TANG. 

•8391 

i-  191  75 

.86929 

I-I5037 

.9004 

1.  11061 

•93252 

1.072  37 

•8396 

.19105 

.8698 

.14969 

•  90093 

.10996 

•93306 

.071  74 

.84009 

•87031 

.14902 

.90146 

.10931 

•9336 

.071  12 

.84059 

.18964 

.87082 

.14834 

.90199 

.  108  67 

•93415 

.07049 

.84108 

.18894 

•87133 

.14767 

.90251 

.  108  02 

.93469 

.06987 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

•93524 

1.06925 

.84208 

.18754 

.87236 

.14632 

•9°357 

.10672 

•93578 

.06862 

.84258 

.18684 

.87287 

•14565 

.9041 

.10607 

•93633 

.068 

.84307 

.18614 

.87338 

.14498 

•  90463 

•10543 

.93688 

.06738 

•84357 

.18544 

•87389 

•1443 

.90516 

.  104  78 

•93742 

.06676 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

•93797 

1.  066  13 

•84457 

.18404 

.87492 

.  142  96 

.90621 

.10349 

•93852 

•06551 

•84507 

•18334 

•87543 

.  142  29 

.90674 

.10285 

.93906 

.06489 

•84556 

.18264 

•87595 

.141  62 

.90727 

.IO2  2 

.93961 

.064  27 

.84606 

.18194 

.87646 

.14095 

.90781 

.10156 

.94016 

.06365 

.84656 

1.18125 

.87698 

1.14028 

.90834 

I.IOOgi 

94071 

1.06303 

.84706 

.18055 

.87749 

.13961 

.90887 

.10027 

•941  25 

.06241 

.84756 

.17986 

.87801 

.13894 

•9°94 

.09963 

.9418 

.06179 

.84806 

.17916 

.87852 

.13828 

.90993 

.09899 

•94235 

.061  17 

.84856 

.17846 

.87904 

.13761 

.91046 

.09834 

.9429 

.06056 

.84906 

1.17777 

•87955 

1.13694 

.91099 

1.0977 

•94345 

1.05994 

.84956. 

.17708 

.88007 

.13627 

•9"  53 

.09706 

•944 

•05932 

.85006 

.17638 

.88059 

.13561 

.91206 

.09642 

•94455 

•0587 

•85057 

•17569 

.8811 

•13494 

.91259 

.09578 

•9451 

.05809 

.85107 

•175 

.88162 

.13428 

•913*3 

.09514 

•94565 

•05747 

•85157 

I-I743 

.88214 

1.13361 

.91366 

1.0945 

.9462 

1.056  85 

.85207 

.17361 

.88265 

.13295 

.91419 

.09386 

.94676 

.056  24 

•85257 

.17292 

.88317 

.13228 

•9M73 

.09322 

•94731 

.05562 

•85307 

.17223 

.88369 

.131  62 

.91526 

.09258 

.94786 

.0550! 

•85358 

.17154 

.88421 

.13096 

.9158 

.09195 

.94841 

•05439 

.85408 

1.17085 

•88473 

1.13029 

•9l633 

1.09131 

.94896 

1-05378 

.85458 

.17016 

.88524 

12963 

.91687 

.09067 

•94952 

•05317 

•85509 

.16947 

.88576 

.12897 

.9174 

.09003 

.95007 

•05255 

•85559 

.16878 

.88628 

.12831 

.91794 

.0894 

.95062 

.051  94 

.85609 

.16809 

.8868 

.12765 

.91847 

.088  76 

.95118 

•051  33 

.8566 

1.16741 

•88732 

1.12699 

.91901 

1.08813 

•95173 

1.05072 

•8571 

.16672 

•12633 

•91955 

.08749 

•95229 

•  0501 

.85761 

.16603 

.88836 

•  12567 

.92008 

.08686 

.95284 

.04949 

.85811 

•I6s35 

.88888 

.12501 

.92062 

.08622 

•9534 

.04888 

.85862 

.16466 

.8894 

•12435 

.921  16 

•08559 

•95395 

.048  27 

.85912 

1.16398 

.88992 

1.12369 

.9217 

1.08496 

•95451 

1.04766 

•85963 
.86014 

.16329 
.16201 

.89045 
.89097 

.12303 
.12238 

.92223 
.92277 

.08432 
.08369 

•955o6 
•95562 

.04705 
.04644 

.86064 

.16192 

.891  49 

.121  72 

•92331 

.08306 

.95618 

•04583 

.861  15 

.16124 

.89201 

.121  06 

•92385 

.08243 

•95673 

.045  22 

.86166 

1.16056 

.89253 

I.I204I 

.92439 

1.081  79 

•957  29 

1.04461 

.86216 

.15987 

.89306 

•"975 

.92493 

.081  16 

•95785 

.04401 

.86267 

.15919 

.89358 

.11909 

•92547 

•  08053 

.95841 

•0434 

.86318 

.158  51 

.8941 

.11844 

.92601 

•0799 

•95897 

.042  79 

.86368 

•15783 

.89463 

.11778 

•92655 

.07927 

•95952 

.042  18 

.86419 

•89515 

1.11713 

.92709 

1.07864 

.96008 

1.041  58 

.8647 

•15647 

.89567 

.11648 

.92763 

.07801 

.96064 

.04097 

.8652! 

•15579 

.8962 

.11582 

.92817 

•077  38 

.961  2 

.  040  36 

.86572 

•155" 

.89672 

•"5  X7 

.92872 

.07676 

.961  76 

•03976 

.86623 

•15443 

.89725 

.11452 

.92926 

.076  13 

.96232 

•03915 

.86674 

I-I5375 

.89777 

1.11387 

.9298 

1-0755 

.96288 

1-03855 

.86725 

.15308 

.8983 

.113  21 

•93034 

•07487 

•96344 

•03794 

.86776 

.1524 

.89883 

.11256 

.93088 

.07425 

.964 

•°3734 

.86827 

•15172 

.89935 

.III  91 

•93M3 

.07362 

•96457 

.03674 

.86878 

.15104 

.89988 

.III  26 

•93197 

.07299 

•96513 

•036  13 

.86929 

.9004 

I.  Ilo6l 

.93252 

1.07237 

.96569 

1-03553 

CO-TANG. 

TANG. 

CO-TANG. 

TANG. 

CO-TING. 

TANG. 

CO-TANG. 

TANG. 

490 

480 

470 

460 

NX* 

426 


NATURAL    TANGENTS   AND    CO-TANGENTS. 


4 

4° 

' 

TANG. 

CO-TANG. 

o 

.96569 

•035  53 

X 

.96625 

•03493 

2 

.96681 

•03433 

3 

.96738 

•03372 

4 

•96794 

.033  12 

5 

.9685 

.032  52 

6 

.96907 

03192 

7 

•96963 

.03132 

8 

.9702 

.03072 

9 

.97076 

.03012 

10 

•971  33 

.02952 

ii 

.97189 

.02892 

12 

.972  46 

.02832 

13 

.97302 

.02772 

14 

•97359 

.02713 

15 

.97416 

.02653 

16 

.97472 

•02593 

i7 

•97529 

•025  33 

18 

.97586 

.024  74 

J9 

•97643 

.024  14 

20 

~ 

•977 

•023  55 

CO-TAUG. 

TANG. 

4 

5° 

4 

4° 

4* 

to 

' 

' 

TANG. 

CO-TANG. 

' 

TANG. 

CO-TANG. 

i 

~6o~ 

21 

•977  56 

.02295 

39 

41 

.98901 

.Oil  12 

»9 

59 

22 

.022  36 

38 

42 

.98958 

.01053 

18 

58 

23 

[9787 

.021  76 

37 

43 

.99016 

.00994 

i7 

57 

24 

•97927 

.021  17 

36 

44 

•99073 

•09935 

16 

56 

25 

.97984 

.02057 

35 

45 

•991  3i 

.00876 

15 

55 

26 

.98041 

.01998 

34 

46 

.99189 

.00818 

J4 

54 

27 

.98098 

.01939 

33 

47 

.99247 

.00759 

13 

53 

28 

•98155 

.01879 

32 

48 

•99304 

.00701 

12 

52 

29 

.98213 

.Ol82 

49 

.99362 

.00642 

II 

30 

.9827 

.01761 

30 

50 

•9942 

.00583 

10 

50 
49 

31 
32 

•98327 
.98384 

.OI7O2 
.01642 

29 
28 

5i 
52 

.99478 
•99536 

•  00525 
.00467 

8 

48 

33 

.98441 

.01583 

27 

53 

•99594 

.00408 

7 

47 

34 

•98499 

•01524 

26 

54 

.99652 

•0035 

6 

46 

35 

•985  56 

.01465 

25 

.00291 

5 

45 

36 

.98613 

.01406 

24 

56 

.99768 

.00233 

4 

44 

37 

.98671 

•01347 

23 

57 

.99826 

•00175 

3 

43 

38 

.98728 

.01288 

22 

58 

.99884 

.001  16 

2 

42 

39 

.98786 

.01229 

21 

59 

.99942 

.00058 

;  .i 

41 

4° 

.98843 

.on  7 

20 

60 

i 

o 

40 

/ 

f 

CO-TANG. 

TANG. 

1 

~7~ 

CO-TANG. 

TANG. 

~T 

4 

50 

4 

50 

Preceding  Table  contains  Natural  Tangents  and  Co-tangents  for  every 
minute  of  the  quadrant,  to  the  radius  of  i. 

If  Degrees  are  taken  at  head  of  columns,  Minutes,  Tangents,  and  Co-tan- 
gents must  be  taken  from  head  also ;  and  if  they  are  taken  at  foot  of  col- 
umns, Minutes,  etc.,  must  be  taken  from  foot  also. 
ILLUSTRATION.— .1974  is  tangent  for  n°  10',  and  co-tangent  for  78°  50'. 
To  Compxite  Tangents   and.  Co-tangents  for  Seconds. 
Ascertain  tangent  or  co-tangent  of  angle  for  degrees  and  minutes  from 
Table ;  take  difference  between  it  and  tangent  or  co-tangent  next  below  it. 

Then  as  60  seconds  is  to  difference,  so  are  seconds  given  to  result  required, 
frhich  is  to  be  added  to  tangent  and  subtracted  from  co-tangent. 
ILLUSTRATION.— What  is  the  tangent  and  co- tangent  of  54°  40'  40"? 
SSSSX  S$#  P"?bto  =  'i;£j$}  .-8,  Serene, 
Then  60  :  .00087  '•''  4°  :  -00° 58,  which,  added  to  1.41061  =  1.41119  tangent. 
Co-tangent  of  54°  40',  per  Table—  .70801  )  ,.«. 

Co-tangent  of  5X1,        »        = . 708  48  }  •  °°° «  difference. 
Then  60°  :  .00043  : :  4°  :  .000  29,  which,  subt'd  from  .70891  =  .70862  co-tangent. 
To    Compute    Tangent   or   Co-tangent   of*    any   A.ngle    in 

Degrees,  IMinxites,  and    Seconds. 

Divide  Sine  by  Cosine  for  Tangent,  and  Cosine  by  Sine  for  Co-tangent. 
EXAMPLE.— What  is  tangent  of  25°  18'? 

Sine  =  .427  36 ;  cosine  =  .904 08.     Then  '4273   =  .4727  tangent. 

.90408 

To  Compute  Number  of*  Degrees,  Minutes,  and  Seconds 

of  a   given    Tangent   or   Co-tangent. 

When  Tangent  is  given. — Proceed  as  by  Rule,  page  402,  for  Sines,  substi- 
tuting Tangents  for  Sines. 
EXAMPLE. — What  is  tangent  for  1.411 19? 

Next  less  tangent  is  1.41061,  arc  for  which  is  54°  40'.     Next  greatest  tangent  is 
1.411  48,  difference  between  which  and  next  less  is  .00087. 

Difference  between  less  tabular  tangent  and  one  given  is  1.41061  — 1.411 19  =  .00058. 
Then  .00087  :  -00058  ::  60  :  40,  which,  added  to  54040  =54°  40'  40". 

When  Co-tangent  is  given.— Proceed  as  by  Rule,  page  402,  for  Cosines, 
substituting  Co-tangents  for  Cosines. 


AEROSTATICS.  427 

AEROSTATICS. 

Atmospheric  Air  consists,  by  volume,  of  Oxygen  21,  and  Nitrogen  79 
parts;  and  in  10000  parts  there  are  4.9  parts  of  Carbonic  acid  gas. 
By  weight,  it  consists  of  23  parts  of  Oxygen,  and  77  of  Nitrogen. 

One  cube  foot  of  Atmospheric  Air  at  surface  of  Earth,  when  barome- 
ter is  at  30  ins.,  and  at  a  temperature  of  32°,  weighs  565.0964  grains  = 
,080  728  Ibs.  avoirdupois,  being  773.19  times  lighter  than  water. 

Specific  gravity  compared  with  water,  at  62.418  =  .obi  293  345. 

Mean  weight  ot  a  column  of  air  a  foot  square,  and  of  an  altitude 
equal  to  height  of  atmosphere  (barometer  30  ins.),  is  2124.6875  Ibs.  = 
14.7548  Ibs.  per  sq.  inch  =  support  of  34.0393  feet  of  water. 

Standard  pound  is  computed  with  a  mercurial  barometer  at  30  ins. ;  hence, 
as  a  cube  inch  of  mercury  at  60°  weighs  .4907769  Ibs.,  pressure  of  atmos- 
phere at  60°  =  14.723307  Ibs.  per  square  men. 

12.3873  cube  feet  of  air  weigh  a  pound,  and  its  weight  varies  about 
I  gr.  for  each  degree  of  heat. 

Extreme  height  of  barometer  in  latitude  30°  to  35°  N.  =  3o.2i  his. 

Rate  of  expansion  of  Air,  and  all  other  Elastic  Fluids  for  all  temperatures, 
is  essentially  uniform.  From  32°  to  212°  they  expand  from  i  to  1.3665 
volumes  =  .002036  or  ^[rY^th  part  of  their  bulk  for  every  degree  of  heat 
From  212°  to  680°  they  expand  from  1.3665  to  2.3192  =  .002036  for  each 
degree  of  heat. 

Thus,  if  volume  of  air  at  132°  is  required,  132°  —  32°  =  100,  and  i  + 
100  X  .002036  =  1.2036  volumes. 

Height,  at  Equator  is  estimated  at  300  feet  greater  than  at  Poles,  its 
mean  height  at  45°  latitude. 

In  like  latitudes,  air  loses  i°  for  every  340  feet  in  height  above  level 
of  sea. 

Below  surface  of  Earth,  temperature  increases. 

Elasticity  of  air  is  inversely  as  space  it  occupies,  and  directly  as  its  density. 

When  altitude  of  air  is  taken  in  arithmetical  proportion,  its  Rarity  will  be 
in  geometric  proportion.  Thus,  at  7  miles  above  surface  of  Earth,  ah*  is  4 
times  rarer  or  lighter  than  at  Earth's  surface;  at  14  miles,  16  times;  at  21 
miles,  64  times,  and  so  on. 

Density  of  an  aeriform  fluid  mass  at  32°  and  at  t°  will  be  to  each  other 
as  i  +  .002  088  (t°  —  32°)  is  to  i. 

For  Volume,  Pressure,  and  Density  of  Air,  see  Heat,  page  521. 

Altitude  of  Atmosphere  at  ordinary  density  is  =  a  column  of  mercury  30 
ins.  in  height,  divided  by  specific  gravity  of  air  compared  with  mercury. 

Hence  30  ins.  =  2.5  feet,  which,  divided  by  .000094987,  specific  gravity 
of  air  compared  with  mercury,  =  263 igfeet=:  4.985  miles. 

Gay  Lussac,  Humboldt,  and  Boussingault  estimated  it  at  a  minimum  of 
30  miles,  Sir  John  Herschell  83,  Bravais  66  to  100,  Dalton  102,  and  Liais  at 
1 80  or  204  miles. 

The  aqueous  vapor  always  existing  in  air,  in  a  greater  or  less  quantity, 
being  lighter  than  ah*,  diminishes  its  weight  in  mixing  with  it ;  and  as,  other 
things  equal,  its  quantity  is  greater  the  higher  the  temperature  of  the  air,  its 
effect  is  to  be  considered  by  increasing  the  multiplier  of  t  by  raising  it  to 
.002  22. 

Glaisher  and  Coxwell,  in  i8624  ascended  hi  a  balloon  to  a  height  of  37  ooo 
feet 


428 


AEROSTATICS. 


At  temperature  of  32°,  mean  velocity  of  sound  is  1089  feet  per  second.  It 
is  increased  or  diminished  about  one  foot  for  each  degree  of  temperature 
below  or  above  32°. 

Velocity  of  sound  in  water  is  estimated  at  4750  feet  per  second. 

Velocity  of  /Sound  at  Various  Temperatures. 


0 

Per  Second. 

0 

Per  Second. 

0 

Per  Second. 

0 

Per  Sec.oi 

5 
i4 
23 

Feet. 
1056 
1070 
1079 

32 
50 
59 

Feet. 
1089 

IIO2 
III2 

68 

g 

Feet. 
1122 
1132 
1142 

95 
104 

«3 

Feet. 
1152 
1161 
1171 

Motions  of  air  and  all  gases,  by  force  of  gravity,  are  precisely  alike  to 
those  of  fluids. 

Sensation  of  hearing,  or  sound,  cannot  exist  in  an  absolute  vacuum.  The 
human  voice  can  be  heard  a  distance  of  3300  feet. 

Echo. — At  a  less  distance  than  100  feet  there  is  not  a  sufficient  interval 
between  the  delivery  of  a  sound  and  its  reflection  to  render  one  perceptible. 

To    Compute  Distances  "by  Velocity   of  Sound    in    Air. 

1089  X  T  V1  +  [.002  088  (t  —  32°)]  =  distance  in  feet  per  second,  T  representing  time 
before  report  was  heard,  and  t  temperature  of  air. 

ILLUSTRATION. — Flash  of  a  caunou  from  a  vessel  was  observed  13  seconds  before 
report  was  heard;  temperature  of  air  60°;  what  was  distance  to  vessel? 

1089  X  13 V i +  [-002088(60°  —  32°)]—  1089  X  13  X  1.029  —  J4  567-  5$feet=2.j6  miles. 
Theoretical  velocity  with  which  air  will  flow  into  a  vacuum,  if  wholly  un- 
obstructed, is  Vvgh  =  1347.4 /eetf  per  second.    In  operation,  however,  it  is 
1347.4  X  .707  =  952.6i  feet. 

To   Compute  "Velocity  of  A.ir    FloAving   into    a  "Vacuum. 

•\/2  g  h  X  c  =  v  in  feet  per  second,  c  representing  coefficient  of  efflux. 

Coefficients  for  openings  are  as  follows : 

Circular  aperture  in  a  thin  plate 65  (0.7 

Cylindrical  adjutage 92    |    Conical  adjutage 93 

Velocity   of  Sound,   in    Several    Solids. 
Velocity  in  Air=i. 

Lead 3.9  I  Zinc 9.8  I  Pine 12.5  I  Glass 11.9  I  Steel 14.3 

Gold 5.6  I  Oak 9.9  |  Copper  ...  11.2  |  Pine 12.5  |  Iron 15.1 

To    Compute    Elevations    "by    a    Barometer. 

Approximately  *  60  ooo  (log.  B  —  log.  6)  C  —  height  in  feet ;  B  and  b  representing 
heights  of  barometer  at  lower  and  upper  stations,  and  C  correction  due  to  T  -}- 1  or 
temperatures  of  lower  and  upper  stations. 

Values    of  C   or  T-f  t. 


0 

C 

0 

C 

0 

C 

o 

C 

0 

C 

0 

C 

O 

C 

4° 

•973 

60 

.996 

80 

.018 

oo 

.04 

20 

1.062 

140 

1.084 

1  60 

.  06 

42 

.976 

62 

.998 

82 

.02 

02 

.042 

22 

1.064 

142 

1.087 

162 

.  08 

44 

.978 

64 

84 

.022 

04 

.044 

24 

1.067 

144 

1.089 

164 

.  ii 

46 

.98 

66 

.002 

86 

.024 

06 

.047 

26 

1.069 

146 

1.09! 

166 

•  13 

48 

.982 

68 

.004 

88 

.027 

08 

.049 

28 

1.071 

148 

1-093 

168 

So 

.984 

70 

.007 

90 

.029 

IO 

.051 

30 

1-073 

ISO 

1.096 

170 

•  *7 

52 

•  987 

72 

.009 

92 

.031 

12 

•053 

32 

1.076 

1*2 

1.098 

172 

.  2 

54 

.989 

74 

.Oil 

Q4 

•033 

14 

.0*6 

134 

1.078 

r^4 

i.i 

I.  22 

56 

.991 

76 

.013 

96 

.036 

16 

1.058 

136 

1.  08 

is6 

I.IO2 

176 

I.  24 

58 

•993 

7» 

.016 

98 

.038 

118 

i.  06 

1.082 

i5y 

I.IO4 

178 

I.  26 

••  For  more  exact  formulas,  see  Tables  and  Formulas,  by  Capt.  T.  S.  Lee,  U.  S.  Top.  Eng.,  1853. 


AEROSTATICS. 


429 


Their  values  vary  approximately  .001 1  per  degree. 


Lower  Station. 
77.6 
30.05 

=  77.6  +  70.4  —  1.093,    log.  6  =  1.4778,    log.  6  =  1.374. 
Then  6ooooX  (1.4778  —  1.374)  X  1.093  =  6807.2/6^. 


Upper  Station. 

ILLUSTRATION. — Thermometer  70.4 

Barometer  23.66 


To   Compute    Elevations   Toy    a   Thermometer. 

520  B  +  B2  X  C  =  height  in  feet.  B  representing  temperature  of  water  boiling  at 
elevated  station  deducted  from  212°. 

Correction  for  temperatures  of  air  at  lower  and  upper  stations,  or  T  +  t,  to  be  taken 
from  table,  page  428,  as  before. 

ILLUSTRATION.— Temperature  of  water  boiling  at  upper  station  192°;  temperature 
of  air  50°  and  32°.  C  =  1.02. 

Then  520X212  —  192  +  212  —  192  X  1.02=:  io8o8/eefc 

To  Compute  Capacity  of  a  Balloon,  etc.,  see  page  218. 

B  ar  oxnet  er . 
Elevations   by   Barometer   Readings,     (Astronomer  Royal.) 

Mean  Temperature  of  Air  50°. 
For  correction  for  temperature,  see  note  at  foot. 


Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

o 

31 

600 

30.325 

1500 

29-34 

4000 

26.769 

7000 

23-979 

50 

30-943 

650 

30.269 

1600 

29.  233 

4250 

26.  524 

7  5°° 

23-543 

too 

30.886 

700 

30.214 

1750 

29.072 

4500 

26.  282 

8000 

23.115 

150 

30-83 

750 

30.159 

1800 

29.019 

4750 

26  042 

8500 

22.695 

2OO 

30-773 

800 

30.  103 

2000 

28.807 

5000 

25.804 

9000 

22.282 

250 

30.717 

850 

30.048 

2250 

28.544 

5250 

25-569 

9500 

21.877 

300 

30.661 

900 

29.993 

25OO 

28.283 

5500 

25-335 

0000 

21.479 

35° 

30.604 

IOOO 

29.883 

2750 

28.025 

5750 

25.  104 

0500 

21.089 

400 

30.548 

1  100 

29.774 

3000 

27.769 

6000 

24.875 

IOOO 

20.706 

450 

30.492 

I2OO 

29.665 

3250 

27-5I5 

6250 

24.648 

1500 

20.329 

500 

30-436 

1300 

29-556 

3500 

27.264 

6500 

24.423 

2000 

19.959 

550 

30-381 

1400 

29.448 

375° 

27.015 

6750 

24.2 

2  500 

I9-952 

Barometer. 

Correction  for  Capillary  Attraction  to  be  added  in  Inches. 


Diameter  of  tube 

Correction,  unboiled 

Correction,  boiled 


•  5 

.007 
004 


•4 

•55 

•3 

•25 

.2 

.1 

.014 

.02 

.025 

04 

.059 

.087 

.007 

OI 

.014 

.02 

029 

.044 

To    Compute   Heignt. 

RULE.— Subtract  reading  at  lower  station  from  reading  at  upper  station,  difference 
is  height  in  feet. 

Table  assumes  mean  temperature  of  atmosphere  to  be  50°  F.  or  10°  C.  For  other 
temperatures  following  correction  must  be  applied. 

Add  together  temperatures  at  upper  and  lower  station  If  this  sum,  in  degrees 
in  P.,  is  greater  than  100°,  increase  height  by  1  fa  Q  part  for  every  degree  of  excess 
above  100°;  if  sum  is  less  than  100°,  diminish  height  by  10100  part  for  every  degree 
of  defect  from  100°.  Or  if  sum  in  C°  is  greater  than  20°,  increase  height  by  ^1^ 
part  for  every  degree  of  excess  above  20°;  if  sum  is  less  than  20°,  diminish  height 
by  giff  part  for  every  degree  of  defect  from  20°. 

Barometer    Indications. 

Increasing  storm.— If  mercury  falls  during  a  high  wind  from  S.  W.,  S.  S.  W.,  W., 
or  S. 

Violent  but  short —If  fall  be  rapid. 

Less  violent  but  of  longer  continuance.— If  fall  be  slow. 

Snow. — If  mercury  falls  when  thermometer  is  low. 

Improved  weather.— When  a  gradual  continuous  rise  of  mercury  occurs  with  a 
falling  thermometer. 


43O  AEROSTATICS.^ 

Heavy  gales  from  N.—  Soon  After  first  rise  of  mercury  from  a  very  low  point 
Unsettled  weather. — With  a  rapid  rise  of  mercury. 
Settled  weather.— With  a  slow  rise  of  mercury. 

Very  tine  weather.— With  a  continued  steadiness  of  mercury  with  dry  air. 
Stormy  weather  with  rain  (or  snow).— With  a  rapid  and  considerable  fall  of  men 
cury. 

Threatening,  unsettled  weather.— With  an  alternate  rising  and  falling  of  mercury 
Lightning  only.—  When  mercury  is  low,  storm  being  beyond  horizon. 
Fine  weather. — With  a  rosy  sky  at  sunset. 
Wind  and  rain.— When  sky  has  a  sickly  greenish  hue. 
Rain. — When  clouds  are  of  a  dark  Indian  red. 
Foul  weather  or  much  wind.— When  sky  is  red  in  morning. 

"Weather   GJ-lasses. 

Explanatory-   Card.     Vice- Admiral  Fitzroy,  F.  R.  S. 
Barometer  Rises  for  Northerly  wind  (including  from  N.  W.  by  N.  to  E.),  for  dry, 
or  less  wet  weather,  for  less  wind,  or  for  more  than  one  of  these  changes — 
Except  on  a  few  occasions  when  rain,  hail,  or  snow  comes  from  N.  with  strong  wind. 
Barometer  Falls  for  Southerly  wind  (including  from  S.  E.  by  S.  to  W.),  for  wet 
weather,  for  stronger  wind,  or  for  more  than  one  of  these  changes — 
Except  on  a  few  occasions  when  moderate  wind  with  rain  (or  snow)  comes  from  N. 
For  change  of  wind  toward  Northerly  directions,  a  Thermometer  falls. 
For  change  of  wind  toward  Southerly  directions,  a  Thermometer  rises. 
Moisture  or  dampness  in  air  (shown  by  a  Hygrometer)  increases  before  rain,  fog, 
or  dew. 

Add  one  tenth  of  an  inch  to  observed  height  for  each  hundred  feet  Barometer  is 
above  half- tide  level. 

Average  height  of  Barometer,  in  England,  at  sea-level,  is  about  29.94  inches;  and 
average  temperature  of  air  is  nearly  50  degrees  (latitude  of  London). 

Thermometer  falls  about  one  degree  for  each  300  feet  of  elevation  from  ground, 
but  varies  with  wind. 

"  When  the  wind  shifts  against  the  sun, 

Trust  it  not,  for  back  it  will  run." 

First  rise  after  very  low  Long  foretold— long  last, 

Indicates  a  stronger  blow.  Short  notice — soon  past. 

Rarefaction  of  Air. 

In  consequence  of  rarefaction  of  air,  gas  loses  of  its  illuminating  power  i  cube 
Inch  for  each  2.69  feet  of  elevation  above  the  sea.  (M.  Bremond.) 

Clouds. 

Classification. — i.  Cirrus — Like  to  a  feather,  commonly  termed  Mare's 
tails.  2.  Cirro-cumulus —  Small  round  clouds,  termed  mackerel  sky. 
3.  Cirro-stratus — Concave  or  undulated  stratus.  4.  Cumulus — Conical, 
round  clusters,  termed  wool-packs  and  cotton  balls.  5.  Cumulo-stratus — 
Two  latter  mixed.  6.  Nimbus — A  cumulus  spreading  out  m  arms,  and 
precipitating  rain  beneath  it.  7.  Stratus — A  level  sheet. 
NOTE. — Cirrus  is  most  elevated. 

Height. — Clouds  have  been  seen  at  a  greater  height  than  37000  feet. 

Velocity. — At  an  apparent  moderate  speed,  they  attain  a  velocity  of  80 
miles  per  hour. 

Lightning. 

Classification. — i.  Striped  or  Zigzag — Developed  with  great  rapidity. 
2.  Sheet — Covering  a  large  surface.  3.  Globular — When  the  electric 
fluid  appears  condensed,  and  it  is  developed  at  a  comparatively  lower 
Telocity.  4.  Phosphoric  —  When  the  flash  appears  to  rest  upon  the 
edges  of  the  clouds. 


AEROSTATICS. — ATMOSPHERIC    AIR. 


431 


WEATHER  INDICATIONS. 


Sky. 

Gray  in  morning  and  light, 
delicate  tints  and  low  dawn. 

High  dawn,  and  sunset  of  a 
bright  yellow. 


Sunset  of  a  pale  yellow. 
Orange  or  copper  color. 

Gaudy  unusual  hues. 


Weather.  Clouds. 

Fine  and  Soft  or  delicate  looking  and  in- 

Fair,  definite  outlines. 

Wind.  Hard  •  edged,  oily  -  looking,  and 

tawny  or  copper-colored,  and  the 
more  hard,  "greasy,"  and  ragged, 
the  more  wind. 
Wind  only.  Light  scud  alone. 

Rain.  Small  and  inky. 

Wind  and  Light  scud  driving  across  heavy 

Rain.  masses. 

Rain  and  Hard  defined  outlines. 

Wind. 

Change  of  High  upper,  cross  lower  in  a  di- 

Wind.  rection  different  to  their  course  or 

that  of  wind. 

G-eneral* 

Fair.— When  sea-birds  fly  early  and  far  out,  when  dew  is  deposited,  and  when  a 
leech,  confined  in  a  bottle  of  water,  will  curl  up  at  the  bottom. 

Rain. — Clear  atmosphere  near  to  horizon  and  light  atmospheric  pressure,  or  a 
good  "hearing  day,"  as  it  is  termed. 

Storm.— When  sea-birds  remain  near  to  shore  or  fly  inland. 

Rain,  Snow,  or  Wind.— When  a  leech,  confined  in  a  bottle  of  water,  will  rise  ex- 
citedly to  the  surface. 

Thunder.— When  a  leech,  confined  as  above,  will  be  much  excited  and  leave  the 
water. 

Value   of  Indications   of  Fair   Weather,  in   Days,  Coin- 
pared,    to   one   of  Rain. 

From  an  extended  series  of  observations.     (Lowe.\ 
Profuse  Dew. 4. 5  I  Mock  Sun  or  Moon. . 


White  Stratus  in  a  valley 7. 2  |  Stars  falling  abundant 3.2 

Colored  Clouds  at  sunset 2.9    Stars  bright 3.4 


Solar  Halo. . 


1.9 

Sun  red  and  rayless 10. 3 

Sun  pale  and  sparkling i 

White  Frost 4.2 

Lunar  Halo i 

Lunar  burr,  or  rough-edged 2.8 


Moon  dim 2 

Moon  rising  red 7 


Stars  dim.. 

Stars  scintillated. .........  .......    6 " " 

Aurora  borealis., 


Toads  in  evening. 2. 4 

Landrails  noisy.   13 

Ducks  and  Geese  noisy 2.3 


Fish  rising i. 5 

Smoke  rising  vertically. 5 

For  weather-foretelling  plants,  see  page  185. 


ATMOSPHERIC    AIR. 

Very  pure  air  contains  Oxygen  20.96,  Nitrogen  79,  and  Carbonic  Acid  .04. 

Air  respired  by  a  human  being  in  one  hour  is  about  15  cube  feet,  produc- 
ing 500  grains  of  carbonic  acid,  corresponding  to  137  grains  carbon,  and 
during  this  time  about  200  grains  of  water  will  be  exhaled  by  the  lungs. 

During  this  period  there  would  be  consumed  about  415  grains  of  oxygen. 

In  one  hour,  then,  there  would  be  vitiated  73  cube  feet  pure  air. 

A  man,  weighing  150  Ibs.,  requires  930  cube  feet  of  air  per  hour,  in  order 
that  the  air  he  breathes  may  not  contain  more  than  i  per  1000  of  carbonic 
acid  (at  which  proportion  its  impurity  becomes  sensible  to  the  nose):  he 
ought,  therefore,  to  have  800  cube  feet  of  well  ventilated  space. 


432 


ATMOSPHERIC    AIR. — ANIMAL    POWER. 


An  adult  human  being  consumes  in  food  from  145  to  165  grains  of  carbon 
per  hour,  and  gives  off  from  12  to  16  cube  feet  of  carbonic  acid  gas. 

An  assemblage  of  1000  persons  will  give  off  in  two  hours,  in  vapor,  8.5 
gallons  water,  and  nearly  as  much  carbon  as  there  is  in  56  Ibs.  of  bitumi- 
nous coal. 

Proportion    of  Oxygen    and.   Carbonic   A.cid.    at    following 
.Locations. 

Pure  Air  represented  by  Oxygen  20.96. 


Street  in  Glasgow 20. 895 

Regent  Street,  London 20. 865 

Centre  Hyde  Park 21.005 


Metropolitan  Railway  (underground)  .  .  20.6 
Pit  of  a  Theatre  ......................  20.74 

Gallery  of  a  Theatre  ..................  20.63 


Carbonic  Acid  .04  Per  cent. 


Open  field,  Manchester 0383 

Churchyard 0323 

Market,  Smithfield 0446 

Factory  mills 283 

School-rooms 097 

Pitt  of  theatre,  n  P.  M 32 

Boxes      "         12     "     218 

Gallery    "         10    "     101 

*  Roscoe. 


Top  of  Monument,  London 0398 

Hyde  Park 0334 

Metropolitan  Railway  (underground)..  .338 

Lake  of  Geneva 046 

Boys1  school 31* 

Girls'      "      723! 

Horse  stable 7 

Convict  prison 045 

t  Peltenhoffer. 


Consumption   of  Atmospheric   A.ir.     (Coathupe.) 

One  wax  candle  (three  in  a  Ib.)  destroys,  during  its  combustion,  as  much 
oxygen  per  hour  as  respiration  of  one  adult. 

A  lighted  taper,  when  confined  within  a  given  volume  of  atmospheric  air, 
will  become  extinguished  as  soon  as  it  has  converted  3  per  cent,  of  given 
volume  of  air  into  carbonic  acid. 

Carbonic  Acid  Exhaled  per  Minute  by  a  Man.     (Dr.  Smith.) 
During  sleep  4.99  per  cent.,  lying  down  5.91,  walking  at  rate  of  2  miles 
per  hour  18.1,  at  3  miles  25.83,  hard  labor  44.97. 


ANIMAL   POWER. 
"Work. 

Work  is  measured  by  product  of  the  resistance  and  distance  through 
which  its  point  of  application  is  moved.  In  performance  of  work  by 
means  of  mechanism,  work  done  upon  weight  is  equal  to  work  done  by 
power. 

Unit  of  Work  is  the  moment  or  effect  of  i  pound  through  a  distance 
of  i  foot,  and  it  is  termed  a  foot-pound. 

In  France  a  kilogrammetre  is  the  expression,  or  the  pressure  of  a 
kilogramme  through  a  distance  of  i  meter  =  7.233  foot-pounds. 

Result  of  observation  upon  animal  power  furnishes  the  following  as  maximum 
daily  effect: 

1.  When  effect  produced  varied  from  .2  to  .33  of  that  which  could  be  produced 
without  velocity  during  a  brief  interval. 

2.  When  the  velocity  varied  from  .16  to  .25  for  a  man,  and  from  .08  to  .066  for  a 
horse,  of  the  velocity  which  they  were  capable  for  a  brief  interval,  and  not  involv- 
ing any  effort. 

3.  When  duration  of  the  daily  work  varied  from  .33  to  .5  for  a  brief  interval, 
during  which  the  work  could  be  constantly  sustained  without  prejudice  to  health 
of  man  or  animal;  the  time  not  extending  beyond  18  hours  per  day,  however  lim- 
ited may  be  the  daily  task,  so  long  as  it  involved  a  constant  attendance. 


ANIMAL   POWER.  433 

Men. 

Mean  effect  of  power  of  men  working  to  best  practicable  advantage,  is 
raising  of  70  Ibs.  i  foot  high  in  a  second,  for  10  hours  per  day  =  4200  foot- 
pounds per  minute. 

Windlass. — Two  men,  working  at  a  windlass  at  right  angles  to  each  other, 
can  raise  70  Ibs.  more  easily  than  one  man  can  30  Ibs. 

Labor.— PL  man  of  ordinary  strength  can  exert  a  force  of  30  Ibs.  for  m 
hours  in  a  day,  with  a  velocity  of  2.5  feet  in  a  second  =  4500  Ibs.  raised  one 
foot  in  a  minute  =  .2  of  work  of  a  horse. 

A  man  can  travel,  without  a  load,  on  level  groundj  during  8.5  hours  a  day, 
at  rate  of  3.7  miles  an  hour,  or  31.45  miles  a  day.  He  can  carry  in  Ibs. 
1 1  miles  in  a  day.  Daily  allowance  of  water,  i  gallon  for  ah1  purposes ;  and 
he  requires  from  220  to  240  cube  feet  of  fresh  air  per  hour. 

A  porter  going  short  distances,  and  returning  unloaded,  can  carry  135  Ibs. 
7  miles  a  day,  or  he  can  transport,  hi  a  wheelbarrow,  150  Ibs.  10  miles  in  a 
day. 

Crane. — The  maximum  power  of  a  man  at  a  crane,  as  determined  by  Mr. 
Field,  for  constant  operation,  is  15  Ibs.,  exclusive  of  frictional  resistance, 
which,  at  a  velocity  of  220  feet  per  minute  =  3300  foot-pounds,  and  when 
exerted  for  a  period  of  2.5  minutes  was  17.329  foot-pounds  per  minute. 

Pile-driving. — G.  B.  Bruce  states  that,  in  average  work  at  a  pile-driver,  a 
laborer,  for  10  hours,  exerts  a  force  of  16  Ibs.,  plus  resistance  of  gearing,  and 
at  a  velocity  of  270  feet  per  minute,  making  one  blow  every  four  minutes. 

Rowing. — A  man  rowing  a  boat  i  mile  in  7  minutes,  performs  the  labor 
of  6  fully-worked  laborers  at  ordinary  occupations  of  10  hours  per  day. 

Drawing  or  Pushing. — A  man  drawing  a  boat  in  a  canal  can  transport 
1 10  ooo  Ibs.  for  a  distance  of  7  miles,  and  produce  156  times  the  effect  of  a 
man  weighing  154  Ibs.,  and  walking  31.25  miles  in  a  day ;  and  he  can  push 
on  a  horizontal  plane  20  Ibs.  with  a  velocity  of  2  feet  per  second  for  10  hours 
per  day. 

Tread-mill. — A  man  either  inside  or  outside  of  a  tread-mill  can  raise  30 
Ibs.  at  a  velocity  of  1.3  feet  per  second  for  10  hours,  =  1 404  ooo  foot-pounds. 

Pulley. — A  man  can  raise  by  a  single  pulley  36  Ibs.,  with  a  velocity  of  .8 
of  a  foot  per  second,  for  10  hours. 

Walking. — A  man  can  pass  over  12.5  times  the  space  horizontally  that  he 
can  vertically,  and,  according  to  J.  Robison,  by  walking  in  alternate  directions 
upon  a  platform  supported  on  a  fulcrum  in  its  centre,  he  can,  weighing  165 
Ibs.,  produce  an  effect  of  3  984  ooo  foot-pounds,  for  10  hours  per  day. 

Pump,  Crank,  Bell,  and  Rowing. — Mr.  Buchanan  ascertained  that,  in  work- 
ing a  pump,  turning  a  crank,  ringing  a  bell,  and  rowing  a  boat,  the  effective 
power  of  a  man  is  as  the  numbers  100,  167,  227,  and  248. 

Pumping. — A  practised  laborer  can  raise,  during  10  hours,  i  ooo  ooo  Ibs. 
water  i  foot  in  height,  with  a  properly  designed  and  constructed  pump. 

Crank. — A  man  can  exert  on  the  handle  of  a  screw-jack  of  n  inches  ra- 
dius for  a  short  period  a  force  of  25  Ibs.,  and  continuously  15  Ibs.,  a  net 
power  of  20  Ibs.  Mr.  J.  Field's  tests  gave  11.5  Ibs.  as  easily  attained,  17.3  as 
difficult,  and  27.6  with  great  difficulty. 

Mowing. — A  man  can  mow  an  acre  of  grass  in  i  day. 

Reaping. — A  man  can  reap  an  acre  of  wheat  in  2  days. 

Ploughing. — A  man  and  horse  .8  of  an  acre  per  day. 
Oo 


434 


ANIMAL    POWER. 


Day's   Work.     (D.  K.  Clark.) 

Laborer.—  Carrying  bricks  or  tiles,  net  load  106  lbs.=  6oo  Ibs.  i  mile. 

Carrying  coal  in  a  mine,  net  load  95  to  115  Ibs.  =  342  Ibs.  i  mile. 

Loading  coke  into  a  wagon,  net  load  100  Ibs.  =  270  Ibs.  i  mile. 

Loading  a  boat  with  coal,  net  load  190  lbs.=  1230  Ibs.  i  mile,  or  20  cube  yards  of 
earth  in  a  wagon. 

Digging  stubble  land  .055  of  an  acre  per  day,  or  2000  cube  feet  of  superficial  earth. 

Breaking  1.5  cube  yards  hard  stone  into  2  inch  cubes. 

Quarrying.— A  man  can  quarry  from  5  to  8  tons  of  rock  per  day. 

A  foot-soldier  travels  in  i  minute,  in  common  time,  90  steps  =  70  yards. 

He  occupies  in  ranks  a  front  of  20  inches,  and  a  depth  of  13,  without  a  knapsack: 
interval  between  the  ranks  is  13  inches. 

Average  weight  of  men,  150  Ibs.  each,  and  five  men  can  stand  in  a  space  of  i 
square  yard. 

Effective    iPovsrer   of   !M!en    for   a    Short    JPeriod.. 


Manner  of  Application. 

Force. 

Manner  of  Application. 

I  orce. 

Lbs. 

Screw-driver  one-hand 

Lbs. 

8. 

Drawing-knife  or  Auger  

IOO 

5° 

Small  screw-driver  

H 

Hand  saw.  .. 

afi 

Windlass  or  Pincers  .  .  . 

& 

The  muscles  of  the  human  jaw  exert  a  force  of  534  Ibs. 

Mr.  Smeaton  estimated  power  of  an  ordinary  laborer  at  ordinary  work  was  equiv- 
alent to  3762  foot-pounds  per  minute.  But,  according  to  a  particular  case  made  by 
him  in  the  pumping  of  water  4  feet  high,  by  good  English  laborers,  their  power  was 
equivalent  to  3904  foot-pounds  per  minute;  and  this  he  assigned  as  twice  that  of 
ordinary  persons  promiscuously  operated  with. 

Mr.  J.  Walker  deduced  from  experiments  that  the  power  of  an  ordinary  laborer,  in 
turning  a  crank,  was  13  Ibs.,  at  a  velocity  of  320  feet  per  minute  for  8  hours  per  day. 

A.ixioiir».t   of  Labor   prod. viced,  toy   a    JVtaii.     (Morin.) 
For  10  hours  per  day. 


MANNER  OF  APPLICATION. 

Power. 

Velocity 
per 
Second. 

Weight 
raised. 
Feet  per 
Minute. 

H> 

for 
Period 
given. 

Throwing  earth  with  a  shovel,  a  height  of  5  feet.  . 
Wheeling  a  loaded  barrow  up  an  inclined  plane, 

!   tO  12    

Lbs. 
6 

132 
6 

13 
132 

H3 

26 
18 
140 
26 

88 

140 

140 

44 
n  an  indiv 
foot  per  s 

Feet. 
i-33 

.625 
2.25 

2-5 

I 
•5 

2 

2-5 

•5 
5 

2-5 

i-75 

.2 

•5 
idual  case 
econd;  h« 

Lbs. 
480 

4950 
810 
1950 
7920 

4290 
3  120 

2790 

4  200 
7800 

13200 

14700 

1680 
1320 

,  at  140  Ib 
nee  70-7- 

No. 
8-7 

9° 
14.7 

35-5 
144 

62 

45-2 
39 
61.1 

"3 

160.5 

160.5 

*9 
14.4 

s.,  at  a  ve- 
1.3  feet  ae 

Raising  and  pitching  earth  in  a  shovel  13  feet 
horizontally    

Pushing  and  drawing  alternately  in  a  vertical 
direction                       

Transporting  weight  upon  a  barrow,  and  return- 
ing unloaded           

FOR  8  HOURS  PER  DAY. 
Ascending  a  slight  elevation  unloaded 

Walking,  and  pushing  or  drawing  in  a  horizontal 

FOR  7  HOURS  PER  DAY. 
Walking  with  a  load  upon  his  back           

FOR  6  HOURS  PER  DAY. 
Transporting  a  weight  upon  his  back,  and  return- 
ing unloaded  

Transporting  a  weight  upon  his  back  up  a  slight 
elevation  and  returning  unloaded                     . 

Raising  a  weight  by  his  hands         

*  Morin  gives  amount  of  labor  of  a  man  upon  tread-mill,  i 
locity  of  .5  feet  per  second  for  8  hours  per  day  =  70  Ibs.  at  i 

ANIMAL   POWEE. 


435 


To   Compute   Number   of  M.en   to  Perform  Work:   upon 
a   Tread-mill   or   file-driver. 

RULE. — To  product  of  weight  to  be  raised  and  radius  of  crank,  add  fric- 
tion of  wheel,  and  divide  sum  by  product  of  power  and  radius  of  wheel. 

EXAMPLE. — How  many  men  are  required  upon  a  tread-mill,  20  feet  in  diameter, 
to  raise  a  weight  of  9233.33  IDS.,  crank  9  inches  in  length,  weight  of  wheel  and  its 
load  estimated  at  5000  Ibs.,  and  friction  at  .015. 

Weight  of  a  man  assumed  at  25  Ibs.     Radius  of  crank  .75  feet. 

Effect  of  a  man  on  a  tread-mill,  page  433,  30  Ibs.  at  a  velocity  of  1.3  feet  per  second, 
=  i.  3  X  60  =  jBfeet  per  minute. 

9233-33  X  .75  +  5000  x  .015  =  7000  Ibs.  resistance  of  load  and  wheel,  and  7000-4- 

X  10  X  30  ==  7000  =  load  and  weight  -l- product  of  power  increased  by  its 

velocity  over  load,  radius  of  wheel  and  power  =  7000  -f- 1.241  x  ioX  30  =  18. 8  men. 

Horse. 

-A.mou.nt  of  Labor  produced  by  a  Morse  under  different 
Circumstances.    (Morin.) 

For  10  hours  per  day. 


MANNER  OF  APPLICATION. 

Power. 

Velocity 
per 
Second. 

Weight 
drawn. 
Feet  per 
Minute. 

IP 

for 
Period 
given. 

Drawing  a  4,-wheeled  carriage  at  a  walk 

Lbs. 

Feet. 

Lbs. 

No. 

With  load  upon  his  back  at  a  walk  

26,, 

5°4 
1080 

Transporting  a  loaded  wagon,  and  returning  un- 

184800 

fi 

FOR  8  HOURS  PER  DAY. 

260  8 

FOR  4.5  HOURS  PER  DAY. 
Upon  a  revolving  platform  at  a  trot  

66 

3 

6   7C 

2l8  7 

Drawing  an  unloaded  4-wheeled  carriage  at  a  trot. 
Drawing  a  loaded  4  wheeled  carriage  at  a  trot  

97 
77° 

7-25 
7-25 

43195 
334  950 

353-5 
2741 

If  traction  power  of  a  horse,  when  continuously  at  a  walk,  is  equal  to  120  Ibs., 
and  grade  of  road  i  in  30,  resistance  on  a  level  being  one  thirtieth  of  load,  he  can 
draw  a  load  of  120  X  30  -4-  2  =  1500  Ibs. 

Street   nails   or  Tramways.     (Henry  Hughes.) 
Cars,  26  Ibs.  per  ton,  or  i  to  86  as  a  mean. 

Performance   of  Horses   in    France.     (M.  CharU-Marsaines.) 


•    SEASON. 

Road. 

Weight 
Horse. 

Speed 
HPour. 

Work  per 
Hour,  drawn 
One  Mile. 

Ratio  of 
Pavement  to 
Macadam. 

Winter  

(  Pavement 

Tone. 
1.306 

Miles. 
2.05 

Ton-miles. 
2.677} 

1.644  tO  i 

Summer  

\  Macadam 
|  Pavement 

.851 
1-395 

1.91 
2.17 

1.625) 
3.027) 

1.229  ^°  x 

I  Macadam 

1.141 

2.l6 

2.464) 

Average  daily  work  of  a  Flemish  horse  in  North  of  France,  where  country  is  flat 
and  loads  heavy,  is,  on  same  authority,  as  follows: 

Winter,    21.82  ton-miles  per  day. }  M       for  th 
Summer,  27.82  ) 

given  in  example  =  53.8  Ibs.,  from  which  a  deduction  is  to  be  made  for  excess  of  amount  of  labor  that 
can  be  performed  in  8  hours  over  10.  Or,  as  10  :  8  ; ;  53.8  :  43.04  lt».,  which  doe*  not  essentially  differ 
from  effect  of  30  Ibs.  for  that  of  an  average  performance. 


436  ANIMAL    POWER. 

Greatest  mechanical  effect  of  an  ordinary  horse  is  produced  in  operating  a 
gin  or  drawing  a  load  on  a  railroad,  when  travelling  at  rate  of  2.5  miles  per 
hour,  where  he  can  exert  a  tractive  force  of  150  Ibs.  for  8  hours  per  day. 

Horse  upon  Turnpike  Road. 

At  a  speed  of  10  miles  per  hour,  a  horse  will  perform  13  miles  per  day  for 
3  years.  In  ordinary  staging,  a  horse  will  perform  15  miles  per  day. 

To  Compute  Tractive  Power  of  a  Horse  Team,  see  Traction,  page  848. 

Assuming  maximum  load  that  a  horse  can  draw  on  a  gravel  road  as  a 
standard,  he  can  draw, 

On  best-broken  stone  road 2  to   3  times. 

On  a  well-made  stone  pavement 3  to   5     " 

On  a  stone  trackway 7  to   8     *' 

On  plank  road 41012     " 

On  a  railway 18  to  20     " 

NOTE. — Track  of  an  iron  railway  compared  with  a  plank-road  is  as  27  to  10. 

To  Comptite  I»ower  of  Draught  of  a  Horse  at  Different 
Elevations. 

Let  ABC  represent  an  inclined  plane,  o  weight 
of  a  horse  which,  being  resolved  into  two  com- 
ponent forces,  one  of  which,  n,  is  perpendicular  to 
plane  of  inclination,  and  other,  r,  is  parallel  to  it. 

Hence,  r  represents  force  which  horse  must  over- 
come to  move  his  own  weight. 

Then,  by  similar  triangles,  A  B  or  I :  B  C  or  h  : :  o  :  r.    Or,  -T-  =  r. 

If  t  represents  tractive  power  of  horse,  upon  a  level,  of  100  Ibs.,  t'  tractive 
power  upon  a  plane  of  inclination,  and  r  that  part  of  force  exerted  by  horse 

which  is  expended  upon  his  own  body,  then  t  =  t  —  r,  or  t  —  —  =  t'  in  Ibs. 

ILLUSTRATION.— If  inclination  is  i  in  50. 

Assume  t  =  100,  weight  of  horse  900  Ibs.,  and  I  =  50.01. 

Then,  100 —  —  100  — 17.99  =  82.01  Ibs. 

Assuming  load  that  a  horse  can  draw  on  a  level  at  100,  he  can  draw  upon 
inclinations  as  follows : 


i  m  ioo  .....  91 


;  in  75. 


70 87 

i  "  60 85 


i  in  50 82 

i  "  45 80 

1  "  40 77 


in  35 74 

"  30 70 

"  25 64 


i  in  20  .....  55 
i  "  15  .....  40 


IO 


On  his  back  a  horse  can  carry  from  220  to  390  Ibs.,  or  about  27.5  per  cent. 

Labor.—  The  work  of  a  horse  as  assigned  by  Boulton  &  Watt,  Tredgold, 
Rennie,  Beardmore,  and  others,  ranges  from  20600  to  39320  foot-pounds  per 
minute  for  8  hours,  a  mean  of  27  750  Ibs. 

A  horse  can  travel,  at  a  walk,  400  yards  in  4.5  minutes  ;  at  a  trot,  in  2 
minutes  ;  and  at  a  gallop,  in  i  minute.  He  occupies  in  ranks,  a  front  of  40 
ins.,  and  a  depth  of  10  feet;  in  a  stall,  from  3.5  to  4.5  feet  front;  and  at  a 
picket,  3  feet  by  9  ;  and  his  average  weight  =  1000  Ibs. 
.  Carrying  a  soldier  and  his  equipments  (225  Ibs.)  he  can  travel  25  miles 
m  a  day  of  8  hours. 

A  draught-horse  can  draw  1600  Ibs.  23  miles  a  day,  weight  of  carriage  in- 
cluded. 


ANIMAL   POWER. 


437 


Ordinary  work  of  a  horse  may  be  stated  at  22  500  Ibs.,  raised  i  foot  in  & 
minute,  for  8  hours  per  day. 

In  a  mill,  he  moves  at  rate  of  3  feet  in  a  second.  Diameter  of  track  should  not 
be  less  than  25  feet. 

Rennie  ascertained  that  a  horse  weighing  1232  Ibs.  could  draw  a  canal-boat 
at  a  speed  of  2.5  miles  per  hour,  with  a  power  of  108  Ibs.,  20  miles  per  day. 
This  is  equivalent  to  a  work  of  23  760  foot-lbs.  per  minute,  He  estimated 
that  the  average  work  of  horses,  strong  and  weak,  is  at  the  rate  of  22  ooo 
foot-lbs.  per  minute. 

From  results  of  trials  upon  strength  and  endurance  of  horses  at  Bedford,  Eng.,  it 
was  determined  that  average  work  of  a  horse  -=^  20000  foot-lbs.  per  minute.  A  good 
horse  can  draw  i  ton  at  rate  of  2.5  miles  per  hour,  from  10  to  12  hours  per  day. 

Expense  of  conveying  goods  at  3  miles  per  hour,  per  horse  teams  being  i,  expense 
at  4.33  miles  will  be  1.33,  and  so  on,  expense  being  doubled  when  speed  is  5. 125  miles 
per  hour. 

Strength  of  a  horse  is  equivalent  to  that  of  5  men,  and  his  daily  allowance  of 
water  should  be  4  gallons. 

-A.inoti.iit  of  La"bor  a  Morse  of  average  Strength  is  capa- 
ble of  performing,  at  different  "Velocities,  on  Canal, 
Railroad.,  and  Toarnpike. 

Traction  estimated  at  83.3  Ibs. 


Veloci- 

Dura- 

Useful Effect,  drawn  i  Mile. 

Veloci- 

Dura- 

Useful Effect,  drawn  i  Mile. 

ty  per 
Hour. 

tion  of 
Work. 

On  a 

Canal. 

On  a  Rail- 
road. 

On  a  Turn- 
pike. 

ty  per 
Hour. 

tion  of 
Work. 

On  a 

Canal. 

On  a  Rail- 
road. 

On  a  Turn- 
pike. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

2-5 

"•5 

520 

"5 

14 

6 

2 

30 

48 

6 

3 

8 

243 

92 

12 

7 

1-5 

19  0 

41 

5-1 

4 

4-5 

102 

72 

9 

8 

I.I25 

12.8 

36 

4-5 

5 

2.9 

52 

57 

7-2 

10 

•75 

6.6 

28.8 

3-6 

Actual  labor  performed  by  horses  is  greater,  but  they  are  injured  by  it. 

Tractive  Power  of  a  horse  decreases  as  his  speed  is  increased,  and  within  limits 
of  low  speed,  or  up  to  4  miles  per  hour,  it  decreases  nearly  in  an  inverse  ratio. 

For  10  Hours  per  Day. 


Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction. 

Per  Hour. 
75 

i 

1-25 

Lbs. 
330 
250 
200 

Per  Hour. 
1-5 
r-75 

2 

Lbs. 
i65 
140 
«5 

Per  Hour. 
2.25 
2-5 

2-75 

Lbs. 
no 

100 
QO 

Per  Hour. 
3 
3-5 
4 

Lbs. 
82 
70 
62 

Fc 

Miles  per  hour 
Power  in  Ibs.  .  . 

r  Ordinal 

y  or  Short  Periods.     (Molesworth.  ) 

2                33-54            4-5              5 
166             121;             104            83          62                41 

M:\ale.    (D.K.Clark.) 

Load  on  back,  170  to  220  Ibs.  day's  work  =  6400  Ibs.  i  mile  ;  400  Ibs.  at  2.9 
miles  per  hour  =,  5300  Ibs.  i  mile,  and  330  Ibs.  at  2  miles  per  hour  =  5000  Ibs. 
i  mile. 

Upon  a  revolving  platform,  at  a  velocity  of  3  feet  per  second,  =  n  880  Ibs.  raised 
one  foot  per  minute,  or  172.2  IP  for  8  hours  per  day 


Load  on  back,  176  Ibs.  carried  19  miles  day's  work  =  3300  Ibs.  i  mile. 
In  Syria  an  ass  carries  450  to  550  Ibs.  grain. 

Upon  a  revolving  platform,  at  a  velocity  of  2.75  feet  per  second,  =  5280  Ibs.  raised 
one  foot  per  minute,  or  76.5  IP  for  8  hours  per  day. 

Go* 


438 


ANIMAL    POWER. 


Ox. 

An  Ox,  walking  at  a  velocity  of  2  feet  in  a  second  (1.36  miles  per  hour), 
exerts  a  power  of  154  lbs.,  =  18480  Ibs.  raised  one  foot  per  minute,  or 
268.8  IP  for  8  hours  per  day. 

A  pair  of  well-conditioned  bullocks  in  India  have  performed  work  =  8000  foot-lbs. 
per  minute. 

Camel. 

Load  on  back,  550  Ibs.  carried  30  miles  per  day  for  4  days,  4  days'  work 
16  500  Ibs.  i  mile,  for  5  days  13  ooo  Ibs.  i  mile  =  44  IP  for  10  hours  per  day, 

Load  of  a  Dromedary,  770  Ibs. 

Llama. 

Load  on  back,  no  Ibs.,  day's  work  2000  to  3000  Ibs.  i  mile  =  .5  to  .75  H> 
for  10  hours  per  day. 

Birds    and    Insects. 

Area  of  their  wing  surface  is  in  an  inverse  ratio  to  their  weight. 

Assuming  weight  of  each  of  the  following  Birds  to  be  one  pound,  and  each  Insect 
one  ounce,  the  relative  area  of  their  wing  surface  proportionate  to  that  of  their  act- 
ual weight  would  be  as  follows  (M.  De  Lucy) : 


Sq.  ft. 

Swallow 4.85 

Sparrow  ....  2. 7 
Turtle-dove..  2.13 


Sq.  ft. 

Pigeon 1.27 

Vulture 82 

Crane,  Australia,   .41 


Sq.  ft. 

Gnat 3.05 

Dragon-fly,  sm'll,  1.83 
Lady-bird 1.66 


Sq.  ft. 

Cockchafer ...  32 

Bee 33 

Meat-fly 35 


Crocodile    and    IDog. 

The  direct  power  of  their  jaws  is  estimated  at  120  Ibs.  for  the  former  and 
44  for  the  latter,  which,  with  the  leverage,  will  give  respectively  6000  and 
1500  Ibs. 

PERFORMANCES   OF   MEN,  HORSES,   ETC. 
Following  are  designed  to  furnish  an  authentic  summary  of  the  fastest  or 
most  successful  recorded  performances  in  each  of  the  feats,  etc.,  given. 

MAN.      Walking. 

1874,  Wm.  Perkins,  London,  Eng.,  .5  mile,  in  2  min.  56  sec.;  i,  in  6  min.  23  sec.; 
1877,  20,  in  2  hours  39  min.  57  sec. 

1881,  C.  A.  Harriman,  Chicago,  111.,  530  miles,  in  5  days  20  hours  47  mm. 

1878,  W.  Howes,  London,  Eng.,  50  miles,  in  7  hours  57  min.  44  sec.;  1880,  57  miles, 
in  13  hours  7  min.  27  sec.,  and  100,  in  18  hours  8  min.  15  sec. 

1801,  Capt.  R.  Barclay,  Eng.,  country  road,  90  miles,  in  20  hours  22  min.  4  sec.,  in- 
cluding rests;  1803,  .25  mile,  in  56  sec.,  and  Charing  Cross  to  Newmarket,  64,  in  10 
hours,  including  rests;  1806,  100,  in  ighours,  including  i  hour  y>min.  in  rests,  1809, 
1000,  in  looo  consecutive  hours,  walking  a  mile  only  at  commencement  of  each  hour. 

1877,  D.  O'Leary,  London,  Eng.,  200  miles,  in  45  hours  21  min.  33  sec. 

1818,  Jos.  Eaton,  Stowmarket,  Eng.,  4032  quarter  miles,  in  4032  consecutive  quar- 
ter hours. 

1877,  Wm.  Gale,  London,  Eng.,  1500  miles,  in  1000  consecutive  hours,  1.5  miles 
each  hour  ;  and  4000  quarter  miles,  in  4000  consecutive  periods  of  10  minutes. 

1882,  Chas.  Rowell,  New  York,  N.  Y.,  and  running,  89  miles  1640  yards,  in  12  hours. 

1882,  Geo.  Hazael,  New  York,  N.  Y.,  and  running,  600  miles  220  yards,  in  6  days. 

1883,  J.  W.  Raby,  London,  Eng.,  2  miles  in  13  min.  14  sec.;  3,  in  20  min.  21.5  sec.; 
4,  in  27  min.  38  sec.  ;  5,  in  35  min.  10  sec.;  and  10,  in  i  hour  14  min.  45  sec. 

1882,  John  Meagher,  New  York,  N.  Y.,  8  miles  in  58  min.  37  sec. 
W.  Franks,  London,  Eng.,  25  miles  in  3  hours  35  min.  14  sec. 

1885,  W.  Cummings,  London,  Eng.,  10  miles  in  51  min.  6.6  sec. 

1884,  J.  E.  Dixon,  Birmingham,  Eng.,  40  miles  in  4  hours  46  min.  54  sec. 

1883,  Peter  Golden,  Brooklyn,  N.  Y.,  50  miles  in  7  hours  29  min.  47  sec. 

R,  tinning. 

1844,  Geo.  Seward,  of  U.  S.,  Manchester,  Eng.,  flying  start,  100  yards,  in  5.25  sec. 
1864,  Jos.  Nutall,  Manchester,  Eng.,  600  yards,  in  i  min.  13  sec. 
1881,  L.  E.  Myers,  New  York,  N.  Y.,  1000  yards,  in  2  min.  13  sec. 
1863,  Wm.  Lang,  Newmarket,  Eng.,  i  mile,  in  4  min.  2  sec.,  descending  ground; 
Manchester,  2,  in  9  min.  11.5  tec.;  1865,  n  miles  1660 yards,  in  i  hour  2  min.  2.5  see. 


ANIMAL   POWER.  439 

1852,  Wm.  Howilt,  "  American  Deer,"  London,  Eng.,  15  miles  in  i  hour  22  min. 
1863,  L.  Bennett,  ''  Deerfoot,"  Hackney  Wick,  Eng.,  12  m.,  in  i  hour  2  min.  2.5  sec. 

1879,  Patrick  Byrnes,  Halifax,  N.  S.,  20  miles,  in  i  hour  54  min. 

1880,  D.  Donovan,  Providence,  R.  L,  40  miles,  in  4  hours  48  min.  22  sec. 
17—,  ^4  Courier,  East  Indies,  102  miles,  in  24  Jiowrs. 

1889,  H.  M.  Johnson,  Denver,  Col.,  50  yards,  in  5  sec. 

1884,  M.  K.  Kittleman,  Oakland,  Cal.,  150  yards  (twice),  in  14  min.  6  sec. 

1890,  James  Grant,  Cambridge,  Mass.,  5  miles  in  25  min.  22.25  sec. 

Jumping,   Leaping,   etc. 

1854,  J.  Howard,  Chester,  Eng.,  i  jump,  board  raised  4  ins.  in  front,  running  start, 
with  dumb-bells,  5  Ibs.,  29  feet  7  ins. 

1868,  Geo.  M.  Kelley,  Corinth,  Mass.,  running,  and  from  a  spring  board,  leaped 
oyer  17  horses  standing  side  by  side. 

1879,  G.  W.  Hamilton,  Romeo,  Mich.,  dumb-bells,  22  Ibs.,  standing  jump,  14  feet 
5.5  ins. 

1886,  J.  Purcell,  Dublin,  running  long  jump,  -23  feet  11.5  ins. 

1889,  J.  Darby,  Ashton- under- Lyne,  Eng.,  two  standing  jumps,  with  weights,  26 
feet  8. 5  ins. 

H.  M.  Johnson,  St.  Louis,  Mo. ,  without  weights,  22  feet  6.75  ins.  10  standing  long 
jumps,  without  weights,  114  feet  8.5  ins. 

J.  F.  Kearny,  Walpole,  Mass.,  3  standing  long  jumps,  with  weights,  4-2  feet  3  ins.; 
without  weights,  at  Boston,  Mass. ,  35  feet  6  ins.  Boston,  Mass. ,  running  high  jump, 
with  weights,  6  feet  5.25  ins.;  backward  jump,  with  weights,  heel  to  toe,  izfeet  1.25 
ins.  Oak  Island,  Mass.,  standing  high  leap,  with  weights,  5  feet  9.5  ins. 

Lifting. 

1825,  Thomas  Gardner,  of  New  Brunswick,  N.  S.,  a  barrel  of  pork,  320  tes.,  under 
each  arm ;  also  transported  across  a  pier  an  anchor,  1200  Ibs. 
1868,  Wm.  B.  Curtis,  New  York,  N.  Y.,  3239  Ibs.,  in  harness. 
1883,  D.  L.  Dowd,  Springfield,  Mass.,  by  hands,  1442.25  Ibs. 

Throwing    ^Weights. 

1870,  D.  Dinnie,  New  York,  N.  Y.,  light  stone,  18  Ibs.,  43  feet;  heavy  stone,  24  Ibs., 
34  feet  6  ins.;  heavy  hammer,  24  Ibs.,  83  feet  8  ins.;  1872,  Aberdeen,  Scotland,  light 
hammer,  i^Bfeet;  run,  16  Ibs.,  162  feet. 

1887,  Peter  Foley,  Milwaukee,  Wis.,  56  Ibs.,  without  follow,  31  feet  5  in*. 

S%viinrning. 

1835,  S.  Bruck,  15  miles,  in  rough  sea,  in  7  hours  30  min. 

1846,  A  Native,  off  Sandwich  Islands,  7  miles  at  sea,  with  a  live  pig  under  one  arm. 
1870,  Pauline  Rohn,  Milwaukee,  Wis.,  650  feet,  still  water,  in  2  min.  43  sec. 
1872,  J.  B.  Johnson,  London,  Eng.,  remained  under  water  3  min.  35  sec. 
1875,  Capt.  M.  Webb,  Dover,  Eng.,  to  Calais,  France,  23  miles,  crossing  two  full 
and  two  half  tides  ==  35  miles,  in  21  hours  45  min.     1880,  Afloat  6b  hours. 

1886,  J.  Haggerty,  Blackburn  Baths,  Eng.,  100  yards,  4  turns,  in  i  min.  5.5  sec. 

1890,  J.  Nuttall,  London,  Eng.,  1000  yards,  23  turns,  in  13  min.  54.5  sec. 

1885,  J.  J.  Collier,  London,  Eng. ,  i  mile  in  26  min.  52  sec. 

Skating. 

1877,  John  Ennis,  Chicago,  111.,  9  laps  to  a  mile,  100  miles,  in  u  hours  37  min.  45 
sec.;  and  145  inside  of  19  hours. 

1887,  T.  Donoghue,  Jun.,  Newburgh,  N.  Y.,  i  mile,  with  wind,  in  2  min.  12.375  ***• 
1882,  S.  J.  Montgomery,  New  York,  N.  Y.,  50  miles,  in  4  hours  14  min.  36  sec. 

NOTE. — The  Sporting  Magazine,  London,  vol.  ix.,  page  135,  reports  a  man  in  1767  to  have  skated  » 
mile  upon  the  Serpentine,  Hyde  Park,  London,  in  57  seconds. 

HORSE.       Trotting. 

1814,  "Boston  Blue,"  Lynn  turnpike,  one  mile,  sulky,  in  2  min.  54  sec. 

1875,  "Steel  Grey,"  Yorkshire,  Eng.,  10  nrles,  saddle,  in  27  min.  56.5  sec. 

1867,  "John  Stewart,"  Boston,  Mass.,  half-mile  track,  20  miles,  harness,  in  58 
min.  5.75  sec.,  and  20.5  miles  in  59  min.  31  sec. 

1830,  "Top  Gallant,"  Philadelphia,  Penn.,  12  miles,  harness,  in  38  min. 

1829,  "Torn  Thumb,"  Sunbury  Common,  Eng.,  16.5  miles,  harness,  248  Ibs.,  in  56 
min.  45  sec.;  and  100  miles,  in  10  hours  7  min.,  including  37  min.  in  rests. 

1869,  "Morning  Star,"  Doncaster,  Eng.,  18  miles,  harness  (sulky  100  Ibs.),  in  57 
min.  27  sec. 

1835,  "  Black  Joke,"  Providence,  R.  L,  50  miles,  saddle,  175  Ibs.,  in  3  hourt  57  min. 


44O  ANIMAL    POWER. 

1855,  "Spangle,"  Long  Island,  N.  Y.,  50  miles,  wagon  and  driver  400  Ibs.,  in  3 
hours  59  min.  4  sec. 

1837,  u Mischief, "  Jersey  City,  N.  J.,  to  Philadelphia,  Penn.,  84.25  miles,  harness, 
very  hot  day  and  sandy  road,  in  8  hours  30  min. 

1853,  "Conqueror,"  Long  Island,  N.  Y.,  100  miles,  harness,  in  8  hours  55  min.  53 
sec.,  including  15  short  rests. 

1873,  M.  Delaney's  mare,  St.  Paul's,  Minn.,  200  miles,  race  track,  harness,  in  44 
hours  20  min.,  including  15  hours  49  min.  in  rests. 

1834,  "Master  Burke "  and  "  Robin,"  Long  Island,  N.  Y.,  100  miles,  wagon,  in  10 
hours,  17  min.  22  sec.,  including  28  min.  34  sec.  in  rests. 

Stage— coaching. 

1750,  By  the  Duke  of  Queensberry,  Newmarket,  Eng.,  19  miles,  in  53  min.  24  sec. 
1830,  London  to  Birmingham,  Eng.,  "Tally-ho,"  109  miles,  in  7  hours  50  min. 
including  stop  for  breakfast  of  passengers. 

Leaping.* 

1821,  A  horse  of  Mr.  Mane,  at  Loughborough,  Leicestershire,  Eug.,  173  Ibs.,  over  a 
hedge  6  feet  in  height,  35  feet. 

1821,  A  horse  of  Lieut.  Green,  Third  Dragoon  Guards,  at  Inchinnan,  Eng.,  ridden 
by  a  heavy  dragoon,  over  a  wall  6  feet  in  height  and  ifoot  in  width  at  top. 

1847,  "Chandler,""  Warwick,  Eng.,  over  water,  37" feet. 

1901,  "  Heather  bloom,"  Chicago,  111.,  over  a  bar,  7  feet  4-5  ins. 

NOTE.— The  maximum  stride  of  a  horse  is  estimated  to  be  28  feet  g  ins.  ;  "  Eclipse"  has  covered  25 
fett.  The  maximum  stride  of  an  elk  is  34  feet,  and  of  an  elephant  14  feet. 

R-u  lining. 

1701,  Mr.  Sinclair,  on  the  Swift  at  Carlisle,  a  gelding,  1000  miles,  in  1000  consecu- 
tive hours. 

1731,  Geo.  Osbaldeston,  Newmarket,  156  Ibs.,  100  miles,  by  16  horses,  in  4  hours  19 
mm.  40  sec.,  and  200,  by  28  horses,  in  8  hours  39  min.,  including  i  hour  2  min.  56  sec. 
in  rests;  i  horse,  "Tranby,"  16  miles,  in  33  min.  15  sec. 

1752,  Spedding's  mare,  100  miles,  in  12  hours  30  min.,  for  2  consecutive  days. 

1754,  A  Galloway  mare  of  Daniel  Corker's,  Newmarket,  300  miles,  by  one  rider, 
67  Ibs.,  in  64  hours  20  min. 

1761,  John  Woodcock,  Newmarket,  100  miles  per  day,  by  14  horses,  one  each  day, 
for  29  consecutive  days. 

1814,  An  Officer  of  i^lh  Dragoons,  Blackwater,  12  miles,  i  horse,  in  25  min.  n  sec. 

1868,  N.  H.  Mowry,  San  Francisco,  Cal.,  race  track,  160  Ibs.,  300  miles,  by  30  horses 
(Mexican),  in  14  hours  g  min.,  including  40  minutes  for  rests;  the  first  200,  in  8 
hours  2  min.  48  sec.,  and  the  fastest  mile  in  2  min.  8  sec. 

1869,  Nell  Coher,  San  Pedro,  Texas,  61  miles,  in  2  hours  55  min.  15  sec.,  including 
rests. 

1870,  John  Faylor,  Carson  City,  Nevada,  50  miles,  by  18  horses,  in  i  hour  58  min. 
33  sec.;  and  Omaha,  Neb.,  56  miles,  in  2  hours  26  min.,  including  rests. 

1876,  John  Murphy,  New  York,  N.  Y.,  155  miles,  by  20  horses,  in  6  hours  45  min. 
7  sec. 

1878,  Capt.  Salvi,  Bergamo  to  Naples,  Italy,  580  miles,  in  10  days. 

1880,  "  Mr.  Brown,"  Rancocas,  N.  J.,  aged,  160  Ibs.,  10  miles,  in  26  min.  18  sec. 

1828,  "Chapeau  de  Paille"  (Arabian),  India,  1.5  miles,  115  Ibs.,  in  2  min.  53  sec. 

183-  Capt.  Home  (Arabians),  Madras  to  Bungalore,  India,  200  miles,  in  less  thaa 
10  hours. 

DOGS.      Ccmrsing    and    Chasing. 

A  Greyhound  and  Hare  ran  12  miles  in  30  min. 

1794,  A  Fox,  at  Brende.  Eng.,  ran  50  miles  in  6.5  hours. 

A  Greyhound,  at  Bushy  Park,  Eng.,  leaped  over  a  brook  30  feet  6  ins. 

BIRDS.       Flying. 

In  Miles  per  hour :  Swallow,  65 ;  Marten,  60 ,  Carrier  Pigeon  and  Seal  Duck, 
50;  Wild  Goose,  45;  Quail,  38;  Crow,  25. 

1870,  Carrier  Pigeons,  Pesth  to  Cologne,  Germany,  600  in  8  hours.  1875,  D"n(iee 
Lake  to  Paterson,  N.  J.,  3  in  3  min.  24  sec. 

NOTE.  —At  50  miles  the  pressure  on  a  plane  surface  is  12.5  Ibs.  per  so.,  foot ;   and  at  *oo,  50  Ibs. 
*  A  Salmon  can  leap  ix  dam  nfeet  in  height.— Sporting  Magazine,  London,  vol.  xii.,  page  79. 


HOKSE-POWER. — BELTS   AND    BELTING. 


441 


HORSE -POWER. 

Horse-power. — IP  is  the  principal  measure  of  rate  at  which  work  is  per- 
formed. One  horse-power  is  computed  to  be  equivalent  to  raising  of  33  ooo 
Ibs.  one  foot  high  per  minute,  or  550  Ibs.  per  second.  Or,  33000  foot-lbs.  of 
work,  and  it  is  designated  as  being  Nominal,  Indicated,  or  Actual. 

A  IP  in  work  is  estimated  at  33000  Ibs.,  raised  i  foot  in  a  minute;  but  as  a  horse 
can  exert  that  force  (or  only  6  hours  per  day,  one  work  IP  is  equivalent  to  that  of 
4.5  horses,  at  a  rate  of  3  miles  per  hour. 

Cheval-vapeur  of  France  is  computed  to  be  equivalent  to  75  kilogram- 
meters  of  work  per  second,  or  7.233  foot-lbs.,  or  75  x  7.233  =  542.5  foot-lbsn 
which  is  1.37  per  cent,  less  than  American  or  English  value. 


BELTS  AND  BELTING. 

Capacity  of  belts  to  transmit  power  is  determined  by  extent  of  their 
adhesion  to  surface  of  pulley,  and  it  is  very  limited  in  comparison  with 
tensile  strength  of  belt. 

Resistance  of  a  belt  to  slipping  depends  essentially  upon  character 
of  surface  of  pulley,  its  degree  of  tension,  and  width,  and  as  adhesion 
is  in  proportion  to  pressure  on  surface  of  pulley,  long  belts,  by  having 
greater  weight,  give  greater  adhesion. 

Ultimate    Tensile    Strength,    of   Belting    per    Sq..    Inch 
of     Section. 

Merchantable  Oak- tanned,  of  first  quality.     Belts,  6  ins.  in  width. 
Single,  .2  inch  in  thickness,  918  Ibs.  per  lineal  inch,  and  4536  Ibs.  per  sq.  inch  of 
section.     Double,  .35  inch,  1396  Ibs.  per  lineal  inch,  and  4101  per  sq.  inch. 

Ratio  of  single  to  double,  918  -4- 1396  =  .658. 

Elongation  in  two  inches  of  length  and  four  in  width  for  a  load  of  2000  Ibs.  Sin- 
gle, 9.09;  double,  5.79. 

The  resistance  and  elongation  of  double  belting  is  more  uniform  than  that  of  sin- 
gle, from  the  irregularities  in  each  layer  counteracting  each  other. 


Length  of 
Lap. 

Rivets. 

Belt    J 

Destructive 

Stress. 

"ointing. 

Sti 
per  sq.  inch  of 
section. 

ess 
per  lineal  inch 
of  width. 

Elongation. 

Ins. 

lit 

7.2 
5-i 
Cemented..  . 

No. 
6 
6 

I 

Lbs. 
4170 
3610 
4520 
2420 
4^8o 

Lbs. 
394<> 
3000 
3545 
1792 
ss6o 

Lbs. 
709 
611 

762 
407 

III2 

Per  cent. 
7-5 
7-5 
7-5 
7-5 

7 

Riveted  joints  failed  at  rivet  holes.  Riveting  of  double  belts  was  shown  to  be 
objectionable.  Riveted  joints  of  single  belts  have  one-third  less  strength  than  the 
average  of  different  manners  of  lacing. 

A  double  staggered  laced  joint,  i  strand  only  in  each  hole  (5  of  .  1875  inch  punched), 
broke  in  belt  at  a  stress  equal  to  that  of  resistance  of  it  per  area  of  section. 


Transmission    of   3?o\ver. 


D  to  4  ooo  Ibs.  per  sq.  inch) 
of  50  Ibs.  per  sq.  inch, 

case,  including  friction,  was  30.6  Ibs.  per  sq.  inch. 

—From  Elements  of  Prof .  Chas.  H:  Benjamin. 


442  BELTS    AND    BELTING. 

Computation  of  IP. 

t>(S-s)  868(88X4-22X4)      ,       .„> 

•  -  =  EP  =  -  =  6.  94  HP. 

33000  33000 

v  representing  velocity  of  belt  in  feet  per  minute,  S  an  s  stress  on  belt  per  lineal 
inch  of  width  on  upper  or  driving  side  and  underneath  or  returning  side. 

To    Compute    Width    of   a    Leather    Belt. 

Assuming  a  well-defined  case  (where  limit  of  adhesion  was  ascertained), 
a  belt  of  ordinary  construction  (laced),  and  9  inches  in  width,  transmitted 
the  power  of  15  horses  over  a  pulley  4  feet  in  diameter,  at  a  velocity  of  1800 
feet  per  minute,  with  an  arc  of  adhesion  of  210°,  or  of  .6  or  7.54  feet  of  cir- 
cumference, and  with  an  area  of  95  square  feet  of  belt  per  H?. 

Hence,—  -  ^  -  =  w;  w  representing  width  of  belt  in  inches,  d  di- 

et v 
ameter  of  pulley  in  feet,  and  v  velocity  of  belt  in  feet  per  minute. 

NOTE.  —  Thickness  of  belt  should  be  added  to  diameter  of  pulley.  Applying  these 
elements  to  the  formulas  of  13  different  authors,  the  result  varies  from  7.85  to  13.5 
ins.,  mean  of  which  is  10  675.  For  double  belting  width  =  .66  w. 

ILLUSTRATIONS.—  If  IP  25,  and  velocity  of  belt  —  2250  feet  per  minute,  what 
should  be  width  of  belt,  diameter  of  pulley  4  feet? 

-  —  —  '==  12.5  ins.  for  ordinary  thickness  of  1875  in. 
To    Compute    Elements    of   Belting. 


_=.    IP  33  OOP...  -p.  33  OOP  IP  =  w.  Wyto  =  ir.  —  =S'  —=t 

'  '  ' 


_  —  — 

1000*  vw  v  '  33000          '    t          '    S 

*  looo  laced,  550  riveted  (for  a  thickness  of  .1875  inch),  with  variations  according 
to  the  character  and  condition  of  the  belt,  diameter  of  pulley,  and  arc  of  adhesion 
of  belt.  P  representing  power  transferred,  W  weight  or  stress  in  Ibs.  ,  t  thickness  of 
belt  in  ins.,  v  velocity  of  it  in  feet  per  minute,  and  S  stress  on  belt  per  lineal  inch  of 
width  w,  in  Ibs. 

Single  belts  at  their  relative  thickness  with  double,  of  .2  to  .35  inch,  will  sustain 
one-tenth  more  stress  per  sq.  inch  of  belt. 

To  Compute  tne  Angle   of  the  Arc  of  Contact  of  a  Belt. 

Sin£-  (R  —  r-r-d)  x  2-f-i8o°  for  large  pulley  or  driver  and  —  180°  for  small. 

R  and  r  representing  radii  of  pulleys,  d  distance  between  their  centres  —  all  in  feet 
or  inches. 

ILLUSTRATION.—  Assume  pulleys  11.2  feet  and  4  feet  in  diameter  and  distance 
apart  15  feet. 

Sin/.r  I'2~4-r-  15  Jx  2+  i8o°  =  207°  46'  for  large  pulley  and  152°  14'  for  small. 

India    Ptn"b"ber    Belting.    (Vulcanized.) 

Results  of  Experiments  upon  Adhesion  of  India  Rubber  and  Leather  Belting.— 
(J.  H.  Cheever). 


Rubber  belt  slipped  on  iron  pulley  at   90 
"  "  leather     "      128 


Leather  belt  slipped  on  iron  pulley  at  48 
leather    "       64 


Hence  it  appears  that  a  Rubber  Belt  for  equal  resistances  with  a  Leather'  Belt 
may  be  reduced  respectively  46,  50,  and  30  per  cent. 

tron  Wire. — A  wire  rope  .375  inch  in  diameter,  over  a  pulley  4  feet  in  di- 
ameter, running  at  a  velocity  of  1250  feet  per  minute,  will  transmit  4.5  IP. 

In  order  to  avoid  undue  bending  of  wires,  diameter  of  pulley  should  not  be  less 
than  140  times  diameter  of  rope. 

By  Experiments  of  H.  R.  Towne  and  Mr.  Kirkaldy.     (England.) 
Tensile  strength  of  Single  leather  belting  per  square  inch  of  section. 
Laced,  966  Ibs.  =  i.     Riveted,  1740  Ibs.  =  1.8.     Solid,  3080  Ibs. 


BELTS    AND    BELTING. BLASTING.  443 

By  the  experiments  of  F  W  Taylor,  M.  R,  the  tensile  strength  of  belts 

Per  Square  Inch  of  Section. 
Oak-tanned 192  to  229  Ibs.     Raw  hide 253  to  284  Ibs. 

Greneral    Notes. 

Leather  Belts— Are  best  when  oak-tanned,  should  be  frequently  oiled  *  and  when 
run  with  hair  side  over  pulley  will  give  greatest  adhesion. 

Ordinary  thickness  .1875  inch,  and  weight  60  Ibs.  per  cube  foot. 

Relative  effect  of  different  pulleys  and  belts : 
Leather  surface.,  i.     Rough  iron. ...  41    Turned  iron...  .64    Turned  wood. ..  .7 

Morin  assigned  50  Ibs.  as  a  proper  stress  per  inch  of  width  of  good  belting. 

Presence  of  small  holes  in  a  belt  will  prevent  its  slipping  or  squealing. 

To  increase  adhesion,  coat  driving  surface  with  boiled  oil  or  cold  tallow,  and  then 
apply  powdered  chalk. 

When  new,  cut  them  .1875  inch  short  for  each  foot  in  length  required,  to  admit 
of  the  stretch  that  occurs  in  their  early  operation. 

Belts  should  be  set  as  nearly  horizontal  as  practicable,  in  order  that  the  sag  may 
increase  adhesion  on  pulley,  and  hence  power  should  be  communicated  through 
under  side. 

The  "creeping"  or  lost  speed  by  belts  is  about  .006  per  cent.,  hence,  to  maintain 
a  uniform  or  required  speed,  driver  must  be  increased  in  diameterpro  rata  with  slip. 

A  double  belt,  75  ins.  in  width  and  153.5  feet  in  length,  transmitted  650  IIP. 
(See  page  989). 


BLASTING. 

In  Blasting,  rock  requires  from  .25  to  1.5  Ibs.  gunpowder  per  cube 
yard,  according  to  its  degree  of  hardness  and  position.  In  small  blasts 
2  cube  yards  have  been  rent  and  loosened,  and  in  very  large  blasts  2  to 
4  cube  yards  have  been  rent  and  loosened,  by  i  Ib.  of  powder. 

Tunnels  and  shafts  require  1.5  to  2  Ibs.  per  cube  yard  of  rock. 

G-tin  pcrwcler  has  an  explosive  force  varying  from  40000  to  90000 
/bs.  per  sq.  inch.  That  used  for  blasting  is  much  inferior  to  that  used  for 
projectiles,  the  proportion  being  fully  one  third  less. 

Nitro-gly  cerine  is  an  unctuous  liquid,  which  explodes  by  concussion, 
an  extreme  pressure  (2000  Ibs.  per  sq.  inch),  or  a  temperature  exceeding  600° 
if  quickly  applied  to  it ;  it  will  inflame,  however,  and  burn  gradually. 

At  a  temperature  below  40°  it  solidifies  in  crystals. 

Its  explosion  is  so  instantaneous  that  in  rock-blasting  tamping  is  not  nec- 
essary ;  its  explosive  power  by  weight  is  from  4  to  5  times  that  of  gun- 
powder. 

Dynamite  is  nitre-glycerine  75  parts,  absorbed  in  25  parts  of  a  sili- 
ceous earth  termed  kieselguhr;  it  also  explodes  so  instantaneously  as  to 
render  tamping  in  blasting  quite  unnecessary. 

It  is  insoluble  in  water,  and  may  be  used  in  wet  holes ;  it  congeals  at  40°, 
is  rendered  ineffective  at  212°,  and  has  an  explosive  force  by  weight  of  3 
times  that  of  gunpowder,  and  by  bulk  4.25  times. 

GKm.  -  cotton  is  insoluble  in  water,  and  has  an  explosive  force  by 
weight  of  from  2.75  to  3  times  that  of  gunpowder,  and  by  bulk  2.5  times. 
It  may  be  detonated  in  a  wet  state  with  a  small  quantity  of  dry  material. 

Tonite  is  nitrated  gun-cotton,  and  is  known  also  as  cotton  powder.  It 
is  produced  in  a  granulated  form. 

Litlio-fractexir  is  a  nitro-glycerine  compound  in  which  a  portion  of 
the  base  or  absorbent  material  is  made  explosive  by  the  admixture  therein 
of  nitrate  of  baryta  and  charcoal. 

*  See  Cements,  etc.,  page  871,  for  compositions,  etc. 


444 


BLASTING. 


Cellulose  Dynamite  is  when  gun-cotton  is  used  as  the  absorbent 
for  nitre-glycerine ;  it  will  explode  frozen  dynamite,  and  is  more  sensitive  tc 
percussion  than  it. 

To  Compxite  Charge  of  Q-tnipo^vcler  for  Roclr  Blasting. 
RULE. — Divide  cube  of  line  of  least  resistance  by  25,  as  for  limestone,  tc 
32  for  granite,  and  quotient  will  give  charge  of  powder  in  Ibs. 

Or,  L3  -i.  32  =  Ibs. 
EXAMPLE. — When  line  of  least  resistance  is  6  feet,  what  is  charge  required? 

63-r-32  =  6.75  Ibs. 
Line  of  least  resistance  should  not  exceed  .5  depth  of  hole. 

Tamping. — Dried  clay  is  the  most  effective  of  all  materials  for  tamping;  Broken 
Brick  the  next,  and  Loose  Sand  the  least. 

Relative  Costs  of  a  Tunnel  and  Shaft  in  England.     (Sir  John  Burgoyne.) 


Smitl 
Fuses 

Diam. 

is  and  coa 

1  6 

18 

rials 
Inch  < 
Pow 
or  G 
cottc 

I 

'n  L 
>J'L 
tor 

m- 
n. 

..abor  48.8 

lolesofL 
zngth. 

Dynamite. 

>iffercnl 

Diam. 

Diamete 

Powder 
or  Gun- 
cotton. 

IOO 

*S. 
Dynamite 

Weight  oj 

Powder 
or  Gun- 
cotton. 

f  Esplosii 

Dynamite. 

e  Mate 
Per 

Diam. 

Ins. 
I 
1.25 

Oz. 

.419 

.654 
.942 

Oz. 

1.046 
I-507 

Ins. 

2 

2.25 

Oz. 

1.283 
1-675 

2.12 

Oz. 

2-053 
2.68 
3-392 

Ins. 

2-5 
2-75 
3 

Oz. 

2.618 
3.166 

Oz. 

4.189 
5.066 
6.03 

Diam. 
of 
Jumper. 

Depth 
of 
Hole. 

Men. 

Depth  bored 
per  Day. 

Ins. 
I 
i-75 

2 

Ins. 
I      to  2 
2.  5  to  6 
4     to  7 

No. 
I 
3 
3 

Feet. 
8 

12 

8 

Boring  Holes  in  Granite. 


14 

Drill.— Width  of  bit  compared  to  stock  .625. 


Diam. 
of 
Jumper. 

Depfth 
Hole. 

Men. 

Ins. 
2.25 
2-5 

3 

Ins. 
5  to  10 
9  to  12 
9  to  15 

No. 
3 
3 
3 

Depth  bored 
per  Day. 

!.  Feet. 
6 
5 
4 


Lbs. 

16 

16 

18 


Charges    of   Powder. 

Usual  practice  of  charging  to  one  third  depth  of  hole  is  erroneous,  inasmuch  as 
volume  of  charge  increases  as  square  of  diameter  of  hole.  Hence  holes  of  1.5  and 
2  inches,  although  of  equal  depths,  would  require  charges  in  proportion  of  2.25  and  4. 


Line  of 
least  re- 
sistance. 

Powder. 

Line  of 
least  re- 
sistance. 

Powder. 

Line  of 
least  re- 
sistance. 

Powder. 

Line  of 
least  re- 
sistance. 

Powder. 

Feet. 

I 
2 

Oz. 

•75 
4 

Feet. 
3 
4 

Lbs.     Oz. 
13-5 
2 

Feet. 

1 

Lbs.     Oz. 
3     14-5 

6      12 

Feet. 

I 

Lbs.  Oz. 
10  11.5 

16 

Effects. 

Gunpowder.  —  From  its  gradual  combustion,  rends  and  projects  rather  than 
shatters. 

A  hole  5.5  ins.  in  diameter  and  IQ  feet  7  ins.  in  depth,  filled  to  8  feet  10  ins.  with 
75  Ibs.  powder,  has  removed  and  rent  1200  cube  yards,  equal  to  2400  tons.  The 
labor  expended  was  that  of  3  men  for  14  days. 

Temperature  of  gases  of  explosion  4000°. 

Gun-cotton. — From  the  rapidity  of  its  combustion,  shatters. 

Dynamite.— From  the  greater  rapidity  of  its  combustion  over  gun  cotton,  ie  more 
shattering  in  ite  explosion. 


BLASTING. — BLOWING    ENGINES.  445 

Drilling. 

Churn-drilling. — A  churn-driller  will  drill,  in  ordinary  hard  rock,  from  8  to  12 
feet,  2  inch  holes  of  2.5  feet  depth,  per  day,  and  at  a  cost  of  from  12  to  18  cents  per 
foot,  on  a  basis  of  ordinary  labor  at  $i  per  day.  Drillers  receiving  $2.50. 

One  man  can  bore,  with  a  bit  i  inch  in  diameter,  from  50  to  100  inches  per  day 
of  10  hours  in  granite,  or  300  to  400  inches  per  day  in  limestone. 

Tamping. — Two  strikers  and  a  holder  can  bore,  with  a  bit  2  inches  in  diameter, 
10  feet  in  a  day  in  rock  of  medium  hardness. 

Composition  for  waterproof  charger  or  fuse  consists  by  weight  of  Pitch,  8  parts; 
Beeswax  and  Tallow  each  i  part. 

Mining.     (Lefroy's  Handbook.) 

In  demolition  of  walls  line  of  least  resistance  L  =  half  thickness,  and  C  is  a  co- 
efficient depending  on  structure. 

Charge  in  Ibs.  =  C  X  L3. 

In  a  wall  without  counterforts,  where  interval  between  the  charge  is  2  L,  C  =  .i5. 

In  a  wall  with  counterforts  the  charge  to  be  placed  in  centre  of  each  counterfort 
at  junction  with  wall,  C  — . 2. 

Where  the  charge  is  placed  under  a  foundation,  having  equal  support  on  both 
sides,  C  =  .4. 

A  leather  bag,  containing  50  to  60  Ibs.  powder,  hung  or  supported  against  a  gate 
or  like  barrier,  will  demolish  it. 
For  ordinary  mines  in  average  rock  charge  in  ounces  =  L3  -f- 160. 


BLOWING  ENGINES. 
For  Smelting. 

Volume  of  oxygen  in  air  is  different  at  different  temperatures.  Thus, 
dry  air  at  85°  contains  10  per  cent,  less  oxygen  than  when  it  is  at  tern- 
perature  of  32°;  and  when  it  is  saturated  with  vapor,  it  contains  12 
per  cent.  less.  If  an  average  supply  of  1500  cube  feet  per  minute  is 
required  in  winter,  1650  feet  will  be  required  in  summer. 

Smelting  of  Iron  Ore. 

Coke  or  Anthracite  Coal. — 18  to  20  tons  of  air  are  required  for  each  ton 
of  Pig  Iron,  and  with  Charcoal  17  to  18  tons  are  re'quired. 

(i  ton  of  air  at  34°  =  29  751,  and  at  60°  =  31 366  cube  feet.) 

Pressure. — Pressure  ordinarily  required  for  smelting  purposes  is  equal  to 
a  column  of  mercury  from  3  to  10  inches,  or  a  pressure  of  1.5  to  5  Ibs.  per 
square  inch. 

Reservoir. — Capacity  of  it,  if  dry,  should  be  15  to  20  times  that  of  cylin- 
der if  single  acting,  and  10  times  rf  double  acting. 

Pipes.— Their  area,  leading  to  reservoir,  should  be  .2  that  of  blast  cylinder, 
and  velocity  of  the  air  should  not  exceed  35  feet  per  second. 

A  smith's  forge  requires  150  cube  feet  of  air  per  minute.  Pressure  of 
blast  .25  to  2  Ibs.  per  square  inch.  A  ton  of  iron  melted  per  hour  in  a  cu- 
pola requires  3500  cube  feet  of  air  per  minute.  A  finery  forge  requires 
100  ooo  cube  feet  of  air  for  each  ton  of  iron  refined.  A  blast  furnace  re- 
quires 20  cube  feet  per  minute  for  each  cube  yard  capacity  of  furnace. 

A  Ton  of  Pig  Iron  requires  for  its  reduction  from  the  ore  310000  cube 
feet  of  air,  or  5.3  cube  feet  of  air  for  each  pound  of  carbon  consumed 
Pressure,  .7  Ib.  per  square  inch. 

P  P 


446  BLOWING    ENGINES.        N 

rive    a  Blowing 


Compute    IPower    Required.    to    Drive    a  Blowing 

Engine. 


=  x  / 

V  . 


t?  representing  velocity  of  air  in  feet  per  sec- 


_    _ 

.93  x  .7854  x  _  . 

ond,  d  and  d!  diameters  of  pipe  and  of  nozzle  in  feet,  =V  .gjx  .7854  X  500 
=  .309. 

ILLUSTRATION.—  What  should  be  power  of  a  steam  engine  to  drive  35  cube  feet  of 
air  at  a  velocity  of  500  feet  per  second,  through  a  pipe  i  foot  in  diameter  and  300 
feet  in  length? 

c  —  ratio  between  power  employed  and  effect  produced  by  it  =  ina  well-constructed 
engine  .  5,  and  C  =  .93.  d  =  .  2974,  assumed  at  .  3. 

l^°5oj>  x  ^3  f3™  +  4«\  60-4-33000  =  22631.625  X  60-5-33000  =  41-15  H». 

To   Compute   Required   Power   of  a,   Blowing   Engine. 
P-r/X  av_  jp     p  ygpregenting  pressure  of  blast  in  Ibs.  per  sq.  inch-f 

a  area  of  cylinder  in  sq.  ins.  ;  v  velocity  of  piston  injeet  per  minute;  f  fric- 
tion of  piston  and  from  curvatures,  etc.,  estimated  at  1.25  per  sq.  inch,  of 
piston. 
NOTE.—  If  cylinder  is  single  acting,  divide  result  by  2. 

ILLUSTRATION.  —  Assume  area  of  blast  cylinder  5600  sq.  ins.,  pressure  of  blast  2.25 
Ibs.  per  sq.  inch,  and  velocity  of  piston  96  feet  per  second. 

2.25  +  1.25X5600X96^1  88!  600  h(y).ses  the  exact  power  developed  in 

-----  33000 


To   Compute   Dimensions   of*  a   Driving   Engine. 

RULE  i.  —  Divide  power  in  Ibs.  by  product  of  mean  effective  pressure  upon 
piston  of  steam  cylinder  in  Ibs.  per  sq.  inch,  and  velocity  of  piston  in  feet 
per  minute,  and  quotient  will  give  area  of  cylinder  in  sq.  ins. 

2.  —  Divide  velocity  of  piston  by  twice  number  of  revolutions,  and  quotient 
will  give  stroke  of  piston  in  feet. 

Volume  of  air  at  atmospheric  density  delivered  into  reservoir,  in  consequence  of 
escape  through  valves,  and  partial  vacuum  necessary  to  produce  a  current,  will  be 
about  .2  less  than  capacity  of  cylinder. 

EXAMPLE.—  Assume  elements  of  preceding  case,  with  a  pressure  of  50  Ibs.  steam, 
cut  off  at  .375,  and  with  12  revolutions  of  engine  per  minute,  what  should  be  area 
of  cylinder  of  a  non-condensing  engine? 

Mean  effective  pressure  of  steam  with  5  per  cent,  clearance  —  50  Ibs.,  and  50  — 
/*  +  14.7  =  50  —  2.5  -J-  3.33-!-  14.7  =  29.47  Ibs.y  and  velocity  of  piston  =  192  feet. 


5600  Xa.3S  +  i.»5X96  =  1881600  _i9£_ 

29.47  X  192  5658  12  X  2 

Area  of  cylinder  in  this  case  was  324  sq.  ins. 

For  Volume,  Pressure,  and  Density  of  Air,  see  Heat,  page  521. 

*  See  formula  and  note  for  power  of  non-condensing  engine,  page  733. 


BLOWING   ENGINES.  447 

To   Compute   Elements   of  a   Blo^wing    Kngine. 

Single  Stroke. 


V  representing  volume  of  air  in  cube  feet  per  minute,  P  pressure  of  air  and 
f  fractional  resistance  in  Ibs.  per  sq.  inch,  A  area  of  cylinder  and  a  area 
of  its  valves  in  sq.  ins.,  s  stroke  of  piston  in  feet,  n  number  of  single  strokes 
of  piston  per  minute,  L  length  of  air-pipe  from  reservoir  to  discharge  in  feet, 
d  diameter  of  air  or  blast  pipe  and  1)  diameter  of  cylinder  in  ins.,  v  velocity 
of  blast  in  feet  per  second,  and  t  temperature  of  blast  consequent  upon  com- 
pression in  degrees. 

ILLUSTRATIONS.  —  Assume  blowing  cylinder  50  ins.  in  diam.,  stroke  of  piston  10 
feet,  number  of  single  strokes  10  per  minute,  pressure  by  mercurial  manometer 
6.12  ins.,  frictional  resistance  .4  lb.,  length  of  pipe  25.25  feet,  and  area  of  valves 
95  sq.  ins. 

V  =  1  363.  54  cube  feet,         P  =  3  Ibs.  ,     -  A  =  1963.  5  sq.  ins. 

Then  ^3J±|L3S  =  20.I6H>,  and    -963.5  X  '°  ^°  XI£i  =  ^  g. 


To  Compute  "Volume  of  Air  transmitted,  by  an  Engine. 
When  Pressure,  Temperature,  etc.,  are  given. 


£-  J  \/  *  (*  A4°H*   )  ^  =  v'    Then  °  r  *  60  =  V  m  cube  feet  per  minute. 

tl  and  h  representing  height  of  barometer  and  pressure  of  blast  in  ins.  of 
mercury  ;  t  temperature  of  blast  ;  and  v  velocity  in  feet  per  second. 

ILLUSTRATION.  —  A  furnace  having  2  tuyeres  of  5  ins.  diameter,  pressure  and  tem- 
perature of  blast  3  ins.  and  350°,  and  barometer  30  ins.  ;  what  is  volume  of  air  trans- 
mitted per  minute? 

C  for  a  conical  opening  —  .94. 

34-5X/3  (^^  X  .94  =  34.5/3~(f)  =  34-5  X  -467  X  .94  =  *«4 
fat  velocity  per  second. 

Then,  area  5  ins.=  19.635,  which  X  2  =  39.27  ins.,  and  39.27  X  15.  14  X  6o-=-  144.-= 
r  47.73  cube  feet. 

To  Compute  Pressure  of  Blast  from  "Water  or  Mercurial 
Grauge. 

RULE.  —  Divide  Water  and  Mercurial  Gauge  in  ins.  by  27.67  and  2.04  re- 
spectively, and  quotient  will  give  pressure  in  Ibs.  per  sq.  inch. 


Proportions  of  Parts.  Blades.—  Their  width  and  length  should  be  at  least 
equal  to  .4  or  .5  radius  of  fan. 

Openings.  —  Inlet  should  be  equal  to  radius  of  fan  ;  and  outlet,  or  dis- 
charge, should  be  in  depth  not  less  than  .125  diameter,  its  width  being  equaJ 
to  width  of  fan. 

Eccentricity.  —  .1  of  diameter  of  fan.    Journals,  4  diameters  of  shaft. 


448  BLOWING    ENGINES. 

By  the  experiments  of  Mr.  Buckle,  he  deduced 

1.  That  velocity  of  periphery  of  blades  should  be  .9  that  of  their  theoretical 
velocity  ;  that  is,  velocity  a  body  would  acquire  in  falling  height  of  a  homo- 
geneous column  of  air  equivalent  to  required  density. 

2.  That  a  diminution  of  inlet  from  proportions  here  given  involved  a 
greater  expenditure  of  power  to  produce  same  density. 

3.  That  greater  the  depth  of  blade,  greater  the  density  of  air  produced 
with  same  number  of  revolutions. 

To    Compute   Elements   of  a   Fan-blower. 

,/H  av6o      ,r  dav 


. 

v  representing  velocity  of  periphery  of  fan  in  feet  per  second,  d  inches  of 
mercury,  V  volume  of  air  in  cube  feet,  and  a  area  of  discharge  in  sq.  ins. 

ILLUSTRATION.—  Assume  velocity  of  periphery  of  fan  123  feet  per  second,  density 
of  blast  .25  inch,  volume  of  air  1845  cube  feet,  and  area  of  discharge  40  sq.  ins. 

244-^.25  =  122/6^.      —  —  *~  -  -  =  1845  cub.  ft. 


—  —  =  2.97  IB?,  independent  of  friction  of  blast  in  pipes  and  tuyeres. 

To   Compute   Power  of*  a   Centrifxigal   ITan. 
V2  -i-  97  300  =  P.    V  representing  velocity  of  tips  of  fan  in  feet  per  second. 

(See  also  p.  1018.) 
Memoranda. 

Operation  of  a  blower  requires  about  2.5  per  cent,  of  power  of  attached 
boiler. 

An  increase  in  number  of  blades  renders  operation  of  fan  smoother,  but 
does  not  increase  its  capacity. 

Pressure  or  density  of  a  blast  is  usually  measured  in  ins.  of  mercury,  a 
pressure  of  i  Ib.  per  sq.  inch  at  60°  =  2.0376  ins. 

When  water  is  used,  a  pressure  of  i  Ib.  =  27.671  ins. 

Cupola  blast  .8  Ibs.,  and  Smith's  forge  .25  to  .3  Ibs.  per  sq.  inch. 

An  ordinary  Eccentric  Fan,  4  feet  in  diameter,  with  5  blades  10  ins.  wide 
and  14  in  length,  set  1.5  ins.  eccentric,  with  an  inlet  opening  of  17.5  ins.  in 
diameter,  and  an  outlet  of  12  ins.  square,  making  870  revolutions  per  min- 
ute, will  supply  air  to  40  tuyeres,  each  of  1.625  ms-  H1  diameter,  and  at  a 
pressure  per  sq.  inch  of  .5  inch  of  mercury. 

An  ordinary  eccentric  fan  blower,  50  ins.  in  diameter,  running  at  1000  revolutions 
per  minute,  will  give  a  pressure  of  15  ins.  of  water,  and  require  for  its  operation  a 
power  of  12  horses.  Area  of  tuyere  discharge  500  sq.  ins. 

A  non-condensing  engine,  diameter  of  cylinder  8  ins.,  stroke  of  piston  i  foot,  press- 
ure of  steam  18  Ibs.  (mercurial  gauge),  and  making  100  revolutions  per  minute,  will 
drive  a  fan,  4  feet  by  2,  opening  2  feet  by  2,  500  revolutions  per  minute. 

Such  a  blower  was  applied  as  an  exhausting  draught  to  smoke  pipe  of  steamer 
Keystone  State,  cylinder  80  ins.  by  8  feet,  and  evaporation  was  doubled  over  that 
of  wiien  wind  was  calm. 

In  French  blowing  engines,  volume  of  air  discharged  75  per  cent,  that  of 
volume  of  piston  space  in  cylinder,  stroke  equal  diameter  of  cylinder,  and 
velocity  of  piston  from  100  to  200  feet  per  minute. 

Area  of  admission  valves  from  .066  to  .083  of  that  of  cylinder  for  speeds 
of  100  to  150  feet  per  minute,  and  from  .1  to  .in  for  higher  speeds.  Area 
of  exit  valves  from  .066  to  .05  of  cylinder.  (M.  Claudel.) 


BLOWING   ENGINES. — CENTRAL   FORCES.  44^ 

By  some  experiments  lately  concluded  in  England  with  boilers  of  two 
iteamers,  to  determine  relative  effects  of  natural  and  forced  draught  furnaces, 
the  results  were  as  follows  (/?.  J.  Butler) : 

Per  Sq.  Foot  of  Grate  Surface. — Natural  Draught,  10  to  10.87  IH?»  Steam 
Blast,  12.5  to  13 ;  Forced  or  Blast  Draught,  15  to  16. 

Heating  Surface  per  IIP. — Natural  Draught,  2.44  to  2.61 ;  Steam  Blast, 
1.71  to  2.86;  Forced  or  Blast  Di*aught,  1.56  to  2.5. 

Tube  Surf  ace  per  IIP  in  Sq.  Feet. — Natural  Draught,  2.03  to  2.18;  Steam 
Blast,  2.02  to  2.08;  Forced  01*  Blast  Draught,  1.3  to  2.8. 

IIP  per  Sq.  Foot  of  Grate  in  these  Trials.  —  Natural  Draught,  10.15  to 
10.87;  Steam  Blast,  12.76  to  13.1 ;  Forced  or  Blast  Draught,  10.6  to  16.9. 

Root's  Rotary  Blower — Is  constructed  from  .125  to  14  nominal  IP,  supplying 
from  1 50  to  10  800  cube  feet  of  air  per  minute.  Delivery  pipe  2.5  to  19  ins. 
in  diameter.  Efficiency  65  to  80  per  cent,  of  power. 

For  Ventilation  of  Mines — From  40  to  280  revolutions  per  minute,  equal 
to  discharge  of  12  500  to  200000  cube  feet  of  air  per  minute.    15.5  to  189  H>. 
Steam  cylinder  from  14  x  18  ins.  to  28  x  48  ins. 
For  other  details  of  Blowing  Engines  see  page  898. 


CENTRAL  FORCES. 

All  bodies  moving  around  a  centre  or  fixed  point  have  a  tendency  to 
fly  off  in  a  straight  line:  this  is  termed  Centrifugal  Force;  it  is  op- 
posed to  a  Centripetal  Force,  or  that  power  which  maintains  a  body  in 
/ts  curvilineal  path. 

Centrifugal  Force  of  a  body,  moving  with  different  velocities  in  same 
eircle,  is  proportional  to  square  of  velocity.  Thus,  centrifugal  force  of 
a  body  making  10  revolutions  in  a  minute  is  4  times  as  great  as  centrif- 
ugal force  of  same  body  making  5  revolutions  in  a  minute.  Hence,  in 
equal  circles,  the  forces  are  inversely  as  squares  of  times  of  revolution. 

If  times  are  equal,  velocities  and  forces,  are  as  radii  of  circle  of  revolution. 

The  squares  of  times  are  as  cubes  of  distances  of  centrifugal  force  from 
axis  of  revolution. 

Centrifugal  forces  of  two  unequal  bodies,  having  same  velocity,  and  at  same  dis- 
'.Ance  from  central  body,  are  to  one  another  as  the  respective  quantities  of  matter 
"n  the  two  bodies. 

Centrifugal  forces  of  two  bodies,  which  perform  their  revolutions  in  same  time, 
the  quantities  of  matter  of  which  are  inversely  as  their  distances  from  centre,  are 
equal  to  one  another. 

Centrifugal  forces  of  two  equal  bodies,  moving  with  equal  velocities  at  different 
distances  from  centre,  are  inversely  as  their  distances  from  centre. 

Centrifugal  forces  of  two  unequal  bodies,  moving  with  equal  velocities  at  different 
distances  from  centre,  are  to  one  another  as  their  quantities  of  matter,  multiplied  by 
their  respective  distances  from  centre. 

Centrifugal  forces  of  two  unequal  bodies,  having  unequal  velocities,  and  at  differ- 
ent distances  from  their  axes  are  in  compound  ratio  of  their  quantities  of  matter, 
squares  of  their  velocities,  and  their  distances  from  centre. 

Centrifugal  force  is  to  weight  of  body,  as  double  height  due  to  velocity  is  to  radius 
of  rotation. 

A  Radius  Vector  is  a  line  drawn  from  centre  of  force  to  moving  body. 
PP* 


450 


CENTRAL   FORCES. 


To    Compute   Centrifugal    Force   of  any   Body. 
RULE  i.  —  Divide  its  velocity  in  feet  per  second  by  4.01,  also  square  of 
quotient  by  diameter  of  circle  ;  this  quotient  is  centrifugal  force,  assuming 
the  weight  of  body  as  i.     Then  this  quotient,  multiplied  by  weight  of  body, 
will  give  centrifugal  force  required. 

EXAMPLE.—  What  is  the  centrifugal  force  of  the  rim  of  a  fly-wheel  having  a  diam- 
tter  of  10  feet,  and  running  with  a  velocity  of  30  feet  per  second? 

3o-r-  4.01  =  7.  48,  and  7.  482  -t-  10  =  5.  59,  or  times  weight  of  rim, 

W  n2  A/R2-h»*2 
Or,  --  !  -  =  C.    r  representing  radius  of  inner  diameter  of  ring. 

NOTE.  —  Diameter  of  a  fly-wheel  should  be  measured  from  centres  of  gravity  of  rim, 
When  great  accuracy  is  required,  ascertain  centre  of  gyration  of  body,  and 
take  twice  distance  of  it  from  axis  for  diameter. 

RULE  2.  —  Multiply  square  of  number  of  revolutions  in  a  minute  by  diam- 
eter of  circle  of  centre  of  gyration  in  feet,  and  divide  product  by  constant 
number  5217  ;  quotient  is  centrifugal  force  when  weight  of  body  is  i.  Then, 
as  in  previous  Rule,  this  quotient,  multiplied  by  weight  of  body,  is  centrif- 
ugal force  required. 

ri2  d 
Or,  —  -  x  W.    n  representing  number  of  revolutions  per  minute,  d  diameter  of 

circle  of  gyration  in  feet,  and  W  weight  of  revolving  body  in  Ibs. 

EXAMPLE.—  What  is  centrifugal  force  of  a  grindstone  weighing  1200  Ibs.,  42  inches 
in  diameter,  and  turning  with  a  velocity  of  400  revolutions  in  a  minute? 

Centre  of  gyration  =  rad.  (42-^-2)  X  .7071  =  14.85  ins.,  which  -7-12  and  X2  = 
2.475  feet  =  diameter  of  circle  of  gyration.  Then  -  -  '-^^  X  1200  =  91  080  Ibs. 

Formulas   to   Determine   "Various   Elements. 


2925 


29300  ^  Wv2     .  72930  C  .  /CR  32.166 

-W^;      =3^66"C;      W=V~vnT'  V  -  W  -  ;     : 

C  representing  centrifugal  force,  W  mass  or  weight  of  revolving  body,  both  in  Ibs., 
R  radius  of  circle  of  revolving  body  in  feet,  n  number  of  revolutions  per  minute,  and 
v  and  v'  linear  or  circumferential  and  angular  velocities  of  body  in  feet  per  second. 

ILLUSTRATION.—  What  is  centrifugal  force  of  a  sphere  weighing  30  Ibs.,  revolving 
around  a  centre  at  a  distance  of  5  feet,  at  30  revolutions  per  second? 


=  = 

Centrifugal  forces  of  two  bodies  are  as  radii  of  circles  of  revolution  directly,  and 
as  squares  of  times  inversely. 

ILLUSTRATION.—  If  a  fly-wheel,  12  feet  in  diameter  and  3  tons  in  weight,  revolves 
in  8  seconds,  and  another  of  like  weight  revolves  in  6,  what  should  be  the  diameter 
of  the  second  when  their  centrifugal  forces  are  equal  ? 

12       a;  12  X  62  .   . 

Then  3  :  3  ::  —  :  —  ;  or  *=  —  —  —  =6.75/6^,  a?  =  unknown  element. 

Centrifugal  forces  of  two  bodies,  when  weights  are  unequal,  are  directly  as  squares 
of  times. 

ILLUSTRATION.—  What  should  be  the  ratio  of  the  weights  of  the  wheels  in  the  pre- 
ceding case,  their  forces  being  equal  ? 


Then  3  :  x  ::  62  :  82,  or  a?  =  =  =  5-333  tons. 

Molesworth  gives  .000  34  W  R  n2  =  C. 

*  Thi«  i»  termed  the  Vi»  Viva,  or  living  force. 


CENTRAL    FORCES.  —  FLY-WHEEL.  451 

FLY-WHEEL. 

A  FLY-WHEEL  by  its  inertia  becomes  a  reservoir  as  well  as  a  regulator 
of  force,  and,  to  be  fully  effective,  it  should  have  high  velocity,  and  its  diam- 
eter be  from  3  to  4  times  that  of  stroke  of  driving  engine. 

Coefficient  of  fluctuation  of  its  energy  ranges  from  .015  to  .035. 

Weight  of  a  fly-wheel  in  engines  that  are  subjected  to  irregular  mo- 
tion, as  in  a  cotton-press,  rolling-mill,  etc.,  must  be  greater  than  in  others 
where  so  sudden  a  check  is  not  experienced,  and  its  diameter  should  range 
from  3.5  to  5  times  length  of  the  stroke  of  the  piston. 

A  single-acting  engine  requires  a  weight  of  wheel  about  2.5  times  greater 
than  that  for  a  double-acting,  and  5  times  for  double  engines  of  double  action. 

To   Compute   \Veiglit   of  Rim   of*  a   Fly-wheel. 

RULE.  —  Multiply  mean  effective  pressure  upon  piston  in  Ibs.  by  its  stroke 
in  feet,  and  divide  product  by  product  of  square  of  number  of  revolutions, 
diameter  of  wheel,  and  .000  23. 

NOTE.—  If  a  light  wheel,  multiply  by  .0003  ;  and  if  a  heavy  one,  by  .000  16. 

EXAMPLE  i.—  A  non  condensing  engine  (double-acting),  having  a  diameter  of  cyl. 
inder  of  14  ins.,  and  a  stroke  of  piston  of  4  feet,  working  full  stroke,  at  a  pressure 
of  65  Ibs.  mercurial  gauge,  and  making  40  revolutions  per  minute,  develops  about 
65  IP  ;  what  should  be  the  weight  of  its  fly-wheel  when  adapted  to  ordinary  work  f 

Area  of  cylinder  154  sq.  ins.  Mean  pressure  assumed  50  Ibs.  per  sq.  inch.  Diam- 
eter of  wheel  =  4  feet  stroke  of  piston  X  3.5,  assumed  as  above,  =  14  feet. 

50  X  154  X  4  =  30  800,  which  -j-  4o2  X  14  X  .000  23  —  5978  Ibs. 
2.  _  If  a  fly-wheel,  16  feet  in  diameter  and  4  tons  in  weight,  is  sufficient  to  regulate 
an  engine  (double-acting),  it  revolving  in  4  seconds,  what  should  be  the  weight  of  a 
wheel  of  12  feet,  revolving  in  2  seconds,  so  that  it  may  have  like  centrifugal  force? 

NOTE.—  The  centrifugal  forces  of  two  bodies  are  as  the  radii  of  the  circles  of  revo- 
lution directly,  and  as  squares  of  times  inversely. 


,2 
4Z  22  12  X42  12X16 

To   Compute    Dimensions   of*  Rim. 

RULE.  —  Multiply  weight  of  wheel  in  Ibs.  by  .1,  and  divide  product  by 
mean  diameter  of  rim  in  feet  ;  quotient  will  give  sectional  area  of  rim  in 
square  inches  of  cast  iron. 

Assume  elements  of  example  i.     5978  X  .1  -5-  13-25  =  45.  12  sq.  ins. 

Or,  —  —  =  W,  and  -  =  A.     P  representing  pressure  on  piston  and  W  weight  of 

•4  if  10  D 

wheel  in  Ibs.,  S  stroke  of  piston,  and  D  mean  diameter  of  wheel,  both  in  feet,  and  A 
area  of  section  of  rim  in  sq.  ins. 

Or,  Ijl  -  =  W.  C  coefficient,  varying  from  3  to  4  ordinarily,  increasing 
to  6  with  great  regularity  of  speed  and  n  number  of  revolutions  per  minute. 

NOTE.—  Maximum  safe  velocity  for  cast  iron  is  assumed  at  80  feet  per  second. 

For  engines  at  high  expansion  of  steam,  or  with  irregular  loads,  as  with  a  rolling- 
mill,  multiply  W  by  i.  5,  or  put  W  100  Ibs.  for  each  IP.  (MoUsworth.  ) 

In  corn  or  like  mills,  the  velocity  of  periphery  of  fly-wheel  should  exceed  that  of 
the  stones,  to  arrest  backlash, 


452   CENTRAL  FORCES. — GOVEKNOKS. — PENDULUMS* 

GOVERNORS. 

A  GOVERNOR  or  CONICAL  PENDULUM  in  its  operation  depends  upon  tha 
principles  of  Central  Forces. 

When  in  a  Ball  Governor  the  balls  diverge,  the  ring  on  vertical  shaft 
raises,  and  in  proportion  to  the  increase  of  velocity  of  the  balls  squared, 
or  the  square  roots  of  distances  of  ring  from  fixed  point  of  arms,  cor- 
responding to  two  velocities,  will  be  as  these  velocities. 

Thus,  if  a  governor  makes  6  revolutions  in  a  second  when  ring  is  16 
ins.  from  fixed  point  or  top,  the  distance  of  ring  will  be  5.76  ins.  when 
speed  is  increased  to  10  revolutions  in  same  time. 

For  10  :  6  : :  V  16  :  2.4,  which,  squared  =  5.76  ins.,  distance  of  ring 
from  top.  Or,  62  ;  10'*  : :  5.76  :  16  ins. 

A  governor  performs  in  one  minute  half  as  many  revolutions  as  a 
pendulum  vibrates,  the  length  of  which  is  perpendicular  distance  be- 
tween plane  in  which  the  balls  move  and  the  fixed  point  or  centre  of 
suspension. 

To  Compnte  Nximtoer  of  Revolutions  of  a  Ball  Q-overnor 
per  IM.in.ixte  to  maintain   JBalls  at  any  given   Height. 

188  -T-  v/H  =  revolutions.  H  representing  vertical  height  between  plane  of  balls 
and  points  of  their  suspension  in  ins. 

ILLUSTRATION. —If  the  rise  of  the  balls  of  a  centrifugal  governor  is  22  ins.,  what 
are  the  number  of  revolutions  per  minute  ? 

188  -r-  ^22  =  40.09  revolutions. 

To    Compute   Vertical   Height  between    Plane    of  Balls 
and.   their    Points    of  Svispension. 

(i  88  -j-  r) 2  =  vertical  height  in  ins.   r  representing  number  of  revolutions  per  minute. 
ILLUSTRATION.—  If  number  of  revolutions  of  a  centrifugal  governor  is  100,  what 
will  be  rise  of  balls? 


i88-7-ioo=i.882  =  3.53  ins. 

To   Compute   Angle   of  A.rms    or    Plane    of    Balls    -with 

Centre    Shaft. 

r -1-1  =  sin.  /__.  r  representing  distance  of  balls  fi'om  plane  of  centre  shaft,  and.  I 
distance  between  balls  and  point  of  suspension  measured  in  plane  of  shaft. 

ILLUSTRATION.— Distance  of  balls  from  plane  of  centre  shaft  is  10  inches,  and 
their  distance  from  point  of  suspension  is  25 ;  what  is  the  angle  ? 
10  -T-  25  =  .4,  and  gin.  .4  =  23°  35'. 

When  Number  of  Revolutions  are  given.        [~~T ~  cos-  £-• 

ILLUSTRATION. — Revolutions  of  a  governor  per  minute  are  50,  and  length  of  its 
arms  2  feet;  what  is  their  angle  with  plane  of  shaft? 

(54'I6^50)2  =  1IZ3  =  . 5865  =  cos.  54°  6'. 

2  2 

PENDULUMS. 

Pendulums  are  Simple  or  Compound,  the  former  being  a  material 
point,  or  single  weight  suspended  from  a  fixed  point,  about  which  it 
oscillates,  or  vibrates,  by  a  connection  void  of  weight ;  and  the  latter, 
a  like  body  or  number  of  bodies  suspended  by  a  rod  or  connection. 
Any  such  body  will  have  as  many  centres  of  oscillation  as  there  are 
given  points  of  suspension  to  it,  and  when  any  one  of  these  centres  are 
determined  the  others  are  readily  ascertained. 


CENTRAL  FORCES. — PENDULUMS.        453 

Thus,  s  o  X  s  g  =  a  constant  product,  and  s  r  =  Vs  o  x  s  g,  s  g  o  and  r 
representing  points  of  suspension,  gravity,  oscillation,  and  gyration. 

Or,  any  body,  as  a  cone,  a  cylinder,  or  of  any  form,  regular  or  irregular, 
so  suspended  as  to  be  capable  of  vibrating,  is  a  compound  pendulum,  and 
distance  of  its  centre  of  oscillation  from  any  assumed  point  of  suspension  is 
considered  as  the  length  of  an  equivalent  simple  pendulum. 

The  Amplitude  of  a  simple  pendulum  is  the  distance  through  which  it 
passes  from  its  lowest  position  to  its  farthest  on  either  side. 

Complete  Period  of  a  pendulum  in  motion  is  the  time  it  occupies  hi  making 
two  vibrations. 

All  vibrations  of  same  pendulum,  whether  great  or  small,  are  performed 
very  nearly  in  same  time. 

Number  of  Oscillations  of  two  different  pendulums  in  same  time  and  at 
same  place  are  in  inverse  ratio  of  square  roots  of  their  lengths. 

T^ength  of  a  Pendulum  vibrating  seconds  is  hi  a  constant  ratio  to  force  of 
gravity. 

Time  of  Vibration  is  half  of  a  complete  period,  and  it  is  proportional  to 
square  root  of  length  of  pendulum.  Consequently,  lengths  of  pendulums  for 
different  vibrations  are — 

Latitude  of  Washington. 

39.0958  ins.  for  one  second.  I          4.344    for  third  of  a  second. 

9. 774    ins.  for  half  a  second.  2.4435  for  quarter  of  a  second. 

^engths  of  Pendulums  vibrating    Seconds   at   "Level   of* 
the    Sea  in   several   Places. 


Ins. 

Equator 39.0152 

Washington 39-0958 


New  York 39. 1017 

Lat.  45° 39. 127 


In*. 


Paris 39-1284 

London 39-1393 


To  Compute  Length,  of*  a  Simple  Pendulum  for  a  given 
Latitude. 

39. 127  —  .099  82  cos.  2  L  =  I.     L  representing  latitude. 

ILLUSTRATION.— Required  the  length  of  a  simple  pendulum  vibrating  seconds  iu 
the  latitude  of  50°  31'. 

L  =  50°  31"  cos.  2L  =  2X5o°  31'  =  cos.  180°  —  50°  31' X  2  =  cos.  78°  58'  = .  191  38 
—  39. 127  -f- .  191  38  x  .099  82  (two  —  or  negative  =  an  affirmative  or  -f-)  =  39. 1461  ins. 

To  Compute  Length  of*  a  Simple  Pendulum  for  a  given 
Number   of  "Vibrations. 

L  tz  =  I.    L  representing  length  for  latitude,  t  time  in  seconds,  and  I  length  of  pen- 
dulum in  ins. 

ILLUSTRATION.— Required  vibrations  of  a  pendulum  in  a  minute  at  New  York,  are 
60;  what  should  be  its  length? 

39. 1017  x  i2=39- 1017.  Or,  —  =  I.   n  representing  number  of  vibrations  per  second. 

To  Compute  Number  of  Vibrations  of  a  Simple  Pendu- 
lum   in    a   given    Time. 

•  •       =  n,  —  representing  time  of  one  vibration  in  seconds. 
v  *  ** 

To  Compute  Centre  of  Gravity  of  a  Compound  Pend.u«» 
lum    of  T\vo    Weights    connected    in    a.    Right    Line. 

When  Weights  are  both  on  one  Side  of  Point  of  Suspension. 

lW-\-l'w 

—     ~\ —  o  =  distance  oj  centre  of  gravity  from  point  of  suspension. 

W  -|-  w 


454 


CENTBAL  FORCES. — PENDULUMS. 


When  Weights  are  on  Opposite  Sides  of  Point  of  Suspension. 

-_"~  —  —  =  c  =  distance  of  centre  of  gravity  of  greater  weight  from  point  ofsus- 
W  -j-  w 
pension. 

NOTE.—  To  obtain  strictly  isochronous  vibrations,  the  circular  arc  must  be  sub- 
stituted for  the  cycloid  curve,  which  possesses  the  property  of  having  an  inclina- 
tion, the  sine  of  which  is  simply  proportional  to  distance  measured  on  the  curve 
from  its  lowest  point. 

For  construction  of  a  Cycloidal  pendulum,  see  Deschaniel's  Physics,  Part  I.,  pp. 
71-2. 


To  Compute  Length  of  a,  Simple  !Pendul  vim,  Vib  ration  s 
of  xvhich  -will  be  same  in.  Number  as  Inches  in  its 
Length.. 

2  =  I  in  inches. 


ILLUSTRATION.—  What  will  be  length  of  a  pendulum  in  New  York,  vibrations  of 
which  will  be  same  number  as  the  ins.  in  its  length  ? 

V  (>/39.  1017  X  6o)2  =  7.21  12  =  52  ins. 

To  Compute  Time  of  Vibration  of  a  Simple  I>endvalum, 
Length   "being   given. 


Vl  -^L  =  tin  seconds. 
dulum 

156.4 


ILLUSTRATION.— Length  of  a  pendulum  is  156.4  ins. :  what  is  the  time  of  its  vibra- 
tion in  New  York  ? 


J: 


-  —  2  seconds. 


39-  IQi7 

Or,  */—  X  3. 1416  =  t.  I  representing  length  of  a  pendulum  vibrating  seconds  in 
ins.,  g  measure  of  force  of  gravity,  and  t  time  of  one  oscillation. 

ILLUSTRATION. — Length  of  a  simple  pendulum  vibrating  seconds,  and  measure  of 
force  of  gravity  at  Washington,  are  39.0958  ins.,  and  32.155  feet. 


3.1416    /—-jT,  ^—  —  3- Ml6  X  VX-OX3  =  3-1416  X  .3183  =  i  second. 

To   Compute   Number   of  Vibrations   of  a   Simple  Pen- 
dulum  in   a  given   Time. 
'L 

—  x  t  =  n.    n  representing  number  of  vibrations. 

ILLUSTRATION.— The  length  of  a  pendulum  in  New  York  is  156.4  ins.,  and  time  of 
its  vibration  is  2  seconds;  what  are  number  of  its  vibrations? 


X  2  =  X  2  =  .5  X  2  =  z  vibration.     Hence,  ,  X      =  30  r»- 

brationsper  minute. 

To  Compute  Measure  of  Gravity,  Length  of  IPendulum 
and.  Number  of  its  "Vibrations  'being  given. 

.82246?  w2 

— =  g.    g  representing  measure  of  gravity  in  feet. 

To  Compute  Number  of  Revolutions  of  a  Conical  Pen- 
dulum   per   iMinute. 

%/ — ^—  =  n.  h  representing  distance  between  point  of  suspension  and  plane  oj 
revolutions  in  ins. 

NOTE.— Number  of  revolutions  per  minute  are  constant  for  any  given  height,  and 
the  time  of  a  revolution  is  directly  as  square  root  of  height. 


CRANES. 


455 


When    IPost    is 


CRANES. 

Usual  form  of  a  Crane  is  that  of  a  right-angled  triangle,  the  sides 
being  post  or  jib,  and  stay  or  strut,  which  is  hypothenuse  of  triangle. 

When  jib  and  post  are  equal  in  length,  and  stay  is  diagonal  of  a  square, 
this  form  is  theoretically  strongest,  as  the  whole  stress  or  weight  is  borne  by 
stay,  tending  to  compress  it  in  direction  of  its  length ;  stress  upon  it,  com- 
pared to  weight  supported,  being  as  diagonal  to  side  of  square,  or  as  1.4142 
to  i.  Consequently,  if  weight  borne  by  crane  is  1000  Ibs.,  thrust  or  com- 
pression upon  stay  will  be  1414.2  Ibs.,  or  as  a  e  to  e  W,  Fig.  i. 

Supported    at   tooth    Head   and    Foot,  as 
Fig.  1. 

a  Weight  W  is  sustained  by  a  rope  or  chain, 

and  tension  is  equal  upon  both  parts  of  it ;  that 
is,  on  two  sides  of  square,  i  a  and  e  W.  Conse- 
quently jib,  i  a,  has  no  stress  upon  it,  and  serves 
merely  to  retain  stay,  a  e. 

If  foot  of  stay  is  set  at  w,  thrust  upon  it,  as 
tv  compared  with  weight,  will  be  as  an  to  aw ; 
and  if  chain  or  rope  from  i  to  a  is  removed,  and 
weight  is  suspended  from  a,  tension  on  jib  will 
be  as  i  a  to  a  W. 

If  foot  of  stay  is  raised  to  o,  thrust,  as  compared  with  weight,  will  be  as 
line  a  o  is  to  a  W,  and  tension  on  jib  will  be  as  line  ar. 

By  dividing  line  representing  weight,  as  a  W  or  a  w,  into  equal  parts,  to 
represent  tons  or  pounds,  and  using  it  as  a  scale,  stress  upon  any  other  part 
may  be  measured  upon  described  parallelogram. 

Thus,  a?  length  of  a  W,  compared  to  a  e,  is  as  i  to  1.4142 :  if  a  W  is  di- 
vided into  10  parts  representing  tons,  a  e  would  measure  14.142  parts  or  tons. 

'When   Post   is    Supported   at   Foot   only. 

If  post  is  wholly  unsupported  at  head,  and  its  foot  is  secured  up  to  line 
o  W,  then  W,  acting  with  leverage,  e  W,  will  tend  to  rupture  post  at  e,  with. 
Bailie  effect  as  if  twice  that  weight  was  laid  upon  middle  ef  a  beam  equal  to 
twice  length  of  e  W,  e  being  at  middle  of  beam,  which  is  assumed  to  be  sup- 
ported at  both  ends,  and  of  like  dimensions  to  those  of  post. 

Or,  force  exerted  to  rupture  post  will  be  represented  by  stress,  W,  multi- 
plied by  4  times  length  of  lever,  e  W,  divided  by  depth  of  post  in  line  of 
stress,  squared,  and  multiplied  by  breadth  of  it  and  Value  *  of  its  material. 

Post  of  such  a  crane  is  in  condition  of  half  a  beam  supported  at  one  end, 
weight  suspended  from  other ;  consequently,  it  must  be  estimated  as  a  beam 
of  twice  the  length  supported  at  both  ends*  stress  applied  in  middle. 

To  Compute  Stress  on  Jib,  and  on  Stay  or  Struts-Fig.  Q. 
On  diagram  of  crane,  Fig.  2,  mark  off  on  line  of 
chain,  a  W,  a  distance,  a  £,  representing  weight  on 
chain;  from  point  b  draw  a  line,  b  c,  parallel  to  jib, 
a  e,  and  where  this  intersects  stay  or  strut,  draw  a 
vertical  line,  c  o,  extending  to  jib,  and  distances 
from  a  to  points  b  c  and  o  c,  measured  upon  a  scale 
of  equal  parts,  will  represent  proportional  strain. 

ILLUSTRATION.— In  figure,  weight  being  10  tons,  stress 
on  stay  or  strut  compressing,  a  c,  will  be  31  tons,  and 
on  jib  or  tension-rods,  a  o,  26  tons. 

*  For  Value  of  Materials,  see  page  77^ 


Fig.  2. 


456 


CRANES. 


To   Compnte   Dimensions   of  Post   of  a   Crane. 

When  Post  is  Supported  at  Feet  only.  RULE. — Multiply  weight  or  stress 
to  be  borne  in  Ibs.  by  length  of  jib  in  feet  measured  upon  a  horizontal 
plane ;  divide  product  by  Value  of  material  tc  >e  used,  and  product,  divided 
by  breadth  in  ins.,  will  give  square  of  depth,  aiso  in  ins. 

EXAMPLE. — Stress  upon  a  crane  is  to  be  22400  Ibs.,  and  distance  of  it  from  centre 
of  post  20  feet;  what  should  be  dimension  of  post  if  of  American  white  oak? 

Value  of  American  white  oak  50.    Assumed  breadth  *"  ins. 


When  Post  is  /Supported  at  both  Ends.    RULE. — Multiply  weight  or  stress 
to  be  borne  in  Ibs.  by  twice  length  of  jib  in  feet  measured  upon  a  horizontal 
plane ;  divide  product  by  Value  of  material  to  be  used,  and  product,  divided 
by  four  times  breadth  in  ins.,  will  give  square  of  depth,  also  in  ins. 
EXAMPLE.— Take  same  elements  as  in  preceding  case.    Assumed  breadth  10  ins. 

22400X20X2  17020 

Then  — =  17  920,  —     —  =  448,  and  ^448  =  21. 166  ins. 

50  4  X  10 

In  Fig. 3,  angle  abe  and  e  b  c  being  equal,  chain  or  rope  is  represented 
by  a  b  c,  and  weight  by  W ;  stress  upon  stay  b  a,  as 
compared  with  weight,  is  as  b  d  to  a  b  or  b  c. 

In  practice,  however,  it  is  not  prudent  to  consider 
chain  as  supporting  stay ;  but  it  is  proper  to  disrep  d 
chain  or  rope  as  forming  part  of  system,  and  crane 
should  be  designed  to  support  load  independent  of  it. 
It  is  also  proper  that  angles  on  each  side  of  diagonal 
stay,  in  this  case,  should  not  be  equal.  If  side  a  b  is 
formed  of  tension-rods  of  wrought  iron,  point  a  should 
be  depressed,  so  as  to  lengthen  that  side,  and  decrease 
angle  a  b  e ;  but  if  it  is  of  timber,  point  a  should  be 
raised,  and  angle  abe  increased. 

Fig.  4.  g      Fig.  4  shows  a  form  of  crane  very  generally  used; 

,<  angles  are  same  as  in  Fig.  3,  and  weight  suspended  from 
fS';!c  it,  being  attached  to  point  d,  is  represented  by  line  b  d. 
'',''/  The  tension,  which  is  equal  to  weight,  is  shown  by  leugt! 
/  of  line  b  c,  and  thrust  by  length  of  line  b  a,  measured  by 
/  a  scale  of  equal  parts,  into  which  line  bd,  repiesentiag 

weight,  is  supposed  to  be  divided. 

But  if  &  e  be  direction  of  jib,  then  b  g  will  show  ten- 
sion, and  bf  the  thrust  (df  being  taken  parallel  to  b  e), 
both  of  them  being  now  greater  than  before ;  line  b  d 
representing  weight,  and  being  same  in  both  cases. 

To  Ascertain   Stress  on  Ji"b,  on  Strut 
of  a  Crane.— Fig.  5. 

Through  a  draw  a  s,  parallel  to  jib  or  tension -rod 
o  r,  and  also  s  u  parallel  to  strut  a  r ;  then  r  s  is  a 
diagonal  of  parallelogram,  sides  of  which  are  equal  to  ra  and  r  u. 

If  then  r  s  represents  a  stress  of  20  Ibs., 
the  two  forces  into  which  it  is  decom- 
posed are  shown  by  r  u  and  r  a ;  or  is 
equal  to  r  u,  as  each  of  them  is  equal  to 
a  s,  and  r  s  is  equal  to  o  a.  Hence,  20 
represented  by  a  o,  stress  on  jib  will  be 
represented  by  o  r,  and  that  on  strut  by 
ra. 

Assuming  then  or  3  feet,  a  r  3.5,  and 
o  a  i,  stress  on  jib  will  be  60  Ibs.,  and  on  strut  70. 


CEANES. 


457 


Thus,  in  all  cases,  stress  on  jib  or  tension-rod  and  on  strut  can  be  deter- 
mined by  relative  proportions  of  sides  of  triangle  formed. 

To    Compttte    Stress   -upon    Strnt   of*  a    Crane. 

RULE.— Multiply  length  of  strut  in  feet  by  weight  to  be  borne  in  Ibs. ;  di- 
vide product  by  height  of  jib  from  point  of  bearing  of  strut  in  fee%  and 
quotient  will  give  stress  or  thrust  in  Ibs. 

EXAMPLE.— Length  of  strut  of  a  crane  is  28.284  feet>  height  of  post  is  26.457  feet, 
and  weight  to  be  borne  is  22  400  Ibs. ;  what  is  stress  ? 

28.284X22400  =  633561.6  =  lbg 

26.457  26.457 

Chains   and.   Ropes. 

Chains  for  Cranes  should  be  made  of  short  oval  links,  and  should  not  ex- 
ceed i  inch  in  diameter. 

Short -linked  Crane  Chains  and  Ropes  showing  T^i- 
mensions  and  "Weight  of  each,  and  3?roof  of  Chain 
in  Tons. 


Diam. 
of 
Chains. 

Weight 
per 
Fathom. 

Proof 
Strain. 

Circumf. 
of 
Rope. 

Weight 
of  Rope 
per  Path. 

Diam. 
of 

Chains. 

Weight 
Fathom. 

Proof 
Strain. 

Circnmf.  1  Weight 
of          of  Rope 
Rope,     j  per  Path. 

Ins. 

I/'S. 

Tons. 

Ins. 

Lba. 

Ing. 

Lbs. 

Tons. 

Ins. 

Lbs. 

•3125 

6 

•75 

2-5 

i-5 

.6875 

28 

6.5 

7 

10.5 

•375 

8-5 

i-5 

3-25 

2-5 

•75 

32 

7-75 

7-5 

12 

•4375 

ii 

2-5 

4 

3-75 

.8125 

36 

925 

8.25 

15 

•  5 

H 

3-5 

4-75 

5 

•875 

44 

10-75 

9 

17-5 

•5625 

18 

4-5 

5-5 

7 

•9375 

50 

12.5 

9-5 

19-5 

.625 

24 

5-25 

6.25 

8.7 

i 

56 

H 

10 

22 

Ropes  of  circumferences  given  are  considered  to  be  of  equal  strength  with 
the  chains,  which,  being  short-linked,  are  made  without  studs. 
A  crane  chain  will  stretch,  under  a  proof  of  15  tons,  half  an  inch  per  fathom. 

Machinery   of  Cranes. 

To  attain  greater  effect  of  application  of  power  to  a  crane,  the  wheel-work 
must  be  properly  designed  and  executed. 

If  manual  labor  is  employed,  it  should  be  exerted  at  a  speed  of  220  feet 
per  minute. 

Proportions. — Capacity  of  Crane,  5  tons. 

Radius  of  winch  or  handle  15  to  1 8  ins.    Height  of  axle  from  flou  36  to  39. 

2d  pinion,  12  teeth,  1.5  ins.  pitch. 
2d  wheel,  96     "       "     "       u 

Barrel  8  ins.  x  1 1  teeth  x  12  teeth  X  1 1  200  Ibs.  =  30  800  ., 

— - =  20. 7  5  I os.  =  statical  re- 
Winch  17  ins.  x  89  teeth  x  96  teeth  x  4  men  =  1513 
sistance  to  each  of  the  4  men  at  winches. 

An  experiment  upon  capacity  of  a  crane,  geared  i  to  105,  develor  ad  that 
a  strong  man  for  a  period  of  2.5  minutes  exerted  a  power  of  27  562  foot- 
pounds per  minute,  which,  when  friction  of  crane  is  considered,  is  fully  equal 
to  the  power  of  a  horse  for  one  minute. 

In  practice  an  ordinary  man  can  develop  a  power  of  15  Ibs.  upon  a  crane, 
handle  moved  at  a  velocity  of  220  feet  per  minute,  which  is  equivalent  to 
3300  foot-pounds. 

For  Treatise  on  Cranes,  see  Weales'  Series,  No.  33. 


ist  pinion,  n  teeth,  1.25  ins.  pitch, 
ist  wheel,  89     "     1.25    "       " 


458 


COMBUSTION. 


COMBUSTION. 

Combustion  is  one  of  the  many  sources  of  heat,  and  denotes  combi- 
nation of  a  body  with  any  of  the  substances  termed  Supporters  of  Com- 
bustion ;  with  reference  to  generation  of  steam,  we  are  restricted  to  but 
one  of  these  combinations,  and  that  is  Oxygen. 

All  bodies,  when  intensely  heated,  become  luminous.  When  this  heat 
is  produced  by  combination  with  oxygen,  they  are  said  to  be  ignited ; 
and  when  the  body  heated  is  in  a  gaseous  state,  it  forms  what  is  termed 
Flame. 

Carbon  exists  in  nearly  a  pure  state  in  charcoal  and  in  soot.  It  com- 
bines with  no  more  than  2.66  of  its  weight  of  oxygen.  In  its  combus- 
tion, i  Ib.  of  it  produces  sufficient  heat  to  increase  temperature  of  14  500 
Ibs.  of  water  i°. 

Hydrogen  exists  in  a  gaseous  state,  and  combines  with  8  times  its 
weight  of  oxygen,  and  i  Ib.  of  it,  in  burning,  raises  heat  of  50  ooo  Ibs. 
of  water  i0.* 

An  increase  in  the  rapidity  of  combustion  is  accompanied  by  a  dimi- 
nution in  the  evaporative  efficiency  of  the  combustible. 

Mr.  D.  K.  Clark  furnishes  the  following:  When  coal  is  exposed  to  heat  in  a  fur- 
nace, the  carbon  and  hydrogen,  associated  in  various  chemical  unions,  as  hydrocar- 
bons, are  volatilized  and  pass  off.  At  lowest  temperature,  naphthaline,  resins,  aud 
fluids  with  high  boiling-points  are  disengaged;  at  a  higher  temperature,  volatile 
fluids  are  disengaged;  and  still  higher,  defiant  gas,  followed  by  light  carburetted 
hydrogen,  which  continues  to  be  given  off  after  the  coal  has  reached  a  low  red  heat. 
As  temperature  rises,  pure  hydrogen  is  also  given  off,  until  finally,  in  the  fifth  or 
highest  stage  of  temperature  for  distillation,  hydrogen  alone  is  discharged.  What 
remains  after  distillatory  process  is  over,  is  coke,  which  is  the  fixed  or  solid  carbon 
of  coal,  with  earthy  matter  or  ash  of  the  coal. 

The  hydrocarbons,  especially  those  which  are  given  off  at  lowest  temperatures, 
being  richest  in  carbon,  constitute  the  flame-making  and  smoke-making  part  of  the 
coal.  When  subjected  to  heat  much  above  the  temperatures  required  to  vaporize 
them,  they  become  decomposed,  and  pass  successively  into  more  and  more  perma- 
nent forms  by  precipitating  portions  of  their  carbon.  At  temperature  of  low  red- 
ness none  of  them  are  to  be  found,  and  the  oleflant  gas  is  the  densest  type  that 
remains,  mixed  with  carburetted  and  free  hydrogen.  It  is  during  these  trans- 
formations that  the  great  volume  of  smoke  is  made,  consisting  of  precipitated  car- 
bon passing  off  uncombined.  Even  olefiant  gas,  at  a  bright  red  heat,  deposits  half 
its  carbon,  changing  into  carburetted  hydrogen;  and  this  gas,  in  its  turn,  may 
deposit  the  last  remaining  equivalent  of  carbon  at  highest  furnace  heats,  and  be 
converted  into  pure  hydrogen. 

Throughout  all  this  distillation  and  transformation,  the  element  of  hydrogen 
maintains  a  prior  claim  to  the  oxygen  present  above  the  fuel;  and  until  it  is  satis- 
fied, the  precipitated  carbon  remains  unburned. 

Sximmary  of  J?rodTu.cts  of  Decomposition,  in  the  ITnrnace. 

Reverting  to  statement  of  average  composition  of  coal,  page  485,  it  ap- 
pears that  the  fixed  carbon  or  coke  remaining  in  a  furnace  after  volatile 
portions  of  coal  are  driven  off,  averages  61  per  cent,  of  gross  weight  of  the 
coal.  Taking  it  at  60  per  cent.,  proportion  of  carbon  volatilized  in  com- 
bination with  hydrogen  will  be  20  per  cent.,  making  total  of  80  per  cent,  of 
constituent  carbon  in  average  coal. 

Of  the  5  per  cent,  of  constituent  hydrogen,  i  part  is  united  to  the  8  per 
cent,  of  oxygen,  in  the  combining  proportions  to  form  water,  and  remaining 
4  parts  of  hydrogen  are  found  partly  united  to  the  volatilized  carbon,  and 
partly  free. 


COMBUSTION.  459 

These  particulars  are  embodied  in  following  summary  of  condition  of 
elements  of  100  Ibs.  of  average  coal,  after  having  been  decomposed,  and  prior 
to  entering  into  combustion — 

100  Lbs.  of  Average  Coal  in  a  Furnace. 

Composition  Lbs.  Lbs.  Decomposition. 

60      fixed  carbon. 

24      hydrocarbons  and  free  hydrogen 
1.25  sulphur. 


Hydrogen  .............  5 

Sulphur  ...............  1.25 

Oxygen  ...............  8 

Nitrogen  ..............  1.2 

Ash,  etc  ...............  4-55 


forming 


85-25 

9      water  or  steam. 
1.2    nitrogen. 
4.55  ash,  etc. 


IOO 


showing  a  total  useful  combustible  of  85.25  per  cent.,  of  which  25.25  per 
cent,  is  volatilized.  While  the  decomposition  proceeds,  combustion  proceeds, 
and  the  25.25  per  cent,  of  volatilized  portions,  and  the  60  per  cent,  of  fixed 
carbon,  successively,  are  burned. 

It  may  be  added  that  the  sulphur  and  a  portion  of  the  nitrogen  are  dis- 
engaged in  combination  with  hydrogen,  as  sulphuretted  hydrogen  and  am~ 
monia.  But  these  compounds  are  small  in  quantity,  and,  for  the  sake  of 
simplicity,  they  have  not  been  indicated  in  the  synopsis. 

Volume  of  Air  chemically  consumed  in  complete  Combustion  of  Coal. 
Assume  100  Ibs.  of  average  coal    Then,  by  following 

(o\       _ 
5  —  -J  +  -4  X  1.25  X  152  =  14060  cube  feet  of  air  at  62°  for  100  Ibs.  coal 

For  volatilized  portion,  Hydrogen  (H),  4      Ibs.  X  457=  1828  cube  feet 
Carbon  (C),      20        "   X  152=  3040    "       " 
Sulphur  (S),      1.25  "   X    57=  _  ij_    "       " 

4939     "       " 
For  fixed  portion,  Carbon,  60      Ibs.x  152=  9120    "      " 

Total  useful  combustible,  85.25   u  14059    "       "  for  com- 

plete combustion  of  100  Ibs.  coal  of  average  composition  at  62°. 

To  Comprite  "Volume  of  Air  at  62°,  nnder  One  At- 
mosphere, chemically-  consumed,  in.  Complete  Com* 
iDxistion  of*  1  Lt>.  of*  a  given  Fuel. 

RULE.  —  Express  constituent  carbon,  hydrogen,  oxygen,  and  sulphur,  as 
percentages  of  whole  weight  of  fuel  ;  divide  oxygen  by  8,  deduct  quotient 
from  hydrogen,  and  multiply  remainder  by  3  ;  multiply  sulphur  by  .4  ;  add 
products  to  the  carbon,  and  multiply  sum  by  1.52.  Final  product  is  volume 
of  air  in  cube  feet. 

To  compute  weight  of  air  chemically  consumed.  —  Divide  volume  thus  found 
by  13.14;  quotient  is  weight  of  air  in  Ibs. 

Or,  1.52  (C  +  3  (H--|)  +.4  S)  =  Air.     0  Oxygen. 

NOTE.—  In  ordinary  or  approximate  computations,  sulphur  may  be  neglected. 
EXAMPLE.—  Assume  i  Ib.  Newcastle  coal       0  =  82.24,  1^  =  5.42,  0  =  6.44,  and 
8  =  1.35- 

-^  =  .805,  5.42  —  .805  =  4.615  X  3  =  13-845,  1.35  X  .4  =  -  54,  13-845  +  -54+8-2.24 
=  96.625,  and  96.625  X  1-52  =  146.87  cube  feet. 

Then  146.  87  -r-  13.  14  =  1  1.  18  Ibs. 


460 


COMBUSTION. 


To  Compxite  Total  Weight  of  Gaseous  Products  of  Com- 
plete   Combustion,   of  1    I-*b.  of  a   given    Fuel. 
RULE. — Express  the  elements  as  per-centages  of  fuel ;  multiply  carbon  by 
.126,  hydrogen  by  .358,  sulphur  by  .053,  and  nitrogen  by  .01,  and  add  prod- 
ucts together.    Sum  is  total  weight  of  gases  in  Ibs. 

Or,  .126  C  +  .358  H-f. 053  S  +  .oi  N  =  Weigkt. 
EXAMPLE.  —Assume  as  preceding  case.    N  =  1,61. 

82.24  X  .1264- 5-42  X. 358  + 1.35  X  053  +  1.61  X  .01  =  12.39  Ibs. 

To  Compute  Total  "Volume,  at  62°,  of  Q-aseous  Products 
of  Complete  Combustion    of  1    .Lb.  of  given    Fuel. 

RULE. — Express  elements  as  per-centages ;  multiply  carbon  by  1.52,  hy- 
drogen by  5.52,  sulphur  by  .567,  and  nitrogen  by  .135,  and  add  products 
together.    Sum  is  total  volume,  at  62°  F..  of  gases,  in  cube  feet. 
Or,  1.52  C-f-5.52  H  +  .s67  S  -f .  135  N  =  Volume. 

To    Compute   "Volume    of  the    several   O-ases    separately 

from  their   Respective   Quantities. 

RULE. — Multiply  weight  of  each  gaseous  product  by  volume  of  i  Ib.  in 
cube  feet  at  62°,  as  below. 

Volume  of  i  Lb.  of  Gases  at  62°  under  a  Pressure  of  14.7  Lbs. 

Cube  feet.  Cube  feet.  Cube  feet. 

Aqueous  Vapor  or)  I  Oxygen 11.887  I  Nitrogen 13.501 

Gaseous  Steam .  j  ,       5  |  Hydrogen 190        |  Carbonic  Acid 8. 594 

Air 13. 141  cube  feet. 

For  a  Ib.  of  oxygen  in  combustion,  4.35  Ibs.  air  are  consumed;  or,  by  volume,  for 
a  cube  foot  of  oxygen  4.76  cube  feet  of  air  are  consumed. 

i  Ib.  Hydrogen  consumes 34. 8  Ibs. ,  or  457  cube  feet,  at  62°. 

i  "  Carbon,  completely  burned,  consumes n.6    "       u  152     "      "     "   " 

i  "        "      partially          "  u        ....    5.8    "       "    76     "      "     "   " 


Composition    and    Equivalents   of  Gases,  combined   in 

Combustion   of  Fuel. 

OASES. 

Elements. 

By 

Weight. 

GASES. 

Elements. 

By 

Weight. 

ELEMENTS. 

Equ  v- 
alents. 

COMPOUNDS. 

Equiv- 
alents. 

0. 
H. 

8 

Light  Carburetted 
Hydrogen 

C.  2 

H     A. 

•;} 

Hydrogen                . 

Carbon  

C. 

S. 

6 
16 

Carbonic  Oxide  

ii.  4 

0.  i 
C.  i 

4  ) 
81 

6  j 

Nitrogen  

N. 

14 

Carbonic  Acid  

O.    2 

C.  i 

16) 
6J 

COMPOUNDS. 

OlefiantGas(Bi-car-) 

C.  4 

24) 

"•Atmospheric  Air 

0.23 

8     ) 

buretted  Hyd.  .  .  .  j 

H.4 

4} 

(mech.  mixture)  .  . 
Aqueous    Vapor    or 
Water.  .  . 

S:7,7 

H.   i 

26.8} 

Sulphurous  Acid.  .  . 

0.    2 

8.   z 

16) 
16 

Weights  of  products  in  combustion  of  i  Ib.  of  given  fuel,  are— 

0^.0366.     H  =  .o9.     S  =  .o2.     N  =  .08930  +  . 268  H  +  . 0335  S+. 01  N. 


Cube  Feet. 

.0366  X     8.59=     .315  volume  carbonic 

acid. 
.09     X 190      =17.1 


Cube  Feet. 

.02  X  5.85  =  .117  volume  sulph.  acid. 
.0893  4-  -268  -f-  .0335  -f-  .01  X  13-501  — 
5.409  volume  nitrogen. 


Volume  of  Air  or  Gases  at  higher  temperatures  than  here  given  (62°)  is  ascer- 
tained by,     V  =  V.     V  representing  volume  of  air  or  gas  at  temperature  t, 

t  +461 
and  V  at  temperature  t'. 


*  By  Volume  i  Oxygen,  3.762  Nitrogen. 


COMBUSTION. 


Chemical    Composition    of    some     Compound.    Com- 
t>u.stil>les. 


COMBUSTIBLK. 

Combi 
Car. 

Ding  equiv 
Hyd. 

alents. 
Oxy. 

In  100 
Car. 

parts  by  weight. 
Hyd.     |     0*y. 

Carbonic  oxide    

I 

— 

I 

Per  Cent. 
42.9 

Per  Cent. 

Per  Cent. 
57-i 

21.7 
34-8 

4-5 
9-4 
9-3 

Oleflant  gas,  Bicarburetted  hyd. 

4 
4 

4 

20 

4 
5 
6 
16 

I 
2 

III 
£1 

81.6 
77.2 
79 

14-3 
13-5 

£• 

i3-9 
i3-4 

ii.  7 

Wax  

Olive  oil    

Tallow  .  .  . 

Heating  powers  of  compound  bodies  are  approximately  equal  to  sum  of 
heating  powers  of  their  elements. 

Thus,  carburetted  hydrogen,  which  consists  of  two  equivalents  of  carbon  and  four 
of  hydrogen,  weighing  respectively  2X6=12  and  i  x  4  =  4,  in  proportion  of  3  to  i, 
or  .75  Ib.  of  carbon  and  .25  Ib.  of  hydrogen  in  one  Ib.  of  gas.  Elements  of  heat  of 
combustion  of  one  Ib.  are,  then — 

Units  of  heat. 

For  carbon 14  544  X  -75  =  10  908 

For  hydrogen 62  032  X  .25  =  15508 

Total  heat  of  combustion,  as  computed. 26  416 

Total  heat,  by  direct  trial 23513 

Heating  IPowers   of  Com.~biistit>les. 

(MM.  Favre  and  Silbermann,  D.  K.  Clark  and  others.) 


i  LB.  OF 
COMBUSTIBLE. 

Oxygen 
consumed 
per  Ib.  of 
Com- 
bustible. 

Weight  and  Volume 
of  Air  consumed  per 
Ib.  of  Combustible. 

Total  Heat 
of  Combus- 
tion of  i  Ib. 
of  Combus- 
tible. 

Equivalent  evaporative 
Power  of  i  Ib.  of  Com- 
bustible, under  one  At' 
mosphere. 

Lbs. 

Lbs. 

Cube  Feet 
at  62°. 

Units. 

Lbs.  of  wa- 
ter at  62°. 

Lbs.  of  wa- 
ter at  212". 

8 

34-8 

457 

62032 

55-6 

64.2 

Carbon,    making  ) 
carbonic  oxide,  j 

i-33 

5-8 

76 

4452 

4 

4.61 

Carbon,     making  ) 
carbonic  acid..  } 

2.66 

n.6 

152 

14500 

13 

15 

Carbonic  oxide  

•57 

2.48 

33 

4325 

3.88 

4.48 

Light  carburetted  ) 
hydrogen  J 

4 

17.4 

229 

23513 

21.07 

24.34 

Oleflant  gas    

•a  A"i 

je 

196 

21  343 

IQ.  12 

22.09 

Sulphuric  ether.  .  .  . 

•3-  43 
2.6 

"•3 

149 

16249 

y   * 
14.56 

16.82 

Alcohol           .   .  • 

2  78 

12.  1 

12  Q2Q 

11.76 

13.38 

Turpentine  

Ui  /W 

3  2Q 

14-  3 

188 

*•*  v^y 

19  534 

17.5 

20.22 

Sulphur      .  .  •  .  . 

O'^y 

A.  35 

57 

4  °32 

2 
7.61 

4.  17 

Tallow  

2  0^ 

12.83 

169 

18028 

J.V» 

16.15 

18.66 

Petroleum 

•^•yo 

17  Q^ 

235 

27  531 

28.5 

Coal  (average)  

4-12 
2.46 

*-/'yj 
10.7 

141 

14  '33 

12.67 

14.62 

Coke,  desiccated.  .  . 

2-5 

IO.Q 

143 

13550 

12.14 

14.02 

Wood,  desiccated  .  . 

1.4 

6.1 

80 

7792 

6.98 

8.07 

Wood  -  charcoal,  ) 
desiccated  J 

2.25 

9.8 

129 

13309 

11.92 

13-13 

Peat,  desiccated  

i-75 

7.6 

IOO 

9951 

8.91 

10.3 

peat-  charcoal,  de-  ) 

2.28 

9.9 

129 

12325 

11.04 

12.76 

2.03 

8.85 

116 

11678 



12.  X 

Asphalt... 

2.73 

11.87 

156 

16655 

— 

I7-*4 

When  carbon  is  not  completely  burned,  and  becomes  carbonic  oxide,  it  producei 
less  than  a  third  of  heat  yielded  when  it  is  completely  burned.  For  heating  powei 
of  carbon  an  average  of  14  500  units  is  adopted. 

Uu* 


COMBUSTION. 


To  Compute   Heating    Power  of*  1  I_/b.  of  a   given  Corn- 
ID  ustifole. 

When  proportions  of  Carbon,  Hydrogen,  Oxygen,  and  Sulphur  are  given. 
RULE.— Ascertain  difference  between  hydrogen  and  .125  of  oxygen;  multi- 
ply remainder  by  4.28 ;  multiply  sulphur  by  .28,  add  products  to  the  carbon, 
multiply  sum  by  14  500,  divide  by  100,  and  product  is  total  heating  power 
hi  units  of  heat. 

Or,  145  (€  +  4.28  H  —  Ox.  125  -f .  288)  =  heat 
ILLUSTRATION.— Assume  as  preceding  case. 


5.42  'v  82.28  X.  125  X  4-28  -f- 1. 35  X  . 28  +  82.28  X  14500-1-100^:15005. 

To    Compute    Evaporative    IPo\ver   of  1    Lt>.  of  a    Griven 
Combustible. 

When  Proportions  of  Carbon,  Hydrogen,  Oxygen,  and  Sulphur  are  given. 
RULE. — Ascertain  difference  between  hydrogen  and  .125  of  oxygen,  multiply 
remainder  by  4.28 ;  multiply  sulphur  by  .28,  add  products  to  the  carbon,  and 
multiply  sum  by  .13,  when  water  is  supplied  at  62°,  and  .15  when  at  212° ; 
product  is  evaporative  power  in  Ibs.  of  water  at  212°. 

Or,  When  total  heating  power  is  known,  divide  it  by  1116  when  water  is 
at  62°,  or  996  when  at  212°. 

ILLUSTRATION.—  By  table,  heating  power  of  Tallow  is  18028  units. 

Hence,  18028  -5- 1116  =  16.15  Ms.  water  evaporated  at  62°. 

Tem.peratu.re   of  ComlmstiorL. 

Temperature  of  combustion  is  determined  by  product  of  volumes  and 
specific  heats  of  products  of  combustion. 

ILLUSTRATION.— i  Ib.  carbon,  when  completely  burned,  yields  3.66  Ibs.  carbonic 
acid  and  8.94  of  nitrogen.  Specific  heats  .2164  and  .244. 

3.66  X  .2164  =  .792  units  of  heat  for  i°. 
8. 94  X.  244  =  2.181  "  "  "  i°. 
12.6  2.973  "  "  "  i°. 

Consequently,  products  of  combustion  of  i  Ib.  carbon  absorbs  2.973  units  of  heat 
in  producing  i°  temperature. 

Weignt  and   Specific   Heat  of  Products  of  Comtmstioii, 
and   Temperature   of  Combustion .     (D.  K.  Clark.) 
Gaseous  Products  for  i  Lb.  of  Combustible. 


i  LB.  OF  COMBUSTIBLE. 

Weight. 

Mean 
specific 
Heat. 

Heat  to  raise 
the  Tempera- 
ture i°. 

Temperature  ol 
Combustion. 

Lbs. 
35-8 
11.97 

15-9 
13.84 
11.94 

12.6 

15-21 

10.09 

18.4 

5-35 
12.  18 
22.64 

Water  =  i. 
.302 
.256 

•257 
.256 
.240 
.236 

•257 
.27 
.268 

.211 

•257 
.242 

Units. 
10.814 
3.063 
4.089 
3-54 
2-935 
2-973 
3-9M 
2.68 

4-933 
1.128 
3-127 
5-478 

0 

5744 
5305 
5219 
•5093 
4879 

4877 
4826 
4825 
4766 
3575 
3470 
2614 

Ratio. 

100 

92 

Is, 

85 

1S 
84 

84 
83 
62 
60 

45 

Sulphuric  ether  

Olefiant  gas  (Bi-carburetted  hyd.) 
Tallow  

Carbon  or  pure  coke.  

Wax       

Light  carburetted  hydrogen  

Turpentine     

Coal,  with  double  supply  of  air.  . 

Whence  it  appears,  that  mean  specific  heat  of  products  of  combustion,  omitting 
hydrogen  .302  and  sulphur  .211,  is  about  .25. 

Hence.  To  Ascertain  Temperature  of  Combustion. — Divide  total  heat  of 
combustion  in  units  by  units  of  heat  for  i°,  and  quotient  will  give  tem- 
perature. 


COMBUSTION.  463 

ILLUSTRATION'.  •  What  is  temperature  of  combustion  of  coal  of  average  composi- 
tion? 

Gaseous  products  as  per  preceding  table  11.94,  which  x  .246  specific  heat  =  2.935 
units  of  heat  at  1°. 

Hence,  14 133  units  of  combustion  (from  table,  page  461)  -7-2.935  =  4812°  temper- 
ature of  combustion  of  average  coal. 

If  surplus  air  is  mixed  with  products  of  combustion  equal  to  volume  of  air  chem- 
ically combined,  total  weight  of  gases  for  ore  Ib.  of  this  coal  is  increased  to  22.64. 
See  following  table,  having  a  mean  specific  heat  ol  .242. 

Then  22.64  X  .242  =  5.478  units  for  1°. 

Hence,  14133  total  heat  of  combustion -4-5. 478  =  2580°  temperature  of  combus- 
tion, or  a  little  more  than  half  that  of  undiluted  products. 

Taking  averages,  it  is  seen  that  the  evaporative  efficiency  of  coal  varies 
directly  with  volume  of  constituent  carbon,  and  inversely  with  volume  of 
constituent  oxygen ;  and  that  it  varies,  not  so  much  because  there  is  more  or 
less  carbon,  as,  chiefly,  because  there  is  less  or  more  oxygen.  The  per-cent- 
ages  of  constituent  hydrogen,  nitrogen,  sulphur,  and  ash,  taking  averages, 
are  nearly  constant,  though  there  are  individual  exceptions,  and  their  united 
effect,  as  a  whole,  appears  to  be  nearly  constant  also. 

Heat  of  Combustion. 

Or,  number  of  times  in  combustion  of  a  substance,  its  equivalent  weight  of  water 
would  be  raised  1°,  by  heat  evolved  in  combustion  of  substance. 

Alcohol 12  930  I  Ether 16  246  I  Olefiant  gas 21  340 

Charcoal 14  545  I  Olive  oil 17  750  |  Hydrogen 62  030 

Com'bnstion    of  IPuel. 

Constituents  of  coal  are  Carbon,  Hydrogen,  Azote,  and  Oxygen. 

Volatile  products  of  combustion  of  coal  are  hydrogen  and  carbon,  the 
unions  of  which  (relating  to  combustion  in  a  furnace)  are  Carburetted 
hydrogen  and  Bi-carburetted  hydrogen  or  Olefiant  gas,  which,  upon  com- 
bining with  atmospheric  air,  becomes  Carbonic  acid  or  Carbonic  oxide, 
Steam,  and  uncombined  Nitrogen. 

Carbonic  oxide  is  result  of  imperfect  combustion,  and  Carbonic  acid 
that  of  perfect  combustion. 

Perfect  combustion  of  carbon  evolves  heat  as  15  to  4.55  compared 
with  imperfect  combustion  of  it,  as  when  carbonic  oxide  is  produced. 

i  Ib.  carbon  combines  with  2.66  Ibs.  of  oxygen,  and  produces  3.66  Ibs. 
of  carbonic  acid. 

Smoke  is  the  combustible  and  incombustible  products  evolved  in  combustion  of 
fuel,  which  pass  off  by  flues  of  a  furnace,  and  it  is  composed  of  such  portions  of 
hydrogen  and  carbon  of  the  fuel  gas  as  have  not  been  supplied  or  combined  with 
oxygen,  and  consequently  have  not  been  converted  either  into  steam  or  carbonic 
acid;  the  hydrogen  so  passing  away  is  invisible,  but  the  carbon,  upon  being  sepa- 
rated from  the  hydrogen,  loses  its  gaseous  character,  and  returns  to  its  elementary 
state  of  a  black  pulverulent  body,  and  as  such  it  becomes  visible. 

Bituminous  portion  of  coal  is  converted  into  gaseous  state  alone,  carbonaceous 
portion  only  into  solid  state.  It  is  partly  combustible  and  partly  incombustible. 

To  effect  combustion  of  i  cube  foot  of  coal  gas,  2  cube  feet  of  oxygen  are  required; 
and,  as  10  cube  feet  of  atmospheric  air  are  necessary  to  supply  this  volume  of  oxy- 
gen, i  cube  foot  of  gas  requires  oxygen  of  10  cube  feet  of  air. 

In  furnaces  with  a  natural  draught,  volume  of  air  required  exceeds  that 
when  the  draught  is  produced  artificially. 

An  insufficient  supply  of  air  causes  imperfect  combustion ;  an  excessive 
supply,  a  waste  of  heat. 


464  COMBUSTION. 


Volume  of  atmospheric  air  that  is  chemically  required  for  combustion  of 
b.  of  bituminous  coal  is  150.35  cube  feet.    Of  this,  44.64*  cube  feet  corn- 


Volume  < 
i  lb. 

bine  with  the  gases  evolved  "from  the  coal,  and  remaining  105.71  cube  feet 
combine  with  the  carbon  of  the  coal. 

Combination  of  gases  evolved  by  combustion  gives  a  resulting  volume 
proportionate  to  volume  of  atmospheric  air  required  to  furnish  the  oxygen, 
as  ii  to  10.  Hence  the  44.64  cube  feet  must  be  increased  in  this  proportion, 
and  it  becomes  44.64  -f-  4.46  =  49.1. 

Gases  resulting  from  combustion  of  the  carbon  of  coal  and  oxygen  of  the 
atmosphere,  are  of  same  bulk  as  that  of  atmospheric  air  required  to  furnish 
the  oxygen,  viz.,  105.71  cube  feet.  Total  volume,  then,  of  the  atmospheric 
air  and  gases  at  bridge  wall,  flues,  or  tubes,  becomes  105.71  +  49.1  =  154.81 
cube  feet,  assuming  temperature  to  be  that  of  the  external  air.  Conse- 
quently, augmentation  of  volume  due  to  increase  of  temperature  of  a  fur- 
nace is  to  be  considered  and  added  to  this  volume,  in  the  consideration  of  the 
capacity  of  flue  or  calorimeter  of  a  furnace. 

There  is  required,  then,  to  be  admitted  through  the  grates  of  a  furnace  for 
combustion  of  i  lb.  of  bituminous  coal  as  follows  : 

Coal  containing  80  per  cent,  of  carbon,  or  .7047  per  cent,  of  coke. 

i  lb.  coal  x  44-64  cube  feet  of  gas =   44.64 

7047  lb.  carbon  x  150  cube  feet  of  air  ...  =  105.71 

150.35  cube  feet. 

For  anthracite,  by  observations  of  W.  R.  Johnston,  an  increase  of  30  per 
cent,  over  that  for  bituminous  coal  is  required  =  195.45  cube  feet. 

Coke  does  not  require  as  much  air  as  coal,  usually  not  to  exceed  108  cube 
feet,  depending  upon  its  purity. 

Heat  of  an  ordinary  furnace  may  be  safely  considered  at  1000° ;  hence  air 
entering  ash-pit  and  gases  evolved  in  furnace  under  general  law  of  expan- 
sion of  permanently  elastic  fluids  of  ^-^ths  of  its  volume  (or  .002087)  for 
each  degree  of  heat  imparted  to  it,  the  154.81  is  increased  in  volume  from 
100°  (assumed  ordinary  temperature  of  air  at  ash-pit)  to  1000°  =  900° ;  then 
9oox  .002087  =  1.8783  times,  or  154.81  +  154.81  x  1.8783  =  445.59  cube  feet. 

If  the  combustion  of  the  gases  evolved  from  coal  and  air  was  complete, 
there  would  be  required  to  give  passage  to  volume  of  but  445.59  cube  feet 
over  bridge  wall  or  through  flues  of  a  furnace ;  but  by  experiments  it  ap- 
pears that  about  one  half  of  the  oxygen  admitted  beneath  grates  of  a  furnace 
passes  off  uncombined ;  the  area  of  the  bridge  wall,  or  flues  or  tubes,  must  con- 
sequently be  increased  in  this  proportion,  hence  the  445.59  becomes  891.18. 

Velocity  of  the  gases  passing  from  furnace  of  a  proper-proportioned  boiler 


may  be  estimated  at  from  30  to  36  feet  per  second.    Then 


891.18 


60' x  60"  x  36" 

.00687  sq.  feet,  or  .99  sq.  ins.,  of  area  at  bridge  wall  for  each  lb.  of  coal  con- 
sumed per  hour. 

A  limit,  then,  is  here  obtained  for  area  at  the  bridge  wall,  or  of  flues  or 
tubes  immediately  behind  it,  below  which  it  must  not  be  decreased,  or  com- 
bustion will  be  imperfect.  In  ordinary  practice  it  will  be  found  advan- 
tageous to  make  this  area  .014  sq.  feet,  or  2  sq.  ins.  for  every  lb.  of  bitu- 
minous coal  consumed  per  sq.  foot  of  grate  per  hour,  and  so  on  In  proportion 
for  any  other  quantity. 

Volumes  of  heat  evolved  are  very  nearly  same  for  same  substance,  what- 
ever temperature  of  combustible. 

*  By  experiment,  4.464  cube  feet  of  gas  are  evolved  from  i  lb.  of  bituminous  coal,  requiring  44.64 
cube  feet  of  air. 


COMBUSTION. 


465 


Relative  Volumes  of  Air  required  for  Combustion  of  Fuels. 

Lbs.     1  Lbs. 

Warlich's  patent. . .  13.  i    I  Anthracite  Coal ....  12. *-. 

Charcoal 11.16    Bituminous  "    ....  io.gl 

Coke 11.28  1  Bitum.  Coal,  average  10.7 


Lba. 
Bitum.  Coal,  lowest. .  5.92 

Peat,  dry 7.08 

Wood,  dry..., 6 


Perfect  combustion  of  i  Ib.  of  carbon  requires  11.18  Ibs.  air  at  62°,  and 
total  weight  =  1  2.39.»/fo.  Total  heat  of  combustion  of  i  Ib.  carbon  or  char- 
coal is  14  500  thermal  units  ;  mean  specific  heat  of  products  of  combustion 
is  .25,  which,  multiplied  by  12.39  as  above  =  3.0975,  and  14  500*  -r-  3.0975  = 
4681°  temperature  of  a  furnace,  assuming  every  atom  of  oxygen  that  was 
ignited  in  it  entered  into  combination. 

If,  however,  as  in  ordinary  furnaces,  twice  volume  of  air  enters,  then 
products  of  combustion  of  i  Ib.  of  coal  will  be  12.39  H~  11.18  =  23.57,  which, 
multiplied  by  its  specific  heat  of  .25  as  before,  and  if  divided  into  14  500, 
quotient  will  be  2461  °,  which  is  temperature  of  an  ordinary  furnace. 

Ratio  of  Combustion.  —  Quantity  of  fuel  burned  per  hour  per  sq.  foot  of 
grate  varies  very  much  in  different  classes  of  boilers.  In  Cornish  boilers  it 
is  3.5  Ibs.  per  sq.  foot  ;  in  ordinary  Land  boilers,  10  to  20  Ibs.  ;  (English)  13 
to  14  Ibs.  ;  in  Marine  boilers  (natural  draught),  10  to  24  Ibs.  ;  (blast)  30  to 
60  Ibs.  ;  and  in  Locomotive  boilers,  80  to  120  Ibs. 

Volumes  of  air  and  smoke  for  each  cube  foot  of  water  converted  into 
steam,  is  for  coal  and  coke  2000  cube  feet,  for  wood  4000  cube  feet  ;  and  for 
each  Ib.  of  fuel  as  follows  : 
Coal  ..........  207  1  Cannel  coal  ...  315  |  Coke  .........  216  |  Wood  .........  173 

Calorific  power  of  i  Ib.  good  coal  =  14  ooo  X  772  =  10  808  ooo  Ibs. 

Relative  Evaporation  of  Several  Combnstil>les  in  L"bs. 


of  Water,  Heated    1° 


1    Ll>.  of  Material. 


Combiwtible. 

Composition. 

Water. 

Combustible. 

Composition. 

Water. 

Alcohol  ....     812 

(Hyd.   .12) 

Lbs. 

8  I2O 

Olive  oil  

(Hyd.   .13) 

Lbs. 

Bituminous  coal.  .  . 

iCarb.  .45} 
]  Hyd.  .04) 

o  830 

Peat,  moist  

\Carb.  .77} 
(Hyd.   .04) 

3481 

\Carb.  .75J 

14220 

"    dry  

(Carb.  .43) 
(Hyd.    .06) 

3  900 

Coke  

Hydrogen  (mean).  . 

Oak  wood,  dry  .... 
"      "     green... 

Carb.  .84 

/Hyd.  .06) 
iCarb.  .53} 
(Hyd.   .08) 
)Carb.  .r7f 

9028 
50854 

6018 
5662 

Pine  wood,  dry.... 

Sulphuric  ether.  .7 
Tallow.  .  . 

(Carb.  .58) 
(Hyd.   .06) 
{Carb.  .7   } 
(Hyd.  .13) 
\Carb.  .6  } 

3618 
8680 

14  ^60 

i  Ib.  Hydrogen  will  evaporate  62.6  Ibs.  water  from  212°  =  60.509  Ibs.  heated  i°. 

i  Ib.  Carbon  "  14.6  Ibs.  "  212°,  or  raise  12  Ibs.  water  at 

60°  to  steam  at  120  Ibs.  pressure. 

i  Ib.  of  Oxygen  will  generate  same  quantity  of  heat  whether  in  combustion  with 
hydrogen,  carbon,  alcohol,  or  other  combustible. 

Relative  Volumes  of  Gases  or  Products  of  Combustion  per  Lb.  of  Fuel. 


Temp. 
Airf 

Supply  ( 
12  Ibs. 
Volume 
per  Ib. 

>f  Air  per  Ib. 
x8  Ibs. 
Volume 
per  Ib. 

of  Fuel. 
24  Ibs. 
Volume 
per  Ib. 

Temp. 
Air. 

Supply  < 
12  Ibs. 
Volume 
per  Ib. 

f  Air  per  Ib. 
1  8  Ibs. 
Volume 
per  Ib. 

of  Fuel. 
24  Ibs. 
Volume 
per  Ib. 

o 

II 

104 

212 

392 

Cube  Feet. 
ISO 
161 
172 
205 
259 

Cube  Feet. 
225 
241 
258 
3°7 
389 

Cube  Feet. 
300 
322 
344 
409 

519 

0 

572 
752 

III2 

1472 
2500 

Cube  Feet. 
314 

369 

9*1 

Cube  Feet. 
471 
553 
718 
882 
1359 

Cube  Feet. 
628 
738 
957 
1176 
18x2 

*  Mean  of  all  experiments  13964. 


466      COMBUSTION. — EXCAVATION    AND    EMBANKMENT. 

To   Compute   Consumption    of  Fuel    to   Heat  Air. 

RULE. — Divide  volume  of  air  to  be  heated  by  volume  of  i  Ib.  of  it,  at  its 
temperature  of  supply ;  multiply  result  by  number  of  heat-units  necessary 
to  raise  i  Ib.  air  through  the  range  of  temperature  to  which  it  is  to  be  heated, 
and  product,  divided  by  number  of  heat-units  of  fuel  used,  will  give  result 
in  Ibs.  per  hour. 

EXAMPLE.— What  is  required  consumption  per  hour  of  coal  of  an  average  compo- 
sition to  heat  776400  cube  feet  of  air  at  54°  to  114°? 

Coal  of  an  average  composition  (Table,  page  461)  =  14 133  heat-units.  Volume  of 
i  Ib.  air  at  54°  (see  formula,  page  522)  =  4  '  g  4  =  12.94  cube  feet,  i  X  114  — 54 
X  .2377  (specific  heat  of  air)  =  14. 262  heat-units. 

776400 

••  X  14. 262  -r- 14 133  =  60. 55  Us. 

Loss  of  heat  by  conduction  of  it  to  walls  of  apartment  is  to  be  added  to  this. 


EXCAVATION   AND   EMBANKMENT. 
Labor  and  "Work  npon  Excavation   and  Embankment. 

Elements  of  Estimate  of  Work  and  Cost. 
Per  Day  of  10  Hours. 

Cart. — One  horse.  Distance  or  lead  assumed  at  100  feet,  or  200  feet  fat 
a  trip,  at  a  speed  of  200  feet  per  minute. 

Earths. — Of  gravelly,  loam,  and  sandy,  a  laborer  will  load  per  day  into  a 
cart  respectively  10, 12,  and  14  cube  yards  as  measured  in  embankment,  and 
if  measured  in  excavation,  .11  more  is  to  be  added,  in  consequence  of  the 
greater  density  of  earth  when  placed  in  embankment  than  in  excavation. 

NOTE.— Earth,  when  first  loosened,  increases  in  volume  about  .2,  but  when  settled 
in  embankment  it  has  less  volume  than  when  in  bank  or  excavation. 

Carting. — Descending,  load  .33  cube  yard,  Level,  .28,  and  Ascending  .25, 
measured  in  embankment ;  and  number  of  cart-loads  in  a  cube  yard  of  em- 
bankment are,  Gravelly  earth  3,  Loam  3.5,  and  Sandy  earth  4. 

Loosening. — Loam,  a  three-horsed  plough  will  loosen  from  250  to  800  cube 
yards  per  day. 

Trimming. — Cost  of  trimming  and  superintendence  i  to  2  cents  per  cube 
yard. 

Scooping.  —  A  scoop  load  measures  about  .1  cube  yard  in  excavation; 
time  lost  in  loading,  unloading,  and  turning,  1.125  minutes  per  load ;  in 
double  scooping  it  is  i  minute.  Time  occupied  for  every  100  feet  of  dis- 
tance from  excavation  to  embankment,  1.43  minutes. 

Time. — Time  occupied  in  loading,  unloading,  awaiting,  etc.,  4  minutes  per 
load. 

To  Compute  Number  of  Loads  or  Trips  in  Cube  Yards 
per    Cart   per    Day. 

(= ^— : — }  h  -r-  y  =  n.     E  representing  average  distance  of  carting  from  em- 
E  -r-IOO-j-4/ 

bankment  in  stations  of  loofeet  each,  y  number  of  cart-loads  to  cube  yard  ofexcava 
tion,  and  n  number  of  cube,  yards  in  embankment,  hauled  by  a  cart  per  day  to  dis 
fancc  E. 


EXCAVATION    AND    EMBANKMENT.  467 

ILLUSTRATION.  —  What  is  number  of  cube  yards  of  loam  that  can  be  removed  by 
one  cart  from  an  embankment  on  level  ground  for  an  average  distance  of  250  feet  ? 
E  =  250  -i-  100  =  2.5,  and  y  =  3.  5. 

—  T—  X  10  -i-  3.  5  =  —  X  10  -7-3.5  =  26.  37  cube  yards. 

Substituting  for  3,  3.5,  and  4  number  of  cart-loads  in  a  cube  yard  of  embank' 
ment,  20,  17.14,  and  15,  =  60  minutes,  divided  respectively  by  these  numbers. 

r=—j  —  =.  n,  in  descending  carting  ;    —.  ~  -  -  =  n,  in  level,  and  -^  -  •=.  n.  in 


ascending,    h  representing  number  of  hours  actually  at  work. 

To   Compute   Cost  of  Excavating  and.    Embaiilsiiig   per 
Cube   Yard. 

--  1  ---  |-  J  +  s  =  V.     L  representing  pay  of  laborers,  v  value  or  result  of  loading 

in  different  earths,  as  10,  12,  and  14,  c  of  one  cart  and  driver  per  day,  I  cost  of  loosen- 
ing material  per  cube  yard,  and  s  cost  of  trimming  and  superintendence,  both  per 
cube  yard,  and  all  in  cents. 

ILLUSTRATION.  —  Volume  of  excavation  in  loam  30000  cube  yards.  Level  carting 
650  feet  =  6.  5  trips  or  courses.  Loosening  by  plough  1.7  cents  per  cube  yard, 
laborers  106  cents  per  day,  carts  160,  and  trimming  and  superintendence  1.5  cents 
per  cube  yard. 

v  =  12,    and  l-^~-   —  =  16.  33,  number  of  loads  per  day  by  preceding  formula. 

Then  ^-+  ^+  1-7  +  1-5  =  8.833  +  9.797  +  1.7  +  1.5  =  21.83  cents  per  cube 

yard. 

Earthworlz  . 

By  Carts.  —  A  laborer  can  load  a  cart  with  one  third  of  a  cube  yard  of  sandy 
earth  in  5  minutes,  of  loam  in  6,  and  of  heavy  soil  in  7.  This  will  give  a  result,  for 
a  day  of  10  hours,  of  24,  20,  and  17.2  cube  yards  of  the  respective  earths,  after  de- 
ducting the  necessary  and  indispensable  losses  of  time,  which  is  estimated  at  .4. 

It  is  not  customary  to  alter  the  volume  of  a  cart-load  in  consequence  of  any  dif- 
ference in  density  of  the  earths,  or  to  modify  it  in  consequence  of  a  slight  inclina- 
tion in  the  grade  of  the  lead. 

In  a  lead  of  ordinary  length  one  driver  can  operate  4  carts.  With  labor  at  $i 
per  day,  the  expense  of  a  horse  and  cart,  including  harness,  repairs,  etc.,  is  $1.25 
per  day. 

A  laborer  will  spread  from  50  to  100  cube  yards  of  earth  per  day. 

The  removal  of  stones  requires  more  time  than  earth. 

The  cost  of  maintaining  the  lead  in  good  order,  the  wear  of  tools,  superintend- 
ence, trimming,  etc.,  is  fully  2.5  cents  per  cube  yard. 

By  Wheel-barrows.  —  A  laborer  in  wheeling  travels  at  the  rate  of  200  feet  per  min- 
ute, and  the  time  occupied  in  loading,  emptying,  etc.,  is  about  1.25  minutes,  with- 
out including  lead.  The  actual  time  of  a  man  in  wheeling  in  a  day  of  10  hours  is  .9 
or  2.25  minutes  per  lead  of  100  feet.  Hence, 

To   Compute  Number   of  Barrow-Loads   removed  by  a 
Laborer  per   Day. 

i2_*  -  *l2  —  n.    n'  representing  number  of  leads  ofioo  feet. 
i.25  +  n' 

A  barrow-load  is  about  .04  of  a  cube  yard 

Rock. 

By  Carte.—  Quarried  rock  will  weigh  upon  an  average  4250  Ibs.  per  cube  yard, 
and  a  load  may  be  estimated  at  .2  cube  yard,  and  weighing  a  very  little  more  than 
a  load  of  average  earth. 

Hence,  the  comparative  cost  of  carting  earth  and  rock  is  to  be  computed  on  the 
basis  of  a  cube  yard  of  earth  averaging  3.5  loads  and  one  of  rock  5  loads,  with  the 
addition  of  an  increase  in  time  of  loading,  and  wear  of  cart. 


468 


EXCAVATION   AND   EMBANKMENT. 


For  labor  of  a  man,  see  Animal  Power,  pp.  433-34. 

By  Wheel-barrow.  —  A  barrow-load  may  be  assumed  at  175  Ibs.  —  2  cube  feet  of 
space. 

Blasting When  labor  is  $  i  per  day,  hard  rock  in  ordinary  position  may  be 

blasted  and  loaded  for  45  cents  per  cube  yard. 

The  cost,  however,  in  consequence  of  condition,  position,  etc.,  may  vary  from  20 
cents  to  $ i 

See  Blasting  page  443. 

17  cube  yards  of  hard  rock  may  be  carted  per  day  over  a  lead  of  100  feet,  at  a  cost 
of  7.29  cents  per  yard. 

The  preceding  elements  are  essentially  deduced  from  notes  furnished  by  EHwood 
MorriSy  C.E.,and  the  valuable  treatise  of  John  C.  Trautwine,  C.E.,  Phila.,  1872. 

Stone. 

Hauling  Stone.— A  cart  drawn  by  horses  over  an  ordinary  road  will  travel  1.15 
miles  per  hour  of  trip  =  2.3  miles  per  hour. 

A  four-horse  team  will  haul  from  25  to  36  cube  feet  of  stone  at  each  load. 

Time  expended  in  loading,  unloading,  etc.,  including  delays,  averages  35  minutes 
per  trip.  Cost  of  loading  and  unloading  a  cart,  using  a  horse-crane  at  the  quarry, 
and  unloading  by  hand,  when  labor  is  $  i  25  per  day,  and  a  horse  75  cents,  is  2f 
cents  per  perch  —  24. 75  cube  feet—  i  cent  per  cube  foot. 

Work  done  by  an  animal  is  greatest  when  velocity  with  which  he  moves  is  .125 
of  greatest  with  which  he  can  move  when  not  impeded,  and  force  then  exerted  .4^ 
of  utmost  force  the  animal  can  exert  at  a  dead  pull. 

Eartli  worlr.    (Molesworth.) 

Proportion  of  Getters,  Fillers,  and  Wheelers  in  different  soils,  Wheelers  being  cal- 
culated at  50  yards  run. 


In  loose  earth,  sand,  etc. 

"  Compact 

"  Marl 


Gett's.  I  Fill's. 

Wheel's. 

Gett's.  Fill's. 

Wheel's. 

i 
I 
i 

i 

2 
2 

I 
2 
2 

In  Hard  clay  
"  Compact  gravel 
"  Rock,  from.  ..  . 

1 

1.25 

i 
i 

1.25 

i 

Average   "Weight   of  Eartlis, 
Per  cube  yard. 


Lbs. 

Gravel.  .  .  . 
Mud.  .  . 

...  3360 
...  2800 

Marl. , 


Lbs. 
,.  2912 

Clay 3472 

Chalk 4032 


Sandstone . 


Lbs. 
4368 


Shale  ........  4480 


Quartz.  .  .  . 


4492 


Lbs. 

Granite 4700 

Trap. 


..  4700 
Slate 4710 


of  Rock:,  Eartliworlz,  etc.,  Original   Excavation 
ass  vim  ed.    at    1. 

When  in  Embankment. 


Rock,  large 1.5  1.6  1.7 

Medium 1.25  to  17 

Metal 1.2    to  1.8 


Sand  and  gravel 1.07 

Clay  and  earth  after  subsidence  . . .  1.08 
u  "        before       u        ...  1.2 


In  small  stones,  per  cent,  of  interstices  to  total  volume  is  44  to  48,  which  is  an 
increase  in  volume  of  solid  rock  to  fragments  of  79  and  92  per  cent. 

The  relative  proportions  of  Earth  in  Bank  and  Embankments,  as  given  by  differ- 
«nt  authorities,  are  so  varied  and  so  opposite  that  it  is  evident  the  difference' is  acci- 
dental, depending,  primarily,  upon  the  season,  location,  and  character  or  condition 
of  the  earth,  and  then  upon  the  height  of  the  embankment,  the  manner  and  dura- 
tion of  time  of  raising  it. 

Thus,  Ellwood  Morris,  ante  p.  466,  makes  the  embankment  less,  and  Molesworth, 
•8  above,  gives  it  greater. 


FRICTION.  469 

FRICTION. 

Friction  is  the  force  that  resists  the  bearing  or  movement  of  one  sur- 
face over  another,  and  it  is  termed  Sliding  when  one  surface  moves 
over  another,  as  on  a  slide  or  over  a  pin ;  and  Rotting  when  a  body  ro- 
tates upon  the  surface  of  some  other,  as  a  wheel  upon  a  plane,  so  that 
new  parts  of  both  surfaces  are  continually  being  brought  in  contact  with 
each  other. 

The  force  necessary  to  abrade  the  fibres  or  particles  of  a  body  is 
termed  Measure  of  friction  ;  this  is  determined  by  ascertaining  what 
portion  of  the  weight  of  a  moving  body  must  be  exerted  to  overcome 
the  resistance  arising  from  this  cause. 

Coefficient  of  Friction  expresses  ratio  between  pressure  and  resistance  of 
one  surface  over  or  upon  another,  or  of  surfaces  upon  each  other. 

Angle  of  Repose  is  the  greatest  angle  of  obliquity  of  pressure  between 
two  planes,  consistent  with  stability,  the  tangent  of  which  is  the  coefficient 
of  friction. 

Experiments  and  Investigations  have  adduced  the  following  observations 
and  results : 

1.  Amount  of  friction  in  surfaces  of  like  material  is  very  nearly  propor- 
tioned to  pressure  perpendicularly  exerted  on  such  surfaces. 

2.  With  equal  pressure  and  similar  surfaces,  friction  increases  as  dimen- 
sions of  surfaces  are  increased. 

3.  A  regular  velocity  has  no  considerable  influence  on  friction ;  if  velocity 
is  increased  friction  may  be  greater,  but  this  depends  on  secondary  or  inci- 
dental causes,  as  generation  of  heat  and  resistance  of  the  air. 

M.  Morin's  experiments  afford  the  principal  available  data  for  use.  Though  con- 
stancy of  friction  holds  good  for  velocities  not  exceeding  15  or  16  feet  per  second, 
yet,  for  greater  velocities,  resistance  of  friction  appears,  from  experiments  of  M. 
Poiree,  in  1851,  to  be  diminished  in  same  proportion  as  velocity  is  increased. 

4.  Similar  substances  excite  a  greater  degree  of  friction  than  dissimilar. 
If  pressures  are  light,  the  hardest  bodies  excite  least  friction. 

5.  In  the  choice  of  ungnepts,  those  of  a  viscous  nature  are  best  adapted  for 
rough  or  porous  surfaces,  as  tar  and  tallow  are  suitable  for  surfaces  of  woods, 
and  oils  best  adapted  for  surfaces  of  metals. 

6.  A  rolling  motion  produces  much  less  friction  than  a  sliding  one. 

7.  Hard  metals  and  woods  have  less  friction  than  soft. 

8.  Without  unguents  or  lubrication,  and  within  the  limits  of  33  Ibs.  press- 
ure per  sq.  inch,  the  friction  of  hard  metals  upon  each  other  may  be  esti- 
mated generally  at  about  one  sixth  the  pressure, 

9.  Within  limits  of  abrasion  friction  of  metals  is  nearly  alike. 

10.  With  greatly  increased  pressures  friction  increases  in  a  very  sensible 
ratio,  being  greatest  with  steel  or  cast  iron,  and  least  with  brass  or  wrought 
iron. 

11.  With  woods  and  metals,  without  lubrication,  velocity  has  very  little 
influence  in  augmenting  friction,  except  under  peculiar  circumstances. 

12.  When  no  unguent  is  interposed,  the  amount  of  the  friction  is,  in  every 
case,  independent  of  extent  of  surfaces  of  contact ;  so  that,  the  force  with 
which  two  surfaces  are  pressed  together  being  the  same,  their  friction  is  the 
same,  whatever  may  be  the  extent  of  their  surfaces  of  contact. 

13.  Friction  of  a  body  sliding  upon  another  will  be  the  same,  whether  the 
body  moves  upon  its  face  or  upon  its  edge. 

RR 


470 


FRICTION. 


14.  When  fibres  of  materials  cross  each  other,  friction  is  less  than  when 
they  run  in  the  same  direction. 

15.  Friction  is  greater  between  surfaces  of  the  same  character  than  be- 
tween those  of  different  characters. 

1 6.  With  hard  substances,  and  within  limits  of  abrasion,  friction  is  as 
pressure,  without  regard  to  surfaces,  time,  or  velocity. 

17.  The  influence  of  duration  of  contact  (friction  of  rest)  varies  with  the 
nature  of  substances ;  thus,  with  hard  bodies  resting  upon  each  other,  the 
effect  reaches  a  maximum  very  quickly ;  with  soft  bodies,  very  slowly ;  with 
wood  upon  wood,  the  limit  is  attained  in  a  few  minutes ;  and  with  metal  on 
wood,  the  greatest  effect  is  not  attained  for  some  days. 

Coefficients    of  Friction    and.    Angles    of   Repose. 
The  Coefficient  of  Friction  is  the  tangent  of  the  angle  of  repose  from  a  horizontal 
plane. 


MATERIAL. 

Coefficient. 

Angle. 

Cotangent  of  Angle. 
Exponent  of 
Stability. 

Belt  on  wood  dry          .   ....... 

•  47 



Clay  damp    

I 

45° 

I 

"    wet    

.25  to  .31 

14°  tO  17° 

3.23  to  4 

Earth  

.i    to  .25 

14°  to  43° 

i  to  4 

'  '     dry  

.81 

1.23 

3.23 

17° 

.31 

Gravel  

.81  to  i.  ii 

39°  to  48° 

.9  to  1.23 

•  53 

1.89 

.-3-1 

18°  30' 

3 

Sand  fine               

6 

3*° 

i.67 

Timber  on  stone  

•4 

22° 

2-5 

.25  to  .5 

14°  to  26°  30' 

2  to  4 

**       "        <j       soaped  

.04  tO  .  2 

2°  tO  11°  30' 

2.8  to  4.9 

Metal  on  Metal  wet  

,a 

1  6°  30' 

3  to  3.3 

t  «       u      1  1      dry              .  • 

.  1  5  to   2 

8°  to  n°  30' 

4.9  to  7 

"       "      "       lubricated       .... 

,08 

4° 

.14 

.4  to  .6 

9°  tO  22° 

25  to  6 

•4 

22°  tO  25° 

2.1  to  2.5 

SURFACES. 

Pressure  =  i. 
Lubrication. 

Coefficient. 

.  .  .  .           Soap 

.16 

Wet 

.26 

Soap 

.21 

Wet 

.22 

.  .  .  .           Soap 

.19 

Leather  belt  on  oak.  .  . 

Dry 

.27 

Wheel  Gearing.  Grooves  of  wheel,  V  angle  50°,  Compared  with  leather  belts, 
under  a  pressure  equal  to  the  tension  of  the  belts,  has  proved  to  have  greater  ad- 
hesion, equal  to  30  per  cent,  in  one  instance. 

Leather  belts  over  wood  drums  .47  of  pressure,  and  over  turned  cast-iron  pulleys 
.28  of  pressure. 

Coefficients   of  Friction   of  Motion. 

Condition  of  Surfaces  and  Unguents. 


SUBSTANCES. 

i 

1 

„• 

c 

oj 

s 

3 

3 

(  On  wood 

•45 

•33 

_ 

— 

— 

•15 

.19 

Metal  upon  wood  Mean  .  .  . 
Sole-leather,  smooth,  upon  wood  f  Raw.  .  .  . 

.18 
•54 

31 

.07 
.16 

.09 

.09 

.2 

or  metaj  \  Dry  

•34 

•31 

.14 

— 

.14 

Wood  upon  metal  Mean  .  .  . 
Wood  uoon  wood  ...                

.42 
.36 

.24 

•  25 

.06 

.07 
.07 

.08 

.07 

FRICTION. 


471 


Relative   Value   of  TLJngxieiits   to    Reduce    Friction. 


UNGUENTS. 

Wood 
Wood. 

Wood 
upon 
Metals. 

Metals 
upon 
Metals. 

UNGUENTS. 

Wood 
upon 
Wood. 

Wood 
upon 
Metals. 

Metala 
upon 
Metals. 

Dry  soap  

•  4 

•32 

.27 

Olive  oil  

Lard 

Ro 

85 

Tallow 

g 

Lard  and  oJumbago. 

.67 

.06 

Water  .  .  . 

.22 

.24 

.18 

To   Determine   Coefficient   of  Friction   of  Bodies. 

Place  them  upon  a  horizontal  plane,  attach  a  cord  to  them,  and  lead  it  in 
a  direction  parallel  to  the  plane  over  a  pulley,  and  suspend  from  it  a  scale  in 
which  weights  are  to  be  placed  until  body  moves. 

Then  weight  that  moves  the  body  is  numerator,  and  weight  of  body  moved 
is  denominator  of  a  fraction,  which  represents  coefficient  required. 

ILLUSTRATION.— If,  by  a  pressure  of  320  Ibs.  friction  amounts  to  80  IDS.,  its  coeffi- 
cient of  friction  in  this  case  would  be  80 -7-320  =  .25. 

Hence,  if  coefficient  of  friction  of  a  wagon  over  a  gravel  road  was  .  25,  and  the  load 
840x3  Ibs.,  the  power  required  to  draw  it  would  be  8400  X  -25  =  2100  Ibs. 

Coefficients   of  Axle   Friction.     (M.  Morin.) 

Condition  of  Surface*  and  Unguents. 


SUBSTANCES. 

Dry  and 
a  little 
Greasy. 

Greasy 
and  wet 
with 
Water. 

Oil,  Tallov 

In  usual 
way. 

r,  or  Lard. 

Continu- 
ously. 

Very  soft 
and  puri- 
fied Car- 
riage 
Grease. 

Cast  iron  upon  bell  metal  

.194 

'iBs 

.161 
.079 

•075 
•075 
.1 
•075 
•075 

.12^ 

•054 
•054 
.092 
•054 
•054 

.065 

.109 
.09 

Cast  iron  upon  cast  iron  
Cast  iron  upon  lignum  -vitse 

Wrought  iron  upon  bell  metal  

.251 

.189 

Wrought  iron  upon  lignum-vitae.  .  . 

.188 

Friction  of  a  journal  of  an  axle  which  presses  on  one  side  only,  as  in  a 
worn  bearing,  is  less  than  when  it  presses  at  all  points,  the  difference  being 
about  .005. 

Friction  of  Axles.  —  With  axles,  friction  of  motion  has  alone  been  experi- 
mented upon.  When  weight  upon  axle  and  radius  of  its  journal  is  given, 
mechanical  effect  of  friction  may  be  readily  determined. 

The  mechanical  effect  absorbed  by,  or  of  friction,  increases  with  pressure 
or  weight  upon  journal  of  axle  and  number  of  revolutions. 

Friction  of  an  axle  is  greater  the  deeper  it  lies  in  its  bearing. 

If  journal  of  an  axle  lies  in  a  prismatic  bearing,  as  in  a  triangle,  etc., 
friction  is  greater,  as  there  is  more  pressure  on,  and  consequently  greater 
friction  in  contact  :  in  a  triangular  bearing  it  is  about  double  that  of  a  cyl- 
indrical bearing. 

To  Compute  Mechanical   Effect  of  Friction  on  Journal 
of  an 


—  —  —  -  =  F.    n  representing  number  of  revolutions,  and  r  radius  of  journal 


in 

ILLUSTRATION.— Weight  of  a  wheel,  with  its  axle  or  shaft  resting  on  its  journals, 
is  360  Ibs. ;  diameter  of  journals  2  ins. ;  and  number  of  revolutions  30;  what  is  me- 
chanical effect  of  the  friction,  the  coefficient  of  it  being  .  16  ? 

3.1416  X  30  X  .16  X  360  X  i  -r- 12  _  452.4  __      og  Ws 
30  "30          5' 


472  FEICTION. 

By  application  of  friction-wheels  (rollers)  friction  is  much  reduced,  and 
mechanical  effect  then  becomes,  when  weights  of  friction-wheels  are  disre- 
garded, 

pnfWr  ^ r  _  F     ^  repre$en^ng  ra^a  Of  axles  of  friction-wheels, 

3°  a'  cos.  a-r-2 

a'  radii  of  friction- wheels,  and  a  angle  of  lines  of  direction  between  axis  of  roller 
and  axis  of  friction-wheels. 

When  a  single  friction-wheel  is  used,  2^n  X  /  W  =  F,  and  -p-~^  =  F'.  F 
representing  mechanical  effect. 

ILLUSTRATION.— A  wheel  and  its  shaft,  making  5  revolutions  per  minute,  weighs 
30 ocx)  Ibs. ;  its  diameter  and  that  of  its  journals  are  32  feet  and  10  ins.  The  journals 
rest  upon  a  friction -wheel,  the  radius  of  which  is  5  times  greater  than  its  axle. 

i.  What  is  the  power  at  circumference  of  wheel  necessary  to  overcome  friction? 
2.  What  is  mechanical  effect  of  the  friction  ?  3.  What  is  reduction  of  friction  by 
use  of  the  friction- wheel  ? 

i.  32~2  X  12  _         circum.  of  wheel  =  38.4  limes  that  of  axle. 

IO-7-2 

Coefficient  of  friction  assumed  at  .075.     Hence  3000° — ^^  =  58. 59  Ibs. = power 

at  circum.  to  overcome  friction  at  axle.     2.  — — 3'141    =  2. 6 1 8  feet  =  distance  passed 
by  friction. 
Consequently,  ^-^- =  .2181  feet  —  distance  passed  by  friction  in  one  second. 

DO 

Hence,  .2181  X  2250  (30000  X  -075)  =  490.725.  3.  i  -4- 5  =  .2  —  radius  offi-iction- 
axle-r-by  radius  of  friction- wheel,  and  38.4  X  .2  =  7.68  =  friction  referred  to  circum. 

of  wheel,  and  12£lZ£5  __  g8  I45  _  mechanical  effect  by  application  of  friction-wheel 
=  a  reduction  of  four  fifths. 

Friction   of  Pivots. 

Friction  on  Pivots  is  independent  of  their  velocity,  increases  in  a  greater 
degree  than  their  pressures,  and  approximates  very  near  to  that  of  sliding 
and  axle  friction. 

Friction  on  Conical  Bearings  is  greater  than  with  like  elements  on  plane 
surfaces. 

Figure  of  point  of  a  pivot,  as  to  its  acuteness,  affects  friction  :  with  great 
pressure  the  most  advantageous  angle  for  the  figure  ranges  from  30°  to  45° ; 
with  less  pressure  it  may  be  reduced  to  10°  and  12°. 

Relative   "Valne   of  Angles   of  Pivots. 

6° i    |    15° 66    |    45° 39 

Relative  "Valnes  of  different  Materials  for  -use  as  Pivots. 

Agate 83  I  Granite i       I  Tempered  steel 44 

Glass 55  |  Rock  crystal 76  | 

Friction   and.   Rigidity   of  Cordage. 

Experiments  by  Amonton  and  Coulomb,  with  an  apparatus  of  Amonton's, 
furnish  the  following  deductions : 

1.  That  resistance  caused  by  stiffness  of  cords  about  the  same  or  like  put 
leys  varies  directly  as  the  suspended  weight. 

2.  That  resistance  caused  by  stiffness  of  cords  increases  not  only  in  direct 
proportion  of  suspended  weights,  but  also  in  direct  proportion  of  diametei 
of  the  cords. 


FEICT10N. 


473 


Consequently,  that  resistance  to  motion  over  the  same  or  like  pulleys, 
arising  from  stiffness  of  cords,  is  in  direct  compound  proportion  of  suspend- 
ed weight  and  diameter  of  cords. 

3.  That  resistance  to  bending  varied  inversely  as  diameter  of  sheave  or 
drum. 

S  I  C  T 

4.  That  complete  resistance  is  represented  by  expression  -^ — .    S  rep- 
resenting constant  for  each  rope  and  sheave,  expressing  stiffness  of  rope ;  T 
tension  of  rope  which  is  being  bent,  expressed  by  C  T ;  C  constant  for  each 
rope  and  sheave;  and  d  diameter  of  sheave,  including  diameter  of  rope. 

5.  That  stiffness  of  tarred  ropes  is  sensibly  greater  than  that  of  white  ropes. 

Extending  results  obtained  by  Coulomb,  Morin  furnishes  following  for- 
mulas : 

For  White  Ropes:  12  n-r-d  (.002  i5-j--ooi  77  n-{-.ooi2  W)  =  R.  For  Tarred 
Ropes :  12  n-r-d  (.010 54  +  .0025  n  -f  .0014  W)  =  R.  R  representing  rigidity  in  Ibs. , 
n  number  of  yarns,  d  diameter  of  sheave  in  ins.  and  rope  combined,  and  W  weight 
in  Ibs. 

ILLUSTRATION.— What  is  value  of  stiffness  or  resistance  of  a  dry  white  rope  hav- 
ing a  diameter  of  60  yarns,  which  runs  over  a  sheave  6  ins.  in  diameter  in  the 
groove,  with  an  attached  weight  of  1000  Ibs.  ? 

Assume  diameter  for  60  yarns  to  be  7.2  in*.  Then  —  —  (.002 is-j-.ooi  77  x 
60 +  .0012  X  1000)  =  100  X  1.30835=  130.835  Ibs. 

Value  of  natural  stiffness  of  ropes  increases  as  the  square  of  number  of 
threads  nearly,  and  value  of  stiffness  proportional  to  tension  is  directly  as 
number  of  threads,  being  a  constant  number.  Hence,  having  the  rigidity  for 
any  number  of  threads,  the  rigidity  for  a  greater  or  lesser  number  is  readily 
ascertained. 

Wire  Popes. 

Weisbach  deduced  from  his  experiments  on  wire  ropes  that  their  rigidity 
for  diameters  capable  of  supporting  equal  strains  with  hemp  ropes  is  con- 
siderably less. 

Wire  ropes,  newly  tarred  or  greased,  have  about  40  per  cent,  less  rigidity 
than  untarred  ropes. 

Rolling  Friction. 

Rolling  Friction  increases  with  pressure,  and  is  inversely  as  diameter  of 
rolling  body. 

For  rolling  upon  compressed  wood, /=. 019  to  .031. 

When  a  Body  is  moved  upon  Rollers  and  Power  applied  at  the  Base  of  the  Body, 

W 
(/+/')  —  =  F.  fandf  representing  coefficients  of  friction  of  two  surfaces  upon 

which  rollers  act. 

When  Power  is  applied  at  Circumference  of  Roller,  fW -r- r  =  F. 
When  Power  it  applied  at  Axis  of  Roller,  f  W  -r-r-r-2  =  F. 

Bearings  for   Propeller   Shaft.    (Mr.  John  Penn.) 


BKABINGS. 

Pressure 
per 
Sq.Inch. 

Time 
of  Op- 

e  rat  ion. 

BEARINGS. 

Pressure 
per 
Sq.  Inch. 

Time 
of  Op- 
eration. 

Babbit's  metal  on  iron*.  .  . 
Box  on  brass      .   

Lbs. 
1600 
4480 
448 
448 
448 

Min. 
8 
5 
30 
30 
30 
t  Ab 

Rf 

Lbs. 
675 
4480 
4000 
4000 
1250 
ast. 

Min. 
60 

5 

2160 

Brass  on  iron  %          ... 

Lignum-vitse  on  brass  .  . 
Snake-wood  on  brass  .  .  . 
Lignum-vitae  on  iron  .  .  . 

raded.                                %  Set  i 

I* 

Brass  on  iron   

«  Rolled  oat. 

474 


FKICTION. 


Result  of  Experiments  -upon  Friction  of  Several  Instru- 
ments.    (R.  S.  Ball.) 


INSTRUMENT. 

Friction. 

Velocity  ratio. 

Mechanical 
efficiency. 

Useful 
effect. 

Pulley,  single  

F 
2.21     - 

2.36     - 
3-87     * 
.0        - 
.09     - 

.66   - 
.204- 

2.46   - 

.0       - 

.18-?-! 

L 

-•5453 
-.238 

-•151 
-.014 
L-55 
-.007 
U.043 
-.169 

-.21 
-.056 
-.008 

2 

6 
16 
193 
3-4 
414 

35.95 

23 

137 

1.8 

6.1 
70 
1.72 
116 

22 

5-55 
4.1 
18 
87 

Per  Cent. 

9° 
64 

38 
36 

^8 

70 

93 
5' 
78 
6^ 

"       differential      

Inclined  plane,  angle  17°  2'.... 
Screw  Jack  

Wheel  and  Axle  

"        u    Barrel   

u        "    Pinion  

Crane....          ...          . 

F  representing  friction,  and  L  load. 

ILLUSTRATION  L—  If  it  is  required  to  ascertain  power  necessary  to  raise  200  Ibs. 
2  feet,  by  a  single  movable  pulley,  200  x  .5453  +  2.21  =  111.27  Ms.,  which  must  be 
applied  as  power  to  raise  200  Ibs.  2  feet.  111.27  X  2  =  222.  54  Ibs.  Hence,  for  appli- 
cation of  222.54  Ibs.,  200  or  89.87  per  cent,  are  usefully  or  effectively  employed. 

2.  —  If  it  is  required  to  raise  100  Ibs.  by  a  three-sheave  pulley,  then  iooX  -2384- 
2.36  =  26.  16  Ibs,  which  must  be  applied  as  power  to  raise  100  Ibs.  6  feet  (3X2  =  6). 
26.16  X  6  =  156.96  Ibs.     Hence,  for  application  of  156.96  Ibs.,  100  or  63.71  per  cent. 
are  effectively  employed. 

3.  —  The  velocity  ratio  of  a  crane  being  137,  and  its  mechanical  efficiency  87,  a 
man  applying  26  Ibs.  to  it  can  raise  87  x  26  =  2262  Ibs. 


Application    of*  preceding    Resxxlts. 

ILLUSTRATION.  —If  a  vessel,  including  cradle,  weighing  1000  tons,  is  to  be  drawn 
upon  an  inclined  plane  having  a  rise  of  10  feet  in  100  of  its  length,  what  will  be  the 
resistance  to  be  overcome,  the  cradle  being  supported  on  wrought-  iron  axles  in  cast- 
iron  rollers,  running  on  cast-iron  rails  ? 


-  =  loo  tons  =:  power  required  to  draw  vessel  independent  of  friction. 

IOO 

Ratio  of  friction  to  pressure  of  wrought  iron  on  cast,  in  an  axle  and  its  bearing, 
.075.  Ratio  of  ditto  of  cast  iron  upon  cast,  say  .005. 

Hence  .075  -f  .005  =  .08  of  1000  tons  =  80  tons,  which,  added  to  100  tons  before  de- 
ducted, gives  1  80  tons,  or  resistance  to  be  overcome. 

Power  or  effect  lost  by  friction  in  axles  and  their  bearing  may  be  ex- 
pressed by  formula 

—  —  —  r  =  P.  /  representing  coefficient  offi-iction,  d  diameter  of  axle  in  ins.,  and 
r  number  of  revolutions  per  minute. 

ILLUSTRATION.  —  Pressure  on  piston  of  a  steam-engine  is  20000  Ibs.,  number  of 
revolutions  20,  and  diameter  of  driving  shaft  of  wrought  iron  in  a  brass  journal  is 
8  ins.  ;  what  is  the  effect  of  friction  ? 

20000  X  -07X  8X20 

23o  -=  97*9'  «* 

Hence  P  v  -4-  33  ooo  =  IP.  v  representing  circumference  of  shaft  infect  X  by  revo- 
lutions per  minute. 

The  power  or  effect  lost  by  friction  in  guides  or  slides  may  be  expressed 
by  following  formula: 

-  -  /     .r  -  57=  P.    *  representing  stroke  of  cross-head,  and  I  length  of  con- 
oo  X  v(5  '    —  s  ) 
necting  rod  in  feet. 


FRICTION. 


475 


liVLotional  Resistances. 
Friction    of*  Steam-engines. 

.Friction   of  Condensing   Engines   in.    Ltos.   per   Sq..  Inolx 
of  Piston. 


Diameter 
of 
Cylinder. 

Oscillating 
and 
Trunk. 

Beam 
and 
Geared. 

Direct- 
acting  and 
Vertical. 

Diameter 
of 
Cylinder. 

Oscillating 
and 
Trunk. 

Beam 
and 
Geared. 

Direct- 
acting  and 
Vertical. 

10 

5 

6 

7 

50 

2-5 

2.7 

3-3 

15 

4 

5 

6 

60 

2.4 

2.6 

3 

20 

3-5 

4 

5 

70 

2-3 

2-5 

2-7 

25 

3 

3-6 

4-5 

80 

2 

2.3 

2.6 

30 

3 

3-5 

4 

100 

1.6 

2.2 

2-5 

35 

2.6 

3 

3-5 

no 

I-S 

2 

2.1 

Experiments  upon  different  steam-engines  have  determined  that  friction, 
when  pressure  on  piston  is  about  12  Ibs.  per  sq.  inch,  does  not  exceed  1.5  Ibs., 
or  about  one  tenth  of  power  exerted. 

Friction  of  double  cylinder  (so-inch  diam.)  direct-acting  condensing  pro- 
peller engine  is  1.25  Ibs.  per  sq.  inch  of  piston  =  10.3  per  cent,  of  total  power 
developed ;  friction  of  load  is  .9  Ibs.  per  sq.  inch  of  piston  =  7.5  per  cent,  of 
total  pressure;  and  friction  of  propeller  is  1.3  Ibs.  per  sq.  inch  of  piston  = 
10.8  per  cent,  of  total  power  =  28.6  per  cent. 

Friction  of  double  cylinder  (yo-inch  diam.)  inclined  condensing  water- 
wheel  engine  with  its  load  is  15  per  cent,  of  total  power  developed. 

In  general,  when  engines  are  in  good  order,  their  efficiency  ranges  from  80 
per  cent,  for  small  engines  tc  93  per  cent,  for  large. 

Power  required  to  work  air-pumps  is  5  per  cent.,  and  to  work  feed-pumps 
i  per  cent. 

Results    of  Experiments   xipon   Friction    of  Machinery. 

(Davison. ) 

Steam-engine,  vertical  beam,  one  tenth  its  power ;  190  feet  horizontal,  and 
1 80  feet  vertical  shafting,  with  34  bearings,  having  an  area  of  3300  sq.  ins., 
with  1 1  pair  of  spur  and  bevel  wheels ;  7.65  H?. 

Set  of  three-throw  Pumps,  6  ins.  in  diam.,  delivering  5000  gallons  per  hour 
at  an  elevation  of  165  feet;  4.7  IP,  or  about  13  per  cent. 

Two  pair  iron  Rollers  and  an  elevator,  grinding  and  raising  320  bushels 
malt  per  hour ;  8.5  IP. 

Ale-mashing  Machine,  800  bushels  malt  at  a  time ;  5.68  IP. 

Archimedes  Screw  (ninety-five  feet),  15  ins.  in  diameter,  and  an  elevator 
conveying  320  bushels  malt  per  hour  to  a  height  of  65  feet ;  3.13  IP. 

Friction  Clutch. — Driven  by  a  leather  belt  14  ins.  in  width ;  face  of  clutch 
5  ins.  deep ;  broke  a  cast-iron  shaft  6.5  ins.  in  diameter. 

Flax  Mill  (M.  Cornut,  1872). — Two  condensing  engines,  cylinders,  12.9 
ins.  x  44.3  ins.  stroke,  and  22  ins.  x  59.8  ins.  stroke.  Pressure  of  steam, 
50  Ibs.  per  sq.  inch ;  revolutions,  25  per  minute.  Friction  of  entire  machin- 
ery, 20  per  cent. 

With  vegetable  oil  and  hand  oiling  a  steam  pressure  of  62  Ibs.  per  sq. 
inch  was  required,  and  with  mineral  oil  and  continuous  oiling  a  pressure  of 
50  Ibs.  only  was  required. 

^  By  continuous  oiling,  a  saving  of  44  per  cent,  was  effected  over  hand 
oiling. 


476 


FRICTION. 


Flax   Mill. 


Power  required   to    Drive   Engine,  Shafting,  and   entire 
Machinery.     (M.  Cornut.) 


PARTS. 

Total. 

Indicated  H 
One  M 

at  work. 

orse-power. 
achine 
empty. 

Effect  of 
Machines. 

30.41 
8.42 

7.19 

2.22 

7.78 

47-5 
46.59 

2.105 
•0934 

.02627* 
.032  i* 

.022  4* 

*  Per  100 

I-423 
.0794 

•151 
2-434 

2-5I5 
1.613 

spindles. 

32 
IS 
78 
7-3 

21.6 

*9 

4  Cctl'iis                

14  drawing  frames  (29  heads  or  156) 

4  combing  machines.  

6  roving  frames  (330  spindles)  

20  spinning  frames. 
Dry  (1480  spindles)  

Wet  (2080       "       )  

Total  150.11  IP. 

Estimate  of  Horse's  Power.— 2080  spindles,     wet,     34.4  per  H?,  long  fibre. 
640       "  dry,     20.1    u     u     u       " 

840       "  "       14.5   "     "   tow. 

3560       "        average,"^  "     u 
The  IP  per  100  spindles  varies  inversely  as  sq.  root  of  their  number. 

Winding    Engine  (G.  H.  Daglish). 

Shafts  738  to  1740  feet  in  depth;  cylinder  65  X  84  ins.  stroke;  pressure  of  steam 
iglbs.per  sq.  inch;  revolutions  12.5  per  minute;  mean  diameter  of  drum,  26  feet. 
IP  313.4;  effect  235  =  75  percent. 

Tools.    (Dr.  Har-tig). 
Single  shearing,   i  -f-  — -  =  IP  to  drive  tool    n  representing  number  of 

cuts  per  minute,  t  thickness  of  plate,  and  — =  IP  to  shear,     a  representing 

area  of  surface  cut  or  punched  per  hour  in  sq.  ins.,  and  F  (1166  -f-  1691  t)  a  factor  ex- 
pressing work  required  to  cut  or  shear  a  surface  of  i  inch  square. 

ILLUSTRATION.— A  shearing  machine  cutting  4648  sq.  ins.  of  surface  per  hour,  in 
plates  .4  inch  thick,  required  .68  IP  to  run  and  4.3  to  operate  it,  equal  to  5  horses. 

S$ooobt*l      n.  ,ii3oo6<2/ 

Iron    Plate-bending.     -  =  Pfor  cold  plates,  and  — 

—  Pfor  red-hot  plates.    6,  t,  and  I  representing  breadth,  thickness,  and  length  of  plate, 
r  radius  of  curvature,  all  in  ins.,  and  P  net  power  of  bending. 
Power  for  large  rolls  when  running  only  .5  to  6  IP. 

Ordinary    Cutting    Tools,  in    Metal. 

Materials  of  a  brittle  nature,  as  cast  iron,  are  reduced  most  economically  in  power 
consumed,  by  heavy  cuts;  while  materials  which  yield  tough  curling  shavings  are 
more  economically  reduced  by  thinner  cuttings.  Following  formulas  apply  to  light 
cutting  work: 

Power  required  to  plane  cast  iron  is— 

Planing  Cast  iron,  W  (.0155  -\ j  =  IP.    W  representing  weight  of  cast 

iron  removed  per  hour,  in  Ibs.,  and  s  average  sectional  area  of  shavings,  in  sq.  ins. 

Steel,  Wrought  iron,  and  Gun-metal,  with  cuts  of  an  average  character- 
Steel 112  W=  IP  |  Wrought  iron,  .052  W=  IP  |  Gun-metal,    .0127  W  =  IP 

Planing   and    Molding.  —  Run  without  cutting.      =  IP.     N  rep 

2000 
resenting  sum  of  revolutions  of  all  the  shafts  per  minute. 


FRICTION.  477 


Molding.  —  Pine,  .0566  +  '°2*  6S,  and  Red  Beech,  088  95  +  '^~-  =  W.  h  rep- 
resenting depth  of  wood  cut  down  to  form  molding. 

Tnr  n  ing.  —  Steel,  .047  W  =  H»;  Wrought  iron,  .0327  W  =  E?;  Cast  iron, 
.03  14  W  =  IP. 

For  turning  off  metals,  power  required  is  less  than  for  planing,  and  it  is  ascer- 
tained that  greater  power  is  required  for  small  diameters  than  large. 

Light  Lathes,  .05  -f-  .0005  n  =  IP  ;  i  or  2  shafts,  .05  -\-  .0012  n  =  IP  ;  3  or  4  shafts, 
•05  +  -05  »  =  IP.    Heavy  Lathes,  .025  +  .003171;  .  025  +  .  053  n\  .o25-f.i8n. 
n  representing  number  of  revolutions  of  spindle  per  minute. 

Drilling.  —  Power  required  to  remove  a  given  weight  of  metal  is  greater  than 
in  planing.  Volume  being  taken  in  place  of  weight. 

Holes  from  .4  to  2  ins.  in  diameter. 

Castiron,dry.  V  (.oi68  +  '-^^W:EP.    Wrought  iron,  oil.  V  (.0i68-K-^p)  =  IP. 
V  representing  volume  removed  in  cube  ins.  per  hour,  and  d  diameter  of  hole. 

Without  gearing,  .0006  n  +  .0005  n';  with  gearing,  .0006  n  -f-  .001  n'\  radial 
drills  without  gearing,  .0006  n  -f-  .004  n'  ;  radial  drills  with  gearing,  .04  +  .0006  n  -\- 
.004  n'.  n  representing  number  of  revolutions  per  minute  of  gearing  shaft,  and  n' 
of  drill. 

Slotting.—  Stroke  8  ins.  .045  -\  --  =  IP.  n  representing  number  of  strokes 
per  minute,  and  s  stroke  in  ins. 

'Wood-sawing,  Circular.  —A  cube  foot  of  soft  wood  and  half  a  cube 
foot  of  hard,  reduced  to  sawdust,  requires  i  IP. 

Ac  A.  c 

Hard  wood,  —  =  IP'.       Soft  wood,  —  =  H".    A  representing  area  in  sq.  feet 

6  12 

and  IP'  horse-power  per  sq  foot,  both  cut  per  hour,  and  c  width  of  cut  in  ins. 

From  .  4  to  4  ins.  in  diameter.  —Pine.    V  I  ooo  125  -f      ^5  j  =  IP. 

Dry  pine  timber.  .004  28  +  .0065  -^  —  H?'.  S  representing  stroke  of  saw  in  feet, 
and  ffeed  per  cut  in  ins. 

—  —  =  IP  /<w  horse-power  to  run  only  without  cutting,    d  representing  diameter 
32000 
of  saw  in  ins.,  and  n  number  of  revolutions  per  minute. 

Net  power  required  to  cut  with  a  circular  saw  is  proportional  to  volume  of  ma- 
terial removed.  For  a  saw  cutting  hot  iron,  at  a  circumferential  speed  of  7875  feet 
per  minute,  and  making  a  cut  .14  inch  wide,  power  is  expressed  by  formulas  — 

.702  A  =  IP,  for  red-hot  iron.     1.013  A  =  IP,  for  red  hot  steel. 

A  representing  sectional  area  of  surface  cut  through,  in  sq.feet. 

Vertical    Saw.    .004  28  +  .0065  -j  =T&  in  dry  pine  timber  per  sq.  foot 

per  hour.    S  representing  stroJce  of  saw  in  feet,  c  width  of  cut  in  ins.,  and  ffeed  of 
cut  in  ins. 

Band  Saw.      0034  + 


.  005  76  4-  *'127  °™  =  H*'  in  Beech,    v  representing  velocity  of  saw,  and  f  rate  of  feed, 

loooof 
in  feet  per  minute. 


Screw  Ciztting.    Screws,  ^—  —  =  IP.     Taps,  -  =  IP.    d  representing 
diameter  in  ins.  ,  and  I  length  cut  in  feet  per  hour. 
Machine  of  medium  dimensions,  .2  IP. 


FBICTION. 

Q-rindstones.     =  IP.    p  representing  pressure  upon  stone,  v  circus 

ferential  velocity  of  stone  in  feet  per  minute,  and  C  coefficient  of  friction. 

Coefficients  of  Friction  between  Grindstones  and  Metals. 

Cast  iron,  .22  at  high  speed,  .72  at  low  speed;  Wrought  iron,  .44  at  high  speed, 
i  at  low;  Steel,  .29  at  high  speed,  .94  at  low. 

Power  required  to  run  them  alone. 

Large 0000409  d  v  =  IP    I    Small i6-f. 000089  5  d  v  =  U? 

or oooi28d2  n  =EP    |          or i6-f  .00028  d2  n    =  IP 

Grrain    Conveyers. 

Conveyers  of  Grain  horizontally  by  Screws  and  Bands. — A  i2-inch  screw,  having 
4  ins.  pitch,  turning  in  a  trough,  with  a  clearance  of  .25  inch,  revolving  with  a 
speed  of  maximum  effect,  60  turns  per  minute,  will  discharge  6.75  tons  of  grain 
per  hour,  expending  .04  IP  per  foot  run.  Sectional  area  of  body  of  grain  moved 
49  per  cent,  of  that  of  screw.  At  speeds  above  60  turns  per  minute,  the  grain  will 
not  advance,  but  will  revolve  with  screw. 

Steam-engines. 

Friction  of  a  Steam-engine  varies  as  its  principal  dimensions,  and  increases 
slightly  with  the  load. 

Results  of  Tests. — Engine,  cylinder  4  in.,  57  per  cent.;  cylinder  9  in.,  13  to 
22  per  cent.  Corliss  engine,  cylinders  18  and  24  in.,  10  percent.  Worthington 
large  pumping  engine,  9  per  cent.  Compound  engine,  first  cylinders  of  from  12  to 
21  ins.,  80  to  89  per  cent.  (D.  K.  Clark.) 

Engine,  Unloaded.     -~-  —pressure  of  steam  in  Ibs.  per  sq.  inch,  and  D  diame- 
ter of  cylinder  in  ins. 
Marine   Engine.     Vertical  Beam.     (J.  V.  Merrick. )    In  Pressure  of  Steam. 

Weight  of  parts. sib. 

Air-pump  packing 046  to  .092  Ib. 

Average  of  all 165  Ib. 

If  journals  are  kept  constantly  lubricated,  friction  of  weight  will  be  reduced  to 
.33,  and  pressure  from  1.65  —  .33  to  1.32  Ibs.  per  sq.  inch  of  piston  to  operate  en- 
gine without  load.  Friction  of  load,  from  2  to  5  per  cent. 

Screw    Steamer.     ( Vice- Admiral  C.  R.  Moorsom,  R.  N.) 

Hull  moving 07    I  Rotation  of  screw. .     .09    I  Hull  resistance ,606 

Load 063  |  Slip  of  screw 171  |  Total i 

Locomotives  and  Railway  Trains.    See  Railways,  page  682. 

.     Friction,    developed,    in    Launching    of  "Vessels. 

Experiments  made  by  a  committee  of  Franklin  Institute  on  friction  of  launching 
vessels  gave,  when  pressure  or  weight  was  from  2280  to  3560  per  sq.  foot,  a  co- 
efficient of  .0335. 

Marine  Railway.— fo  draw  3000  tons  upon  greased  slides  a  power  of  250  tons  was 
necessary  to  move  it,  but  when  started  150  tons  would  draw  it. 

Woollen  Machinery.  (Dr.  Hartig.)  When  running  empty  8. 15  IIP,  and  at  work 
32-97- 

The  efficiency  of  the  various  machines  averaging  60.5  per  cent. 

Friction    of*  a   Non-condensing   Steam-engine. 

Friction  of  an  Engine.  Diameter  of  cylinder  20  ins.  by  40  ins.  stroke  of  piston. 
Revolutions,  15  to  70  per  minute. 

Engine,  unloaded,  2  Ibs.  per  sq.  inch :.  =  x.86  to    8.69  IP. 

Shafting,  unloaded,  2.5  to  45  Ibs.  per  sq.  inch =  2.36  to  19.61  " 

Total  4.5  to  6.5  Ibs.  per  sq.  inch =  4.22  to  28.3    " 


Air-pump 58510.7      Ib. 

Cylinder  packing 15    ".3       " 

Valves,  etc 169  "  .258   " 


FUEL. 


479 


FUEL. 

With  equal  weights,  where  each  kind  is  exposed  under  like  advan- 
tageous circumstances,  that  which  contains  most  hydrogen  ought,  in  its 
combustion,  to  produce  greatest  volume  of  flame.  Thus,  pine  wood  is 
preferable  to  hard,  and  bituminous  to  anthracite  coal. 

When  wood  is  used  as  a  fuel,  it  should  be  as  dry  as  practicable. 
To  produce  greatest  quantity  of  heat,  it  should  be  dried  by  direct  ap- 
plication of  heat;  usually  it  has  about  25  per  cent,  of  water  combined 
with  it,  heat  necessary  for  evaporation  of  which  is  lost. 

Different  fuels  require  different  volumes  of  oxygen  ;  for  different 
kinds  of  coal  it  varies  from  1.87  to  3  Ibs.  for  each  Ib.  of  coal.  60  cube 
feet  of  air  is  necessary  to  furnish  i  Ib.  of  oxygen  ;  and,  making  a  due 
allowance  for  loss,  nearly  90  cube  feet  of  air  are  required  in  furnace  of 
a  boiler  for  each  Ib.  of  oxygen  applied  to  combustion. 


Anthracites . 


( Hard. 

( Semi  or  gaseous. 


tion       -i  ( Caking. 

of  Coal.        Bituminous { Cherry. 

(Splint. 

Bitniniiioxis    Coal. 

Lignite.  Brown  Coal  or  Bituminous  Wood. — Presents  a  distinct  woody 
structure ;  is  brittle,  and  burns  readily,  leaving  a  white  ash,  and  contains 
and  absorbs  moisture  in  some  cases  fully  40  per  cent. 

Caking. — Fractures  uneven,  and  when  heated  breaks  into  small  pieces, 
which  afterwards  agglomerate  and  form  a  compact  body.  When  the  pro- 
portion of  bitumen  is  great,  it  fuses  into  a  pasty  mass.  This  coal  is  unsuit- 
ed  where  great  heat  is  required,  as  the  draught  of  a  furnace  is  impeded  by 
its  caking.  It  is  applicable  for  production  of  gas  and  coke. 

Splint  or  Hard. — Color  black  or  brown-black,  lustre  resinous  and  glisten- 
ing. It  kindles  less  readily  than  caking  coal,  but  when  ignited  produces  a 
clear  and  hot  fire. 

Cherry  or  Soft. — Alike  to  splint  coal  in  fracture,  but  its  lustre  is  more 
splendent.  Does  not  fuse  when  heated,  is  very  brittle,  ignites  readily,  and 
produces  a  bright  fire  with  a  yellow  flame,  but  consumes  rapidly. 

Cannel. — Color  jet,  or  gray  or  brown-black,  compact  and  even  texture,  a 
shining,  resinous  lustre.  Fractures  smooth  or  flat,  conchoidal  in  every  di- 
rection, and  polishes  readily. 

Experiments  upon  practical  burning  of  this  description  of  coal  in  furnace  of  a 
steam  boiler  give  an  evaporation  of  from  6  to  10  Ibs.  of  fresh  water,  under  a  pressure 
of  30  Ibs.  per  sq.  inch  per  Ib.  of  coal;  Cumberland  (MA,  U.  S.)  coal  being  most  ef- 
fective, and  Scotch  least. 

Limit  of  evaporation  from  212°  for  i  Ib.  of  best  coal,  assuming  all  of  heat 
evolved  from  it  to  be  absorbed,  would  be  14.9  Ibs. 

Coals  that  contain  sulphur,  and  are  in  progress  of  decay,  are  liable  to  spontaneous 
combustion. 

There  are  very  great  variations  in  the  chemical  composition  and  proper- 
ties of  coals. 


American. 
Carbon,  from  75  to  80  per  cent. 
Hydrogen,  from  5  to  6. 
Oxygen,  from  4  to  10. 
Nitrogen,  from  i  to  2. 
Sulphur,  from  .4  to  3. 
Ash,  from  3  to  10. 
Coke,  from  48. 5  to  79. 5. 


For  Volume  of  Air,  etc.,  see  Combustion,  page  465. 


British. 

Carbon,  from  70  to  91  per  cent. 
Hydrogen,  from  3.5  to  nearly  7. 
Oxygen,  from  about .  5  to  20. 
Nitrogen,  from  a  mere  trace  to  2.3. 
Sulphur,  from  o  to  5. 
Ash,  from  .2  to  15. 
Coke,  from  49  to  93. 


480 


FUEL. 


Coal. 

Anthracite. 

Anthracite  or  Glance  Coal,  or  Culm — Is  hard,  compact,  lustrous,  and  some- 
times iridescent,  most  perfect  being  entirely  free  from  bitumen ;  it  ignites 
with  difficulty,  and  breaks  into  fragments  when  heated. 

Evaporative  power,  in  furnace  of  a  steam-boiler  and  under  pressure,  is 
from  7.5  to  9.5  Ibs.  of  fresh  water  per  Ib.  of  coal. 

Coal  from  one  pit  will  sometimes  vary  6  per  cent,  in  evaporative  value. 

Elements  of  Various  A  merican  Coals. 


Specific 
Gravity. 

Fixed 
Carbon. 

Volatile 
Matter. 

Water. 

Moist- 
ure. 

Ash. 

Earthy 
Matter. 

Illinois  Warren  Co    

,23 

Per 
Cent, 
ej  7 

Per 
Cent. 
43  * 

Per 
Cent. 

Per 
Cent. 

Per 

Cent. 

Per 
Cent, 
c  2 

Bureau   "  

•  12 

57  I 

28.8 

II.  2 

2.4 

Mercer    "    

26 

548 

31  2 

8  A. 

r    6 

Indiana  Clay       u  

.28 

54.0 
56.5 

O2.  ? 

8.«J 

2   C 

.28 

50.  ^ 

42.5 

q 

4. 

Pennsyl-  )  Connellsville  
vania  }  Youghiogheny  .  .  . 
Fayette  Co     .  .  . 

.28 

•3 

20 

65 
58.4 

eg 

24 

35 

4-5 

I 

6'I 

5-6 

— 

.  02 

ci 

42.  <» 

2 

A.  e 

Mud  River  . 

28 

C7 

Ohio  Nelsonville  

.27 

IL 

•2-3    QC 

665 

I     Q 

Colorado  Carbon  City  ... 

<6  8 

Washington  Territory  .  . 

.12 

*8.2<; 

11.7? 

7 

1 



Coke. 

Coke. — Coking  in  a  close  oven  will  give  an  increase  of  yield  of  40  per  cent, 
over  coking  in  heaps,  gain  in  bulk  being  22  per  cent.  Coals  when  coked  in 
heaps  will  lose  in  bulk. 

Cannel  and  Welsh  (Cardiff)  coals  when  coked  in  retorts  will  gain  from  10 
to  30  per  cent,  in  bulk  and  lose  36.5  per  cent,  in  weight. 

Relative  costs  of  coal  and  coke  for  like  results,  as  developed  by  an  ex- 
periment in  a  locomotive  boiler,  are  as  i  to  2.4. 

Evaporative  power  in  furnace  of  a  steam-boiler  and  under  pressure,  is 
from  7.5  to  8.5  Ibs.  of  fresh  water  per  Ib. 

Bituminous  coal  will  yield  from  60  to  80  per  cent,  of  coke.  Averaging 
66  per  cent*  It  is  capable  of  absorbing  15  to  20  per  cent,  of  moisture. 

Heat  of  combustion  lost  in  coking  of  bituminous  coal  40  per  cent. 

Charcoal. 

Charcoal,  properly  termed,  is  not  made  below  a  temperature  of  536°.  The 
best  quality  is  made  from  Oak,  Maple,  Beech,  and  Chestnut. 

Wood  will  furnish,  when  properly  burned,  about  23  per  cent,  of  coal. 

Charcoal  absorbs,  upon  an  average  of  the  various  kinds,  from  .8  per  cent, 
of  water  for  Beech,  to  16.3  for  Black  Poplar,  Oak  absorbing  about  4.28,  and 
Pine  8.9. 

Evaporative  power,  in  furnace  of  a  boiler  and  under  pressure,  is  5.5  Ibs. 
of  fresh  water  per  Ib.  of  coal. 

Volume  of  air  chemically  required  for  combustion  of  i  Ib.  of  charcoal  is, 
when  it  consists  of  79  carbon,  129  cube  feet  at  62°. 

138  bushels  charcoal  and  432  Ibs.  limestone,  with  2612  Ibs,  of  ore,  will  pro- 
duce i  ton  of  pig  iron. 


FUEL. 


48l 


Produce  of  Charcoal  from  Various  Woods  dried  at  300°  and  Carbonized 
at  572°.     (M.  Viotette.) 


WOOD. 

Weight. 

WOOD. 

Weight. 

WOOD. 

Weight. 

Cork 

Per  Cent. 
62  8 

Larch     

Per  Cent. 
40.31 

Maple  

Per  Cent. 
33-75 

Oak 

46  cxi 

Chestnut  

36.06 

Willow  

•33.74 

Beech 

44  25 

34-69 

Black  elder  

33-OI 

Pine 

4.1   4.8 

Elm  

•2A.  CQ 

Ash  

33.28 

Poolar  roots.  .  . 

40.0 

Birch... 

34-17 

Pear  

31-88 

Poplar 31. 12  per  cent 

In  a  Green  or  Ordinary  State,     ( Weight  per  cent. ) 


Apple 23.8 

Ash 26.7 

Beech 21.1 


Birch 24.1 

Elm 25.1 

Maple 22.9 


Oak 22.85  I  Red  pine 23 

"  young...  33.3      White  Pine...  23.5 
Poplar. 2c. 5    I  Willow 18.6 


It  appears  from  this  that  cork,  the  lightest  of  woods,  yields  largest  per-centage 
of  charcoal,  about  63  per  cent. ;  and  that  poplar  yields  lowest,  about  31  per  cent. 
There  does  not  appear  to  be  any  definite  relation  between  density  of  wood  and 
volume  of  yield. 

Produce  by  a  slow  process  of  charring  is  very  nearly  50  per  cent,  greater  than  by 
a  quick  process. 

Lignite. 

Lignite  is  an  imperfect  mineral  coal.  It  is  distinguished  from  coal  by 
its  large  proportion  of  oxygen,  being  from  13  to  29  per  cent.  Its  specific 
gravity  ranges  from  1.12  to  1.35. 

-  Elements  of  Various  American  Lignites.    (W.  M.  Barr.) 


LOCATION. 

Spec. 
Grav. 

Fixed 
Carbon. 

Volatile 
Matter. 

Water. 

Ash. 

Total 
Volatile. 

Coke. 

Kentucky  .  .  . 

I  2 

Per  Cent. 
40 

Per  Cent. 
23 

Per  Cent. 
30 

Per  Cent. 
n 

Per  Cent. 
53 

Per  Cent. 
47 

Blandville  .  .  . 
Washington  Terr'y  .... 
Vancouver's  Island  
Colorado,  Carbon  City.  . 
Canon  City  .  . 
Arkansas  

1.  17 

;i 

58-25 
62 

4i-25 
56.8 

34*5 

48 
31-75 

H 

34-2 
28.5 

"•5 
7 
4 
3-5 

4-5 
32 

9-5 
3 
3 
9-25 
4-5 
5 

59-5 
38.75 
35 
49-5 

60,5 

40.5 
61.25 
65 
50-5 
61.3 
39-5 

Texas,  Robertson  Co.  .  . 

1.23 

45 

39-5 

ii 

4-5 

50.5 

49-5 

A  splaal  txim  . 

Asphaltum  contains  1.65  to  10.09  Per  cent-  °^  oxygen. 

Wood. 

Wood,  as  a  combustible,  is  divided  into  two  classes,  the  hard,  as  Oak,  Ash, 
Elm,  Beech,  Maple,  and  Hickory,  and  soft,  as  Pine,  Cotton,  Birch,  Sycamore, 
and  Chestnut. 

Green  wood  subjected  to  a  temperature  ranging  from  340°  to  440°  will 
lose  30  to  45  per  cent,  of  its  weight. 

At  a  temperature  of  300°,  Oak,  Ash,  Elm,  and  Walnut,  in  a  comparatively 
seasoned  state,  lost  from  16  to  18  per  cent. 

Woods  contain  an  average  of  56  per  cent,  of  combustible  matter. 

From  an  analysis  of  M.  Violette  it  appears  that  composition  of  wood  is  about 
same  throughout  the  tree,  and  that  of  the  bark  also ;  that  wood  and  bark  have  about 
same  proportion  of  carbon  (49  per  cent.),  but  that  bark  has  more  ash  than  wood. 
Leaves  and  small  roots  have  less  carbon  than  wood  (45  per  cent.),  and  more  ash, 
5  and  7  per  cent. 

Leaves  when  dried  at  212°  lost  60  per  cent,  of  water,  and  branches  45  per  cent. 
Ss 


482 


FUEL. 


Evaporative  power  of  i  cube  foot  of  pine  wood  is  equal  to  that  of  i  cube 
foot  of  fresh  water ;  or,  in  the  furnace  of  a  steam-boiler  and  under  pressure, 
it  is  4.75  Ibs.  fresh  water  for  i  Ib.  of  wood. 

Northern  Wood. — One  cord  of  hard  wood  and  one  cord  of  soft  wood,  such 
as  is  used  upon  Lakes  Ontario  and  Erie,  is  equal  in  evaporative  effects  to 
2000  Ibs.  of  anthracite  coal. 

Western  Wood. — One  cord  of  the  description  used  by  the  river  steamboats 
is  equal  in  evaporative  qualities  to  12  bushels  (960  Ibs.)  of  Pittsburgh  coal. 
9  cords  cotton,  ash,  and  cypress  wood  are  equal  to  7  cords  of  yellow  pine. 

Solid  portion  (ligniri)  of  all  woods,  wherever  and  under  whatever  circum- 
stances of  growth,  are  nearly  similar,  specific  gravity  being  as  1.46  to  1.53. 

Densest  woods  give  greatest  heat,  as  charcoal  produces  greater  heat  than 
flame. 

For  every  14  parts  of  an  ordinary  pile  of  wood  there  are  1 1  parts  of  space ; 
or  a  cord  of  wood  in  pile  has  71.68  feet  of  solid  wood  and  56.32  feet  of  voids. 

Trees  in  the  early  part  of  April  contain  20  per  cent,  more  water  than  they 
do  in  the  end  of  January. 

Ash. 

Proportion  of  Ash  in  100  Lbs.  of  several  Woods. 
WOODS.  Wood. 


Ash. . . , 
Beech. , 
Birch . , 


•  5 

•35 

•34 


Leaves. 

WOODS. 

Wood. 

Leaves. 

Per  Cent. 

Elm  

Per  Cent. 
1.88 

Per  Cent. 
n.8 

e  A 

Oak  

.21 

=; 

Pitch  Pine.  .  . 

•  2t; 

3.I5 

JPeat. 

Peat  is  the  organic  matter,  or  soil,  of  bogs,  swamps,  and  marshes  ^decayed 
moss,  sedge,  coarse  grass,  etc. — in  beds  varying  from  i  to  40  feet  in  depth. 
That  near  the  surface,  and  less  advanced  in  transformation,  is  light,  spongy, 
and  fibrous,  of  reddish-brown  color ;  lower  down,  it  is  more  compact,  of  a 
darker  brown  color ;  and,  in  lowest  strata,  it  is  of  a  blackish  brown,  or  almost 
black,  of  a  pitchy  or  unctuous  surface,  the  fibrous  texture  nearly  or  alto- 
gether transformed. 

Peat,  in  its  natural  condition,  contains  from  75  to  80  per  cent,  of  water. 
Occasionally  its  constituent  water  amounts  to  85  or  90  per  cent.,  in  which 
case  peat  is  of  the  consistency  of  mire.  It  shrinks  very  much  in  drying ; 
and  its  specific  gravity  varies  from  .22  to  1.06,  surface  peat  being  lightest, 
and  deepest  peat  densest. 

When  peat  is  milled,  so  that  its  fibre  is  broken  up,  its  contraction  in  dry- 
ing is  much  increased,  and  in  this  condition  it  is  termed  condensed. 

When  ordinarily  air  dried,  it  will  contain  20  to  30  per  cent,  of  moisture, 
and  when  effectively  dried  at  least  15  per  cent. 

Products  of  Distillation  of  Peat. 

Water  31.4.    Tar  2.8.     Gas  36.6.     Charcoal  29.2. 

The  distillation  of  the  tar  will  yield  paraffine,  oil,  gas,  water,  and  char- 
coal, and  the  water  acetic  acid,  wood  spirit,  and  chloride  of  ammonia. 

Evaporative  power,  in  furnace  of  a  steam-boiler  and  under  pressure,  is 
from  3.5  to  5  Ibs.  of  fresh  water  per  Ib.  of  fuel. 

Tan. 

Tan,  oak  or  hemlock  bark,  after  having  been  used  in  the  process  of  tan- 
ning, is  combustible  as  a  fuel.  It  consists  of  the  fibre  of  the  bark,  and, 
according  to  M.  Peclet,  5  parts  of  bark  produce  4  parts  of  dry  tan ;  and 
heating  power  of  it  when  perfectly  dry,  or  containing  but  15  per  cent,  of 
ash,  is  6100  units ;  while  that  of  tan  in  an  ordinary  state  of  dryness,  con- 
taining 30  per  cent,  of  water,  is  4284.  Weight  of  water  evaporated  at  212° 
by  i  Ib.,  equivalent  to  these  units,  is  6.31  Ibs.  for  dry,  and  4.44  for  moist. 


tfTJEL. 


483 


Relative   Values   of  different    Fuels. 

1V§ 

ijl 

i1! 

ii 

| 

*"  Nta 

|s§ 

&*1 

fH 

li 

1*0  (i 

DBSOBIPTION. 

Lbs.ofStear 
Water  al 
by  z  Ib.  oi 

*& 

111 
&-' 

•a 

i* 

Relative  F 
ties  of  Ign 

is 

il 

t? 

]]] 

RelatlT* 
Weighta. 

Anthracites. 

Peach  Mountain,  Pa  
Beaver  Meadow  

10.7 
0.88 

i 

•  Q2T. 

i 
.082 

•505 
.207 

.633 
.748 

:r5 

•945 

j 

Bituminous. 

y.  uu 

'y-*3 

»y«nB 

••*"/ 

»/^v 

Newcastle  

8.66 

800 

776 

CQC 

.887 

g 

.004 

Pictou  

8.48 

7  84. 

•  wy 

.792 

.//u 
.738 
.663 

j| 

«' 

.876 
.852 

Caunelton  Ind 

/.uq. 

686 

.6l6 

"^ 

g 

-333 

c7g 

.848 

Scotch  

7-34 

'... 

'fioe: 

•y  4 

-  57° 

Pine  wood.  drv.  .. 

0.95 
^.60 

.049 
.1*6 

•u*a 

.521 

•499 

.049 

.909 

"Weights,  Evaporative    Powers    per   Weight   and    Bulk: 
etc.,  of  different   Fuels.    (W.  R.  Johnson  and  others.) 


FUII.. 

Specific 
Gravity. 

Weight 
per 
Cube  Foot. 

Steam  from 
Water  Ht 
212*  by  i  Ib. 
of  Fuel. 

Clinker 
from  100  Ibs. 

Cube  Feet 
in  a  Ton. 

BITUMINOUS. 
Cumberland,  maximum  
"           minimum  
Duflryn  

•313 

•337 
.-226 

UK 

52.92 
54-29 

Lbs. 
10.7 
9-44 

Lba. 
2.13 
4-53 

No. 

42-3 
41.2 
42  OO 

Caunel  Wigan  

Ae  Q 

15  - 

•224. 

287 

8  O4. 

•1    -3-} 

4O 

'  '          average  

g  ~n 

882 

50  82 

°-39 
8  76 

3l8 

8  A.1 

6  11 

xc 

46  81 

8  2 

478 

<17  3 

Carr's  Hartley  

6 

A7  88 

7-99 

7  8/1 

i  86 

46.7 

Clover  Hill  Va  

.285 

7.04 

7  O7 

3-86 

4Q.9 

Cannelton  Ind      

I  6A 

A7 

Scotch  Dalkeith  

708 

1.04 

e  6q 

438 

Chili  

Japan          .       .          

A&  1 

ANTHRACITE. 

.464 

eo  70 

3,Q-1 

41.6 

Forest  Improvement 

53-79 
eo  66 

06 

81 

41  7 

Beaver  Meadow     

ec  A 

9  88 

6 

• 

on  3 

4.21 

4.8  80 

1.24 

jy-w 
45-8 

Beaver  Meadow,  No.  3  

:ff 

CQ 

54-93 

9.21 
8  o? 

X.OI 

1.08 

40.7 
40.5 

COKE. 
Natural  Virginia  

I.  •227 

46  64 

8.47 

5-3' 

48.  t 

Midlothian  

32.7 

8.6q 

10.  51 

68.5 

31.6 

8.99 

3-55 

70.9 

MISCELLANEOUS. 
Charcoal  Oak    

f  .  e 

24. 

5  5 

Ash. 
3  06 

104 

Peat  .  ..'     

•53 

3O 

5 

75 

Warlich's  fuel  

1.15 

60.  1 

10.4 

2.91 

32.44 

Wy  lam's     "       

6e 

8.9 

Pine  wood.  drv.  .  . 



ai 

4.7 

.•u 

1  06.  6 

FUEL. 


Weights  and  Comparative  Values  of  different  Woods. 


WOODS. 

Cord. 

Value. 

WOODS. 

Co**     t 

Value. 

Shell-bark  Hitkory  .  .  . 
Red-heart  Hickory 

Lbs. 
4469 

i 
81 

New  Jersey  Pine  
Yellow  Pine  

2137 
1904 

•54 
•43 

White  Oak       

3821 

81 

White  Pine  

1868 

.42 

Red  Oak 

60 

Beech     

7 

Virginia  Pine 

2689 

61 

.52 

Southern  Pine  

3375 

•73 

Hemlock  

•44 

Hard  Maple  .  .  , 

2878 

.6 

Cottonwood.  .  . 



•  33 

Hiiq.i3.id.   ITu.els. 
3?etroleu.rxi. 

Petroleum  is  a  hydro-carbon  liquid  which  is  found  in  America  and  Europe. 
According  to  analysis  of  M.  Sainte-Claire  Deville,  composition  of  15  petro- 
leums from  different  sources  was  found  to  be  practically  constant.  Average 
specific  gravity  was  .87.  Extreme  and  average  elementary  composition  was 
as  follows : 

Carbon 82     to  87.  i  per  cent.        Average,  84. 7  pei  cftut. 

Hydrogen 11.21014.8       "  "       13.1       " 

Oxygen 5  to   5.7       "  "         2.2       " 

IOO 

Its  heat  of  combustion  is  20240,  and  its  evaporative  power  at  212*  20.33. 

Petroleum  Oils — Are  obtained  by  distillation  from  petroleum,  and  are  coui' 
pounds  of  carbon  and  hydrogen,  in  average  proportioh  of  72.6  and  27.4. 

Boiling-point  ranges  from  86°  to  495°. 

Schist  Oil — Consists  of  carbon  80.3  parts,  hydrogen  11.5,  and  oxygen  8.2. 

Pine  Wood  Oil — Consists  of  carbon  87.1  per  cent.,  hydrogen  10.4,  and 
oxygen  2.5. 

Coal-gas. 

Coal  Gas — As  furnished  by  Chartered  Gas  Co.  of  London  is  composed  as 
follows : 

Carbon.  | Hydrogen  Oxygen.    Hydrogen.  Nitrogen. 


Oleflant  Gas,     ) 
Bi-carb.hyd. }  " 

Marsh  gas,     ) 
Carb.  hyd.  J 

Carbonic  oxide. .. . 


3-096 
26.445 


•434 
8.815 


Hydrogen . . . 

Oxygen 

Nitrogen 


.08 


•7s 


^^^  Total zoo  parts. 

Heat  of  combustion  at  212°  52961  units,  and  evaporative  power  47.51  Ibs. 

Coal-gas.     (F.  Harcourt.) 


Cnrb. 

Hyd. 

Oxy. 

Nit. 

Carb. 

Hyd. 

Oxy. 

Nit. 

Olefiantgas  
Marsh  gas  
Carbonic  oxide.  . 
Carbonic  dioxide 

Per  ct. 
10.5 
39-7 
5-9 
1.9 

Per  ct. 
i-7 
13.2 

Per  ct. 

7-9 
5 

Per  ct. 

Hydrogen  
Nitrogen  

Per  ct. 
~s8~ 

Perct. 
8.1 

Perct. 
•  3 

Per  ct. 

5~8 

~oT 

Oxygen  

Total  .  .  . 

23 

13.2 

One  Ib.  of  this  gas  had  a  volume  of  30  cube  feet  at  62° ;  heat  of  combus- 
tion 22  684  units ;  and  of  one  cube  foot  756  units,  which  is  equivalent  to 
evaporation  of  .68  Ib.  of  water  from  62°.  or  of  .78  Ib.  from  212°  per  cube  foot. 


FUEL. 


485 


.A/verage    Composition,   of  Fuels. 


BITUMINOUS  COALS. 
Australian  .       

Specific 
Grav- 
ity. 

Carbon. 

Hydro- 
gen. 

Nitro- 
gen. 

Oxygen. 

Sul- 
phur. 

Ash. 

i-3i 
1.28 

1.18 
1.29 

1.23 
I-3I 

i-33 
1.24 
1.32 
1.29 
i-3 
i-3i 

i-5 
i-5 

1.06 
x.  29 
1.25 

1.18 
1.28 

1.2 

7^8 

I-I5 
I.I 

Per  ct. 

££ 

63-94 
70-55 
38.98 
79-23- 
93.8i 

& 

8:2 

78.26 
88.56 

87-73 
82.94 

85 

47-3 
f 
62.25 
91-45 
82.39 
74.82 
82.92 
83-75 
67.6 
84-85 
87-95 
89.78 

?4.97 
66.93 

88.54 
86.17 
96.66 

5o-i7 
48.12 
48.13 
49-95 
49-7 

87.68 
71-36 
70.07 

79.18 
69.02 
60.  18 
74.82 
56.8 
40 
34-5 
61.02 
58.18 
90.02 

7Q.QI 

Per  ct. 

4-74 
4-76 
8.86 
5-76 
4.01 
6.08 
1.82 

4~66 

ft 

5-08 
5-35 
4-5 

5-05 
4-5 
5-32 
6.18 

5-49 
5-66 

5-4 
5-05 
5-24 

S3 

5-32 

^67 
i-35 

6.12 

6.37 
5-25 
6.41 
6.06 

2.83 
5-95 
4.61 

9-3 
5-05 

5-77 
5.96 
5-56 
*.6o 

Per  ct. 
.8 

^6 
•95 
•58 
1.18 

'I5 
144 

y 

(    7 

73 
(    4 
1.27 
(    5 

(10 

(    8 

(12 

(    8 
(5 
2.16 
i-45 

1.02 

(     2 
(     I 

1.05 

':M 

1.05 
(   8 

(20 

(29 

(13 

.81 
1.23 

1.68 

Per  ct. 

Per  ct. 
-5 
i-45 
'•35 
•3* 
,.98 
6.14 
i-43 

^5 
1.25 

1.22 

1.77 

•49 

M3 

.07 

i.  02 
.42 

2.2 

•52 

1.62 
I.2S 

Per  ct. 
8.38 
7-74 

21.22 
7-52 

36.91 
4.84 

1.6 
i-55 

9-25 
5-34 
3.26 
3-96 

2.  19 
1-54 
3-08 

3-5 
22.9 

4 
13-4 

2.04 

'3-91 
1.13 
2-55 
14-57 
1.67 

i-39 
i-5 

si 

15-83 

8.67 
8.56 

1.77 
.48 
1.3 

,:!' 
3.06 

•4 
•43 

2.8 

5-82 
5-57 
4-45 
4-5 
7 
5 

3-43 
2.918 

4.84 

Borneo        

20.75 
18.63 

4-7 
13.24 
I3-38 
7.24 
2-77 

.6 
10.95 
38) 
65) 
63) 

17-54 
05) 
8.32 
09) 
46) 
04) 
43) 
43) 
42) 
•39 
3-5 
8.7 

857 
99) 

40.38 
43-95 
44-5 
43-65 
4i-3 

6-43 
22.19}: 
24.89! 

72) 

12) 
03) 

38) 

32-4 
31-21 

6.6? 

Boghead  dry  average     

Chili  Conception  Bay  

"     Chiriqui 

Cannel  Wigan    

Coke  Garesfield 

44     Durham  

Duffryn 

Formosa,  Island  

44       caking      

44       long  flame  

Indian  average     

44      Kotbec  

Patagonia 

Russian  Miouchif  

Sydney,  S.  W  

Splint,  Wylam  

44     Glasgow  

44     Cannel,  Lancashire 
44          "        Edinburgh 
44     Cherry,  Newcastle. 
44     Caking,  Garesfield  . 
"     Ebbro  Vale,  Welsh. 
14     Llangenneck    44     . 
Vancouver's  Island  

ANTHRACITES. 
Anthracite  

French     ,  

WOODS. 
Beech  

Birch  

Oak  

White  Pine  

Woods,  average  

CHARCOAL. 
Oak  

Pine  

Maple  

MISCELLANEOUS. 
Asphalt  

Lignite,  perfect  ,  

4  4       imperfect  

44       bituminous  

44       Colorado  

44       Kentucky  . 

4  '       Arkansas  

Peat  dense             

41    Irish  average  ,.,.... 

Patent,  Warlich  

44      Wvlam's  .  .  , 

»  Heat  of  Combustion  of  x  Lb.  14  723. 
t  Including  Nitrogen. 


Ss* 


t  Heat  of  Combustion  of  i  Lb.  15651. 
§  Including  Oxygen. 


486 


FUEL. 


Average  Composition  of  Ooals  and  Fuels,  Heat  of*  Com«? 
lonstion,  and.    Evaporative    3r*o\ver. 

Deduced  from  analysis  and  experiments  of  Messrs.  De  La  Bfohe,  Playfair,  and  Pectet. 


COALS  AMD  FUELS. 

020 

Carbon. 

Hydro- 
gen. 

COMPO 

Nitro- 
gen. 

3ITIOX. 

Sul- 
phur. 

Oxy- 
gen. 

Ash. 

Heat  of  Com- 
bustion of  i 
Ib. 

Evaporation 
from  water 
at  212*. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Units. 

Lbs. 

Derbyshire    and  ) 
Yorkshire  .  .  .  .  } 

1.29 

79.68 

4-94 

1.41 

1.  01 

10.28 

2.65 

13860 

14-34 

Lancashire  

1.27 

77-0 

5.32 

.3 

•44 

9.53 

4.88 

13918 

14.  56 

Newcastle  . 

1.26 

/  i'y 
82.12 

5*  31 

•35 

.24 

3.77 

14  820 

I  5  32 

Scotch  

1.26 

78.53 

.11 

9.69 

4-°3 

14  164 

*3-  o^ 
14.77 

Welsh  

I.  32 

83-78 

4-79 

.98 

•43 

4.91 

14858 

15.  52 

Average  of  British. 
Patent  fuels 

M..  $£. 
1.28 

80.4 
83-4 

5-19 

4  O7 

.21 

.08 

:2 

2.7Q 

4-<>5 

5.Q3 

14320 
15  ooo 

*  J-  Sm 

14.82 

15  66 

Van  Diemen's  Land 
Chili 

I.   17 

65.8 

go  e6 

T  y/ 

3-5 

.82 

.1 

•*  /y 
5.58 

j-yj 
22.71 

II  320 

11-83 
11.68 

Lignite,  Trinidad  .  . 

— 

U3ou 

65.2 

5-43 
4-25 

2-5 

.69 

21.69 

6.84 

10438 

10.87 

"    French  Alps 

1.28 

70.02 

5-2 



— 

— 

3.01 

11790 

12.  I 

"    Bitum.,Cuba 

1.2 

75.85 

7-25 



— 

— 

3-94 

14562 

14.96 

"    Wash.Ter.*. 



67 

4-55 

— 

I. 

— 

12538 

12.91 

Asphalt  

1.06 

70.18 

9*3 

___ 

__ 

__ 

2.8 

16655 

17.24 

Petroleum  

.87 

n 
84.7 

13.  i 





2.  2 

20  240 

2O  73 

"         oils.  

•"/ 
•75 

_ 

_ 

_ 

27530 

•»*  JJ 

28.5 

Oak  bark  Tan,  dry. 



— 



— 



15 

6100 

6.3I 

"         "  moist 
Charcoal  at  302°.  .  . 

— 

47-51 

6.12 

(Oa 

ddN  A< 

5.29) 

I5.8 

4284 

8130 

48;r 

"        "572°... 
"         "  810°.  .  . 

1.4 
1.71 

73.24 
81.64 

& 

(OandN  21.96) 
(OandN  15.24) 

liii 

11861 
14916 

12.27 
15-43 

Peat,  dry,  average. 
"    moist,  t  " 

•53 

42 

58.18 

33.38 

5-96 

1-23  1    — 
(  O  and  N  2 
.18  1    — 

31.21 

.08 

3-43 
3-3 

9951 
8917 

52061 

10.3 

9.22 
47.51 

*  Water  7.    Oxygen  and  Nitrogen  17.36. 


t  Moisture  27.8.    Sulphur  .2. 


Elements  of  I^viels  not  included  in  ^Preceding  Tatoles. 


FUEL. 

Heat  of 
Combustion 
of  i  Ib. 

Evaporative 
Power  of  i 
Ib.  at  212°. 

Coke  pro- 
duced. 

Weight 
of  i 
Cub.  Foot. 

Volume  of 
i  Ton. 

Bituminous  Coal. 
Welsh  

Units. 
14858 

Lbs. 
0.05 

Per  cent. 
73 

Lbs. 
82 

Cube  Feet. 

42  7 

14  820 

8.01 

61 

78.3 

45*  3 

13018 

7.04 

58 

70-4 

AC    2 

Scotch  

ijyio 
14  164 

7.7 

54 

78.6 

42 

Boghead  

14478 

787 

•30.04 

British   average      .  .  

8  13 

OI 

7O  8 

Q  85 

oo 

79.0 

OQ  6 

oe  7 

Cumberland  Md         .   .  .... 

837 

84.  O7 

825 

87  54 

14  723 

64.2 

40 

68  27 

Anthracite. 

14  500 

94.82 

OX  78 

42  15 

14038 

88.83 

Miscellaneous. 
Warlich's  fuel      

16  495 

70.  e 

34  5 

Coke  Mickley 

15  600 

80 

Virginia  average    

*3  55® 

14  O2 

45 

60  8 

Charcoal  

12  ^25 

vy.o 
12.70 

Lignite  perfect.        ..  ^  ...... 

ii  678 

12  I 

47 

0814 

I0.l8 

37'  5 

"       Russian  

15837 

Asphalt  

16  555 

17.24 

Q 

_ 

Woods,  dry,  average.  .  . 

7  702 

8.07 



114 

FUEL. — GRAVITATION.  487 

^Miscellaneous. 

Experiments  undertaken  by  Baltimore  and  Ohio  R.  R.  Co.  determined 
evaporating  effect  of  i  ton  of  Cumberland  coal  equal  to  1.25  tons  of  anthra- 
cite, and  i  ton  of  anthracite  to  be  equal  to  1.75  cords  of  pine  wood;  also 
that  2000  Ibs.  of  Lackawanna  coal  were  equal  to  4500  Ibs.  best  pine  wood. 

One  Ib.  of  anthracite  coal  in  a  cupola  furnace  will  melt  from  5  to  10  Ibs.  of  cast 
iron;  8  bushels  bituminous  coal  in  an  air  furnace  will  melt  i  ton  of  cast  iron. 

Small  coal  produces  about  .75  effect  of  large  coal  of  same  description. 

Experiments  by  Messrs.  Stevens,  at  Bordentown,  N.  J.,  gave  following  results: 

Under  a  pressure  0/30  Ibs.,  i  Ib.  pine  wood  evaporated  3.5  to  4.75  Ibs.  of  water, 
i  Ib.  Lehigh  coal,  7.25  to  8.75  Ibs. 

Bituminous  coal  is  13  per  cent,  more  effective  than  coke  for  equal  weights;  and 
in  England  effects  are  alike  for  equal  costs. 

Radiation  from  Fuel. — Proportion  which  heat  radiated  from  incandescent  fuel 
bears  to  total  heat  of  combustion  is, 

From  Wood 29  |  From  Charcoal  and  Peat 5 

Least  consumption  of  coal  yet  attained  is  1.5  Ibs.  per  IIP.  It  usually  varies  in 
different  engines  from  2  to  8  Ibs. 

Volume  of  pine  wood  is  about  5.5  times  as  great  as  its  equivalent  of  bituminous 
coal. 


GRAVITATION. 

GRAVITY  is  an  attraction  common  to  all  material  substances,  and 
they  are  affected  by  it  directly,  in  exact  proportion  to  their  mass,  and 
inversely,  as  square  of  their  distance  apart. 

This  attraction  is  termed  terrestrial  gravity  -,  and  force  with  which  a 
body  is  drawn  toward  centre  of  Earth  is  termed  the  weight  of  that  body. 

Force  of  gravity  differs  a  little  at  different  latitudes  :  the  law  of  variation, 
however,  is  not  accurately  ascertained  ;  but  following  theorems  represent  it 
very  nearly  : 

£  \l  Z  '^  88^Cat'  theses     I  -a        *  representing  force  of  gravity  at  lati* 
I'  [*±:SaSf7|  al  uSSStorp*    **  45°,  and  g  force  at  other  places. 


Or,  32.171  (lat  45°)  (i  +  .  005  133  sin.  L)     i  —  ^-J  =  g.     L  representing  latitude, 
H  height  of  elevation  above  level  of  sea,  and  R  radius  of  Earth,  both  in  feet. 

NOTE.—  If  2  L  exceeds  90°,  put  cos.  180  —  2  L,  and  R  at  Equator  =  20  926  062,  at 
Poles  20853  429>  and  mean  20889  746. 

ILLUSTRATION.—  What  is  force  of  gravity  at  latitude  45°,  at  an  elevation  of  209 
feet,  and  radius  =  20  900000  feet? 

32.171  (i  +  .005  133  sin.  45°)  (i  —  —  l__J:=32.i7i  X  1.00363  X  -999  9s  =  32-  287. 

Gravity  at  Various  Locations  at  Level  of  Sea. 

Equator  ...........  32.088  I  New  York  .........  32.  161  I  London  ...........  32.  189 

Washington  .......  32-155  I  Lat.  45°  ...........  32-171  I  p°les  ......  ........  32.253 

In  bodies  descending  freely  by  their  own  weight,  their  velocities  are  as 
times  of  their  descent,  and  spaces  passed  through  as  square  of  the  times. 

Times,  then,  being  i,  2,  3,  4,  etc.,  Velocities  will  be  i,  2,  3,  4,  etc. 

Spaces  passed  through  will  be  as  square  of  the  velocities  acquired  at  end 
of  those  times,  as  i,  4,  9,  16,  etc.  ;  and  spaces  for  each  tune  as  1,3,  5,  7,  9,  etc. 


483 


GRAVITATION. 


A  body  falling  freely  will  descend  through  16.0833  feet  m  first  second  of 
time,  and  will  then  have  acquired  a  velocity  which  will  carry  it  through 
32. 166  feet  in  next  second. 

If  a  body  descends  in  a  curved  line,  it  suffers  no  loss  of  velocity,  and  the 
curve  of  a  cycloid  is  that  of  quickest  descent. 

Motion  of  a  falling  body  being  uniformly  accelerated  by  gravity,  motion 
of  a  body  projected  vertically  upwards  is  uniformly  retarded  in  same  manner. 

A  body  projected  perpendicularly  upwards  with  a  velocity  equal  to  that 
which  it  would  have  acquired  by  falling  from  any  height,  will  ascend  to 
the  same  height  before  it  loses  its  velocity.  Hence,  a  body  projected  up- 
wards is  ascending  for  one  half  of  time  it  is  in  motion,  and  descending  the 
other  half. 

Various  Formulas  here  given  are  for  Bodies  Projected  Upwards  or 
Falling  Freely,  in  Vacuo. 

When,  however,  weight  of  a  body  is  great  compared  with  its  volume,  and  velocity 
of  it  is  tow,  deductions  given  are  sufficiently  accurate  for  ordinary  purposes. 

In  considering  action  of  gravitation  on  bodies  not  far  distant  from  surface  of  the 
Earth,  it  is  assumed,  without  sensible  error,  that  the  directions  in  which  it  acts  are 
parallel,  or  perpendicular  to  the  horizontal  plane. 

A  distance  of  one  mile  only  produces  a  deviation  from  parallelism  less  than  one 
minute,  or  the  6oth  part  of  a  degree. 

Relation   of  Time,  Space,  and.   "Velocities. 


Time  from 
Beginning  of 
Descent. 

Velocity  acquired 
at  End  of  that 
Time. 

Squares 
of 
Time. 

Space  fallen 
through  in  that 
Time. 

Spaces 
for 
this  Time. 

Space  fallen 
through  in  last 
Second  of  Full. 

Seconds. 

Feet. 

Seconds. 

Feet. 

No. 

Feet. 

I 

32.166 

i 

16.083 

I 

1  6.  08 

2 

64-333 

4 

64.333 

3 

48-25 

3 

96.5 

9 

144-75 

5 

80.41 

4 

5 

128.665 
160.832 

16 
25 

257-33 
402.08 

7 
9 

112.58 
'44-75 

6 

193    f 

36 

579 

ii 

176.91 

7 

225.  166 

49 

788.08 

13 

209.  08 

8 

257-333 

64 

1029.33 

15 

241.25 

9 

10 

32?!  666 

100 

1302.75 
1608.33 

17 

273-42 
305-58 

and  in  same  manner  this  Table  may  be  continued  to  any  extent. 

Velocity  acquired  due  to  given  Height  of  Fall  and 
Height   due  to  given  Velocity. 


8.04  ^/h  =  v 


- — =fc;    and 
64.4 


6.083  t2  =  h 


h  representing  height  of  fall  in  feet,  v  velocity  acquired  in  feet  per  second,  and  t 
time  of  fall  in  seconds. 

To   Compute   Action   of  GJ-ravity. 
Time. 

When  Space  is  given.    RULE.— Divide  space  by  16.083,  and  square  root 
of  quotient  will  give  time. 

EXAMPLE.— How  long  will  a  body  be  in  falling  through  402.08  feet? 
•\/402.o8-r-  16.083  —  5  seconds. 

When  Velocity  is  given.    RULE.  —  Divide  given  velocity  by  32.166,  and 
quotient  will  give  time. 

EXAMPLE.— How  long  must  a  body  be  iu  falling  to  acquire  a  velocity  of  800  feet 
per  second  ?  800  -r-  32. 166  =  24. 87  seconds. 


GRAVITATION.  489 

Velocity. 

When  Space  w  given.    RULE.  —  Multiply  space  in  feet  by  64.333,  and 
square  root  of  product  will  give  velocity. 
EXAMPLE.— Required  velocity  a  body  acquires  in  descending  through  579  feet. 

Vs79  X  64.333  =  i93  fat- 

Velocity  acquired  at  any  period  is  equal  to  twice  the  mean  velocity  during 
that  period. 

ILLUSTRATION.— If  a  ball  fall  through  2316  feet  in  12  seconds,  with  what  velocity 
will  it  strike  ? 

2316  -r- 12  =  193,  mean  velocity,  which  x  2  =  386  feet  =  velocity. 

When  Time  is  given.    RULE. — Multiply  time  in  seconds  by  32.166,  and 
product  will  give  velocity. 

EXAMPLE.—  What  is  velocity  acquired  by  a  falling  body  in  6  seconds? 
32.166  X  6  =  192.996/66*. 

Space. 

When  Velocity  is  given.    RULE. — Divide  velocity  by  8.04,  and  square  of 
quotient  will  give  distance  fallen  through  to  acquire  that  velocity. 
Or,  Divide  square  of  velocity  by  64.33. 

EXAMPLE.  — If  the  velocity  of  a  cannon-ball  is  579  feet  per  second,  from  what 
height  must  a  body  fall  to  acquire  the  same  velocity? 

579  -r-  8.04  •=  72.014,  and  72.oi42  =  5186. 02  feet. 

When  Time  is  given.    RULE.  —  Multiply  square  of  time  in  seconds  by 
16.083,  and  it  will  give  space  in  feet. 
EXAMPLE.— Required  space  fallen  through  in  5  seconds. 

5«  =  25,  and  25  X  16.083  =  402.08  feet. 

Distance  fallen  through  in  feet  is  very  nearly  equal  to  square  of  time  in  fourths 
of  a  second. 

ILLUSTRATION  L— A  bullet  dropped  from  the  spire  of  a  church  was  4  seconds  in 
reaching  the  ground ;  what  was  height  of  the  spire  ? 

4  X  4  =  16,  and  i62  =  256  feet. 
By  Rule,  4  X  4  X  16.0833  =  257.33/66?. 

2.— A  bullet  dropped  into  a  well  was  2  seconds  in  reaching  bottom;  what  is  the 
depth  of  the  well  ? 

Then  2X4  =  8,  and  82  =  64  feet. 
By  Rule,  2  X  2  X  16.0833  =  64.33/66*. 

By  Inversion.— In  what  time  will  a  bullet  fall  through  256  feet? 
•^256  =  16,  and  16  -r-  4  =  4  seconds. 

Space   fallen   through   in   last    Second,   of  Fall. 

When  Time  is  given.    RULE. — Subtract  half  of  a  second  from  time,  and 
multiply  remainder  by  32.166. 

EXAMPLE.— What  is  space  fallen  through  in  last  second  of  time,  of  a  body  falling 
for  10  seconds? 

10  —  .5  X  32. 166  =  305. 58  feet. 

JPromiscuovis    Examples. 

i.  If  a  ball  is  i  minute  in  falling,  how  far  will  it  fall  in  last  second? 
Space  fallen  through  =  square  of  time,  and  i  minute  =  60  seconds. 
6o2  X  16.083  =  57  898  feet  for  60  seconds. 
59*  X  16.083  =  55  984   "     "  59 
1914  i 

a.  Compute  time  of  generating  a  velocity  of  193  feet  per  second,  and  whole  space 
descended. 

193  -r-  32. 166  =  6  seconds ;  62  X  16.083  =  579 /66*. 


4QO  GRAVITATION. 

3.  If  a  body  was  to  fall  579  feet,  what  time  would  it  be  in  falling,  and  how  far 
would  it  fall  in  the  last  second  ? 


/579_X2  __  ^^  _  6  Kconds^  ftnd  6  — .  5  x  32. 166  =  5. 5  X  32. 166  =  1 76. 91  /««t 
V    32. 160 

Formulas   to   determine   the   various    Elements. 
/  S         _  V       _  a_S  /a~S 

''          V-so'    ~  g'    -  V  ''    -V  g> 


/S  X  2  0  ;    =T0;    =2 

T  representing  time  of  fatting  in  seconds,  V  velocity  acquired  in  feet  per  second, 
9  space  or  vertical  height  in  feet,  h  space  fallen  through  in  last  second,  g  32.166  and 
•  5  g  and  .25  ^  representing  16.083  and  8.04. 

Retarded    Motion. 

A  body  projected  vertically  upward  is  affected  inversely  to  its  motion 
when  falling  freely  and  directly  downward,  inasmuch  as  a  like  cause  retards 
it  in  one  case  and  accelerates  it  in  the  other. 

In  air  a  ball  will  not  return  with  same  velocity  with  which  it  started.  In 
vacuo  it  would.  Effect  of  the  air  is  to  lessen  its  velocity  both  ascending  and 
descending.  Difference  of  velocities  will  depend  upon  relative  specific  grav- 
ity of  ball  and  density  of  medium  through  which  it  passes.  Thus,  greater 
weight  of  ball,  greater  its  velocity. 

To  Compute  .Auction  of  Grravity  "by  a  Body  projected 

Upward  or  I3ownward  with,  a  given.  Velocity. 

Space. 

When  projected  Upward.  RULE. — From  the  product  of  the  given  velocity 
and  the  time  in  seconds  subtract  the  product  of  32.166,  and  half  the  square 
of  the  time,  and  the  remainder  will  give  the  space  in  feet. 

Or,  Square  velocity,  divide  result  by  64.33,  an(^  quotient  will  give  space 
in  feet. 

EXAMPLE — If  a  body  is  projected  upward  with  a  velocity  of  96.5  feet  per  second, 
through  what  space  will  it  ascend  before  it  stops  ? 

96.5-7-  32- 166  =  3  seconds  =  time  to  acquire  this  velocity. 

Then,  96. 5  X  3  —  (32- 166  X  ^  =  289.5  — 144.75  =  144.75  feet. 

Time. 

RULE.— Divide  velocity  in  feet  by  32.166,  and  quotient  will  give  time  in 
seconds. 
EXAMPLE.— Velocity  as  in  preceding  example. 

96. 5  -T-  32. 1 66  =  3  secondt. 

"Velocity. 

RULE.— Multiply  time  in  seconds  by  32.166,  and  product  will  give  velocity 
in  feet  per  second. 
EXAMPLE.— Time  as  in  preceding  example. 

3  x  32. 166  =  96. 5  feet  velocity. 

Space  fallen   through   in  last   Second.. 
RULE. — Subtract  .5  from  time,  multiply  remainder  by  32.166,  and  product 
will  give  space  in  feet  per  second. 
EXAM  FIJI. —Time  as  in  preceding  example. 

3  — -5  X  32.166  =  2.5  X  32.166  =  8o-4i6/eet 


GRAVITATION.  49! 

When  projected  Downward. 

Space. 

RULE. — Proceed  as  for  projection  upwards  and  take  sum  of  products. 
EXAMPLE  i.— If  a  body  is  projected  downward  with  a  velocity  of  96.5  feet  per  sec- 
ond, through  what  space  will  it  fall  in  3  seconds? 

96-5  X  3-1-  (32-166  X  ^J  =  289.5-1-144.75  =  434-25/«*. 

Or, t2  X  16.083  +  «X  t  =  *• 

2.  —If  a  body  is  projected  downward  with  a  velocity  of  96.5  feet  per  second, 
through  what  space  must  it  descend  to  acquire  a  velocity  of  193  feet  per  second? 
96.5-7-  32. 166  =.  3  seconds,  time  to  acquire  this  velocity. 
193  -r-  32. 1 66  =  6  seconds,  time  to  acquire  this  velocity. 
Hence  6  —  3  =  3  seconds,  time  of  body  falling. 
Then  96.5  X  3  =  289.5  =  product  of  velocity  of  projection  and  time. 

16.083  X  32  =  144-75  =product  0/32. 166,  and  half  square  of  time. 
Therefore  289.5  + 144.75  =  434.25/6^. 

Titne. 

RULE. — Subtract  space  for  velocity  of  projection  from  space  given,  and 
remainder,  divided  by  velocity  of  projection,  will  give  time. 

EXAMPLE.— In  what  time  will  a  body  fall  through  434.25  feet  of  space,  when  pro- 
jected with  a  velocity  of  96.5  feet? 

Space  for  velocity  of  96.5  =  144.75/^1 

Then,  434. 25  — 144. 75  •+•  96. 5  =  289. 5  -=-  96. 5 = 3  seconds. 
Velocity. 

RULE. — Divide  twice  space  fallen  through  in  feet  by  time  in  seconds. 

EXAMPLE. — Elements  as  in  preceding  example. 

Space  fallen  through  when  projected  at  velocity  of  96. 5  feet  =  144. 75  feet,  and  434. 25 
feet  =  space  fallen  through  in  3  seconds. 

Then,  144.75-1-434.25  =  579  feet  space  fallen  through,  and  Vs79  -*- 16.083  =  6 
seconds. 

Hence,  579  x  2  -4-  6  =  1158  -=-  6  =  193  feet. 

Space    Fallen   through,   in   last   Second.. 

RULE. — Subtract  .5  from  time,  multiply  remainder  by  32.166,  and  product 
will  give  space  in  feet  per  second. 

EXAMPLE.— Elements  as  in  preceding  example. 

6  —  .5  X  32. 166  =  5.5  X  32. 166  =  176.91  feet 

Ascending  bodies,  as  before  stated,  are  retarded  in  same  ratio  that  descending 
bodies  are  accelerated.  Hence,  a  body  projected  upward  is  ascending  for  one  half 
of  the  time  it  is  in  motion,  and  descending  the  other  half. 

ILLUSTRATION  i. — If  a  body  projected  vertically  upwards  return  to  earth  in  12 
seconds,  how  high  did  it  ascend  ? 

The  body  is  half  time  in  ascending.     12  -f-  2  =  6. 

Hence,  by  Rule,  p.  489,  62X  16.083  =  579  feet  =  product  of  square  of  time  and 
16.083. 

2.— If  a  body  is  projected  upward  with  a  velocity  of  96.5  feet  per  second,  it  is 
required  to  ascertain  point  of  body  at  end  of  10  seconds. 

96.5-7-32.166  =  3  seconds,  time  to  acquire  this  velocity,  and  32X  16.083  =  144.75 
feet,  height  body  reached  with  its  initial  velocity. 

Then  10  —  3  =  7  seconds  left  for  body  to  fall  in. 

Hence,  by  Rule,  as  in  preceding  example,  72  X  16.083  =  788.07,  and  788.07  — 
144. 75  =  643. 32  feet  =  distance  below  point  of  projection. 

Or,  io2  X  16.083  =  1608.3  feet,  space  fallen  through  under  the  effect  of  gravity,  and 
96. 5X10  =  965  feet,  space  if  gravity  did  not  act.  Hence  1608. 3  —  965  =  643. 3  feet. 


492  GRAVITATION. 

3.— A  body  is  projected  vertically  with  a  velocity  of  135  feet;  what  velocity  will 
it  have  at  60  feet  ? 

*352-^  64. 33  =  283.3/6^  space  projected  at  that  velocity,  135-7-32.16  =  4.197  sec- 
onds =  time  of  projection,  and  283. 3  —  60  =  223. 3  =  space  to  be  passed  through  after 
attainment  of  60  feet.  Hence,  V 223. 3  X  64.33  =  "9-85  feet  velocity,  and  223.3  +  60 
=  283.3/6^. 

By  Inversion.—  Velocity  119.85.  Hence,  **9'  5  =  223.3  feet  space,  and  283.3  — 
223.3  =  6 


Formvilas  to  Determine  Klements  of  Retarded  Motion. 

7.  »  =  «»  +  —  .          b.  «  =  — 

v  representing  velocity  at  expiration  of  time,  t  any  less  time  than  T,  t'  less  lime  than 
t,  s  space  through  which  a  body  ascends  in  time  t,  V,  T,  S,  and  h  as  in  previous  formu- 
las, page  490. 

ILLUSTRATION.— A  body  projected  upwards  with  a  velocity  of  193  feet  per  second, 
was  arrested  in  5  seconds. 

i  _  o,  t  —  i. 

1.  What  was  its  velocity  when  arrested?    (i.) 

2.  What  was  the  time  of  its  passing  through  562.92  feet  of  space  ?    (8.) 

3.  What  space  had  it  passed  through?    (5.) 

4.  What  was  the  time  of  its  projection,  when  it  had  a  velocity  of  96.5  feet?    (4.) 

5.  What  was  the  height  it  was  projected  in  the  last  second  of  time?    (6.) 


i.  193-^32.166X5  =  32.17/6^.  3.  32.17  +  32.166X5  =  1 

562.92      3*.  166X5  velocit  I9L±»      j£l  =3  seconds. 

52  32.166       32.166 

6.  6-r=l  -.5X32.^66  =  48.25 

7.  S  =  t  v  +  g  t2  -T-  2  =  562.92  feet.  8.  *93  ~     "I7  =  5  seconds. 


GJ-ravity   and    !M!otion   at   an    Inclination. 

If  a  body  freely  descend  at  an  inclination,  as  upon  an  inclined  plane,  by 
force  of  gravity  alone,  the  velocity  acquired  by  it  when  it  arrives  at  ter- 
mination of  inclination  is  that  which  it  would  acquire  by  falling  freely 
through  vertical  height  thereof.  Or,  velocity  is  that  due  to  height  of  in- 
clination of  the  plane. 

Time  occupied  in  making  descent  is  greater  than  that  due  to  height,  in 
ratio  of  length  of  its  inclination,  or  distance  passed,  to  its  height. 

Consequently,  times  of  descending  different  inclinations  or  planes  of  like 
heights  are  to  one  another  as  lengths  of  the  inclinations  or  planes. 

Space  which  a  body  descends  upon  an  inclination,  when  descending  by 
gravity,  is  to  space  it  would  freely  fall  in  same  time  as  height  of  inclination 
is  to  its  length  ;  and  spaces  being  same,  times  will  be  inversely  in  this  pro- 
portion. 

If  a  body  descend  in  a  curve,  it  suffers  no  loss  of  velocity. 

If  two  bodies  begin  to  descend  from  rest,  from  same  point,  one  upon  an  in- 
clined plane,  and  the  other  falling  freely,  their  velocities  at  all  equal  heights 
below  point  of  starting  will  be  equal. 


GRAVITATION.  493 

ILLUSTRATION.—  What  distance  will  a  body  roll  down  an  inclined  plane  300  feet 
long  and  25  feet  high  in  one  second,  by  force  of  gravity  alone? 
As  300  :  25  ::  16.083  •  1.34025/68*. 

Hence,  if  proportion  of  height  to  length  of  above  plane  is  reduced  from  25  to  300 
to  25  to  600,  the  time  required  for  body  to  fall  1.34025  feet  would  be  determined  as 
follows: 

As  25  :  600  ::  1.34025  :  32.166,  and  32.166  =  16.083  X  2  =  twice  time  or  space  in 
which  it  would  fall  freely  required  for  one  half  proportion  of  height  to  length. 

300      600 
Or,  as  -  —  :  —  ::  1.34025  :  32.166,  as  above. 

Impelling  or  accelerating  force  by  gravitation  acting  in  a  direction  paral- 
lel to  an  inclination,  is  less  than  weight  of  body,  in  ratio  of  height  of  in- 
clination to  its  length.  It  is,  therefore,  inversely  in  proportion  to  length  of 
inclination,  when  height  is  the  same. 

Time  of  descent,  under  this  condition,  is  inversely  in  proportion  to  accel- 
erating force. 

If,  for  instance,  length  of  inclination  is  five  times  height,  time  of  making 
freely  descent  at  inclination  by  gravitation  is  five  times  that  in  which  a 
body  would  freely  fall  vertically  through  height  ;  and  impelling  force  down 
inclination  is  .2  of  weight  of  body. 

When  bodies  move  down  inclined  planes,  the  accelerating  force  is  ex- 
pressed by  h  -r-  /,  quotient  of  height  -r-  length  of  plane;  or,  what  is  equivalent 
thereto,  sine  of  inclination  of  plane,  f  .  e.,  sin.  a. 

ILLUSTRATION.—  An  inclined  plane  having  a  height  of  one  half  its  length,  the  space 
fallen  through  in  any  time  would  be  one  half  of  that  which  it  would  fall  freely. 

Velocity  which  a  body  rolling  down  such  a  plane  would  acquire  in  5  seconds  is 
80.416  feet. 

Thus,  32.166  X  5  =  160.833  fee^  an^  an  inclined  plane,  having  a  height  one  half 
of  its  length,  has  an  angle  or  sine  of  30°.  Hence,  sin.  30°  =  .5,  and  160.833  x  .5  = 
Bo.  416  feet. 

Formulas   to  Determine   various   Elements   of  G-ravita* 
tion   on.   an   Inclined.    !Plane. 


=  »qr  gTsin.  a. 


.  a;    =  ^(z  9  S  sin.  a)  ;    =~.          6.    H= 

' 


3-     A  = 

V2 


5.     S  =  VTqr.s  gT2sin.  a.        Or,- 


7  2  g  sin.  a 

v  representing  velocity  of  projection  in  feet  per  second,  S  space  or  vertical  height 
of  velocity,  and  projection,  a  angle  of  inclination  of  plane,  I  length,  and  H  height  of 
plane. 

ILLUSTRATION.  —Assume  elements  of  preceding  illustration.  ¥  =  80.416, 1=5, 
and  H  =  2oi.o4. 

i.    sX32.i66X52X.5  —  zoi.o+feet.  2.  32.166  X  5  X  -5  =  80.416/6^. 


•5X          

If  projected  downward  with  an  initial  velocity  of  16.083  feet  Per  second. 

4.  16.083  +  32.166  X  5X-5  =  96. 5  feet. 

5.  80.4164-16.083  X  5  — -5  X  32-166  X  52  X  .5  =  281.46  feet. 

T  T 


494  GBAVITATIOK. 

ILLUSTRATION.— What  time  will  it  take  for  a  ball  to  roll  38  feet  down  an  inclined 
plane,  the  angle  a=  12°  20',  and  what  velocity  will  it  attain  at  38  feet  from  its  start- 
ing-point? 


X .  2136  =  22. 88  feet  per  second. 

When  a  body  is  projected  upward  it  is  retarded  in  the  same  ratio  that  a 
descending  body  is  accelerated. 

ILLUSTRATION. — If  a  body  is  projected  up  an  inclined  plane  having  a  length  of 
twice  its  height,  at  a  velocity  of  96.5  feet  per  second, 

Then,  T  =  96. 5  -r-  32. 166  =  3  seconds.  S  = .  5  32. 166  x  32  X  •  5  —  72. 375  feet,  v  = 
32. 166  X  3  X  .  5  =  48-  25  feet. 

Inclined   Plane. 

Problems  on  descent  of  bodies  on  inclined  planes  are  soluble  by  formulas 
i  to  9,  page  495,  for  relations  of  accelerating  forces.  As  a  preliminary  step, 
however,  accelerating  force  is  to  be  determined  by  multiplying  weight  of 
descending  body  by  height  of  plane,  and  dividing  product  by  length  of  plane. 

ILLUSTRATION.  —  If  a  body  of  15  Ibs.  weight  gravitate  freely  down  an  inclined 
plane,  length  of  which  is  five  times  height,  accelerating  force  is  15  -r-  5  =  3  Ibs.  If 
length  of  plane  is  ioo  feet  and  height  20,  velocity  acquired  in  falling  freely  from  top 
to  bottom  of  plane  would  be 

v  =  8  V  ^p  =  8  V20  =  35.776/0*. 
Time  occupied  in  making  descent, 

/i5  X  ioo 
t  = .  25  ^ = .  25  V  5°°  =  5-  59  seconds. 

Whereas,  for  a  free  vertical  fall  through  height  of  20  feet,  time  would  be, 
-=  1.118  seconds, 


32.166 
which  is  .2  of  time  of  making  descent  on  inclined  plane. 

Velocities  acquired  by  bodies  in  falling  down  planes  of  like  height  will  all  be 
equal  when  arriving  at  base  of  plane. 

When  Length  of  an  Inclined  Plane  and  Time  of  Free  Descent  are  given. 

RULE.— Divide  square  of  length  by  square  of  time  in  seconds  and  by  16 ; 
the  quotient  is  height  of  inclined  plane. 

EXAMPLE.— Length  of  plane  is  ioo  feet,  and  time  of  descent  is  5.59  seconds;  then 
Tertical  height  of  descent  is 

ioo2  .  , 

=  20  feet. 

5-59   X  16.08 

Accelerated   and   Retarded   Motion. 

If  an  Accelerating  or  Retarding  force  is  greater  than  gravity,  that  is, 
weight  of  the  body,  the  constant,  <jr,  or  32.166,  is  to  be  varied  in  proportion 
thereto,  and  to  do  this  it  is  to  be  multiplied  by  the  accelerating  force,  and 
product  divided  by  weight  of  body. 

Thus,  Let  f  represent  accelerating  force,  and  w  weight  of  body. 

Then,  *±m±t  or  21^/,  Or  l6^3/  become  the  constants. 

The  same  rules  and  formulas  that  have  been  given  for  action  of  gravity  alone 
are  applicable  to  the  action  of  any  other  uniformly  accelerating  or  retarding  force, 
the  numerical  constants  above  given  being  adapted  to  the  force. 


GRAVITATION.  49  5 

Average  "Velocity  of"  a   Moving   Body  -uniformly  Accel- 
erated.  or    Retarded. 

Average  velocity  of  a  moving  body  uniformly  accelerated  or  retarded, 
during  a  given  time  or  in  a  given  space,  is  equal  to  half  sum  of  initial  and 
final  velocities  ;  and  if  body  begin  from  a  state  of  rest  or  arrive  at  a  state  of 
rest,  its  average  speed  is  half  the  final  or  initial  velocity,  as  the  case  may  be. 

Thus,  in  example  of  a  ball  rolling,  initial  speed  or  velocity  is,  in  either 
case,  60  feet  per  second,  and  terminal  speed  is  nothing  ;  average  speed  is 

therefore  —  —  ,  namely,  one  half  of  that,  or  30  feet  per  second. 

When  a  cannon-ball  is  projected  at  an  angle  to  horizon,  there  are  two  forces  act- 
ing on  it  at  same  time—  viz.,  force  of  charge,  which  propels  it  uniformly  in  a  right 
line,  and  force  of  gravity,  which  causes  it  to  fall  from  a  right  line  with  an  accel- 
erated motion;  these  two  motions  (uniform  and  accelerated)  cause  the  ball  to  move 
in  the  curved  line  of  a  Parabola. 

Formulas  for  Flight  of  a  Cannon-ball. 
/P  w  V2 


Va  sin.  a,  cos.  a  V  sin.  a  Va 

-  '  -  ;    t  =  -  :    h  = 


sin.8  a 


9  9*9 

w  representing  weight  of  ball  and  P  of  powder  in  Ibs.  ;  t  time  of  flight  in  seconds; 

b  horizontal  range,  and  h  vertical  height  of  range  of  projection  of  ball  in  feet. 
ILLUSTRATION.—  A  cannon  loaded  to  give  a  ball  a  velocity  of  900  feet  per  second, 

the  angle  a  ==  45°;  what  is  horizontal  range,  the  time  t  and  height  of  range  fc? 


, 

32.166  32.166 

7071  =        Bsecond        fe  =  9oo»  X.  7071'^       feet 


=  =  = 

32.166  2X32.166 

NOTE.—  As  distance  b  will  be  greatest  when  angle  0  =  45°,  product  of  sine  and 
cosine  is  greatest  for  that  angle.    Sin.  45°  X  cos.  45°=  5. 
*4  Ib.  ball  with  a  velocity  of  2000  feet  per  second  at  45°  range  7300  feet 

G-eneral    Formulas    for    Accelerating    and    Retarding 
Forces. 


V      9ft  <*      '  , 

v  =  —  —  .  a.    s  =        •  •    '.  3.    t  =  —  —.  4.    s  = 

w  w  gf 


'•  /*         8-  '=         »  -=- 

NOTE  i.—  When  accelerating  or  retarding  force  bears  a  simple  ratio  to  weight  of 
body,  the  ratio  may,  for  facility  of  calculation,  be  substituted  in  the  quantities  rep- 
resenting modified  constants,  for  force  and  weight.  Thus,  if  accelerating  force  is  a 

tenth  part  of  weight,  then  ratio  is  i  to  10,  and  32jl66  =  3.2166:  or,  l6'°83  =  1.6083, 


and  — -  —  —  6.4333;  and  these  quotients  may  be  substituted  for  16.083,  32.166,  and 

64.333  respectively,  in  formulas  for  action  of  gravity  i  tog,  to  fit  them  for  computa- 
tion in  an  accelerating  or  retarding  force  one-tenth  of  gravity 

2. — Table,  page  488,  giving  relations  of  velocity  and  height  of  falling  bodies,  may 
be  employed  in  solving  questions  of  accelerating  force  general. 

EXAMPLE.— A  ball  weighing  10  Ibs.  is  projected  with  an  initial  velocity  of  60  feel 
per  second  on  a  level  plane,  and  frictional  resistance  to  its  motion  is  i  Ib.    What  dis- 
tance will  it  traverse  before  it  comes  to  a  state  of  rest  ?    By  formula  4 : 
io  Ibs.  x  6o» 
64-333  X  i  Ib. : 


496 


GRAVITATION. 


Again,  same  result  may  be  arrived  at,  according  to  Note  i,  by  multiplying  con- 
stant 64. 333,  in  Rule,  page  494,  for  gravity,  by  ratio  of  force  and  weight,  which  in 
this  case  is  ^,  and  64.333  X  1^  =  6.4333.  Substituting  6.4333  for  64-333  in  tna* 
rule,  formula  becomes 

"6.4333  "6.4333" 

The  question  may  be  answered  more  directly  by  aid  of  table  for  falling  bodies, 
page  488.  Height  due  to  a  velocity  of  60  feet  per  second,  is  55.9  feet;  which  is  to 
be  multiplied  by  inverse  ratio  of  accelerating  force  and  weight  of  body,  or  ££-,  or  10; 

thatis>  55-9  X  10  =  559 /«*• 

If  the  question  is  put  otherwise— What  space  will  a  weight  move  over  before  it 
comes  to  a  state  of  rest,  with  an  initial  velocity  of  60  feet  per  second,  allowing  fric- 
tion to  be  one  tenth  weight?  The  answer  is  that  friction,  which  is  retarding  force, 
being  one  tenth  of  weight,  or  of  gravity,  space  described  will  be  10  times  as  great  as 
is  necessary  for  gravity,  supposing  the  weight  to  be  projected  vertically  upwards  to 
bring  it  to  a  state  of  rest.  The  height  due  to  velocity  being  55.9  feet;  then 
55. 9  X  10  =  559  feet. 

Average  velocity  of  a  moving  body,  uniformly  accelerated  or  retarded  during  a 
given  period  or  space,  is  equal  to  half  sum  of  initial  and  final  velocities. 

To  Compute  "Velocity  of  a  Falling  Stream  of  Water  per 
Second   at   End   of  any  given   Time. 

When  Pei*pendicular  Distance  is  given. 

EXAMPLE.— What  is  the  distance  a  stream  of  water  will  descend  on  an  inclined 
plane  10  feet  high,  and  100  feet  long  at  base,  in  5  seconds? 

5  '2  X  16. 083  =  402.08  feet  =  space  a  body  will  freely  fall  in  this  time. 

Then,  as  100  :  10  ::  402.08  :  40.21  feet  = proportionate  velocity  on  a  plane  of  these 
dimensions  to  velocity  when  falling  freely. 

Miscellaneous   Illustrations. 

x.— What  is  the  space  descended  vertically  by  a  falling  body  in  7  seconda 
S  =  .5gxt2.    Then  16.083  X  72  —  788.067  feet. 

2.— What  is  the  time  of  a  falling  body  descending  400  feet,  and  velocity  acquired 
at  end  of  that  time? 

t  =  - .    Then       °'4  =  4.08  sec.     v  =  Vz  g  X  S.    Then  i 
9  3*166 

3.— If  a  drop  of  rain  fall  through  176  feet  in  last  second  of  its  fall,  how  high  was 
the  cloud  from  which  it  fell? 

h*  _    i762 


4.— If  two  weights,  one  of  5  Ibs.  and  one  of  3,  hanging  freely  over  a  sheave,  are 
set  free,  how  far  will  heavier  one  descend  or  lighter  one  rise  in  4  seconds. 

^-j^X  16.083  X42  =  ^X  257. 328  =  64.33/66*. 

5. — If  length  of  an  inclined  plane  is  100  feet,  and  time  of  descent  of  a  body  is  6 
seconds,  what  is  vertical  height  of  plane  or  space  fallen  through  ? 

IOO2  IOOOO 

— — = =  i7. 27  feet. 

62X-5P        579 

6.— If  a  bullet  is  projected  vertically  with  a  velocity  of  135  feet  per  second,  what 
velocity  will  it  have  at  60  feet? 

Formula  9,  page  492.         TS'-X/S^  ~~ 


GUNNERY.  497 

GUNNERY. 

A  heavy  body  impelled  by  a  force  of  projection  describes  in  its  flight 
or  track  a  parabola,  parameter  of  which  is  four  times  height  due  to 
velocity  of  projection. 

Velocity  of  a  shot  projected  from  a  gun  varies  as  square  root  of 
charge  directly,  and  as  square  root  of  weight  of  shot  reciprocally. 
To   Compute   Velocity  of  a   Shot  or    Shell. 

RULE.— Multiply  square  root  of  triple  weight  of  powder  in  Ibs.  by  1600; 
divide  product  by  square  root  of  weight  of  shot ;  and  quotient  will  give  ve- 
locity in  feet  per  second. 

EXAMPLE.— What  is  velocity  of  a  shot  of  196  Ibs.,  projected  with  a  charge  of  9  Ibs. 
of  powder? 

A/9  X  3  X  1600  -4-  >/I96  =  8320  -*•  14  =  594-  3  feet. 
To  Compute  Range  for  a  Charge,  or  Charge  for  a  Range. 

When  Range  for  a  Charge  is  given.— Ranges  have  same  proportion  as 
charges  of  powder ;  that  is,  as  one  range  is  to  its  charge,  so  is  any  other 
range  to  its  charge,  elevation  of  gun  being  same  in  both  cases.  Consequently, 

To   Compute    Range. 

RULE. — Multiply  range  determined  by  charge  in  Ibs.  for  range  required, 
divide  product  by  given  charge,  and  quotient  will  give  range  required. 

EXAMPLE.  —If,  with  a  charge  of  9  Ibs.  of  powder,  a  shot  ranges  4000  feet,  how  far 
will  a  charge  of  6.75  Ibs.  project  same  shot  at  same  elevation? 
4000  x  6.75 -7-9  =  3000  feet. 

To   Compute   Charge. 

RULE. — Multiply  given  range  by  charge  in  Ibs.  for  range  determined, 

divide  product  by  range  determined,  and  quotient  will  give  charge  required. 

EXAMPLE.— If  required  range  of  a  shot  is  3000  feet,  and  charge  for  a  range  of  4000 

feet  has  been  determined  to  be  9  Ibs.  of  powder,  what  is  charge  required  to  project 

same  shot  at  same  elevation  ? 

3000  X  9  -T-  4000  =  6. 75  Ibs. 

To  Compute  Range  at  one   Elevation,  when   Range  for 

another   is   given. 

RULE.— As  sine  of  double  first  elevation  in  degrees  is  to  its  range,  so  is 
sine  of  double  another  elevation  to  its  range. 

EXAMPLE.— If  a  shot  range  1000  yards  when  projected  at  an  elevation  of  45°,  how 
tar  will  it  range  when  elevation  is  30°  16',  charge  of  powder  being  same  ? 

Sine  of  45°  X  2  — 100  ooo ;  sine  of  30°  16'  X  2  =  87  064. 
Then,  as  100  ooo  :  1000  : :  87  064  :  870. 64  feet 

To  Compute   Elevation    at   one    Range,  when   Elevation 

for    another   is   given. 

RULE. — As  range  for  first  elevation  is  to  sine  of  double  its  elevation,  so 
is  range  for  elevation  required  to  sine  for  double  its  elevation. 

EXAMPLE.— If  range  of  a  shell  at  45°  elevation  is  3750  feet,  at  what  elevation 
must  a  gun  be  set  for  a  shell  to  range  2810  feet  with  a  like  charge  of  powder? 

Sine  of  45°  X  2  =  100000. 
Then,  as  3750  :  100000  ::  2810  :  74933  =  sine  for  double  elevation  —  24°  16'. 

Approximate  Rule  for  Time  of  Flight. 

Under  4000  yards,  velocity  of  projectile  900  feet  in  one  second ;  under 
6000  yards,  velocity  800  feet ;  and  over  6000  yards,  velocity  700  feet. 

Guns  and  Howitzers  take  their  denomination  from  weights  of  their  solid 
shot  in  round  numbers,  up  to  the  42-pounder ;  larger  pieces,  rifled  guns,  and 
mortars,  from  diameter  of  their  bore. 

TT* 


498 


GUNNERY. 


Initial  Velocity    and.    Ranges    of  Shot    and.    Shells. 

The  Range  of  a  shot  or  shell  is  the  distance  of  its  first  graze  upon  a  horizontal 
plane,  the  piece  mounted  upon  its  proper  carriage. 


Project 
Description. 

le. 

Weight. 

Powder. 

Initial 
Velocity. 

Time  of 
Flight. 

Eleva- 
tion. 

Range. 

Grains. 

Grains. 

Feet. 

Seconds. 

0           / 

Yards. 

Elongated,  j     err* 

60 

06? 

Round. 

412 

no 

yuj 
1500 

Lbs. 

LbB. 

«( 

6.15 

1.25 

e 

1523 

I2       "      

i 

12.3 

2.  5 

1  8~2~6 

1.75 

I 

575 

i 

24.25 

65 

1870 

2 

1147 

t 

32.3 

8 

,' 
1040 



I 

7*3 

< 

42.5 
65 

10.5 

IO 

14.19 

5 
15 

1955 
3224 

8  inch  Columbiad... 

10      "                   " 

M 

127.5 

15 

— 

14.32 

15 

3281 

10    "     Mortar.  

Shell. 

98 

IO 

36 

45 

4250 

200 
302 

20 
40 

— 

45 
7 

1948 

15    "     Columbiad... 

15    "            " 

« 

315 

50 

— 

23-29 

25 

4680 

RIFLED. 

io-pounder  Parrott.  . 

" 

9-75 

X 

— 

21 

20 

5000 

20 

u 

i9 

2 

— 

I7-25 

15 

4400 

30 

it 

29 

3-25 

— 

27 

25 

6700 

100 

Elongated. 

100 

10 

— 

29 

25 

6910 

100 

Shell. 

101 

10 

1250 

28 

25 

6820 

2OO 

" 

150 

16 



4 

2200 

1  2-  inch  Rodman  

" 

50 

"54 

5-5 

40 

Hall's  Rockets.  .  . 

vinch. 

16 





47 

1720 

Penetration   of  Shot   and    Shell. 

Experiments  at  Fort  Monroe,  1839,  and  at  West  Point,  1853. 


® 

Mean  Penetration. 

» 

Mean  Penetration. 

ORDNANCB. 

1 

6 

5 

i^l 

1| 

5 

ORDNANCE. 

1 

1 

$ 

|| 

Granite. 

Lbs. 

Yds; 

Ins. 

Ins. 

Ins. 

Lbs. 

Yds. 

Ins. 

Ins. 

Ins. 

32  Lbs.  Shot. 

8 

880 

I5-25 

3-5 

8-inch  Howitz.* 

6 

880 

8-5 

I 

32     "        " 

II 

IOO 

60 

8    "  Columb.if 

12 

200 



— 

42     «       « 
42    "  Shell. 

10.5 
7 

100 
IOO 

54-75 
40-75 

18 

4 

10      "               "          t 
10       "              "         * 

18 
18 

114 
IOO 

63-5 
56.75 

44 

7-75 

1 24  ins.  of  Concrete. 


*  Shell. 


t  Shot. 


Solid  shot  broke  against  granite,  but  not  against  freestone  or  brick,  and  general 
effect  is  less  upon  brick  than  upon  granite. 

Shells  broke  into  small  fragments  against  each  of  the  three  materials. 
Penetration  in  earth  of  shell  from  a  io-inch  Columbiad  was  33  ins. 

Experiments —  England.     (Holley. ) 


ORDNANCE. 

Charge. 

Projectile. 

Weight. 

Velocity. 

Range. 

Target  and  Effects. 

Lbs. 

Lbs. 

Feet. 

Yards. 

n-inch  U.  S.  Navy. 

3<> 

Shot. 

169 

1400 

50 

Iron  plates,  14  ins. 

—  loosened. 

i5-inch  Rodman... 

60 

" 

400 

1480 

50 

Iron  plates,  6  ins.— 

destroyed. 

RIFLED. 

7-inch  Whitworth.  . 

25 

Shot. 

ISO 

1241 

200 

Inglis'st—  destr'd. 

io-5-inch  Armstrong 

45 

" 

3°7 

1228 

200 

u                     a 

i3-inch           " 

90 

*      u 

344-5 

1760 

200 

Sol  id  plates,  n  ins. 

thick—  destr'd. 

*  Steel.  f  8-inch  vertical  and  5-inch  horizontal  slabs,  and  7-inch  vertical  and  $-in.  horizontal 

slabs,  9X5  ins.  ribs  and  3-inch  ribs. 


GCNNBEY. 


499 


Elements  of  Report  of  Board  of  Engineers  for  Fortifications,  U.  S.  A. 
Professional  Papers  No.  25.    (Brev.  Maj.-Gen,  Z.  B.  Tower.) 

Experimental  firings  for  penetration  during  the  past  twenty  years  have 
determined  that  wrought  iron  and  cast  iron,  unless  chilled,  are  unsuitable  for 
projectiles  to  be  used  against  iron  armor ;  that  the  best  material  for  that 
purpose  is  hammered  steel  or  Whitworth's  compressed  steel. 

2.  That  cast-iron  and  cast-steel  armor-plates  will  break  up  under  the  im- 
pact of  the  heaviest  projectiles  now  in  service,  unless  made  so  thick  as  to 
exclude  their  use  in  ship-protection. 

3.  That  wrought-iron  plates  have  been  so  perfected  that  they  do  not  break 
up,  but  are  penetrated  by  displacement  or  crowding  aside  of  the  material  hi 
the  path  of  the  shot,  the  rate  of  penetration  bearing  an  approximately  deter- 
mined ratio  to  the  striking  energy  of  the  projectile,  measured  per  inch  of 
shot's  circumference,  as  expressed  by  the  following  formula : 


2.035  / 

vil 


V2P 


—  =  penetration  in  ins.    V  representing  velocity  in 
n  X  2240  X- 86 

feet  per  second,  P  weight  of  shot  in  Ibs.,  and  r  radius  of  shot  in  ins. 

That  such  plates  can  therefore  be  safely  used  in  ship  construction,  their 
thickness  being  determined  by  the  limit  of  flotation  and  the  protection 
needed. 

4.  That,  though  experiments  with  wrought-iron  plates,  faced  with  steel, 
have  not  been  sufficiently  extended  to  determine  the  best  combination  of 
these  two  materials,  we  may  nevertheless  assume  that  they  give  a  resistance 
of  about  one  fourth  greater* than  those  of  homogenous  iron. 

5.  That  hammered  steel  in  the  late  Spezzia  trials  proved  superior  to  any 
other  material  hitherto  tested  for  armor-plates.    The  icj-mch  plate  resisted 
penetration,  and  was  only  partially  broken  up  by  4  shots,  three  of  which  had 
a  striking  energy  of  between  33  ooo  and  34  ooo  foot-tons  each.     Not  one  shot 
penetrated  the  plate.    Those  of  chilled  iron  were  broken  up,  and  the  steel 
projectile,  though  of  excellent  quality,  was  set  up  to  about  two  thirds  of  its 
length. 

Velocity  and  [Ranges  of  Shot.    (Krupp's  Ballistic  Tables.) 
Penetration,   in   "Wrought   Iron. 


•  / 

tration  in  i 

elocity 
Range. 

3000     6000 

ns.     G 

at 
Muzzle 

ration 
Range 
3000 

6000 

22.04 
23-47 
21-35 
21.89 
12.14 
5-17 

V2 

GUN. 

0X2 

Cali- 
ber. 

rnX2'< 
Powder. 

540  X( 

Shot. 

j—^/CTO 

V 

at 
Muzzle 
per  Sec. 

—  2.53 

Penet 
600 

Tons. 
Armstrong,  100 

Woolwich,     8  1 
Krupp,          71 
"               18 
U.S.*  8-inch.. 

Ins. 
17-75 
17-75 
16 

15-75 

Vs 

Lbs. 

552 
776 

445 
485 
165 
35 

Lbs. 

2O2  2 
2000 
1760 
1715 

474 
180 

Feet. 
1715 
1832 

1657 
1703 

1450 

Yds. 
1424 
1518 
1393 
H34 
1351 
1036 

Yds. 
1191 
1259 
1181 

I2II 

1113 

840 

Ins. 

34-76 
37-52 
32.6 
33-52 
20.42 
10.23 

Ins. 
33-2 
35-8i 
31-23 
32.12 

I9-3I 
9.22 

Ins. 

27-55 
29.66 
26.24 
27.04 
15.46 
6.72 

*  Unchambered. 

Target— For  loo-ton  gun,  steel  plate  22  ins.  thick,  backed  with  28.8  ins.  of  wood, 
2  wrought-iron  plates  1.5  ins.  thick,  and  the  frame  of  a  vessel. 

Effect.— Total  destruction  of  steel  plate,  and  backing  entered  to  a  depth  of  22  ina, 
but  not  perforated. 


500 


GUNNERY. 


Summary  of  Record,  of"  Practice  in  Enrope  with  Heavy 
A.rm  strong,  \Vool\vich,  and.    Krvipp    Grtins. 

Board  of  Engineers  for  Fortifications,  U.  S.  A.,Professional  Papers  No.  25. 


^ 

Energy 

•g  . 

o« 

ii 

. 

•si 

GUN. 

Powder. 

Projectile. 

fl 

II 

]f 

1^    * 

u!^£ 

ARMSTRONG,       "] 
loo  Tons,  caliber    1 

i.  5-  inch  cubes.  . 
Waltham  Abbey 

Shot  

Lbs. 
330 
375 

Lbs. 

2OOO 
2000 

Feet. 
1446 
1543 

Ft.-tons. 
28990 
33000 

Foot-ton 
544-Of 
623 

17  ins.,  bore  30.5    f 
feet. 

400 
776 

2000 
2OOO 

1502 
1832 

31282 
46580 

835.3: 

WOOLWICH,  81      "1 

•  75-inch  cubes. 

//'-' 
170 

1258 

1393 

16922 

371-5 

Tons,  caliber  14.5    V 

1.5      "        " 

220 

1450 

1440 

20842 

457-57 

ins.  ,  bore  24  feet.  J 

2              "             " 

'     .... 

250 

1260 

I523 

20259 

444-7* 

caliber  16  ins  

1-5         "             " 

' 

310 

I466 

1553 

24508 

520.4 

38  Tons,          ] 

1.5         "             " 

Pall,  shell 

130 

800 

1451 

11668 

297.6,! 

caliber  12.5  ins.,    \- 

15         "             " 

11 

2OO 

800 

1421 

II  210 

285.4 

bore  16.5  feet.     J 

1.5      "         " 

" 

1  80 

800 

1504 

12545 

3!9-4 

KRUPP,  71  Tons,    ] 
caliber  15.75  ins.,  \- 
bore  28.  58  feet.    J 

Prism  A  
u     H 

Plain  .  .  . 
Shrapnel 
Shell.  .  .  . 

298 
485 
441 

1707 
1419 

1184 

1703 
1761 

16602 
34503 
30484 

335-4- 
^97-9 
616.1^ 

"     2  inch... 

1  8  Tons,          1 

"     i  hole... 

Plain  .  .  . 

132 

300 

1873 

7298 

246.0 

caliber  9.45  ins.,    V 
bore  17.5  feet.      J 

"     2  inch..  . 

Shrapnel 
Shell.  .  .  . 

145 

474 
300 

1688 
1991 

9307 
8244 

3i5-6( 

277.6( 

Penetration  in  Ball  Cartridge  Paper,  No.  i. 

Musket,  with  134  grains,  at  13.3  yards 653  sheets. 

Common  rifle,  92  grains,  at  13.3  yards 500  sheets. 

Penetration,    of  Lead.    Balls    in    Small   Arms. 
Experiments  at  Washington  Arsenal  in  1839,  and  at  West  Point  in  1837. 


ARM. 

Diameter 
of  Ball. 

Charge 
Powder. 

Distance. 

Weight 
of  Ball. 

Peneti 
White  Oak. 

ation. 

White  Pine, 

Musket  

Inch. 
H4 
(-64 

j-5775 

•5775 

•5775 
•5775 
•55 
•55 

Grains. 
134 
144 
IOO 

92 

IOO 

70 
70 

80 

90* 

IOO* 

51 

60 
70 
40 

60 

55 

Yards. 
9 
5 
5 
9 
5 
9 
5 
5 
5 
5 
5 

200 
200 
200 
2OO 
30 
30 

397-5 
397-5 
219 

219 
219 

219 

219 
500 
730 
500 
450 
463 
350 

Ins. 

1.6 

3 
2.05 
1.8 

2 

.6 

Ji 

i.i 

1.2 

•725 

Ina. 

II 
10.5 
9-33 
5-75 
7.17 
6.15 

Common  Rifle  

Hall's  rifle  

Hall's  carbine,  musket 
caliber  

Pistol     

Rifle  musket  

Altered  musket 

Rifle,  Harper's  Ferry.  . 
Pistol  carbine              . 

Sharpens  carbine  
Burnside's    "      

*  Charges  too  great  for  service. 


Musket  discharged  at  9  yards  distance,  with  a  charge  of  134  grains,  i  ball  and  3 
buckshot,  gave  for  ball  a  penetration  of  1.15  ins.,  buckshot,  .41  inch. 


GUNNERY. 


501 


Loss    of  Force    toy    "Windage. 

A  comparison  of  results  shows  that  4  Ibs.  of  powder  give  to  a  ball  without  wind 
age  nearly  as  great  a  velocity  as  is  given  by  6  Ibs.  having  .14  inch  windage,  which 
is  true  windage  of  a  24-lb.  ball;  or,  in  other  words,  this  windage  causes  a  loss  of 
nearly  one  third  of  force  of  charge. 

Vents.  —  Experiments  show  that  loss  of  force  by  escape  of  gas  from  vent 
of  a  gun  is  altogether  inconsiderable  when  compared  with  whole  force  of 
charge. 

Diameter  of  Vent  in  U.  S.  Ordnance  is  in  all  cases  .2  inch. 


Effect  of  different  Waddings  with  a  Charge  of  77  Grains  of  Powder. 


WAD. 

Velocity  of  Ball 
per  Second. 

Feet. 

i  felt  wad  upon  powder  and  i  upon  ball  

I377 

1482 

i  elastic  wad  upon  powder  and  i  upon  ball  

IIOO 

Felt  wads  cut  from  body  of  a  hat,  weight  3  grains. 
Pasteboard  wads  .1  of  an  inch  thick,  weight  8  grains. 
Cartridge  paper  3X4-5  ins.,  weight  12.82  grains. 

Elastic  wads,  "Baldwin's  indented,"  a  little  more  than  .1  of  an  inch  thick, 
weight  5.127  grains. 

Most  advantageous  wads  are  those  made  of  thick  pasteboard,  or  of  or- 
dinary cartridge  paper. 

In  service  of  cannon,  heavy  wads  over  ball  are  in  all  respects  injurious. 

For  purpose  of  retaining  the  ball  in  its  place,  light  grommets  should  be  used. 

On  the  other  hand,  it  is  of  great  importance,  and  especially  so  in  use  of  small 
arms,  that  there  should  be  a  good  wad  over  powder  for  developing  full  force  of 
charge,  unless,  as  in  the  rifle,  the  ball  has  but  very  little  windage.  (Capt.  Mordecai.) 

Weight   and    Dimensions    of  I.^ead    Balls. 
Number  of  Balls  in  a  Lb.,from  1.67  to  .237  of  an  Inch  Diameter. 


Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Ins. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

1.67 

i 

•75 

ii 

•57 

25 

•  388 

80 

301 

170 

•259 

270 

1.326 

2 

•73 

12 

•537 

30 

•375 

88 

295 

180 

256 

280 

I-I57 

3 

•71 

13 

•5i 

35 

•372 

90 

29 

190 

.252 

290 

1.051 

4 

•693 

14 

•SOS 

36 

•359 

100 

285 

200 

249 

300 

•977 
.919 

.677 
.662 

\l 

.488 
.469 

40 
45 

.348 
•338 

no 

120 

281 
276 

2IO 
220 

247 
244 

310 
320 

•8?3 

7 

.65 

17 

•453 

50 

•329 

130 

272 

230 

.242 

330 

•835 

8 

•637 

18 

.426 

60 

.321 

I40 

268 

240 

•239 

340 

.802 

9 

.625 

19 

•405 

7° 

•3H 

150 

265 

250 

•237 

350 

•775 

10 

.615 

20 

•395 

75 

•307 

160 

262 

260 

Heated  shot  do  not  return  to  their  original  dimensions  upon  cooling,  but  retain 
a  permanent  enlargement  of  about  .02  per  cent,  in  volume. 

Number  of  Pellets  in  an  Ounce  of  Lead  Shot  of  the  different  Sizes. 
B 


A  A 40 

A 50 

BB 58 


No. 


^0-3 i35 

4 i77 

5 218 

No.  14 


No.  6 

7 


Xo.  9 . . 
10. . 


,  1726 
,  2140 


502 


GUNNERY. 


Proportion  of"  Po'wd.er  to  Shot  for  following 
of  Snot. 


No. 

Shot. 

Powder. 

No. 

Shot. 

Powd«r. 

No. 

Shot. 

Powder. 

3 

Oi. 

2 

i-75 

Drains. 
1.625 

4 
5 

Oi. 
i-5 
1-375 

Drams. 
I-875 
2.125 

6 
7 

Oi. 
1.25 
1-125 

Drams. 
2-375 

2.625 

NOTE.— 2  oz.  of  No.  2  shot,  with  1.5  drams  of  powder,  produced  greatest  effect. 
Increase  of  powder  for  greater  number  of  pellets  is  in  consequence  of  increased 
friction  of  their  projection. 

Numbers  of  Percussion  Caps  corresponding  with  Birmingham,  Numbers. 


Eley's. , 


Birmingham.. 


43l44l46 


48 


49 


18 


51  and  52  |  53  and  54  |  55  and  56 


Where  there  are  two  numbers  of  Birmingham  sizes  corresponding  with  only  one 
of  Eley's,  it  is  in  consequence  of  two  numbers  being  of  same  size,  varying  only  in 
length  of  caps. 

Comparison   of  Force   of  a   Charge   in  various  Arms. 


ABM. 

Lock. 

Powder, 

AS- 

Windage. 

Weight 
of  Ball. 

Velocity. 

Percussion 

Grains. 

Inch. 

Grains. 

Feet. 

219 

Hall's  rifle  

Flint 

Percussion 

219 

1687 

Cadet's  musket  

Flint. 

7O 

OAK 

2IQ 

Pistol... 

Percussion. 

•JS 

OI* 

2l8.< 

OA1 

Ranges  for  Small  Arms. 

Musket. — With  a  ball  of  17  to  pound,  and  a  charge  of  no  grains  of  powder,  etc., 
an  elevation  of  36'  is  required  for  a  range  of  200  yards;  and  for  a  range  of  500 
yards,  an  elevation  of  3°  30'  is  necessary,  and  at  this  distance  a  ball  will  pass  through 
a  pine  board  i  inch  in  thickness. 

jRt/Ze.— With  a  charge  of  70  grains,  an  effective  range  of  from  300  to  350  yards  is 
obtained;  but  as  75  grains  can  be  used  without  stripping  the  ball,  it  is  deemed  better 
to  use  it,  to  allow  for  accidental  loss,  deterioration  of  powder,  etc. 

Pistol.— Wiib  a  charge  of  30  grains,  the  ball  is  projected  through  a  pine  board 
i  inch  in  thickness  at  a  distance  of  80  yards. 


GKmp  owder . 

Gunpowder  is  distinguished  as  Musket,  Mortar,  Cannon,  Mammoth,  and 
Sporting  powder ;  it  is  all  made  in  same  manner,  of  same  proportions  of 
materials,  and  differs  only  in  size  of  its  grain. 

Bursting  or  Explosive  Energy.— By  the  experiments  of  Captain  Rodman,  U.  S. 
Ordnance  Corps,  a  pressure  of  45  ooo  Ibs.  per  square  inch  was  obtained  with  10  Ibs. 
of  powder,  and  a  ball  of  43  Ibs. 

Also,  a  pressure  of  185000  Ibs.  per  sq.  inch  was  obtained  when  the  powder  was 
burned  in  its  own  volume,  in  a  cast-iron  shell  having  diameters  of  3.85  and  12  ins. 

Proof  of  IPowder.    (V.  S.  Ordnance  Manual) 

Powder  in  magazines  that  does  not  range  over  180  yards  is  held  to  be  unservice- 
able. 

Good  powder  averages  from  280  to  300  yards;  smaU  grain,  from  300  to  320  yards. 

Restoring  Unserviceable  Powder.  —  When  powder  has  been  damaged  by  being 
stored  in  damp  places,  it  loses  its  strength,  and  requires  to  be  worked  over.  If 
quantity  of  moisture  absorbed  does  not  exceed  7  per  cent.,  it  is  sufficient  to  dry  it 
to  restore  it  for  service.  This  is  done  by  exposing  it  to  the  sun. 

When  powder  has  absorbed  more  than  7  per  cent,  of  water  it  should  be  sent  to  a 
powder  mill  to  be  worked  over. 


GUNNEEY. 


503 


^Properties  i 
1C 

24-Po 
Weight  of  ball  an 
44       *4  powder 
Windage  of  ball 

and    Res 
xperime 

[JNDER  GUN. 

d  wad.  ...  2 

i 

alts 

tits. 

3-i35  i 

311. 

Sul- 
phur. 

of  < 

(Cat 

nch. 

1 

\ 

3-nnpowc 

tain  A.  Mord 

M 

Weight  of  ba 
"  pc 
Windage  of  1 

lanufacture. 
Vhere  from. 

ler,  detern 

ecai,  U.  S.  A.) 

USKET  PENDULI 
til  

ainec 

UM. 
397-5 

1  iDy 

grains. 

iach. 

-8 

)ail     .     

Water  ab- 
sorbed by  ex- 
posure to  Air.  ^ 

GRAIN. 

c 

Salt- 
petre. 

jmpositi 

Char- 
coal. 

i--i 

Relative 
Quickness  of 
Burning. 

Cannon,  large.  .  . 
"       small.  .  . 
Musket  
Rifle  

76 

i 

77 
70 

76 

75 

Slazed. 

'4 
"•S 

13 
15 

15 
15 

12 

12.5 

10    \ 
15    ) 

9 

10 

*  Dupont's  Mills, 
Del. 

t  Dupont's  Mills, 
Del. 
*  Dupont's  Mills, 
Del. 
Loomis,  Hazard, 
&  Co.,  Conn.* 
Waltham  Abbey, 
England.* 

"34 
6174 

5344 
1642 

13  '52 

166 
103 
72808 
295 
2378 

11600 
t  E 

275 
3H 
214 

282 

182 
100 
212 
204 

ough. 

Per  c't. 
2.77 
3-35 

3-55 

2.09 
1.91 

4.42 

.677 

:'S 
.834 

•943 
.78§ 
.756 

i 

.82 
.888 

.865 

Rifle  

Musket  

Rifle  

Cannon,  uneven. 
"       large  .  .  . 

Blasting,  uneven 
Rifle  

Rifle  

*  < 

Manufacture  of  Powder. — Powder  of  greatest  force,  whether  for  cannon  or  small 
arms,  is  produced  by  incorporation  in  the  "cylinder  mills." 

Effect  of  Size  of  Grain.—  Within  limits  of  difference  in  size  of  grain,  which  occurs 
in  ordinary  cannon  powder,  the  granulation  appears  to  exercise  but  little  influence 
upon  force  of  it,  unless  grain  be  exceedingly  dense  and  hard. 

Effect  of  Glazing.  —  Glazing  is  favorable  to  production  of  greatest  force,  and  to 
quick  combustion  of  grains,  by  affording  a  rapid  transmission  of  flame  through 
mass  of  the  powder. 

Effect  of  using  Percussion  Primers.  —Increase  of  force  by  use  of  primers,  which 
nearly  closes  vent,  is  constant  and  appreciable  in  amount,  yet  not  of  sufficient  value 
to  authorize  a  reduction  of  charge. 

Ratio  of  Relative  Strength  of  different  Powders  for  use  under  water  differ 
but  little  from  the  reciprocal  of  the  ratio  between  the  sizes  of  the  grain*, 
showing  that  the  strength  is  nearly  inversely  proportional  thereto.* 
Mammoth,  .08;    Oliver,  .09;   Cannon,  .18;   Mortar,  i;    Musket,  1.57; 
Sporting  2.61,  and  Safety  Compound  30.62. 
Du.alin  is  nitro-glycerine  absorbed  by  Schultze's  powder. 
For  other  powders  and  explosive  materials  see  Blasting,  p.  443. 

Heat   and   Explosive   JPcrwer.     (Capt  Noble  and  F.  A.  Abel.) 
One  gram  of  fired  powder  evolves  a  mean  temperature  of  730°.    Temper- 
ature of  explosion  3970°.    Volume  of  permanent  gas  (which  is  in  an  in- 
verse ratio  to  units  of  heat  evolved)  at  32°  =250. 

The  explosive  power  of  powder,  as  tested  in  Ordnance,  ranges,  for  volumes 
of  expansion  of  1.5  to  50  times,  from  36  to  170  foot-tons  per  Ib.  burned. 

A  charge  of  70  Ibs.  gave  to  an  180  Ibs.  shot  a  velocity  of  1694  feet  per 
second,  equal  to  a  total  energy  of  3637  foot-tons,  and  a  charge  of  100  Ibs. 
gave  a  velocity  of  2182  feet,  and  an  energy  of  5940 foot-tons. 

*  Report  of  Experiments  and  Investigation*  to  develop  a  system  of  submarine  mine*.  Professional 
Papers,  U.  S.  E.,  No.  23. 


504  HEAT. 

HEAT. 

Heat,  alike  to  gravity,  is  a  universal  force,  and  is  referred  to  both  as 
Cause  and  effect. 

Caloric  is  usually  treated  of  as  a  material  substance,  though  its  claims 
to  this  distinction  are  not  decided ;  the  strongest  argument  in  favor  of 
this  position  is  that  of  its  power  of  radiation.  Upon  touching  a  body 
having  a  higher  temperature  than  our  own,  caloric  passes  from  it,  and 
excites  the  feeling  of  warmth ;  and  when  we  touch  a  body  having  a 
lower  temperature  than  our  own,  caloric  passes  from  our  body  to  it,  and 
thus  arises  the  sensation  of  cold. 

To  avoid  any  ambiguity  that  may  arise  from  use  of  the  same  expres- 
sion, it  is  usual  and  proper  to  employ  the  word  Caloric  to  signify  the 
principle  or  cause  of  sensation  of  heat. 

Heat  Unit. — For  purpose  of  expressing  and  comparing  quantities  of 
heat,  it  is  convenient  and  customary  to  adopt  a  Unit  of  heat  or  Thermal 
unit,  being  that  quantity  of  heat  which  is  raised  or  lost  in  a  defined 
period  of  temperature  in  a  defined  weight  of  a  particular  substance. 

Thus,  a  Thermal  unit,  Is  quantity  of  heat  which  corresponds  to  an  interval  0/1°  in 
temperature  of  i  Ib.  of  pure  liquid  water,  at  and  near  its  temperature  of  greatest 
density,  39.1°. 

Thermal  unit  in  France,  termed  Caloric,  Is  quantity  of  heat  which  corresponds 
to  an  interval  ofi°  C.  in  temperature  ofi  kilogramme  of  pure  liquid  water,  at  and 
near  its  temperature  of  greatest  density. 

Thermal  unit  to  Caloric,  3.96832;  Caloric  to  Thermal  unit,  .251 996. 

One  Thermal  unit  or  i°  in  i  Ib.  of  water,  772  foot-lbs. 

One  Caloric  or  i°  C.  in  i  kilogramme  of  water,  423.55  kilogrammetres. 

i°  C.  in  i  Ib.  water,  1389.6  foot-lbs. 

Ratio  of  Fahrenheit  to  Centigrade,  1.8;  of  Centigrade  to  Fahrenheit,  .555. 

Absolute  Temperature,  Is  a  temperature  assigned  by  deduction,  as  an 
opportunity  of  observing  it  cannot  occur,  it  being  the  temperature  corre- 
sponding to  entire  absence  of  gaseous  elasticity,  or  when  pressure  and  vol- 
ume ==o.  By  Fahrenheit  it  is — 461.2°,  by  Reaumur — 229.2°,  and  by  Cen- 
tigrade— 274°. 

Heat  is  termed  Sensible  when  it  diffuses  itself  to  all  surrounding 
bodies ;  hence  it  is  free  and  uncombined,  passing  from  one  substance 
to  another,  affecting  the  senses  in  its  passage,  determining  the  height 
of  the  thermometer,  etc. 

Temperature  of  a  body,  is  the  quantity  of  sensible  heat  in  it,  present 
at  any  moment. 

Heat  is  developed  by  water  when  it  is  violently  agitated. 

Heat  is  developed  by  percussion  of  a  metal,  and  it  is  greatest  at  the  first 
blow. 

Quantities  of  heat  evolved  are  nearly  the  same  for  same  substance,  with- 
out reference  to  temperature  of  its  combustion. 

Mechanical  power  may  be  expended  in  production  of  heat  either  by  fric- 
tion or  compression,  and  quantity  of  heat  produced  bears  the  same  propor- 
tion to  quantity  of  mechanical  power  expended,  being  i  unit  for  power 
necessary  to  raise  i  Ib.  772  feet  in  height.  This  number  of  772  is  termed 
the  mechanical  equivalent  of  heat  (Joules). 


HEAT. 

Specific   Heat. 

Specific  Heat  of  a  body  signifies  its  capacity  for  heat,  or  quantity  re- 
quired  to  raise  temperature  of  a  body  i°,  or  it  is  that  which  is  ab- 
sorbed by  different  bodies  of  equal  weights  or  volumes  when  their 
temperature  is  equal,  based  upon  the  law,  That  similar  quantities  of 
different  bodies  require  unequal  quantities  of  heat  at  any  given  tempera- 
ture. It  is  also  the  quantity  of  heat  requisite  to  change  the  tempera- 
ture of  a  body  any  stated  number  of  degrees  compared  with  that  which 
would  produce  same  effect  upon  water  at  32°. 

Quantity  of  heat,  therefore,  is  the  quantity  necessary  to  change  the  tem- 
perature of  a  body  by  any  given  amount  (as  i°),  divided  by  Quantity  of 
heat  necessary  to  ciiauge  an  equal  weight  or  volume  of  water  at  32"  by  same 
amount. 

NOTE.— Water  has  greater  specific  heat  than  any  known  body. 

Every  substance  has  a  specific  heat  peculiar  to  itself,  whence  a  cliange  of 
composition  will-  be  attended  by  a  change  of  its  capacity  for  heat. 

Specific  heat  of  a  body  varies  with  its  form.  A  solid  has  a  less  capacity 
for  heat  than  same  substance  when  in  state  of  a  liquid;  specific  heat  of 
water,  for  instance,  being  .5  in  solid  state  (ice),  .622  in  gaseous  (stea:n), 
and  i  in  liquid. 

Specific  heat  of  equal  weights  of  same  gas  increases  as  density  decreases ; 
exact  rate  of  increase  js  not  known,  but  ratio  is  less  rapid  than  diminution 
in  density. 

Change  of  capacity  for  heat  always  occasions  a  change  of  temperature. 
Increase  in  former  is  attended  by  diminution  of  latter,  and  contrariwise. 

Specific  heat  multiplied  by  atomic  weight  of  a  substance  will  give 
the  constant  37.5  as  an  average,  which  shows  that  the  atoms  of  all 
substances  have  equal  capacity  for  heat.  This  is  a  result  for  which  as 
yet  no  reason  has  been  assigned. 

Thus:  atomic  weights  of  lead  and  copper  are  respectively  1294.5  and  395.7,  and 
their  specific  heats  are  .031  and  .095.  Hence  1294.5  x  .031  =  40.129,  and  395.7  x 
.095  =  37.591. 

It  is  important  to  know  the  relative  Specific  Heat  of  bodies.  The  most  conve- 
nient method  of  discovering  it  is  by  mixing  different  substances  together  at  dif- 
ferent temperatures,  and  noting  temperature  of  mixture ;  and  by  experiments  it 
appears  that  the  same  quantity  of  heat  imparts  twice  as  high  a  temperature  to 
mercury  as  to  an  equal  quantity  of  water;  thus,  when  water  at  100°  and  mercury 
at  40°  are  mixed  together,  the  mixture  will  be  at  80°,  the  20°  lost  by  the  water 
causing  a  rise  of  40°  in  the  mercury;  and  when  weights  are  substituted  for  meas- 
ures, the  fact  is  strikingly  illustrated;  for  instance,  on  mixing  a  pound  of  mercury 
at  40°  with  a  pound  of  water  at  160°,  a  thermometer  placed  in  it  will  fall  to  155°. 
Thus  it  appears  that  same  quantity  of  heat  imparts  twice  as  high  a  temperature  to 
mercury  as  to  an  equal  volume  of  water,  and  that  the  heat  which  gives  5°  to  water 
will  raise  an  equal  weight  of  mercury  115°,  being  the  ratio  of  i  to  23.  Hence,  if 
equal  quantities  of  heat  be  added  to  equal  weights  of  water  and  mercury,  their 
temperatures  will  be  expressed  in  relation  to  each  other  by  numbers  i  and  23;  or, 
in  order  to  increase  the  temperature  of  equal  weights  of  those  substances  to  the 
same  extent,  the  water  will  require  23  times  as  much  heat  as  the  mercury. 

Capacity  for  Heat  is  relative  power  of  a  body  in  receiving  and  re- 
taining heat  in  being  raised  to  any  given  temperature ;  while  Specific 
applies  to  actual  quantity  of  heat  so  received  and  retained. 

Specific   Heat    of  Air   and.   other   Q-ases. 
Specific  heat,  or  capacity  for  heat,  of  permanent  gases  is  sensibly  constant 
for  all  temperatures,  and  for  all  densities.    Capacity  for  heat  of  each  gas  is 

U  u 


HEAT. 


same  for  each  degree  of  temperature.  M.  Regnault  proved  that  capacity 
for  heat  for  air  was  uniform  for  temperatures  varying  from  —22°  to 
+437°;  consequently,  specific  heat  for  equal  weights  of  air,  at  constant 
pressure,  averaged  .2377. 


Metals  from,  32°  to 

Specific   Heat 
Silver  056 

.     Water  at  32°  =  i. 
Woods. 

Sulphur  2026 

212°. 

Steel  1165 

Oak                    <7 

Antimony        0^08 

Tin  0562 

Pear                  5 

Liiquias. 

Bismuth  0308 

Wrought  iron  .1138 

Pine  65 

Brass  0939 

Zinc  0955 

Copper  092 

MinH  Substances. 

ijinseeu  on  ..  .Oi 

Cast  iron  1298 

Stones. 

Charcoal  2415 

Steam  365 

Gold  0324 

Chalk   2I49 

Lead  03*4 

Limestone...  .2174 

Coke  203 

Mercury           03  7  ^ 

Masonry           2 

Glass                 iQ77 

Nickel  1086 

Marble,  gray.  .2694 

Gypsum  1966 

Solid. 

Platinum  0124 

"     white.    21^8 

Phosphorus..   250^ 

Ice  504 

Air 

Oxygen. 


Air.. 


Gases. 

...  .2377   I  Hydrogen 2356 

...  .2412   I  Carbonic  Acid 3308 

For  Equal  Weights. 

.1688    |    Hydrogen 2.4096 


Oxygen 1559    |    Carbonic  Acid 1714 

Metals  have  least,  ranging  from  Bismuth  .0308  to  Cast  Iron  .1298.  Stones  and 
Mineral  Substances  have  .2  that  of  water,  and  Woods  about  .5.  Liquids,  with  ex- 
ception of  Bromine,  are  less  than  water,  Olive  oil  being  lowest  and  Vinegar  highest. 

ILLUSTRATION.  — If  i  Ib.  of  coal  will  heat  i  Ib.  of  water  to  100°,  — ^-  of  a  Ib.  will 

•°33 
heat  i  Ib.  of  mercury  to  100°. 

To    Compute    Temperature    of  a    Mixture    of   lilie    Sub- 
stances. 

w  (if t) 

— „. \- 1'  =  T.     W  representing  weight 

or  volume  of  a  substance  of  temperature  T,  w  weight  or  volume  of  a  like  substance  of 
temperature  t,  and  t'  temperature  of  mixture  W  +  w. 

ILLUSTRATION  i.  —  When  5  cube  feet  of  water  (W)  at  a  temperature  of  150°  (T)  is 
mixed  with  7.5  cube  feet  (w)  at  50°  (0,  what  is  the  resultant  temperature  of  the 
mixture? 

5X150°  + 7-5X50° 
5  +  7-5 


w(t'-t) 
T  —  t'    ' 


1125 

= =  90°. 

12.5        y 


2.— How  much  water  at  (T)  100°  should  be  mixed  with  30  gallons  (w)  at  60°,  for 
a  required  temperature  of  8o°? 

30(80°  — 60°)      600 
xooo-soo-  =  -£  =  3°  9allon*' 

To    Compute    Temperatxire    of  a    Mixture    of  Unlike 
Substances. 


WST  + 


-  =  <'; 


ws(t—  t') 
S(T  —  t) 


-  =  T.     W  and  w 


YVS-j-ws     "  S(T  —  t)   ~  WS 

representing  weights,  and  S  and  s  specific  heat  of  substances. 

ILLUSTRATION.  —To  what  temperature  should  20  Ibs.  cast  iron  (W)  be  heated  to 
raise  150  Ibs.  (w)  of  water  to  a  temperature  (t)  of  50°  to  60°  ? 


20  X-  1298 


2.596 


_ 

~~    * 


HEAT.  5O7 

To   Compute    Specific   Heat  at   Constant  "Volume. 
When  Specific  Heat  at  Constant  Pressure  is  knoum.      -^-  =  s.     S  represent- 
ing specific  heat  at  constant  pressure,  p  proportion  of  heat  absorbed  at  constant  vol- 
ume, H  total  heat  absorbed  at  constant  pressure,  and  s  specific  heat  at  constant  volume. 

Or,  — '  * — =  s.  t  and  t'  representing  initial  and  final  tempera- 
ture of  the  gas  and  that  to  which  it  is  raised,  and  V  and  v  initial  and  final  volumes 
of  the  gas  under  14.7  Ibs.  per  sq.  inch,  and  of  it  heated  under  constant  pressure  in 
cube  feet. 

ILLUSTRATION.— Assume  i  Ib.  air  at  atmospheric  pressure  and  at  32°,  doubled  in 
volume  by  heat.  S  = .  2377  *,  t  —  t'  =  32°  a,  525°  =  493°  and  V  —  v  =  12. 387  *  cube 
feet. 

.2377X493-(2. 742X12.387)  =  > i688       ^  heat 
493 

For  comparative  volumes  of  other  gases,  see  Table,  page  506. 

To    Compute    Specific    Heat   for    Eq.ua!   Volume   of  Gras 
and    Air. 

RULE. — Multiply  specific  heat  of  the  gas  for  equal  weights  of  gas  and  air 
by  specific  gravity  of  gas,  and  product  is  specific  heat  for  equal  volume. 

EXAMPLE. — What  is  specific  heat  of  air  at  equal  volume  with  hydrogen? 

Specific  heat  of  hydrogen  for  equal  weights  at  constant  volume,  2.4096,  and  speci- 
fic gravity  of  the  gas,  .0692.  (See  Table,  page  506.) 

Then,  2.4096  X  .0692  = .  1667  specific  heat  for  equal  volumes  at  constant  volume. 

Specific  heat  of  steam,  air  at  unity  =  1.281. 

Capacity   for   Heat. 

When  a  body  has  its  density  increased,  its  capacity  for  heat  is  di- 
minished. The  rapid  reduction  of  air  to  .2  of  its  volume  evolves  heat 
sufficient  to  inflame  tinder,  which  requires  550°. 

Relative  Capacity  for  Heat  of  Various  Bodies.    ( Water  at  32°  =  i. ) 


BODIES. 

Equal 
Weights. 

Equal 
Volumes. 

BODIES. 

Equal 
Weights. 

Equal 
Volumes. 

BODIES. 

Equal 
Weights. 

Equal 
Volumes. 

Water.. 
Brass.  .  . 
Copper.. 
Glass... 

I 
.116 
.114 
.187 

i 
.971 
1.027 
.448 

Gold.  .  .  . 
Ice  
Iron  
Lead  .  .  . 

•05 

:?26 

•043 

.966 

•993 
.487 

Mercury 
Silver  .  . 
Tin  
Zinc  

.036 

'.06 
.IO2 

.83*3 

To    Ascertain.    Relative    Capacities    of   Different    Bodies, 
combined,    -with,    experiment. 

RULE. — Multiply  weight  of  each  body  by  number  of  degrees  of  heat  lost 
or  gained  by  mixture,  and  capacities  of  bodies  will  be  inversely  as  products. 

Or,  if  bodies  be  mingled  in  unequal  quantities,  capacities  of  the  bodies 
will  be  reciprocally  as  quantities  of  matter,  multiplied  into  their  respective 
changes  of  temperature. 

ILLUSTRATION.— If  i  Ib.  of  water  at  156°  is  mixed  with  i  Ib.  of  mercury  at  40°, 
resultant  temperature  is  152°. 

Thus,  i  x  156°  — 152°  =  4°,  and  i  x  40°  ou  152°  =  112°.  Hence  capacity  of  water 
for  heat  is  to  capacity  of  mercury  as  112°  to  4°,  or  as  28  to  i. 

Sensible    Heat. 

Sensible  heat  or  temperature  to  raise  water  from  32°  to  212°  =  180.9°,  or 
heat  units. 

*  See  Tables,  pages  506  and  520-21. 


508 


HEAT. 


Latent  Heat. 

Latent  Heat  is  that  which  is  insensible  to  the  touch  of  our  bodies, 
and  is  incapable  of  being  detected  by  a  thermometer. 

When  a  solid  body  is  exposed  to  heat,  and  ultimately  passes  into  the 
liquid  state  under  its  influence,  its  temperature  rises  until  it  attains  the 
point  of  fusion,  or  melting  point.  The  temperature  of  the  body  at  this 
point  remains  stationary  until  the  whole  of  it  is  melted ;  and  the  heat  mean- 
time absorbed,  without  affecting  the  temperature  or  being  sensible  to  the 
touch  or  to  the  indications  of  a  thermometer,  is  said  to  become  latent.  It  is, 
in  fact,  the  latent  heat  of  fusion,  or  the  latent  heat  of  liquidity,  and  its  func- 
tion is  to  separate  the  particles  of  the  body,  hitherto  solid,  and  change  their 
condition  into  that  of  a  liquid.  When,  on  me  contrary,  a  liquid  is  solidified, 
the  latent  heat  is  disengaged. 

If  to  a  pound  of  newly-fallen  snow  were  added  a  pound  of  water  at  172°, 
the  anow  would  be  melted,  and  32°  would  be  resulting  temperature. 

When  a  body  is  fusing,  no  rise  in  its  temperature  occurs,  however  great 
the  additional  quantity  of  heat  may  be  imparted  to  it,  as  the  increased  heat 
is  absorbed  in  the  operation  of  fusion.  The  quantity  of  heat  thus  made 
latent  varies  in  different  bodies. 

A  pound  of  water,  in  passing  from  a  liquid  at  212°  to  steam  at  212°,  re- 
ceives as  much  heat  as  would  be  sufficient  to  raise  it  through  966.6  ther- 
mometric  degrees,  if  that  heat,  instead  of  becoming  latent,  had  been  sensible. 

If  5.5  Ibs.  of  water,  at  temperature  of  32°,  be  placed  in  a  vessel,  communicating 
with  another  one  (in  which  water  is  kept  constantly  boiling  at  temperature  of  212°), 
until  former  reaches  temperature  of  latter  quantity,  then  let  it  be  weighed,  and 
it  will  be  found  to  weigh  6.5  Ibs.,  showing  that  one  Ib.  of  water  has  been  received 
In  form  of  steam  through  communication,  and  reconverted  into  water  by  loweF 
temperature  in  vessel.  Now  this  pound  of  water,  received  in  the  form  of  steam, 
had,  when  in  that  form,  a  temperature  of  212°.  It  is  now  converted  into  liquid 
form,  and  still  retains  same  temperature  of  212°;  but  it  has  caused  5.5  Ibs.  of  water 
to  rise  from  the  temperature  of  32°  to  212°,  and  this  without  losing  any  tempera- 
ture of  itself.  Now  this  heat  was  combined  with  the  steam,  but  as  it  is  not  sensible 
to  a  thermometer,  it  is  termed  Latent. 

Quantity  of  heat  necessary  to  enable  ice  to  resume  the  fluid  state  is  equal 

to  that  which  would  raise  temperature  of  same  weight  of  water  140° ;  and  an 

equal  quantity  of  heat  is  set  free  from  water  when  it  assumes  the  solid  form. 

Su.m   of  Sensible   and.    .Latent   Heats. 

From  Water  at  32°. 


Press- 
ure. 

Latent. 

Sum. 

Press- 
ure. 

Latent. 

Sum. 

Press- 
ure. 

Latent. 

Sum. 

Press- 
ure. 

Latent. 

Sum. 

Lbs. 

0 

o 

Lbs. 

0 

0 

Lbs. 

0 

o 

Lbs. 

o 

0 

14.7 

064-3 

146.1 

26 

943-7 

155-3 

55 

912 

1169 

120 

873-7 

185.4 

16 
17 

962.1 
959-8 

147.4 
148.3 

27 

28 

942.2 
940.8 

155-8 
156.4 

60 
65 

908 
904.2 

1170.7 
1172.3 

130 
140 

869.4 
865-4 

187-3 
189 

18 

957-7 

149.2 

29 

939-4 

157-  1 

70 

900.8 

1173.8 

ISO 

861.5 

190.7 

*9 

955-7 

150.1 

30 

937-9 

157-8 

75 

897-5 

1175.2 

160 

857-9 

192.2 

20 

952.8 

150.9 

32 

935-3 

158.9 

80 

894-3 

1176.5 

170 

854-5 

!93-7 

21 

95i-3 

I5I-7 

35 

931.6 

160.5 

85 

891.4 

1177.9 

1  80 

851-3 

I95-J 

22 

23 

949-9 
948.5 

152-5 
153-2 

37 
40 

929-3 
926 

161.5 
162.9 

90 
95 

888.5 
885.8 

1179s1 
1180.3 

190 

200 

848 
845 

196.5 
197.8 

24 

946.9 

153-9 

45 

920.9 

164.6 

IOO 

883.1 

1181.4 

220 

829.2 

200.3 

25 

945-3 

154.6 

So 

916.3 

167.1 

no 

878.3 

"83.5 

250 

831.2 

203.7 

Latent  Heat  of  Vaporization,  or  Number  of  Degrees  of  Heat  required  to  con- 
vert following  Substances  from  their  'Liquidities  to  Vapor  at  Pressure 
of  Atmosphere. 

Alcohol 364°    Ice 142.6°    Water 966.6° 

Ammonia 860°    Mercury 157°       Zinc 493° 

Ether  (Sulph.) 163°    Carbonic  Acid 298°       Oil  of  Turpentine. .  124° 


HEAT. 


509 


Latent  Heat  of  Fusion  of  Solids.    (Person.) 


Substances. 

Melt- 

A 

Specifi 
Liquid. 

:  Heat. 

Solid. 

In  Heat- 
units  of 
lib. 

Substances. 

Melt- 
ing 
Point. 

Specifi 
Liquid. 

Heat. 
Solid. 

In  Heat- 
unite  of 
ilb. 

0 

0 

o 

c 

0 

o 

Tin  
Bismuth. 
Lead  .... 

442 

507 
617 

.0637 
•  0363 
.0402 

.0562 
.0308 
.0314 

25-6 
22.7 
9.86 

Ice.  
Phosphorus  .... 
Spermaceti  

32 

112 

120 

I 
.2045 

•504 
.1788 

142-85 
,4? 

Zinc  

773 

.0056 

50.6 

Wax  

Silver  .  .  . 

1873 

•°57 

•37.0 

142 

27Q 

234. 

, 

175 

Mercury. 
Cast  iron. 

39 
3400 

•0333 

.0319 
.129 

o/'y 

5 
233 

Nitrate  of  soda.. 
Nit.  of  potassia  . 

*3y 
59i 
642 

'*M 

•413 
•3319 

.2782 
.2388 

17 

'11 

To  Compnte  Latent  Heat  of  Fusion  of  a  Non-metallic 
Substance. 

C  'v  c  (t  +  256°)  =  L.  C  and  c  representing  specific  heats  of  substance  in  solid  and 
liquid  state,  t  temperature  of  fusion,  and  L  latent  heat. 

ILLUSTRATION.—  What  is  latent  heat  of  fusion  of  ice? 

€  =  .504;    c  =  i 


.504  ^  i  X  32  -f-  256  =  142.85°  units. 

NOTE.—  For  Latent  Heat  of  Fusion  of  some  substances,  see  Deschanel's,  New  York, 
1872,  Heat,  part  2. 

Radiation   of  Heat. 

Radiation  of  Heat  is  diffusion  of  heat  by  projection  of  it  in  diverging  right 
lines  into  space,  from  a  body  having  a  higher  temperature  than  space  sur- 
rounding it,  or  body  or  bodies  enveloping  it 

Radiation  is  affected  by  nature  of  surface  of  body  ;  thus,  black  and  rough 
surfaces  radiate  and  absorb  more  heat  than  light  and  polished  surfaces. 
Bodies  which  radiate  heat  best  absorb  it  best. 

Radiant  heat  passes  through  moderate  thicknesses  of  air  and  gas  without 
suffering  any  appreciable  loss  or  heating  them.  When  a  polished  surface 
receives  a  ray  of  heat,  it  absorbs  a  portion  of  it  and  reflects  the  rest.  The 
quantity  of  heat  absorbed  by  the  body  from  its  surface  is  the  measure  of 
its  absorbing  power,  and  the  heat  reflected,  that  of  its  reflecting  power. 

When  temperature  of  a  body  remains  constant  it  is  in  consequence  of 
quantity  of  heat  emitted  being  equal  to  quantity  of  heat  absorbed  by  body. 
Reflecting  power  of  a  body  is  complement  of  its  absorbing  power  ;  or,  sum 
of  absorbing  and  reflecting  powers  of  all  bodies  is  the  same. 

Thus,  if  quantity  of  heat  which  strikes  a  body  =  100,  and  radiating  and  reflecting 
powers  each  90,  the  absorbent  would  be  10. 

Radiating   or   Absorbent    and.    Reflecting    Powers    of 
Substances. 


SUBSTANCES. 

Radiating 
or  Ab- 
sorbing. 

Reflect- 
ing. 

SUBSTANCES. 

Radiating 
or  Ab- 
sorbing. 

Reflect- 
ing. 

IOO 

Wrought  Iron  polished 

Water  

IOO 

Lead  polished 

11 

Carbonate  of  Lead  

IOO 

Zinc  polished     

IQ 

81 

Lead,  white  

IOO 

Steel  polished 

e. 

Writing  Paper  

98 

Platinum  in  sheet 

83 

Ivory,  Jet,  Marble  

03  to  08 

7  to  2 

Tin  .'  !!..' 

I  e 

Re 

Resin  

QO 

Copper  varnished 

86 

Glass           

Brass  dead  polished 

ort 

India  Ink  

£ 

"      bright  polished 

11 

»9 

Ice  

85 

I  e 

Copper  ham'ered  or  cast 

93 

Shellac  

28 

93 

Lead  

45 

ee 

Gold  plated 

93 

Cast  Iron,  bright  polished 

25 

75 

95 

Platinum,  a  little  polish'd 

24 

76 

Silver  polished  .   . 

Mercury  

23 

77 

**       cast  polished 

"u 

u* 

97 

5io 


HEAT. 


Radiating  and  A'bisOr'bing  Power  of  varioxxs  Bodies,  in 
Units  of  Heat  per  Sq..  Foot  per  Hou.r  for  a  Difference 
of  1°.  (Peclet.) 

Unit.  Unit. 

Iron,  ordinary 5662    Woollen  stuff 7522 

Glass 5948    Oil  paint , 7583 

ron,  cast 648      Paper 77o6 


Unit. 

Silver,  polished 0266 

Copper 0327 

Tin 0439 


Brass,  polished 0491 

Iron,  sheet 092 


Wood  sawdust 7225 

Stone,  Brick,  etc 7358 


Lamp-black 8196 

Water 1.0853 


To  Compute  Loss  of  Heat  toy  Radiation  per   So;.  Foot. 

-  =  R.    T  representing  temperature  of  pipe,  which  is  assumed  to  be  .05 


dv 

less  than  that  of  steam;  t  temperature  of  air;  I  length  of  pipe,  and  v  velocity  of  heat 
in  feet  per  second;  d  diameter  in  ins.,  and  R  radiation  in  degrees  per  second. 

ILLUSTRATION.— Assume  temperatures  of  a  steam  pipe,  steam,  212°,  200°,  and  air 
60°,  length  of  pipe  20  feet,  velocity  of  heat  (steam)  15  feet  per  second,  and  diameter 
of  pipe  16  ins. ;  what  will  be  loss  of  heat  by  radiation? 
1.7X20(200-60) 
i6X  15 

Reflection. 

Reflection  of  Heat  is  passage  of  heat  from  surface  of  one  substance 
to  another  or  into  space,  and  it  is  the  converse  of  radiation. 

Heat  is  reflected  from  surface  upon  which  its  rays  fall  in  same  manner  as 
light,  angle  of  reflection  being  opposite  and  equal  to  that  of  incidence.  Met- 
als are  the  strongest  reflectors. 

Reflecting  Power  of  various  Substances. 


Silver. 97 

Gold 95 

Brass 93 


Specular  metal 86  I   Zinc 81 

Tin 85       Iron 77 

Steel 83  I  Lead 6 


Communication   and   Transmission   of  Heat. 

Communication  of  Heat  is  passage  of  heat  through  different  bodies 
with  different  degrees  of  velocity.  This  has  led  to  division  of  bodies 
into  Conductors  and  Non-conductors  ;  former  includes  such  as  metals, 
which  allow  caloric  to  pass  freely  through  their  substance,  and  latter 
comprise  those  that  do  not  give  an  easy  passage  to  it,  such  as  stonesf 
glass,  wood,  charcoal,  etc. 

Velocity  of  cooling,  other  things  being  equal,  increases  with  extent  of  sur- 
face compared  with  volume  of  substance ;  and  of  two  bodies  of  same  mate- 
rial, temperature,  and  form,  but  differing  in  volume. 

Transmission  of  Heat  is  passage  of  heat  through  different  bodies  with  dif- 
ferent degrees  of  intensity.  Gaseous  bodies  and  a  vacuum  are  highest  in 
order  of  transmittents. 

Relative  Power  of  various  Substances  to  Transmit  Heat. 
All  bodies  capable  of  transmitting  heat  are  more  or  less  translucent, 
though  their  powers  of  transmitting  heat  and  light  are  not  in  same  rela- 
tive proportions. 

Nitric  acid 

Rock-crystal  . . 
Rape  seed  oil. . 

Heat  which  passes  through  one  plate  of  glass  is  less  subject  to  absorption 
in  passing  through  a  second  and  a  third  plate.  Of  1000  rays,  451  were  iiy» 
tercepted  by  4  plates  as  follows : 

ist.  381.  2d.  43.  3d.  1 8.  4th.  9. 


Air i 

Alcohol 15 

Crown-glass..     .49 


Flint-glass 67 

Gypsum 2 

Ice 06 


Sulphuric  acid. 

Turpentine 

Water 


HEAT. 


Average  Results  of  Heati 
Steam   in  Copper   Fip 

ng  and.  Evaporating  "Water  "by 
es   and    Boilers.     (D.K.Clark.) 
Steam  condensed                  Heat  transmitted 
Per  sq.  foot  for  i*  difference  per  hour. 
Heating.      Evaporating.      Heating.      Evaporating 

Lbs. 
.077 
.248 
.201 

Lba. 
•  105 
•483 

1.07 

Unit*. 
82 
276 
•U2 

Unite. 

100 

534 

ion 

Copper-plate  surface  •  

Copper-  pipe  surface.  .  . 

Yellow 40 

Orange 44 

Red 53 


Whence. — Efficiency  of  copper-plate  surface  for  evaporation  of  water  is 
double  its  efficiency  for  heating ;  for  copper-pipe  surface  efficiency  is  more 
than  three  times  as  much ;  and  for  cast-iron-plate  surface,  a  fourth  more. 

Efficiency  of  pipe  surface  is  a  fifth  more  than  that  of  plate  surface  for 
heating,  and  more  than  twice  as  much  for  evaporation. 

Generally,  copper -plate  surface  condenses  .5  Ib.  of  steam,  copper-pipe 
i  Ib.,  and  cast-iron-plate  surface  .1  Ib.  per  sq.  foot  per  i°  of  temperature  per. 
hour,  for  evaporation. 

Quantity  of  heat  transmitted  is  at  rate  of  about  1000  units  per  Ib.  of  steam 
condensed. 

Transmission  of  Heat  through  Glass  of  different  Colors. 
Direct  =  100. 

Plate 65. 5   I   Blue,  deep 19 

Window 52  "    light 42 

Violet,  deep 53      |  Green 26 

M.  Peclet  defines  law  of  transmission  of  heat  as :  The  flow  of  heat  which 
traverses  an  element  of  a  body  in  a  unit  of  time  is  proportional  to  its  sur- 
face, and  to  difference  of  temperature  of  the  two  faces  perpendicular  to  direc- 
tion of  flow,  and  is  in  inverse  of  thickness  of  element. 

ri 

Or,  (t —  t')  -  =  H.     t  and  If  representing  temperatures  of  surfaces,  C  constant  for 

material  i  inch  thick,  or  quantity  of  heat  transmitted  per  hour  for  i°  difference  of 
temperature  through  i  unit  of  thickness,  T  thickness,  and  H  quantity  of  heat  in  unitt 
passed  through  plate  per  sq.  foot  per  hour. 

Quantities  of  Heat  transmitted  from  "Water  to  "Water 
throngh  Plates  or  Beds  of  IMetals  and  other  Solid 
Bodies,  1  Inch  thick,  per  Sq[.  Foot. 

For  i°  Difference  of  Temperature  between  the  two  Faces  per  Hour. 
Selected  from  M.  Peclet's  tables.     (D.  K.  Clark.) 

C  or  G  or  C  or 

SUMTANCB.  Quantity  SUBSTANCE.  Quantity  SUBSTANCK. 

of  Heat.  of  Heat. 


Quantity 
of  Heat. 


UniU. 
225 
225 
177 
112 


Marble. , 
Plaster. , 
Glass . . , 
Sand 


UniU. 

•£ 

6.56 
2.16 


Gold 620  Iron 

Platinum 604  Zinc 

Silver 596  Tin 

Copper 555  Lead 

The  conditions  are,  that  the  surfaces  of  conducting  material  must  be  per- 
fectly clean,  that  they  be  in  contact  with  water  at  both  faces  of  different 
temperatures,  and  that  the  water  in  contact  with  surfaces  be  thoroughly  and 
constantly  changed.  M.  Peclet  found  that  when  metallic  surfaces  became 
dull,  rate  of  transmission  of  heat  through  all  metals  became  very  nearly 
the  same. 

To   Compnte   Units   of  Heat   Transmitted. 

ILLUSTRATION  i.  —  If  2000  Ibs.  beet  root  juice  at  40°  are  contained  in  a  copper 
boiler  with  a  double  bottom,  and  heated  to  212°,  with  a  heating  surface  of  25  sq.  feet, 
and  subjected  to  steam  at  a  temperature  of  275°,  for  a  period  of  15  minutes,  what 
will  be  the  total  heat,  and  heat  per  degree  of  difference  transmitted  per  sq.  foot  per 
hour? 


512 


HEAT. 


212°  —  40°  x  60  4- 15  =  688°  per  hour,  and  2000  X  688  -7-25  =  55  040  units  per  sq. 
foot  per  hour. 

(212°  -f  40°)  -4-  2  =  126°  mean  temperature  of  juice,  and  275°  — 126°  =  149°  mean 
difference  of  temperature. 

Hence,  55040-=-  149  =  369.4  units  per  sq.  foot  per  degree  of  difference  per  hour. 

2.—  If  48.2  S}.  feet  of  iron  pipe  1.36  ins.  in  diameter,  is  supplied  with  steam  at  275°, 
and  it  raises  temperature  of  882  Ibs.  water  from  46°  to  212°  in  4  minutes,  what  will 
be  total  heat  per  sq.  foot  per  hour,  total  heat  per  sq.  foot  per  degree,  and  quantity 
condensed  per  sq.  foot  per  degree  per  hour  ? 

212°  —  46°  X  60  -4-  4  =  2490°  in  an  hour  ;  46°  +  212°  -4-  2  =  129°  mean  temper- 
ature, and  275°  —  129°  =  146°  difference  of  temperature. 

2490    X — -  =  45  563  units  per  sq.foot  per  hour,  45  563  -r- 146  =  312.  i  units  per  sq. 

48.2 
foot  per  degree,  and  total  heat  of  steam  above  129°  =  1068°. 

Hence       -I  =  .292  Ibs.  steam  condensed  per  sq.foot  per  degree  per  hour. 

Evaporation. 

Evaporation  or  Vaporization  is  conversion  of  a  fluid  into  vapor,  and 
it  produces  cold  in  consequence  of  heat  being  absorbed  to  form  vapor. 

It  proceeds  only  from  surface  of  fluids,  and  therefore,  other  things  equal, 
must  depend  upon  extent  of  surface  exposed. 

When  a  liquid  is  covered  by  a  stratum  of  dry  air,  evaporation  is  rapid, 
even  when  temperature  is  low. 

As  a  large  quantity  of  heat  passes  from  a  sensible  to  a  latent  state  during 
formation  of  vapor,  it  follows  that  cold  is  generated  by  evaporation. 

Fluids  evaporate  in  vacuo  at  from  120°  to  125°  below  their  boiling-point. 

Heat  required  to  Kvaporate  1  Ito.  \Vater  at  Temperatures 
toelcrw  313°  from,  a  Vessel  in.  open  air  at  3S°. 

(Thomas  Box.) 


13     ^ 

HEAT 

?v.b' 

HEAT 

1 

1-1 

« 

S  . 

\ 

S.-I 

i 

3  . 

11 

rn 

feS"S 

Ml 

'3 

.2 

K; 

11 

**"»! 

§£ 

Srg-S 

IT? 

Hi 

•3 

3 

Ki 

Is' 

"-1  o 
"a-0 

H 

14 

fh 

1 

*fi 
3H 

SS, 

H 

•S  u,  & 

«  S^: 
ES^g 

|l« 

1 

*£ 

3  H 

SI 

0 

Lbs. 

Units. 

Units. 

Units. 

Units. 

0 

Lbs. 

Units. 

Units. 

Units. 

Units. 

32 

.027 

1091 

29 

132 

.706 

182 

202 

1506 

1068 

42 

.04 

270 

424 

1788 

71 

142 

.916 

158 

162 

1445 

1326 

52 

.058 

375 

58i 

2052 

119 

152 

1.178 

137 

127 

1392 

1637 

62 

.083 

4°5 

605 

2110 

174 

162 

I-505 

118 

97 

J346 

2039 

72 

.117 

386 

566 

2055 

239 

172 

I  895 

106 

72 

1312 

2475 

82 

.162 

358 

504 

1968 

319 

182 

2-373 

92 

So 

1279 

3°45 

92 

.223 

3i9 

434 

1862 

415 

192 

2.947 

81 

32 

1253 

3685 

102 

•303 

280 

366 

1758 

533 

202 

3-633 

7i 

14 

1228 

4465 

112 

.406 

245 

3°4 

l664 

671 

212 

4.471 

.    63 

— 

1209 

5397 

122 

.528 

211 

250 

1580 

849 

— 

— 

— 

— 

To    Compute    Surface   of  a    Refrigerator. 

Illustration  of  Table.  —  If  it  is  required  to  cool  20  barrels,  of  42  gallons  each,  of 
beer,  from  202°  to  82°  in  an  hour. 

Result  to  be  attained  is  to  dissipate  42  X  8.33  (Ibs.  TJ.  S.  gallons)  X  20  X  202  —  82 
=  840000  units  of  heat  per  hour. 

At  202°,  4465  units  are  lost,  and  at  82°,  319,  hence,  average  loss  for  each  temper- 
ature between  extremes  =  1850  units  per  sq.foot  per  hour. 

840000 

Then  ~ =  454  sq.feet  in  a  still  air. 

1050 

The  volume  of  air  required  per  hour  in  this  case  would  be  about  100000  cube  feet 


HEAT.  5  I  3 

To   Compute  Area  of*  Q-rate  and  Consumption   of  Fuel 
for    Evaporation. 

Illustration  of  Table.—  It  it  is  required  to  evaporate  6  Beer  gallons  (282  cube  ins.) 
of  liquid  per  hour,  at  a  temperature  not  exceeding  152°. 

6  gallons  =  50  Ibs.     At  152°,  water  evaporated  as  per  table  =  1.  178  Ibs.  per  hour. 

-^-  =  42  sq.  feet.    Heat  required  to  effect  this  =  1392  X  50  =  69  6o°  units- 
1-178  ^^ 

Assuming  6000  units  as  average  economic  value  of  coals,  then  ^^  =  u.6  Ibs. 
coal,  on  a  grate  of  i  sq.foot. 

When  it  is  practicable  to  evaporate  at  a  high  temperature,  as  at  or  above  212°,  it 
is  most  economical. 

Thus,  water  requires  only  1209  units  per  Ib.  if  surface  is  exposed,  but  if  enclosed, 
heat  is  reduced  (1209  —  63)  to  1146  units. 

Evaporative  Powers  of  Different  Tubes  per  Degree  of  Heat,  per  Sq.  Foot  of 

Surface.—  In  Units. 
Vertical  tube,  230;  Double-bottomed  vessel,  330;  Horizontal  tube  or  Worm,  430. 

To   Compute  "Volume   of*  Water   Evaporated  in.  a  given 
Time. 

ILLUSTRATION.—  What  is  volume  evaporated  at  212°,  in  15  minutes  per  sq.  foot  of 
surface,  in  a  double-  bottomed  vessel  having  an  area  of  heating  surface  of  17  feet, 
and  subjected  to  steam  at  a  pressure  of  25  Ibs.  ? 

Temperature  of  steam  at  25  +  14.7  Ibs.  =  269°.  269°  —  212°  =  57°,  and  latent 
heat  —  927. 

Then  33QX57XI7X.5  =  86.2  to*,  water. 
927X60 

When  Water  is  at  a  Lower  Temperature  than  212°. 

If  120  gallons  or  1000  Ibs.  of  water  were  to  be  evaporated  from  42°  in  an 
hour,  from  same  vessel  and  under  like  pressure  as  preceding  : 

There  would  be  requ  ired  1000  X  (2  1  2°  —  42°)  1  70  ooo  units  of  heat.  Mean  tempera- 
tare  of  water  while  being  heated  =  42  +212°  _  I27<j 

Difference  between  temperature  of  steam  and  water  =  267°  —  127°  =  140°. 


Then,  -  10  -  _  2l6  ft(mr  _  time  ^  rai$e  water  fo  2I2o  .  hence  i  —  .216  =. 
330  X  140  X  17 

.784  hour  left  for  evaporation,  and  quantity  evaporated  =  33°  x  57  X  17  X  .784., 

927 
270.4  Z&*.,  or  32.44  gallons. 

Dessieeation.. 

Dessiccation,  or  the  drying  of  a  substance,  is  best  effected  in  a  drying 
chamber,  and  it  is  imperative  that  to  attain  greatest  effect  the  hot  air 
should  be  admitted  at  highest  point  of  exposed  substance  and  dis- 
charged at  its  lowest. 

Wood,  submitted  to  an  average  temperature  of  300°  in  an  enclosed  space 
for  a  period  of  2.5  clays,  will  lose  its  moisture  at  a  consumption  of  i  Ib.  of 
wood  for  10.5  Ibs.  of  wood  dried,  and  evaporating  4  Ibs.  of  water,  equal  to 
2.66  Ibs.  of  water  per  Ib.  of  undried  wood. 

Limit  of  temperature  for  drying  of  wood  is  340°. 


514 


HEAT. 


Kvaporation  of*  "Water  per  S<i.  Foot  of  Surface  per  Honr. 

(Dr.  Dalton.) 


Temperature 
of  Water. 

] 
Calm. 

Evaporation 

a? 

Brisk 
Wind. 

Temperature 
of  Water. 

ii              fc 
Calm. 

vaporation 
Light 
Air. 

Brisk 

Wind. 

0 

Lbs. 

Lbs. 

Lba. 

0 

Lbs. 

Lbs. 

Lbs. 

3« 

.0349 

0448 

•055 

100 

.3248 

.4169 

.5116 

40 

•0459 

0589 

.0723 

125 

.6619 

.8494 

1.043 

50 

•0655 

0841 

.1032 

ISO 

1.296 

1.663 

2.043 

60 

.0917 

"75 

.1441 

175 

2.378 

3-053 

3-746 

70 

.1257 

1616 

.1983 

200 

4.128 

5.298 

6.502 

80 

.1746 

2241 

•2751 

212 

5-239 

6.724 

8.252 

The  rates  of  evaporation  for  these  conditions  of  the  air  when  perfectly  dry  are  as 
i,  1.28,  and  1.57. 

To  Compute  Quantity  of  Water  exposed  to  Air  that  would  be  evaporated  as 
above. — Subtract  tabulated  weight  of  water  corresponding  to  dew-point  from 
weight  of  water  corresponding  to  temperature  of  dry  air,  and  remainder  is 
weight  of  water  that  would  be  evaporated  per  sq.  foot  of  surface  per  hour. 

Distillation. 

Distillation  is  depriving  vapor  of  its  latent  heat,  and,  though  it  may 
be  effected  in  a  vacuum  with  very  little  heat,  no  advantage  in  regard  to 
a  saving  of  iael  is  gained,  as  latent  heat  of  vapor  is  increased  propor- 
tionately to  diminution  of  sensible  heat. 

A  temperature  of  70°  is  sufficient  for  distillation  of  water  in  a  vessel  ex- 
hausted of  air. 

Conduction,  or   Convection   of  Heat. 

Air  and  gases  are  very  imperfect  conductors.  Heat  appears  to  be 
transmitted  through  them  almost  entirely  by  conveyance,  the  heated 
portions  of  air  becoming  lighter,  and  diffusing  the  heat  through  the 
mass  in  their  ascent.  Hence,  in  heating  a  room  with  air,  the  hot  air 
should  be  introduced  at  lowest  part.  The  advantage  of  double  win- 
dows for  retention  of  heat  depends,  in  a  great  measure,  upon  sheet  of  air 
confined  between  them,  through  which  heat  is  very  slowly  transmitted. 

Convection  of  heat  refers  to  transfer  and  diffusion  of  heat  in  a  fluid  mass, 
by  means  of  the  motion  of  the  particles  of  the  mass. 

Relative   Internal   Conducting    Powers   of  Various 

Sn"b  stances. 

Metals. 


Brass       .... 

76      Gold  i 

Porcelain.... 
Silver  

.012 

•97 
.on 

.61 

£ 

Tin  

Wrought  Iron 
Zinc  

Gypsum  
Lime  
Marble  

•3 

.2 
.24 
1.22 

Cast  Iron  
Copper  

Cement  
Chalk  

.89      Platinum  98 
Mim 
.21  1  Coal,  anth'cite  1.92 
.6        "     bitumin.  1.68 
.07    Coke  i.  08 

Terra  Gotta.  . 
•rals. 
Fire  brick  
Fire  clay  
Glass  

Charcoal.  .  . 

Slate.... i  Wood  ash 08 

Woods  with  Birch  =  .41  with  Silver. 

Apple 68  I  Birch i       I  Ebony 5    I  Oak.. 

Ash 73  I  Chestnut 7    |  Elm 73  |  Pine.. 

Hair  and  Fur  with  Air  =  i. 

I  Flannel 2.44  I  Hair 2       |  Silk. . 

I  Hemp  Canvas.    .28  |  Hare's  fur 43  |  Wool. 

Liquids  with  Water. 

Alcohol 93  I  Proof  spirit 85  I  Turpentine 3.1 

Mercury 2.8    |  Sulphuric  acid 1.7    |  Water. ..,.  x 


Cotton 55  I 

Eider  down...    .44  | 


•73 

•73 


•43 
•5 


HEAT.  515 

Practical  Deductions  from  preceding  Remits. 

A$phalt  compositions  are  best  for  resisting  moisture  and  insuring  dryness. 
Being  a  slow  conductor  of  heat,  it  will  help  to  exclude  or  retain  heat  as  de- 
sired. 

Slate  is  a  very  dry  material,  but,  from  its  quick  conducting  power,  it  is 
not  adapted  for  retention  of  heat. 

Cements.  —  Plaster  of  Paris  and  Woods  are  well  adapted  for  lining  of 
rooms,  having  low  conductive  powers,  while  Hair  and  Lime,  being  a  quick 
conductor,  is  one  of  the  coldest  compositions. 

Fire-brick  absorbs  much  heat,  and  is  well  adapted  for  lining  of  fire-places, 
etc. ;  while  Iron,  being  a  high  conductor  of  heat,  is  one  of  the  worst  of  sub- 
stances for  this  purpose.  Common  brick  is  not  a  very  slow  conductor  of  heat. 

Steam  Pipe.— A  wrought-iron  pipe,  4  ins.  in  internal  diameter,  conveying 
steam  at  a  pressure  of  35  Ibs.  per  sq.  inch  (280°)  and  100  feet  in  length, 
will  lose  .84  H». 

Casing  to  Pipes. — A  like  pipe  with  the  above,  cased  with  following  mate- 
rials and  covered  with  canvas,  to  give  like  radiating  power  to  the  outer 
surface,  gave  loss  of  heat  in  units  per  hour,  and  for  the  thickness  given,  as 
follows : 


CASINO. 

.5  Inch. 

i  Inch. 

2  Inches. 

4  Inchw. 

6  Inchei. 

Woollen  Felt  .  .  . 

06 

16 

Sawdust  

IOO 

55 

26 

£ 

•j 

Coal-ashes.  .  . 

172 

no 

60 

27 

16.6 

Condensation. 

Tredgold  ascertained  by  experiment  that  steam  at  pressure  (absolute) 
of  17.5  Ibs.  per  sq.  inch,  221°,  produced  I  cube  foot  of  water  per  hour 
by  condensation  in  182  sq.  feet  of  cast-iron  pipe,  at  a  uniform  and  qui- 
escent temperature  of  60°.  Hence,  condensation  .352  Ib.  water  per 
hour,  or  .0022  Ibs.  per  degree  of  difference  of  temperature  (221  —  60). 

From  experiments  of  Mr.  B.  G.  Nichol  in  England,  1875,  it  was  deduced : 
That  rates  of  transmission  of  heat,  between  temperature  of  steam  and 

that  of  water  of  condensation  at  its  exit,  at  the  rate  of  150  feet  per  minute, . 

may  be  taken  as  380  units  for  vertical  tubes  and  520  for  horizontal. 

Condensation   of  Steam,  in   Cast-iron    Pipes.    (M.  Burnat.) 


Average 
Press,  per 
Sq.  Inch. 

?  I  1 

Steam. 

'emperatur 
Air. 

Difference. 

Condens 
Bare. 

ition  per  sq 
Straw. 

.  foot  of  eit 
per  hour. 
Pipe. 

ernal  surfa 
Waste. 

,e  of  pipe 
Plaster. 

Lba. 
22 

0 

233 

o 
36.5 

0 

196.5 

Lb. 

.581 

Lb. 
.2 

Lb. 
.229 

Lb. 
.286 

Lb. 
•324 

From  these  data,  following  constants  are  deduced  for  an  absolute  pressure  of 
22  Ibs.  per  sq.  inch  of  steam  condensed,  and  heat  passed  off  per  sq.  foot  of  external 
surface  of  pipe  per  hour  of  i°  difference  of  temperature. 


SUHFACE  OF  PIPE. 

Steam 
condensed 
per  Sq.  Foot. 

Heat 
passed 
off. 

SURFACE  OF  PIPE. 

Steam 
condensed 
per  Sq.  Foot. 

Heat 
off. 

Lb. 
.003 

Units. 
2.812 

Cotton  waste  i  inch.  . 

Lb. 
.001  46 

Unite. 
1.384 

Straw  coat     

.001  02 

.068 

Earth  and  hair  

.001  65 

1568 

Cased  with  clav  Dine.  .  . 

.001  i* 

1.108 

White  caint.  .  . 

.001  <^6 

1.486 

Si6 


1IEAT. 


Pipes  were  4.72  ins.  diameter,  .25  inch  thick,  and  had  area  of  58.5  sq.  feet. 
Bare — rough  surface  as  cast.  Straw  coat—  laid  lengthwise  .6  inch  thick  and  bound. 
Pipe — laid  in  clay  pipe  with  an  air  space  between  them,  the  whole  covered  with 
loam  and  straw.  Waste  cotton— i  inch  thick  and  bound  with  twine.  Plaster— 
laid  in  clay  and  hair  2.36  ins.  thick. 

A  wrought-iron  pipe  3.75  ins.  in  external  diameter,  .25  inch  thick,  and  lagged 
with  felt  and  spun  yarn  .5  inch  thick,  condensed  steam  at  245°  at  rate  of  .262  Ib. 
per  sq.  foot  per  hour,  in  an  external  temperature  of  60°. 

Steam  Condensed,  per  Sq..  Foot  and  per  Degree  per  Hoiar. 

Mean  Results  of  several  Experiments  with  bare  Cast-iron  Pipes,  with  Steam 
at  Absolute  Pressure  of  20  Ibs.per  Sq.  Inch. 

.4  Ib.  per  sq.  foot,  and  .002  39  Ib.  per  degree. 

Hence,  to  ascertain  quantity  of  heat  lost  by  condensation  of  .002  39  Ib.  =  —  of  a  Ib. 

Difference  of  total  and  sensible  heats  of  i  Ib.  steam  at  20  Ibs.  absolute  pressure  = 
1151°  +  32°  —  228°  =  955  units,  and  955-7-420  =  2.274  units  =  heat  condensed. 

The  loss  of  heat  from  a  naked  boiler  in  air  at  62°,  under  an  absolute  pressure  of  50 
Ibs.  per  sq.  inch,  was  5.8  units. 

Congelation,   and.   Liquefaction. 

Freezing  water  gives  out  140°  of  heat.  All  solids  absorb  heat  when 
becoming  fluid. 

Particular  quantity  of  heat  which  renders  a  substance  fluid  is  termed 
Ks  caloric  of  fluidity,  or  latent  heat. 

Temperature  of  Solidification  of  Several  Gases.    (Faraday. ) 


Cyanogen 31° 

Carbonic  Acid 72° 


Ammonia 103° 

Sulphurous  Acid. . .  105° 


Sulphuretted  Hydrogen,  123° 
Protoxide  of  Nitrogen..  148° 


Frigorific    ^Mixtures. 


MIXTURES. 

Parts. 

Fall  of 
Temperature. 

MIXTURES. 

Parts. 

Fall  of 
Temperature. 

Sea  salt  

Jl 

H 

il 

81 

10  J 

1! 

!l 

0                 0 
—  1  8  tO  —25 

—5  to  —  1  8 
—34  to  -50 

—40  to  —73 
—68  to  —91 
oto—  34 
o  to  —46 

Nitrate  of  ammonia. 
Water  
Snow  

!} 

[! 

ii 

:} 

0 

+50  to  +4 
—  10  to  —  60 
+50  to   —3 

+50  to   -7 

-{-50  to  —  10 
-f-5o  to  —  12 

-f-20  tO  —48 

-{-32  to  —51 

Nitrate  of  ammonia  .  .  . 
Snow,  or  pounded  ice.  . 
Muriate  of  ammonia  ) 
Nitrate  of  potash        } 
Snow,  or  pounded  ice.  . 
Phosphate  of  soda  
Nitrate  of  ammonia  .  .  . 
Dilute  mixed  acids  .... 
Snow 

Dilute  sulphuric  acid 
Sulphate  of  soda.  .  .  . 
Diluted  nitric  acid  .  . 
Nitrate  of  ammonia. 
Carbonate  of  soda... 
Water  

Sulphate  of  soda.  .  .  . 
Muriate  of  ammonia. 
Nitrate  of  potash  
Dilute  nitric  acid  .  .  . 
Phosphate  of  soda.  .  . 
Dilute  nitric  acid... 
Snow 

Crystallized  muriate  ) 
of  lime  J 

Snow 

Dilute  sulphuric  acid  .  . 
Phosphate  of  soda  
Nitrate  of  ammonia  .  .  . 
Dilute  nitric  acid  

Muriate  of  lime  
Potash 

Dilute  nitric  acid.  .  . 

Snow.  .  .  , 

A  Mixture  of  Solid  Carbonic  Acid  and  Sulphuric  Ether,  under  receiver  of  an  air 
pump,  under  pressures  of  .6  Ibs.  to  14  Ibs.,  exhibited  a  temperature  ranging  from 
—107°  to  — 166°5  which  is  the  most  intense  cold  as  yet  known.  (Faraday.) 


HEAT. 


IVEelt  ing-p  oint  s . 


METALS. 

0 

ALLOYS. 

o 

Aluminum.  

1400 

Lead  i,  Tin  4,  Bismuth  5  

Antimony      .            .   . 

810 

2.     "      3..  

^ 

Arsenic  

36  <; 

3,   "  2,  Bismuth  5     .   . 

476 

3,    '     i  

CC2 

Bronze         .                        .... 

IOQ2 

2,    '     i  (solder)  

Calcium  at  red  heat  

i,    '    2  (soft  solder)  

Copper                                     .  . 

IQO6 

i     *     i  

300 

^68 

Gold,  pure  

J2282 

x,   *     i,  Bism.  4,  Cadm.  i 

"      standard                    .  . 

(2590 
2156 

4       2               '             I        ... 

o76 

!2OOO 

'     8,         '         i  

22^O 

Zinc  i,  Tin  i  

o  nn 

u     ad  melting  

3479* 

!2200 
2450 

Fusible   Plugs. 

3700* 

"     6,    "    2... 

$3 

u    Wrought  

12700 
2QI2 

;;  7,  ;;  

^8i 

"     malleable  forge  

y       ... 
3509* 

*     8,    "   2  

410 

Lead  

608 

Various  Substances. 

Lithium    .                    ... 

^;6 

Ambergris  . 

—  •3Q 

108 

Nickel,  highest  forge  heat  

Glass  

2377 

Potassium         

1  16 

Ice  

Silver 

*J«J 

(1250 

Lard  

95 

Sodium  

11873 

194 

Nitro-  Glycerine  
Phosphorus  

45 

112 

Steel  

2500 

Pitch  

Tin  

446 

Saltpetre  

606 

Zinc  

680 

Spermaceti  

112 

Stearine  

ALLOYS. 

23O 

Lead  2,  Tin  3,  Bismuth  5  

212 

Tallow  

Q2 

"      i,    "     ?.         "         5... 

210 

Wax,  white.  .  . 

142 

Volume  of  Water,  Antimony,  and  Cast  iron,  in  the  solid  state,  exceeds 
that  of  the  liquid,  as  evidenced  by  the  floating  of  ice  on  water,  and  of  cold 
iron  on  iron  in  a  liquid  state. 

33 oiling-points.    ( Under  One  Atmosphere.) 


LIQUIDS. 

0 

LIQUIDS. 

0 

173 

3*5 

Ammonia  . 

140 

Water  

08 

146 

Whale  oil  

630 

Ether  

IOO 

CQ7 

SATURATED  SOLUTIONS. 

Mercury   

648 

Acetate  of  Soda  

255  8 

Milk 

"       "  Potash  

Nitric  acid  s  g  i  42.  

248 

Brine  

226 

U                U            (I        j    £ 

Carbonate  of  Soda 

Oil  of  Turpentine      

qjc 

"          "  Potash  

* 

Petroleum   rectified 

y 

Nitrate  of  Soda 

Phosphorus                      

554 

"       "  Potash 

Sea  water  average  

213.2 

Salt  common  

J 

57° 

Sulphuric  acid,  s.  g.  1.848  

59° 

VARIOUS  SUBSTANCES. 
Coal  Tar 

IOO 

Naphtha  .  .  . 

31 

Pressure  of   Saturated.  Vapors   nnder  Various  Temper- 

atures.    (Regnault.  ) 

Temper- 
ature. 

Water. 

Alcohol. 

Ether. 

Chloro- 
form. 

Temper- 
ature. 

Water. 

Alcohol. 

Ether. 

Chloro- 
form. 

o 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

O 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

32 

.089 

.246 

3-53 

— 

212 

14.7 

32.6 

95-17 

45-54 

50 

.178 

.466 

5-54 

2.52 

230 

20.8 

45-5 

120.9 

58.42 

68 

•337 

.851 

8.6 

3-68 

240.8 

25-37 

— 

137 

Turp'tine. 

86 

.609 

1.52 

12.32 

5-34 

248 

29.88 

62.05 

— 

4-97 

104 

i.  06 

2-59 

17.67 

-7-°4 

266 

39-27 

83.8 

— 

6.71 

122 

1.78 

4.26 

24-53 

10.14 

276.8 

46.87 

— 

— 

140 

2.88 

6.77 

33-47 

14.27 

284 

52.56 

109.1 

— 

8.94- 

158 

4-5i 

10.43 

44.67 

1  8.  88 

302 

69.27 

140.4 

— 

11.7 

I76 

6.86 

I5-72 

57-01 

26.46 

3°5-6 

73-07 

H7-3 

— 

I94 

10.  16 

23.02 

75-41 

35-03 

320 

89.97 

.— 

I3-I 

Boil 

Barom. 

ing-points  oj 

Boiling-point. 

f  Watet 

Barom. 

"  correspond 
62  and 

Boiling-point. 

'nff  to  A 
31  Ins 

Barom. 

[Ititudes  ofL 

Boiling-point. 

taromet 

Barom. 

er  between 

Boiling-point. 

26 
26.5 
27 

o 
204.91 

2°5'79 
206.67 

27-5 
28 
28.5 

0 

207.  55 
208.43 
209.31 

29 
29-5 
30 

0 
210.19 
211.07 
212 

30-5 
32 

0 
212.88 
213.76 

Boiling-point  of  Salt  water,  213.2°.  Water  may  be  heated  in  a  Digester 
to  400°  without  boiling. 

Fluids  boil  in  a  vacuum  with  less  heat  than  under  pressure  of  atmosphere. 
On  Mont  Blanc  water  boils  at  187°  ;  and  in  a  vacuum  water  boils  at  98°  to 
100°,  according  as  it  is  more  or  less  perfect. 

Water  may  be  reduced  to  5°  if  confined  in  tubes  of  from  .003  to  .005  inch  in  diam- 
eter: this  is  in  consequence  of  adhesion  of  water  to  surface  of  tube,  interfering  with 
a  change  in  its  state.  It  may  also  be  reduced  in  its  temperature  below  32°  if  it  is 
kept  perfectly  quiescent. 


Effect  upc 
Wedgewood's  zero  is  i< 
In  designation  of  degrees 
is  above  o;  but  when  below 

Degrees. 
Acetification  ends  ....    88 
Acetous  fermen-  )             g 
tation  begins..  J  *"    7 
Air  Furnace                 3300 

n.  "Various    Bodies 

377°  (Fahrenheit),  and  ea 
of  temperature,  symbol  +  is 
it,  symbol  —  must  be  prefix 
Degrees. 

Highest  natural  tern-  ) 
perature,  Egypt  .  .  j    "7 
India  -  rubber  and    j 
Gutta-percha  vul-  [    293 
canize  ) 

toy   Heat. 
2h  degree  =  130°. 

omitted  when  temperature 
ed. 

Degrees. 

Sea-  water  freezes  28 
Snow  and  Salt,  equal  ) 
parts                      } 

Spirits  Turpen.  freezes    14 
Steel,  faint  yellow.  .  .  .  430 
full        "      470 

Ammonia  (liq.)  freezes  —  46 
Blood  (hum.),  heat  of.    98 
"           freezes.    25 
Brandy  freezes  —  7 
Charcoal  burns  800 
Cold,  greatest  artific.  —  166 
"         "       natural  —56 
Common  fire                 790 

Iron,  bright  red  in) 
the  dark  J    752 

Iron,  red  hot  in  twi-  )    00 
light  j    884 

blue       .     .           55° 

full  blue  560 

Iron,  wrought,  welds.  .  2700 
Ignition  of  bodies  ....  750 
Combustion  of  do.  .  .  800 
Mercury  volatilizes...  680 
Milk  freezes  30 

dark  "         ...  600 

polished,  blue  ..  580 
"  straw  color  460 
Strong  Wines  freeze.  .     20 
S  ulph.  Acid  (sp.  grav.  ) 
1.641)  freezes  j      45 
Sulph.  Ether  freezes.  .—46 
Vinegar  freezes  28 
Vinous  ferment.  ..60  to    77 
Zinc  boils   1872 

Fire  brick.  .  .  .4000  to  5000 
Gutta-percha  softens.  .  145 
Heat,  cherry  red  1500 
"           "    (Daniell)ii4i 
««    bright  red  1860 

"  reifib*by}Io77 

Nitric  Acid  (sp.  grav.  ) 
1.424)  freezes  j      *5 
Nitrous  Oxide  freezes  —  1  50 
Olive-oil  freezes  36 
Petroleum  boils  306 
Proof  Spirit  freezes.  .  .  —7 

ral    LicLuicls    at   the 

Steam. 

hol  528    i  Ether  

44    white  2900 

Wood,  dried  340 

Volnrsae   of  Seve 

Steam.  I 

I  Water  1700  |  x  Alco 

ir    Boiling-point. 

Steam.  1                               Steam. 

.  .  298  1  i  Turpentine  .  .  193 

HEAT.  519 

Height   corresponding  to  Boiling-point   of*  3?u.re  "Water. 

Bailing-point  at  Level  of  Sea  =  212°. 


degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

211 
2IO 

209 
208 

521 

2°7 
206 
205 
204 

2625 
3156 
3689 
4224 

203 
202 
201 
200 

4761 
5300 
5841 
6384 

197 
196 

6929 
8576 

195 
194 

193 
192 

9129 
9684 
10241 
10800 

Correction  for  temperature  of  air  same  as  given  at  page  428  for  Elevation 
by  a  Barometer  by  multiplying  by  C. 
ILLUSTRATION.— If  water  boils  at  a  temperature  of  200°  and  C  =  136°, 

Then  6384  X  1.08  =  6894.72^^. 
"Underground.   Temperature. 

Mean  increase  of  underground  temperature  per  foot,  from  observations  in 
36  mines  in  various  and  extended  localities,  is  .01565°  =  1°  in  64  feet. 

Linear   Expansion  or  Dilatation  of  a   Bar  or  IPrism.  "by" 

Heat. 

For  i°  in  a  Length  of  100  Feet. 
METALS,  MINERALS,  ETC. 


Antimony. 007  22 

Bismuth 009  28 

Brass 012  5 

"     yellow 0126 

Brick ooi  44 

Cast  Iron °°74 

Cement 009  56 

Copper  from  o°  to  212° on  5 

"     from  32°  to  572° 00418 

Fire  brick 003  3 

Glass 005  74 

"    flint 00541 

"    tube 0612 

Gold — Paris  standard  annealed..  .0101 

"          u  "        unannealed  .0103 

Granite 005  25 

Gun  Metal — 16  copper 4-1  tin...  .0127 

"       "         8  copper  +  i  tin...  .0121 

Ice 033  3 

Iron,  forged 008 14 

"     from  o°  to  212° 007  88 


Inch. 

Iron,  from  32°  to  572° 003  26 

Iron  wire 008  23 

Lead 019 

Marble 005  66 

Palladium 006  67 

Platinum 005  71 

"       from  32°  to  572° 00204 

Sandstone 013 

" 00814 

Silver 012  7 

Speculum  metal 013 

Steel,  rod 007  63 

"     cast 0072 

"     tempered 008  26 

"     not  tempered 00719 

Tin 0145 

Water ooo  222  9 

White  Solder— tin  1  +  2  lead. .  .016  7 

Zinc,  forged 0207 

"     sheet 0196 

"     8  -\- 1  tin 017  9 


To  Compute  Linear  Expansion  of  a  Substance. 

Multiply  difference  of  the  temperatures  by  the  decimal  in  the  above  table.  Or, 
Divide  i  by  it,  and  quotient  will  give  proportion. 

Superficial  expansion  is  twice  linear,  and  cubical  is  three  times  linear. 

ILLUSTRATION  i.— A  rod  of  copper  100  feet  in  length  will  expand  between  tem- 
peratures of  32°  and  212°.  212  — 32  =  180  x  .0115  =  2.07  ins. 

2.— A  cube  of  cast  iron  of  i  foot  will  expand  in  volume  between  temperatures  ol 
62°  and  212°.  212  —  62  =  150,  and  150  X  .0074=: i.u,  which -?-ioo  for  i  foot  = 
.oin  foot,  and  .0111  X  3  =  .0333  foot. 

Some  solids,  as  Ice,  Cast  iron,  etc.,  have  more  volume  when  near  to  their  melting- 
point  than  when  melted.  This  is  illustrated  in  the  floating  of  solid  metal  in  a  liquid. 

Expansion    of  "Water. 

Water  expands  from  temperature  of  maximum  density  (see  page  520), 
39.1°,  to  46°,  at  which  degree  it  regains  its  initial  volume  of  32°,  and  from 
thence  it  expands  under  one  atmosphere  to  212°;  and  its  cubical  expansion 
is  .0466,  that  is,  its  volume  is  dilated  from  i  at  32°  to  1.0466  at  212°. 

Its  expansion  increases  in  a  greater  ratio  than  that  of  its  temperature. 


520 


HEAT. 


To  Compute  Density  of  "Water  at  a  given  Temperature. 


62.5X2 


-  =  approximate  density,  t  representing  temperature  of  water. 


500 


62.5X2 


ILLUSTRATION.— What  is  density  »*•**•*  _  ^  or  wetaW  of 

•f  pure  water  at  298°  ?  298  +  461  500          ,  cube  foot. 

500      "*"  298  -J-  461 


Temp. 

T  ] 

Expansion. 

Expan 

Temp. 

sion    of 

Expansion. 

Watei 

Temp. 

•.    (Dalton.) 

Expansion.  II    Temp. 

ExpansiCii. 

0 
22 

1.0009 

X 

i 

0 

52 
72 
92 

I  OOO2I 

i.ooi  8 
1.00477 
*  Greatest  d 

o 

112 
132 
152 

ensity  39.1° 

o 
i.  008  8           172 
1.01367          192 

1.01934     II        212 

1-02575 
1.03265 
1.0466 

Hence,  at  72°,  water  expands =  555.  ssth  part  of  its  original  bulk. 

Expansion   of  Liquids   from   32°  to   S12°.     Volume  at  32°  =  i 


LIQUIDS. 

Vo  ume  at  212°. 

LIQUIDS. 

Volume  at  212°. 

Alcohol  

.  ii 

Olive  oil  

i.  08 

Linseed  oil    . 

08 

i.  06 

Mercury  

.015  4 

"        ether  

1.07 

"           212®  tO  3Q2^ 

I  07 

"       392°  to  572°  .... 

Ol8867  Q 

Water  

i  046  6 

Nitric  acid.  .  . 

.11 

Water  sat.  with  salt.  .  . 

1.05 

Expansion  of  Gases   from   32°  to  S1S°.    Volume  at  32°  =  i. 


Volume 

Volume 

GASES. 

at  212°. 

GASES. 

at  212°. 

Air  i  Atmosphere 

1.36706 

Nitrous  oxide  .  .  .  i  Atmosphere.  . 

1-3179 

Hydrogen  .  .  .  .  i 

1.36964 
1.36613 

Sulphurous  acid,  i           " 
1.16       " 

1.3903 
1.398 

Carbonic  acid,  i 

3.32 

1.36616 
1.37099 
1.384  55 

Carbonic  oxide  ..  i            " 
Cyanogen  i            " 

1.3669 
1.3877 

Expansion  of  Gases  is  uniform  for  all  temperatures. 
Volume  of  One  Pound  of  Various  Gases  at  32°  under  one  Atmosphere. 


Cube  feet. 

Air 12.387 

Carbonic  acid 8. 101 

Ether,  vapor 4. 777 


Cube  feet. 

Hydrogen 178.83 

Nitrogen 12.753 

Olefiant 12.58 


Cube  feet. 

Oxygen 12.205 

Mercury 1.776 

Steam 19-913 


1.002 
1.004 
I.OO7 


Expansion   of  Air.     (Dalton.) 
T.mp. 


1. 021 
1.032 
1.043 
1-055 


1.066 
1.077 
1.089 
1.099 


I.IIO 
I.I2I 
I.I32 
I  142 


100 
2OO 
212 
302 


1-739 
1.912 
2.028 
2.319 


To    Compute   Volume   of  a   Constant   "Weignt   of  A.ir   or 
Permanent    GJ-as    for   any    Pressure. 

When  volume  at  a  given  pressure  is  known,  temperature  remaining  con- 
stant. RULE. — Multiply  given  volume  by  given  pressure  and  divide  by 
new  pressure. 

EXAMPLE. — Pressure  at  212°=  18.92  lbs.  per  sq.  inch,  and  volume  16.91  cube  feet; 
What  is  volume  at  pressure  of  13.86  lbs. 

16.91  X  13.86-1-18.92  =  12.39  cube  feet. 


HEAT. 


521 


Relative  Densities  of  some  Vapors. 
Water  i.    Alcohol  3.59.    Ether  4.  id    Spirits  of  Turpentine  8.06.    Sulphur  3. 59. 

Volume,  [Pressure,  and  Density  of  Air  at  Various  Tem- 
peratures. 

Volume  and  Atmospheric  Pressure  at  62°  =  i. 


Temper- 
ature. 

Volume  of 
lib.  of  air  at 
atmospheric 
pressure  of 
14.7  Ibs. 

Pressure 
of  a  given 
weight  of 
air. 

Density,  or 
weight  of  one 
cube  foot 
of  air  at 
14.7  Ibs. 

Temper. 

ature. 

Volume  of 
lib.  of  air  at 
atmospheric 
pressure  of 
14.7  Ibs. 

Pressure 
of  a  given 
weight  of 
air. 

Density,  or 
weight  of  one 
cube  foot 
of  air  at 
14  7  Ibs. 

0 

Cube  feet. 

Lbs.  per 
Sq.  Inch. 

Lbs. 

0 

Cube  feet. 

Lbs.  per 
Sq.  Inch. 

Lbs. 

0 

"•583 

12.96 

.086331 

360 

20.63 

23.08 

.048476 

32 

12.387 

13-86 

.080728 

38o 

21.131 

23.64 

•047323 

40 

12.586 

14.08 

•079439 

400 

21.634 

24.2 

.046223 

50 

12.84 

14.36 

.077884 

425 

22.262 

24.9 

.04492 

62 

13.141 

14.7 

.076097 

450 

22.89 

25.61 

.043686 

1° 

I3-342 

14.92 

•07495 

475 

23-518 

26.31 

.042  52 

80 

13-  593 

15-21 

•073565 

500 

24.146 

27.01 

.041414 

90 

13-845 

15-49 

.07223 

525 

24-775 

27.71 

.040364 

100 

14.096 

15-77 

.070942 

550 

25.403 

28.42 

•039365 

120 

14.592 

16.33 

.0685 

575 

26.031 

29.12 

.038415 

140 

iS-i 

16.89 

.066221 

600 

26.659 

29.82 

.037  51 

160 

15-603 

17-5 

.064088 

650 

27-9I5 

3J-23 

.035  822 

180 

16.106 

1  8.  02 

.06209 

700 

29.171 

32-635 

.03428 

200 

16.606 

18.58 

.06021 

750 

30.428 

34-04 

.032865 

210 

16.86 

18.86 

.059313 

800 

31-684 

35-445 

.031  561 

212 

16.91 

18.92 

•059  '35 

850 

32.941 

36-85 

.030358 

220 

17.111 

19.14 

.058442 

900 

34-  197 

38-255 

.029242 

2£ 

260 

17.612 
18.116 

19.7 

20.27 

.056774 
•0552 

950 

1000 

36.811 

39.66 
41.065 

.028  206 
.027241 

280 

18.621 

20.83 

.05371 

1500 

49-375 

55-U5 

.020295 

300 

19.121 

21.39 

.052297 

2000 

61.94 

69.165 

.016  172 

320 

19.624 

21-95 

.050959 

2500 

74-  565 

83.215 

013441 

340 

20.126 

22.51 

.049686 

3000 

87-13 

97.265 

.011499 

To  Compxite  Volume  of  a  Constant  "Weight  of  Air  or 
other  Permanent  Q-as  for  any  other  Pressure  and. 
Temperature. 

When  volume  is  known  at  a  given  pressure  and  temperature.  RULE. — Mul- 
tiply given  volume  by  given  pressure,  and  by  new  absolute  temperature, 
and  divide  by  new  pressure,  and  by  given  absolute  temperature. 

EXAMPLE.— Given  volume  16.91  cube  feet,  pressure  13.86  Ibs.,  and  temperature 
32°;  what  is  volume  at  this  temperature? 
Temperature  for  volume  16.91  =  212°. 


16.91  X  I3-86X  32 +  46 1 -hi  3. 86  X  212 +  461  =  12. 39  cube  feet. 

To  Compute  Pressure  of  a  Constant  \7VTeight  of  Air  or 
other  Permanent  G-as  for  any  other  Volume  and 
Temperature. 

When  pressure  is  known  for  a  given  volume  and  temperature.  RULE. — 
Multiply  given  pressure  by  new  absolute  temperature,  and  divide  by  given 
absolute  temperature. 

NOTE.— Absolute  temperature  is  found  by  adding  461°  to  temperature. 

EXAMPLE.— Given  pressure  13.86  Ibs.,  and  temperature  at  this  volume  32°;  what 
is  pressure  at  temperature  of  212°? 

13. 86  X  212  +  461  -r-  32  -j-  461  =  18. 92  Ibs. 
Xx* 


5  22  HEAT. 

To   Compute   "Volume   of  a   Constant   "Weight   of  Air   o* 
other    Permanent    Q-as   at   any    Temperature. 

When  volume  at  a  given  temperature  is  known,  pressure  being  constant. 
RULE.  —  Multiply  given  volume  by  new  absolute  temperature,  and  divide 
by  given  absolute  temperature. 

Absolute  zero-point  by  different  thermometrical  scales  is:  Fahrenheit  —461.2°; 
Reaumur  —  219.2°;  Centigrade  —  274°. 

EXAMPLE.—  Volume  of  i  Ib.  air  at  32°  =  12.387  cube  feet;  what  is  its  volume  at 

212°  ?  _        _ 

12.387  x  2124-461-4-324-  461  =  16.91  cube  feet. 

To   Compute   Increased   "Volume  of  a   Constant  "Weight 
of  -Aar. 

When  initial  volume  at  62°  =  i  under  i  atmosphere.    RULE.  —  To  given 
temperature  add  461,  and  divide  sum  by  523  (62  +  461). 
EXAMPLE.—  Assume  elements  of  preceding  case. 

212°  -f  461  -T-  523  =  1.287  comparative  volume  to  i. 

To  Compute  Pressure  of  a  Constant  Weight  of  Air  or 
other  Gtas  at  63°,  and  at  14.7"  Ifos.  Pressure  per  Sq.  In., 
•with  Constant  "Volnme,  for  a  given  Temperature. 

RULE.  —  Add  461  to  given  temperature,  and  divide  sum  by  35.58. 
EXAMPLE.—  Temperature  is  212°;  what  is  pressure? 
212  +  461-7-35.58  =  18.92  Ibs. 

To    Compute    "Volume,   Pressure,   Temperature,  and 
Density   of  Air. 


a.  7  1     .j       =  D.    t  representing  temperature,  p  pressure  in  Ibs.  per  sq.  inch,  V  vol- 
ume in  cube  feet,  and  D  weight  ofi  cube  foot  at  14.7  Ibs.  per  sq.  inch. 

Product  of  volume  and  pressure  of  a  constant  weight  of  air,  or  any  other 
permanent  gas,  is  equal  to  product  of  absolute  temperature  and  a  coefficient, 
determined  for  each  gas  by  its  density. 
Or,  Vp 


Coefficients,  as  determined  by  volumes  and  consequent  densities.* 


Air  ..................  2.71 

Carbonic  acid  ........  4.  14 

Ether,  vapor  .........  7.02 


Hydrogen  ..........  1875  I  Oxygen  ............    2.09 

Nitrogen  ..........  2.63        Mercury  ............  18.88 

Oleflant  ...........  2.67     |  Steam  ..............    1.68 


*  See  D.  K.  Clark,  London,  1877,  P«J?e  349. 
Decrease   of  Temperature   by   Altitudes. 

In  clear  sky.  With  cloudy  tky. 

From  i  to   i  ooo  feet  ...............  i°  in  139  feet  ...............  i°  in  222  feet. 

I  "  10000  "  ...............  1°  "  288  "  ...............  1°  "  331   " 

I  "  20000  "  ...............  lO  *'  365   "  ...............  1°.  "  468  " 

To  Compute  Temperatnre  to  which  a  Substance  of  & 
given  Length  or  Dimension  must  be  Submitted  or 
Rednced,  to  give  it  a  Q-reater  or  Less  Length  or  "Vol- 
nme  by  Expansion  or  Contraction. 

Lineal.  —  When  Length  is  to  be  increased.     —^—4-t  =  f.    L  and  I  represent' 

c> 

ing  lengths  of  increased  and  primitive  substance  in  like  denominations,  T  and  t  tern 
peratures  o/L  and  I,  and  C  expansion  of  substance  for  each  degree  of  heat, 


HEAT.  523 

ILLUSTRATION.—  A  copper  rod  at  32°  is  100  feet  in  length;  to  what  temperature 
must  it  be  subjected  to  increase  its  length  1.1633  ins.  ? 
Expansion  for  a  length  of  100  feet  of  copper  for  i°  =  .0115. 


. 
.0115  .0115 

When  Length  is  to  be  reduced.    —  --  T  =  L 
ILLUSTRATION.—  Take  elements  of  preceding  case. 

,*>».  I633-  1200  _I33.l60=IOI.l6_I33.l6  =  32Q 
.0115 

To  Reduce  Degrees  of*  Fahrenheit  to  Reaumur  and.  Cen~ 
tigrade,  and    Contrariwise; 

Fahrenheit  to  Reaumur.  If  above  zero.  —  Multiply  difference 
between  number  of  degrees  and  32  by  4,  and  divide  product  by  9. 

Thus,  212°  —  32°  =  180°,  and  180°  X  4  +  9  =  80°. 

If  below  zero.—  Add  32  to  number  of  degrees  ;  multiply  sum  by  4,  and 
divide  product  by  9. 

Thus,  —40°  +  32°  =  72°,  and  72°  X  4  -*-  9  —  —32°- 

Reaumur  to  Fahrenheit.  If  above  freezing-point.  —  Multiply 
number  of  degrees  by  9,  divide  by  4,  and  add  32  to  quotient. 

Thus,  80°  X  9  -P-  4  —  180°,  and  180°  -f  32  =  212°. 

If  below  freezing-point.—  Multiply  number  of  degrees  by  9,  divide  by  4, 
and  subtract  32  from  product. 

Thus,  —32°  X  9  -s-  4  =  72°,  and  72°  —  32  =  —40°. 

Fahrenheit  to  Centigrade.     If  afiove  zero.—  Multiply  difference 
between  number  of  degrees  and  32  by  5,  and  divide  product  by  9. 
Thus,  212°  —  32°  X  5^-9  =  i8o°X5-:-9=ioo0. 

If  below  zero.—  AM  32  to  number  of  degrees,  multiply  sum  by  5,  and 
divide  product  by  9. 

Thus,  —40°  -f  32°  X  5  -?•  9  =  72°  X  5  -=-  9  =  —40°- 

Centigrade  to  Fahrenheit.      If  above  freezing  -point,.  —  Multiply 
number  of  degrees  by  9,  divide  product  by  5,  and  add  32  to  quotient. 
Thus,  100°  x  9-1-5  =  180°,  and  1  80°  +  32  =  212°. 

If  below  freezing-point.  —  Multiply  number  of  degrees  by  9,  divide  product 
by  5,  and  take  difference  between  32  and  quotient. 
Thus,  —10°  X  9  -r-  5  =  1  8°,  and  18°  <v  32  =  14°. 

Reaumur  to  Centigrade.  —  Divide  by  4,  and  add  product. 

Thus,  80°  -r-  4  =  20°,  and  20°  -f-  80°  =  100°. 

Centigrade  to  Reaumur.  —  Divide  by  5,  and  subtract  product. 
Thus,  100°  -j-  5  =  20°,  and  100°  —  20  =  80°. 

Corresponding  Degrees  upon  the  Three  Scales. 


Fahr. 

Cent. 

Reaum.   II     Fahr. 

Cent. 

Reaum.  II     Fahr. 

Cent. 

Reaum. 

212 

100 

80       II       32 

o 

o        [I     —40 

—40 

—32 

To    Compxate    Expansion    of*  Fluids   in    Volume. 

RULE. — Proceed  by  preceding  formulas  for  computing  length  of  a  sub- 
stance.   Substitute  V  and  v  for  volume,  instead  of  L  and  /,  the  lengths. 


524 


HEAT,    VENTILATION,   BUILDINGS,   ETC. 


ILLUSTRATION. — A  closed  vessel  contains  6  cube  feet  of  water  at  a  temperature  of 
40°;  to  what  height  will  a  column  of  it  rise  in  a  pipe  1.152  ins.  in  area,  when  it  is 
exposed  to  a  temperature  of  130°  ? 

1. 152  ins.  =  .008  sq.  foot    C  for  water  =  .000  222  9. 


6  (l  -f-  .OOO  222  9  (130  - 


-  40))  —  6. 120  366.  and  — — =  15-047  lineal  feet 

.008 


Results  of 

Duration 
of  Agitation. 

Tern 

Experiments  w 
Temper 

Increase 
of  Temperature. 

peratxire 

ith  Water  enc 
ature  of  Air,  f 

Duration 
of  Agitation. 

toy   Agita 

losed  in  a  Vess 
>o.50;  of  Wate 

Increase 
of  Temperature. 

tion. 
el  and  violentl 

",59-5°. 
Duration 
of  Agitation. 

y  Agitated. 

Increase 
of  Temperature. 

Hour. 
•5 

I 

o 

JO 

14-5 

Hours. 
3 

O 
19-5 
29-5 

Hours. 

I 

0 

39-5 
42-5 

VENTILATION. 
Buildings,  -A.partirLen.ts,  etc. 

In  Ventilation  of  Apartments. — From  3.5  to  5  cube  feet  of  air  are  required 
per  minute  in  winter,  and  5  to  ip  feet  in  summer  for  each  occupant.  In 
Hospitals,  this  rate  must  be  materially  increased. 

Ventilation  is  attained  by  both  natural  draught  and  artificial  means.  In 
first  case  the  ascensional  force  is  measured  by  difference  in  weight  of  two 
columns  of  air  of  same  height,  the  height  being  determined  by  total  difference 
of  level  between  entrance  for  warm  air  and  its  escape  into  the  atmosphere. 
The  difference  of  weight  is  ascertained  from  difference  of  temperatures  of 
ascending  warm  air  and  the  external  atmosphere,  as  by  Table,  page  521,  or 
by  formula,  page  522. 

Volumes  of  Air  Discharged  through  a  Ventilator  One 
Foot  Sq.uare  of  Opening,  at  Various  Heights  and. 
Temperatures. 


Height  of 
Ventilator 

Excess  of  Temperature  of  Apartment 
above  that  of  External  Air. 

Height  of 
Ventilator 
from 

Excess  of  Temperature  of  Apartment 
above  that  of  External  Air. 

Ba8e-iine. 

5° 

10° 

»5° 

20° 

25° 

3o° 

Base-line. 

5° 

10° 

15° 

20° 

25° 

30° 

Feet. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

Feet. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

C.ft. 

10 

116 

164 

200 

235 

260 

284 

35 

218 

306 

376 

436  486 

53i 

15 

142 

202 

245 

284 

3i8 

348 

40 

235 

329 

403 

465 

5i8 

57° 

20 

164 

232 

285 

330 

368 

404 

45 

248 

348 

427 

493 

55i 

605 

25 

30 

184 

2OI 

260 
284 

3l8 

347 

368 
403 

410 
450 

450 
493 

50 

55 

260 
270 

367 
385 

450 
472 

5i8     579 
54i  1  605 

635 
663 

Velocity  of  draft  having  been  ascertained  for  any  particular  case,  together  with 
volume  of  air  to  be  supplied  per  minute,  sectional  area  of  both  air  passages  may  be 
computed  from  these  data. 

Heating   "by   Hot  "Water. 

One  sq.  foot  of  plate  or  pipe  surface  at  200°  will  heat  from  40  to  100  cube 
feet  of  enclosed  space  to  70°  where  extreme  depression  of  temperature  is 
—10°. 

The  range  from  40  to  100  is  to  meet  conditions  of  exposed  or  corner 
buildings,  of  buildings  less  exposed,  as  intermediate  ones  of  a  cluster  or 
block,  and  of  rooms  intermediate  between  the  front  and  rear. 

When  the  air  is  in  constant  course  of  change,  as  required  for  ventilation 
or  occupation  of  space,  these  proportions  are  to  be  very  materially  increased 
as  per  following  rules. 


HEAT,  VENTILATION,  AND   HEATING.  52$ 

In  determining  length  of  pipe  for  any  given  space  it  is  proper  to  include 
in  the  computation  the  character  and  occupancy  of  the  space.  Thus,  a 
church,  during  hours  of  service,  or  a  dwelling-room,  will  require  less  service 
of  plate  or  length  of  pipe  than  a  hallway  or  a  public  building. 

Reduction  of  Heat  by  Surfaces  of  Glass  or  Metal. — In  addition  to  the 
volume  of  air  to  be  heated  per  minute  for  each  occupant,  1.25  cube  feet  for 
each  sq.  foot  of  glass  or  metal  the  space  is  enclosed  with  must  be  added. 
The  communicating  power  of  the  glass  and  metal  being  directly  proportion- 
ate to  difference  of  external  and  internal  temperature  of  the  air.  Thus,  80 
feet  of  glass  will  reduce  100  feet  of  air  per  minute. 

When  Pipes  are  laid  in  Trenches  in  the  Earth. — The  loss  of  heat  is  es- 
timated by  Mr.  Hood  at  from  5  to  7  per  cent. 

Circulation  of  Water  in  Pipes.— In  consequence  of  the  complex  forms  of 
heating-pipes  and  the  roughness  of  their  internal  surface,  it  is  impracticable 
to  apply  a  rule  to  determine  the  velocity  of  circulation,  as  consequent  upon 
difference  of  weights  of  ascending  and  descending  columns  of  the  water. 

For  a  difference  of  temperature  in  the  two  columns  of  30°  (190°  —  160°) 
and  a  height  of  20  feet,  the  velocity  due  to  the  height  would  be  3.74  feet. 
In  practice  not  .3,  and  hi  some  cases  but  .1,  would  be  attained. 

In  Churches  and  Large  Public  Rooms,  with  ordinary  area  of  doors  and  windows 
and  moderate  ventilation,  a  large  amount  of  heat  is  generated  by  the  respiration 
of  the  persons  assembled  therein. 

In  these  cases  it  is  not  necessary  to  heat  the  air  above  55°,  and  a  rule  that  will 
meet  the  ordinary  ranges  of  temperature  from  10°  is  to  divide  volume  in  cube 
feet  by  200,  and  quotient  will  give  area  of  plate  in  sq.  feet  or  length  of  4-inch  pipe 
in  lineal  feet. 

Volume  of  Air  required  per  Hour  for  each  Occupant  in  an  Enclosed  Space, 
(General  Morin.) 


Cube  Feet. 


Hospitals ....  2100  to  3700 
Workshops  . .  2100  "  3500 


Cube  Feet. 

Lecture-rooms  1000  to  2100 
Theatres: 1400  "  1800 


Cube  Feet. 

Prisons 1800 

Schools 424  to  1060 


To  Compute  Length  of  Iron  IPipe  reqxiired.  to  Heat  Air 
in  an  Enclosed.  Space. 

By  Hot  Water. 

RULE. — Multiply  volume  of  air  to  be  heated  per  minute  in  cube  feet  by 
difference  of  temperatures  in  space  and  external  air,  divide  product  by  differ- 
ence of  temperatures  of  surface  of  pipe  and  space,  multiply  result  by  follow- 
ing coefficients,  and  product  will  give  length  of  pipe  in  feet. 

For  diameter  of  4  ins.  multiply  by  .5  to  .55,  for  3  ins.  by  .7  to  .75,  and  for 
2  ins.  by  i  to  i.i. 

A  pipe  4  ins.  in  diameter,  .375  inch  thick,  and  i  foot  in  length  has  an 
area  of  internal  surface  of  1.05  sq.feet. 

EXAMPLE.— Volume  of  a  room  of  a  protected  dwelling  is  4000  cube  feet;  what 
length  of  4  ins.  pipe,  at  200°,  is  necessary  to  maintain  a  temperature  of  70°,  when 
external  air  is  at  o°  ? 

4000  X  ^  X.  4  =  862  feet. 
200  —  70 

In  computing  length  of  pipe  or  surface  of  plate  it  is  to  be  borne  in  mind 
that  the  coefficients  here  given  and  computation  in  following  table  are  based 
upon  a  ventilation  or  change  of  air  ordinarily  of  3.5  to  5  cube  feet  per 
person,  and  from  5  to  10  cube  feet  in  summer  per  minute.  Hence,  when 
the  ventilation  is  restricted  the  coefficient  may  be  correspondingly  .in- 
creased. 


526 


HEAT   AND    HEATING. 


Ijength.8    of  Four-Inch.   ripe   to   Heat  1OOO   Cube   Feet 
of  Air  per   IMiivute.     (Chas.  Hood.) 

Temperature  of  Pipe  200°. 


Temperature 

Temperature  of  Building. 

External  Air. 

45° 

50° 

55°  |  60° 

65° 

70° 

75° 

80° 

85° 

90° 

0 

Feet. 

Feet. 

Feet,  j  Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

10 

126 

174  |  200 

229 

259 

292 

328 

367 

409 

16 

20 

105 
91 

127 
112 

135 

176 
1  60 

204 
187 

223 
2IO 

265 
247 

300 
281 

337 

378 
358 

26 

69 

9° 

112 

136 

162 

I90 

220 

253 

288 

327 

30 
36 

54 

75 
52 

97 
73 

120 
96 

H5 
1  20 

173 
147 

202 

S 

269 
239 

307 
276 

40 

18 

37 

58 

80 

104 

129 

157 

187 

220 

255 

SO 

— 

19 

40 

62 

86 

112 

140 

171 

204 

Proper   Temperatures   of  Enclosed.    Spaces. 


SPACES. 

Temper- 
ature 
required. 

SPACES. 

Temper- 
ature 
required. 

Work-rooms  manufactories  etc 

O 

Dwelling-rooms  

0 

Churches  and  like  spaces    ... 

Hot-houses 

80 

Schools  lecture-rooms 

5* 

Drying-rooms  when  filled     .  . 

80 

Halls,  shops,  waiting-rooms,  etc. 
Dwelling-rooms  .  .  . 

<£ 

6* 

"          "       for  curing  paper.. 

7° 
1  20 

Boiler. 

Boiler  for  steam-heating  should  be  capable  of  evaporating  as  much  water 
as  the  pipes  or  surfaces  will  condense  in  equal  times.  Mr.  Hood  recom- 
mends that  6  sq.  feet  of  direct  heating-surface  of  boiler  should  be  provided 
to  evaporate  a  cube  foot  per  hour.  Adopt  mean  weight  of  steam  of  5  Ibs. 
above  pressure  of  atmosphere,  or  20  Ibs.  absolute  pressure,  condensed  per  sq. 
foot  of  pipe  per  degree  of  difference  of  temperature  per  hour,  viz.,  .002  35  Ib. 
(as  given  by  D.  K.  Clark),  the  quantity  of  pipe  or  plate  surface  that  would 
form  a  cube  foot  of  condensed  water  per  hour,  weight  of  like  volume  of 
water  62.4  Ibs.,  would  be,  per  i°  difference  of  temperature, 

62. 4 -T-. 002 35  =  26 5 50  sq.feet,  and  for  differences  of  168°,  158°,  148°,  and  108°, 
required  surface  would  be  respectively  (26550-7-168  =  158)  158, 168,  179,  and  246 
sq.  feet. 

Henoe,  assuming,  as  previously  stated,  that  4  sq.  feet  of  direct  and  effec- 
tive heating  boiler-surface,  or  its  equivalent  flue  or  tube  surface,  will  evap- 
orate i  cube  foot  of  water  per  hour,  158  sq.  feet  of  steam-pipe  or  plate  will 
require  4  sq.  feet  of  direct  surface,  etc.,  for  a  temperature  of  60°,  and  cor- 
respondingly for  other  temperatures. 

Boiler-power. — One  sq.  foot  of  boiler-surface  exposed  to  direct  action  of 
fire,  or  3  sq.  feet  of  flue-surface,  will  suffice,  with  good  coal,  for  heating  50 
sq.  feet  of  4-inch,  66  of  3-inch,  and  100  of  2-inch  pipe.  Mr.  Hood  assigns  the 
proportion  at  40  feet  of  4-inch  pipe  for  all  purposes.  Usual  rate  of  com- 
bustion of  coal  is  10  or  n  Ibs.  per  sq.  foot  of  grate-surface,  and  at  this  rate, 
20  sq.  ins.  of  grate  suffice  for  heating  40  feet  of  4-inch  pipe. 

Four  sq.  feet  of  direct  heating  boiler-surface,  or  equivalent  flue  or  tube 
surface,  exposed  to  direct  action  of  a  good  fire,  are  capable  of  evaporating 
i  cube  foot  of  water  per  hour. 

According  to  M.  Grouvelle,  i  sq.  meter  of  pipe-surface  (10.76  sq.  feet),  heated  to 
60°  an  ordinary  room  alike  to  a  library  or  office,  of  from  90  to  100  cube  meters 
(3178  to  3531  cube  feet). 


HEAT,  WARMING   BUILDINGS,  ETC. 


527 


If  a  workshop  to  be  heated  to  a  high  temperature,  i  sq.  meter  (10.76  sq.  feet)  of 
surface  is  assigned  to  70  cube  meters  (2472  cube  feet)  =  4.  35  sq.  feet  or  5.11  lineal 
feet  of  4  inch  pipe  per  1000  cube  feet. 

For  heating  workshops,  having  a  transverse  section  of  260  sq.  feet,  with  a  window- 
surface  of  one  sixth  total  surface,  it  is  customary  in  France  to  assign  1.33  sq.  feet 
of  iron  pipe  surface  per  lineal  foot  of  shop  =.  5.2  sq.  feet  per  1000  cube  feet. 

Illustrations  of  extensive  Heating  by  Steam.     (R.  Briggs,  M.  I.  C.  E.) 
i.  Total  number  of  rooms,  including  halls  and  vaults  ............     286 

"     Area  of  floor  surface  ................................  137  370  sq.  feet. 

"     Volume  of  rooms  ..................................  i  923  500  cube  feet. 

"     Number  of  occupants  .................................     650 

Maximum  average  of  occupants  at  any  time  ..................  1300 

Volume  per  occupant,  excluding  vaults  ......................   1443  cube  feet. 

Boilers.  —  8  with  173  sq.  feet  of  grate  surface  and  8000  sq.  feet  of  heating  surface. 
Furnishing  steam  in  addition  to  the  above,  to  operate  the  elevators  and  electric 
dynamos,  elevating  water,  and  supplying  steam  to  heat  a  distant  building,  requiring 
one  third  of  their  capacity. 

By    Steam. 

To  Compute  .Length  of  Iron  Pipe  req.ui.red.  to  Heat  Air 
in  an  Enclosed.  Space,  with  JSteain  at  5  libs,  per  Sq.. 
Inch  a"bove  Pressure  of  Atmosphere. 

RULE.—  Multiply  volume  of  air  in  cube  feet  to  be  heated  per  minute,  by 
difference  of  temperature  in  space  and  external  air,  divide  product  by  coeffi- 
cients in  preceding  table,  and  quotient  will  give  length  of  4-inch  pipe  in 
lineal  feet,  or  area  of  plate-surface  in  sq.  feet. 

Temperature  of  steam  at  5  Ibs.  -f-  pressure  =  228°.  Hence,  if  temperature  of  space 
required  is  60°,  70°,  80°,  or  120°,  the  differences  will  be  168°,  158°,  148°,  and  108°, 
which  for  a  coefficient  of  .5,  as  given  in  rule  for  hot  water,  would  be  336,  316,  296, 
and  216,  for  a  pipe  4  ins.  in  diameter,  and  for 

60°  70°  80°  120° 

3-inch  pipe  ........  252       237       222       162 

2     '        "    ........  168        158        148        108 

i     *         "    ........  84          79          74          54 

ILLUSTRATION.—  Volume  of  combined  spaces  of  a  factory  is  50000  cube  feet;  what 

surface  of  wrought  iron  plate  at  200°  is  necessary  to  maintain  a  temperature  of  50° 

when  external  air  is  at  o°  ? 


50000  Xo 
200  —  50 


= 


Coal    Consumed   per  Hour   to  Heat   1OO  Feet   of  Pipe. 

(Chas.  Hood.) 
Difference  of  Temperature  of  Pipe  and  Air  in  Space,  in  Degrees. 


ijiam.  01 
Pipe. 

ISO 

MS 

140 

135 

130 

125 

1  20 

"5 

no 

105 

IOO 

95 

9° 

85 

80 

IDB. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

I.I 

i.i 

1.  1 

i 

I 

•9 

•9 

•9 

.8 

.8 

•7 

•7 

•7 

.6 

.6 

2 

2-3 

2.2 

2.2 

2.1 

2 

1.9 

1.8 

1.8 

i-7 

1.6 

i-5 

1.4 

1.4 

i-3 

1.2 

3 

3-5 

3-4 

3-3 

3-i 

3 

2.9 

2.8 

2.7 

2-5 

2.4 

2-3 

2.2 

2.1 

2 

1.8 

4 

4-7 

4-5 

4.4 

4.2 

4.1 

3-9 

3-7 

3-6 

3-4 

3-2 

3-i 

2.9 

2.8 

2.6 

2-5 

To  warm  a  factory,  according  to  M.  Claude!,  43  feet  in  width  by  10.5  high,  a  single 
line  of  hot- water  pipe  6.25  ins.  in  diameter  per  foot  of  length  of  room,  appears  to  be 
sufficient,  temperature  in  pipe  being  from  170°  to  180°.  Also,  water  being  at  180°, 
and  air  at  60°,  making  a  difference  of  120°,  it  is  convenient  to  estimate  from  1.5 
to  1.75  sq.  feet  of  water-heated  surface  as  equivalent  to  one  sq.  foot  of  steam-heated 
surface,  and  to  allow  from  8  to  9  sq.  feet  of  hot-water  pipe-surface  per  1000  cube 
feet  of  room. 

M.  Grouvelle  states  that  4  sq.  feet  of  cast-iron  pipe-surface,  whether  heated  by 
steam  or  by  water  at  176°  to  194°,  will  warm  1000  cube  feet  of  workshop,  main- 
taining a  temperature  of  60°.  Steam  is-  condensed  at  rate  of  .328  Ib.  per  sq.  foot 
per  hour. 


528 


HEAT,  WASHING   BUILDINGS,  ETC. 


2.  (R.  L.  Greene.)  Length  of  fronts  of  buildings. . . . , 2  ooo  lineal  feet 

Total  volume  of  rooms '. 2  574  084  cube  feet. 

Radiating  surfaces,  direct,  10804 j  ,4 100  sq  feet. 

indirect,  23  296 >  " 

Boilers Grate-surface 180 

Heating  surface 5  863        " 

Volnxne  of  Air  Heated  toy  Radiators  ;    Consumption  of 
Coal ;    A.reas  of  GJ-rate  arid  Heating-svirfaee  of  J3  oiler. 

(RoPt  Briggs.) 
Per  100  Sq.  Feet  of  Warming-surface  of  Radiator. 


Pressure  of  steam  per  sq  inch  -f- 
atmosphere  in  Ibs 

_ 

3 

10 

30 

60 

Heat  from  radiators  per  minute 
in  units                                      . 

456 

486 

537 

642 

74i 

Volume  of  air  heated  i°  per  min- 
ute in  cube  feet 

25110 

26772 

29570 

35352 

40803 

Efficiency  of  radiators  in  ratio.  .  .  . 
Coal  consumed  per  hour  in  Ibs.... 
Area  of  grate  consuming  8  Ibs. 

i 
304 
•38 

i.  066 
3-24 
.405 

1.178 

3.58 
.448 

1.408 

4.28 

1.625 
4-94 

do    12  Ibs  ....          

.208 

•  357 

.412 

Heating  surface  of  boiler  ;  coal 
consumed  per  hour  Xa.8  in  sq.feet 
8  Ibs.  X28  

8.512 

22  4 

9.072 

22  A, 

10.02 
22.4 

11.98 

13-83 

12  Ibs.  X  2.  8... 

^.6 

n.6 

^.6 

By  Hot-j^ir   Furnaces   or   Stoves. 

A  square  foot  of  heating  surface  in  a  hot-air  furnace  or  stove  is  held  to 
be  equivalent  to  7  sq.  feet  of  hot  water  pipe. 

M.  Peclet  deduced  that  when  the  flue-pipe  of  a  stove  radiated  its  heat 
directly  to  air  of  a  space,  the  heat  radiated  per  sq.  foot  per  hour,  for  i° 
difference  of  temperature,  were,  for:  Cast  iron,  3.65  units;  Wrought  iron,  1.45 
units,  and  Terra  cotta  .4  inch  thick,  1.42  units. 

In  ordinary  practice,  i  sq.  foot  of  cast  iron  is  assigned  to  328  cube  feet 
of  space. 

Open   Fires. 

According  to  M.  Claudel,  the  quantity  of  heat  radiated  into  an  apart- 
ment from  an  ordinary  fireplace  is  .25  of  total  heat  radiated  by  combustible. 

For  wood  the  heat  utilized  is  but  from  6  to  7  per  cent.,  and  for  coal  13  per 
cent. 

In  combustion  of  wood,  chimney  of  an  ordinary  open  fireplace  draws 
from  looo  to  1600  cube  feet  of  air  per  pound  of  fuel,  and  a  sectional  area 
of  from  50  to  60  sq.  ins.  is  sufficient  for  an  ordinary  apartment. 

Proportions  of  fuel  required  to  heat  an  apartment  are :  For  ordinary  fire- 
places, 100 ;  metal  stoves,  63 ;  and  open  fires,  13  to  16. 

Furnaces. 

By  D.  K.  Clark,  from  investigations  of  Mr.  J.  Lothian  Bett. 

Cupola. — M.  Peclet  estimates  that  in  melting  pig-iron  by  combustion 
of  30  per  cent,  of  its  weight  of  coke,  14  per  cent,  only  of  the  heat  of  combus- 
tion is  utilized. 

Metallurgical. — According  to  Dr.  Siemens,  i  ton  of  coal  is  consumed 
in  heating  1.66  tons  of  wrought  iron  to  welding-point  of  2700°,  and  a  ton 
of  coal  is  capable  of  heating  up  39  tons  of  iron ;  from  which  it  appears  that 
only  4.5  per  cent,  of  whole  heat  is  appropriated  by  the  iron.  Similarly,  he 
estimates  1.5  per  cent,  of  whole  heat  generated  is  utilized  in  melting  pot 


HEAT    AND    HEATING. HYDRAULICS.  529 

steel  in  ordinary  furnaces,  whilst,  in  his  regenerative  furnace,  i  ton  of  steel 
is  melted  by  combustion  of  1344  Ibs.  of  small  coal,  showing  that  6  per  cent, 
of  the  heat  is  utilized. 

Blast-furnace. — Mr.  Bell  has  formed  detailed  estimates  of  appro- 
priation of  the  heat  of  Durham  coke  in  a  blast-furnace ;  from  which  is  de- 
duced following  abstract : 

Durham  coke  consists  of  92.5  per  cent,  of  carbon,  2.5  of  water,  and  5  of 
ash  and  sulphur.  To  produce  i  ton  of  pig-iron,  there  are  required  1232  Ibs. 
of  limestone,  and  5388  Ibs.  of  calcined  iron-stone ;  the  iron-stone  consists  of 
2083  Ibs.  of  iron,  1008  Ibs.  of  oxygen,  and  2509  Ibs.  of  earths.  There  is 
formed  813  Ibs.  of  slag,  of  which  123  Ibs.  is  formed  with  ash  of  the  coke, 
and  690  Ibs.  with  the  limestone.  There  are  2397  Ibs.  of  earths  from  the  iron- 
stone, less  93  Ibs.  of  bases  taken  up  by  the  pig-iron  and  dissipated  in  fume, 
say  2314  Ibs.  Total  of  slag  and  earths,  3127  Ibs. 

Mr.  Bell  assumes  that  30.4  per  cent,  of  the  carbon  of  the  fuel,  which  es- 
capes in  a  gaseous  form,  is  carbonic  acid;  and  that,  therefore,  only  51.27 
per  cent,  of  heating  power  of  fuel  is  developed,  and  remaining  48.73  per 
cent,  leaves  tunnel-head  undeveloped.  He  adopts,  as  a  unit  of  heat,  the 
heat  required  to  raise  the  temperature  of  112  Ibs.  of  water  33.8°. 


HYDRAULICS. 

Descending  Fluids  are  actuated  by  same  laws  as  Falling  Bodies. 

A  Fluid  will  fall  through  i  foot  in  .25  of  a  second,  4  feet  hi  .5  of  a 
second,  and  through  9  feet  in  .75  of  a  second,  and  so  on. 

Velocity  of  a  fluid,  flowing  through  an  aperture  in  side  of  a  vessel, 
reservoir,  or  bulkhead,  is  same  that  a  heavy  body  would  acquire  by  fall- 
ing freely  from  a  height  equal  to  that  between  surface  of  fluid  and 
middle  of  aperture. 

Velocity  of  a  fluid  flowing  out  of  an  aperture  is  as  square  root  of 
height  of  head  of  fluid.  Theoretical  velocity,  therefore,  in  feet  per  sec- 
ond, is  as  square  root  of  product  of  space  fallen  through  in  feet  and 

64.333  =  ^  19 h',  consequently,  for  one  foot  it  is  1/64.333  =  S.ozfeet. 
Mean  velocity,  however,  of  a  number  of  experiments  gives  5.4  feet, 
or  .673  of  theoretical  velocity. 

In  short  ajutages  accurately  rounded,  and  of  form  of  contracted  vein, 
(vena  coniracta),  coefficient  of  discharge  =  .974  of  theoretical. 

Fluids  subside  to  a  natural  level,  or  curve  similar  to  Earth's  convexity;  apparent 
level,  or  level  taken  by  any  instrument  for  that  purpose,  is  only  a  tangent  to  Earth's 
circumference;  hence,  in  leveling  for  canals,  etc.,  difference  caused  by  Earth's  cur 
vature  must  be  deducted  from  apparent  level,  to  obtain  true  level. 

Deductions    from.    Experiments    on  Discharge  of  IFlnids 
from.    Reservoirs, 

1.  That  volumes  of  a  fluid  discharged  in  equal  times  by  same  apertures 
from  same  head  are  nearly  as  areas  of  apertures. 

2.  That  volumes  of  a  fluid  discharged  in  equal  times  by  similar  apertures, 
under  different  heads,  are  nearly  as  square  roots  of  corresponding  heights 
of  fluid  above  surface  of  apertures. 

3.  That,  on  account  of  friction,  small-lipped  or  thin  orifices  discharge  pro- 
portionally more  flaid  than  those  which  are  larger  and  of  similar  figure, 
under  same  height  of  fluid. 

YY 


530 


HYDRAULICS. 


4.  That  in  consequence  of  a  slight  augmentation  which  contraction  of  the 
fluid  vein  undergoes,  in  proportion  as  the  height  of  a  fluid  increases,  the  flow 
is  a  little  diminished. 

5.  That  if  a  cylindrical  horizontal  tube  is  of  greater  length  than  its  di- 
ameter, discharge  of  a  fluid  is  much  increased,  and  may  be  increased  with 
advantage,  up  to  a  length  of  tube  of  four  times  diameter  of  aperture. 

6.  That  discharge  of  a  fluid  by  a  vertical  pipe  is  augmented,  on  the  prin- 
ciple of  gravitation  of  falling  bodies ;  consequently,  greater  the  length  of  a 
pipe,  greater  the  discharge  of  the  fluid. 

7.  That  discharge  of  a  fluid  is  inversely  as  square  root  of  its  density. 

8.  That  velocity  of  a  fluid  line  passing  from  a  reservoir  at  any  point  is 
equal  to  ordinate  of  a  parabola,  of  which  twice  the  action  of  gravity  (2  g) 
is  parameter,  the  distance  of  this  point  below  surface  of  reservoir  being  the 
abscissa.*    Or,  velocity  of  a  jet  being  ascertained,  its  curve  is  a  parabola, 
parameter  of  which  =  4  A,  due  to  velocity  of  projection.! 

9.  Volume  of  water  discharged  through  an  aperture  from  a  prismatic 
vessel  which  empties  itself,  is  only  half  of  what  it  would  have  been  during 
the  time  of  emptying,  if  flow  had  taken  place  constantly  under  same  head 
and  corresponding  velocity  as  at  commencement  of  discharge ;  consequently, 
the  time  in  which  such  a  vessel  empties  itself  is  double  the  time  in  which 
all  its  fluid  would  have  run  out  if  the  head  had  remained  uniform. 

10.  Mean  velocity  of  a  fluid  flowing  from  a  rectangular  slit  in  side  of  a 
reservoir  is  two  thirds  of  that  due  to  velocity  at  sill  or  lowest  point,  or  it  is 
that  due  to  a  point  four  ninths  of  whole  height  from  surface  of  reservoir. 

11.  When  a  fluid  issues  through  a  short  tube,  the  vein  is  less  contracted 
than  in  preceding  case,  in  proportion  of  16  to  13 ;  and  if  it  issues  through 
an  aperture  which  is  alike  to  frustum  of  a  cone,  base  of  which  is  the  aper- 
ture, the  height  of  frustum  half  diameter  of  aperture,  and  area  of  small  end 
to  area  of  large  end  as  10  to  16,  there  will  be  no  contraction  of  the  vein. 
Hence  this  form  of  aperture  will  give  greatest  attainable  discharge  of  a  fluid. 

12.  Velocity  of  efflux  increases  as  square  root  of  pressure  on  surface  of  a 
fluid. 

13.  In  efflux  under  water,  difference  of  levels  between  the  surfaces  must 
be  taken  as  head  of  the  flowing  water. 

14.  To  attain  greatest  mechanical  effect,  or  vis  viva,  of  water  flowing 
through  an  opening,  it  should  flow  through  a  circular  aperture  in  a  thin 
plate,  as  it  has  less  frictional  surface. 

From    Conduits    or    Pipes.     (Bossut.) 

1.  Less  diameter  of  pipe,  the  less  is  proportional  discharge  of  fluid. 

2.  Discharges  made  in  equal  times  by  horizontal  pipes  of  different  lengths, 
but  of  same  diameter,  and  under  same  altitude  of  fluid,  are  to  one  another 
in  inverse  ratio  of  sq.  roots  of  their  lengths. 

3.  In  order  to  have  a  perceptible  and  continuous  discharge  of  fluid,  the 
altitude  of  it  in  a  reservoir,  above  plane  of  conduit  pipe,  must  not  be  lees 
than  .082  ins.  for  every  100  feet  of  length  of  pipe. 

4.  In  construction  of  hydraulic  machines,  it  is  not  enough  that  elbows  and 
contractions  be  avoided,  but  also  any  intermediate  enlargements,  the  in- 
jurious effects  of  which  are  proportionate,  as  in  following  Table,  for  like 
volumes  of  fluid,  under  like  heads  in  pipes,  having  a  different  number  of 
enlarged  parts. 


No. 
of  Parts. 


II      No. 
Velocity.       Of  part8. 


Velocity. 


•741 


No. 
of  Parts. 


Velocity. 


.569 


No. 
of  Parts. 


5 


Velocity. 


'454 


*  See  D'Aubulsson,  page  66.  t  Humber,  page  57. 


HYDKAUL1CS. 


531 


Friction. 

Flowing  of  liquids  through  pipes  or  in  natural  channels  is  materially  af- 
fected by  friction. 

If  equal  volumes  of  water  were  to  be  discharged  through  pipes  of  equal 
diameters  and  lengths,  but  of  following  figures : 

Fig.  i.  Fig.  2.  Fig.  3. 


Figs.  i.  2.  3. 

The  times  would  be  as i,  i.n,  and  1.55. 

And  velocities  AS i,  .72,  and     .64. 

Discharges    from    Compound,   or   Divided.    Reservoirs. 

Velocity  in  each  may  be  considered  as  generated  by  difference  of  heights 
in  contiguous  reservoirs ;  consequently,  square  root  of  difference  will  rep- 
resent velocities,  which,  if  there  are  several  apertures,  must  be  inversely  as 
their  respective  areas. 

NOTE.— When  water  flows  into  a  vacuum,  32.166  feet  must  be  added  to  height  of 
it;  and  when  into  a  rarefied  space  only,  height  due  to  difference  of  external  and 
internal  pressure  must  be  added. 

VELOCITY  OF  WATER  OK  OF  FLUIDS. 
Coefficients  of  Discharge. 

Coefficient  of  Discharge  or  Efflux  is  product  of  coefficients  of  Contraction 
and  Velocify. 

It  is  ascertained  in  practice  that  water  issuing  from  a  Circular  Aperture 
in  a  thin  plate  contracts  its  section  at  a  distance  of  .5  its  diameter  from 
aperture  to  veiy  nearly  .8  diameter  of  aperture,  so  as  to  reduce  its  area 
from  i  to  about  .61.*  Velocity  at  this  point  is  also  ascertained  to  be  about 
.974  times  theoretical  velocity  due  to  a  body  falling  from  a  height  equal 
to  head  of  water.  Mean  velocity  in  aperture  is  therefore  .974,  which,  X 
.61  =  .594,  theoretical  discharge ;  and  in  this  case  .594  becomes  coefficient  of 
discharge,  which,  if  expressed  generally  by  C,  will  give  for  discharge  itself 


.  .3  —  V.     a  representing  area  of  aperture,  and  V  volume  discharged  per 
second.     Or,  4. 97  a  -\Jh  =.  V.    Or,  3.91  d2  ^/h  •=.  V.    d  representing  diameter  in  feet. 
Hence,  for  cube  feet  per  second,  4.97  a-^/h,  or  3.91  d2  ^/h. 

ILLUSTRATION.— Assume  head  of  water  10  feet,  diameter  of  opening  1.127  feet, 
area  i  sq.  foot,  and  C  =  .62. 

Then  i  V*  g  10  X  .62  =  15.72  cube  feet.  4.97  X  i  X  V10  =  I5-72  CM&e  fee*">  and 
3.91  X  i.i272  X  VIO  =  I5-7  cube  feet. 

For  square  aperture  it  is  .615,  and  for  rectangular  .621. 

Volume  of  water  or  a  fluid  discharged  in  a  given  time  from  an  aperture 
of  a  given  area  depends  on  head,  form  of  aperture,  and  nature  of  approaches. 

_.2 

h  representing  height  to  centre  of  opening  in  feet. 


64.333  h  =  v2,  and  -  -  =  h. 


NOTE.  —  Head  ,  or  height,  h,  may  be  measured  from  surface  of  water  to  centre  of 
aperture  without  practical  error,  for  it  has  been  proved  by  Mr.  Neville  that  for  cir- 
cular apertures,  having  their  centre  at  the  depth  of  their  radius  below  the  surface, 
and  therefore  circumference  touching  the  surface,  the  error  cannot  exceed  4  pel 
cent,  in  excess  of  the  true  theoretical  discharge,  and  that  for  depths  exceeding  threa 


*  Bayer,  .61.    Observed  discharge!  of  water  coincide  nearer  to  unit  of  Bayer  than  that  of  all  others. 


532  HYDRAULICS. 

times  the  diameter,  the  error  is  practically  immaterial.  For  rectangular  apertures 
it  is  also  shown  that,  when  their  upper  side  is  at  surface  of  the  water,  as  in  notches, 
the  extreme  error  cannot  exceed  4.17  per  cent,  in  excess;  and  when  the  upper  is 
three  times  depth  of  aperture  below  the  surface,  the  excess  is  inappreciable. 

For  notches,  weirs,  slits,  etc.,  however,  it  is  usual  to  take  full  depth  for  head,  when 
.666  only  of  above  equation  must  be  taken  to  ascertain  the  discharge. 

Experiments  show  that  coefficient  for  similar  apertures  in  thin  plates,  for 
small  apertures  and  low  velocities,  is  greater  than  for  large  apertures  and 
high  velocities,  and  that  for  elongated  and  small  apertures  it  is  greater  than 
for  apertures  which  have  a  regular  form,  and  which  approximate  to  the 
circle. 

When  Discharge  of  a  Fluid  is  under  the  Surface  of  another  body  of  a 
like  Fluid.—  The  difference  of  levels  between  the  two  surfaces  must  be  taken 
as  the  head  of  the  fluid. 

Or,  ^2g(h  —  h')  =  v. 

When  Outer  Side  of  opening  of  a  discharging  Vessel  is  pressed  by  a  Force. 
—  The  difference  of  height  of  head  of  fluid  and  quotient  of  pressures  on  two 
sides  of  vessel,  divided  by  density  of  fluid,  must  be  taken  as  heads  of  fluid. 


Or,  ^/2  g(h  —  te""g)Xl*A  =  v.     s  representing  density  of  fluid. 

ILLUSTRATION.—  Assume  head  of  water  in  open  reservoir  is  12  feet  above  water- 
line  in  boiler,  and  pressures  of  atmosphere  and  steam  are  14.7  and  19.7  Ibs. 


Then  =        4.333XI2_.M    =  5.56/ee, 


When  Water  flows  into  a  rarefied  Space,  as  into  Condenser  of  a  Steam- 
engine,  and  is  either  pressed  upon  or  open  to  Atmosphere.  —  The  height  due  to 
mean  pressure  of  atmosphere  within  condenser,  added  to  height  of  water 
above  internal  surface  of  it,  must  be  taken  as  head  of  the  water. 

Or,  V  2  g  (h  -f-  hr)  =  v. 

ILLUSTRATION.—  Assume  head  of  water  external  to  condenser  of  a  steam-engine  to 
feet,  vacuum  gauge  to  indicate  a  colum 
a  column  of  water  of  13  Ibs.  =  29.  g  feet. 


be  3  feet,  vacuum  gauge  to  indicate  a  column  of  mercury  of  26.467  ins.  (=  13  Ibs.), 


Then  -v/2  g  (3  +  29.9)  =  1/64.333  X  32-9  =  V21 16- 57  = 

Relative  "Velocity  of  Discharge  of  "Water  through  differ- 
ent Apertures  and  xinder  like  Heads. 

Velocity  that  would  result  from  direct,  unretarded  action  of  the  column  of 

water  which  produces  it,  being  a  constant,  or i 

Through  a  cylindrical  aperture  in  a  thin  plate 625 

A  tube  from  2  to  3  diameters  in  length,  projecting  outward 8125 

A  tube  of  the  same  length,  projecting  inward 6812 

A  conical  tube  of  form  of  contracted  vein 974 

Wide  opening,  bottom  of  which   is  on   a  level  with  that  of  reservoir; 

sluice  with  walls  in  a  line  with  orifice;  or  bridge  with  pointed  piers 96 

Narrow  opening,  bottom  of  which  is  on  a  level  with  that  of  reservoir; 

abrupt  projections  and  square  piers  of  bridges 86 

Sluice  without  side  walls 63 

Discharge  or  Efflux  of  "Water  for  various  Openings  and 
Apertvires. 

Rectangular   "Weir. 

Weirs  are  designated  Perfect  when  their  sill  is  above  surface  of  natural 
stream,  and  Imperfect,  Submerged,  or  Drowned  when  it  is  below  that  surfaco. 


HYDRAULICS.  533 

Height  measured  from  Surface  of  Water  to  Sill.   (Jas.  B.  Francis.) 


Mean  Head. 

Length  of  Opening. 

Mean  Discharge  per  Second. 

Mean  Coefficient. 

.62  to  1.55  feet. 

10  feet. 

32.9  cube  feet. 

.623 

Principal  causes  for  variation  in  coefficients  derived  from  most  experi- 
ments giving  discharge  of  water  over  weirs  arises  from, 

1.  Depth  being  taken  from  only  one  part  of  surface,  for  it  has  been  proved 
that  heads  ora,  ctf,  and  above  a  weir  should  be  taken  in  order  to  determine 
true  discharge. 

2.  Nature  of  the  approaches,  including  ratio  of  the  water-way  hi  channel 
above,  to  water-way  on  weir. 

When  a  weir  extends  from  side  to  side  of  a  channel,  the  contraction  is 
less  than  when  it  forms  a  notch,  or  Poncelet  weir,  and  coefficient  sometimes 
rises  as  high  as  .667. 

When  weir  or  notch  extends  only  one  fourth,  or  a  less  portion  of  width, 
coefficient  has  been  found  to  vary  from  .584  to  .6. 

When  wing-boards  are  added  at  an  angle  of  about  64°,  coefficient  is  greater 
than  even  when  head  is  less. 

Computation,   of  "Volume   of  Discharge. 

Mean  velocity  of  a  fluid  issuing  through  a  rectangular  opening  in 
side  of  a  vessel  is  two  thirds  of  that  due  to  velocity  at  sill  or  lower 
edge  of  opening,  or  it  is  that  due  to  a  point  four  ninths  of  whole  height 
from  surface  of  fluid. 

Height  measured  from  Surface  of  Head  of  Water  to  Sill  of  Opening. 

RULE.—  Multiply  square  root  of  product  of  64.333  and  height  or  whole 
depth  of  the  fluid  in  feet,  by  area  in  feet,  and  by  coefficient  for  opening,  and 
two  thirds  of  product  will  give  volume  in  cube  feet  per  second. 


t  representing  time  in  seconds  and  V  volume  in  cube  feet. 

EXAMPLE.—  Sill  of  a  weir  is  i  foot  below  surface  of  water,  and  its  breadth  is  10 
feet;  what  volume  of  water  will  it  discharge  in  one  second? 

C  =  .  623,    V64-33X  i  X  10  X  i  =  80.  2,  and  £  80.  2  X  .  623  =  33.  32  cube  feet. 

NOTE.—  Mean  coefficient  of  discharge  of  weirs,  breadth  of  which  is  no  more  than 
Ihird  part  of  breadth  of  stream,  is  two  thirds  of  .6  =  .4  ;  and  for  weirs  which  extend 
Vhole  width  of  stream  it  is  two  thirds  of  .666  =  .444. 

Or,  214  VP=  V  in  cube  feet  per  minute.     When  h  is  in  ins.,  put  5.  15  for  214. 
Or,  C  6  h  VTgh  —  V.    C  for  a  depth  .  i  of  length  =  .  417,  and  for  .  33  of  length  =  .  4. 

3 

Or,  by  formula  of  Jas.  B.  Francis:  3.33  (L  —  .1  n  H)  H*  =  V. 

L  representing  length  of  weir  and  H  depth  of  water  in  canal,,  sufficiently  far  from 
weir  to  be  unaffected  by  depression  caused  by  the  current,  both  in  feet,  and  n  number 
of  end  contractions. 

NOTE.—  When  contraction  exists  at  each  end  of  weir,  n  =  2;  and  when  weir  is  of 
width  of  canal  or  conduit,  end  contraction  does  not  exist,  and  n  =  o. 

This  formula  is  applicable  only  to  rectangular  and  horizontal  weirs  in  side  of  a 
dam,  vertical  on  water-side,  with  sharp  edges  to  current;  for  if  bevelled  or  rounded 
off  in  any  perceptible  degree,  a  material  effect  will  be  produced  in  the  discharge; 
it  is  essential  also  that  the  stream,  after  passing  the  edges,  should  in  nowise  be 
restricted  in  its  flow  and  descent. 

Y  Y* 


534  HYDRAULICS. 

In  cases  in  which  depth  exceeds  one  third  of  length  of  weir,  this  formula  is  not 
applicable.  In  the  observations  from  which  it  was  deduced,  the  depth  varied  from 
7  to  nearly  19  ins. 

With  end  contraction,  a  distance  from  side  of  canal  to  weir  equal  to  depth  on 
weir  is  least  admissible,  in  order  that  formula  may  apply  correctly. 

Depth  of  water  in  canal  should  not  be  less  than  three  times  that  on  weir  for  ac- 
curate computation  of  flow. 

ILLUSTRATION.— If  an  overfull  weir  has  a  length  of  7.94  feet  and  a  depth  of  .986 
(as  determined  by  a  hook  gauge),  what  volume  will  it  discharge  in  24  hours? 


3-33  (7-94  — -2  X- 986)   .986^  =  3. 33  X  7-94  — -I972  X- 979°7  =  3-33X7-7428  X 
.97907  =  25.243  875,  which  X  60  X  60  X  24  =  2 181 061  cube  feet. 
By  Logarithms.— Log.  3. 33  =  .522444 

7.7428  =  .888898 

.986^  =  7.993877 

3 

2)  1.981631 

1.990  815  =  1.990815 
1.403  157 

Log.  24  hours  =  86  400  seconds.  4. 936  514 

6.338671 
Log.  6. 338  67  =  2 181 073  cube  feet.        C  in  this  case  =  .615. 

Or.  2i4V/lP  and  s.i5v/P  =  V,  if  stream  above  the  sill  is  not  in  motion.  H 
representing  height  of  surface  of  water  above  sill  in  feet,  h  in  inches;  and 
214  VH3-J--035  v2  H3  —  V,  if  in  motion,  v  representing  velocity  of  approach  of 
water  in  feet  per  secondhand  V  volume  in  cube  feet  discharged  over  each  lineal  foot 
of  sill  per  minute. 

In  gauging,  waste-board  must  have  a  thin  edge.  Height  measured  to  level  of  sur- 
face not  affected  by  the  current  of  overfall.  (Molesworth.) 

To  Compute   Depth,   of  Flow   over   a  Sill  that  will  Dis- 
charge  a  given.  "Volume   of*  "Water. 

—  -j-  K$  \¥  —  k  =  d.    k  —  —  representing  height  due  to  velocity  (v)  as  it 

2C&V20  /  29 

Hows  to  the  weir. 

NOTE.— When  back-water  is  raised  considerably,  say  2  feet,  velocity  of  water  ap- 
proaching weir  (k)  may  be  neglected. 

Rectangular  Notches,  or  "Vertical  Apertures  or  Slits. 

A  Notch  is  an  opening,  either  vertical  or  oblique,  in  side  of  a  vessel,  reser- 
voir, etc.,  alike  to  a  narrow  and  deep  weir. 

Vertical  Apertures  or  Slits  are  narrow  notches  or  weirs,  running  to  or 
near  to  bottom  of  vessel  or  reservoir. 

Coefficient  for  opening,  8  ins.  by  5,  mean  .606  (Poncelet  and  Lesbros). 
Coefficient  increases  as  depth  decreases,  or  as  ratio  of  length  of  notch  to 
its  depth  increases. 

When  sides  and  under  edge  of  a  notch  increase  in  thickness,  so  as  to  be  converted 
into  a  short  open  channel,  coefficients  reduce  considerably,  and  to  an  extent  beyond 
what  increased  resistance  from  friction,  particularly  for  small  depths,  indicates. 

Poncelet  and  Lesbros  found,  for  apertures  8x8  ins.,  that  addition  of  a  horizontal 
•hoot  21  ins.  long  reduced  coefficient  from  .604  to  .601,  with  a  head  of  about  4.  feet; 
but  for  a  head  of  4.5  ins.  coefficient  fell  from  .572  to  .483. 

For  Rule  and  Formulas,  see  preceding  page. 


HYDRAULICS.  535 

Rectangular  Openings  or  Sluices,  or  Horizontal  Slits. 

Height,  measured  from  Surface  of  Head  of  Water  to  Upper  Side  and  to  Sill 
of  Opening. 

(Opening,  i  inch  by  i  inch.    Head,  7  to  23  feet.  =  .621. 
"       3    "     "  3    "  7  "  23    "    =.614. 

"        2  feet    "  i  foot.  i  "    a    "    =.641. 

Poncelet  and  Lesbros  deduced  that  coefficient  of  discharge  increases  with  small 
and  very  oblong  apertures  as  they  approach  the  surface,  and  decreases  with  large 
and  square  apertures  under  like  circumstances. 

Coefficients  ranged,  in  square  apertures  of  8  by  8  ins.,  under  a  head  of  6  ins.  to 
rectangular  apertures,  8  by  4  ins.  ;  under  a  head  of  10  feet,  from  .572  to  .745. 

In  a  Thin  Plate,  C  =  .6x6  (Bossut)  ;  C  =  .61  (Michelotti). 
To   Compute   Discharge. 

RULE.  —  Multiply  square  root  of  64.333  afld  breadth  of  opening  in  feet,  by 
coefficient  for  opening,  and  by  difference  of  products  of  heights  of  water  and 
their  square  roots,  and  two  thirds  of  whole  product  will  give  discliarge  in 
cube  feet  per  second. 

Or,—  b^Tg(h^h—h'^/h')  C  =  V;      -  -  —  -  -  -  =  <;      and 
3  f  bVTj  (h^/h-h'^/h')  C 

y 

r—r  —  rr  =  v.    h  and  h'  representing  depth  to  sill  and  opening  infect,  and  v  velocity 

b  (fi  —  li) 

in  feet  per  second. 

EXAMPLE.—  Sill  of  a  rectangular  sluice,  6  feet  in  width  by  5  feet  in  depth,  is  9  fe«t 
below  surface  of  water;  what  is  discharge  in  cube  feet  per  second? 

C  =  .625,    9  —  5  =  4,  and  —  A/2~0X6x.625X(9-\/9—  4X  V4)  =  38°-95  cube  feet. 

Or,  Vzgd  a  C  =  V.    d  representing  depth  to  centre  of  opening  in  feet. 
d  =  9  —  2.5  =  6.5,    a  =  6  X  5  =  30,  and  ^64.33  X  6.5  X  30  X  .625  =  383.44  cube  ft. 

Sluice   "Weirs  or   Sluices. 

Discharge  of  water  by  Sluices  occurs  under  three  forms  —  viz.,  Unimpeded, 
Impeded,  or  Partly  Unimpeded. 

To   Compute   Discharge  when.   Unimpeded. 

C  d  b  V^  gh-=V.  d  representing  depth  of  opening  and  h  taken  from  centre  of 
•pening  to  surface  of  water. 


If  velocity,  &,  with  which  water  flows  to  sluice  is  considered, 

.,      d\ 

7) 


V  V 

v  —  -d-      and    -  =d. 


n.  ., 

o»v* 

h'  representing  height  to  which  water  is  raised  by  dam  above  sill 

ILLUSTRATION.—  How  high  must  the  gate  of  a  sluice  weir  be  raised,  to  discharge 
150  cube  feet  of  water  per  second,  its  breadth  being  24  feet  and  height,  h't  5  feet? 
C  by  experiment  =  .6.    d  approximately  =  i. 


=  1.0204.0* 


5-i)       I4'4  X  I 
To  Compute  Discharge  when   Impeded. 

CdbVTgh  =  V,    and    -  -  -  =  d. 

CbVzgh 
h  representing  difference  of  level  between  supply  and  back-water. 


HYDRAULICS. 


536 

To    Compute   Discharge    -when   partly    Impeded. 

C  b  A/2T0  (d  */h 1-  d'VM  =  V.    d'  representing  depth  of  back-water  above 

upper  edge  of  sill. 


ILLUSTRA 


RATION.  —  Dimensions  of  a  sluice  are  18  feet  in  breadth  by  .5  in  depth; 
height  of  opening  above  surface  of  water  .7  feet,  and  difference  between  levels  of 
supply  and  surface  water  is  2  feet;  what  is  discharge  per  second? 


.6  X  18  X  8.02 


}  =86.62  X.  896  +  .707  =  138.85  cube  feet. 


%%Z%%&%%&%22 


Coefficients   of  Circular   Openings   or    Sluices. 

Height  measured  from  Surface  of  Head  of  Water  to  Centre  of  Opening. 

Contraction  of  section  from  i  to  .633,  and  reduction  of  velocity  to  .974;  hence 
•633  X  -974  =  -617  (Neville). 

In  a  Thin  Plate,  C  =  .666  (Bossut);  .631  (Fenfrm);  .64  (Eytelwein). 

Cylindrical  Ajutages,  or  Additional  Tubes,  give  a  greater  discharge  than 
apertures  in  a  thin  side,  head  and  area  of  opening  being  the  same ;  but  it 
is  necessary  that  the  flowing  water  should  entirely  fill  mouth  of  ajutage. 

Mean  coefficient,  as  deduced  by  Castel,  Bossut,  and  Eytelwein,  is  .82. 

Short   Tubes,  Mouth-pieces,  and   Cylindrical    Prolonga- 
tions   or   A j  \itages. 

Fjg-  4-  If  an  aperture  be  placed  in  side  of  a      ft          Fig.  5. 

vessel  of  from  1.5  to  2.5  diameters  in 
thickness,  it  is  converted  thereby  into  a 
short  tube,  and  coefficient,  instead  of  being 
reduced  by  increased  friction,  is  increased 
from  mean  value  up  to  about  .815,  when 
opening  is  cylindrical,  as  in  Fig.  4 ;  and 
when  junction  is  rounded,  as  in  Fig.  5,  to  form  of  contracted  vein,  coefficient 
increases  to  .958,  .959,  and  .975  for  heads  of  i,  10,  and  15  feet. 

Conically   Convergent   and    Divergent   TxVbes. 

Fig.  6.  In  conically  divergent  tube,  Fig.  6,  coeffi- 

cient of  discharge  is  greater  than  for  same 
tube  placed  convergent,  fluid  filling  in  both 
miirir—  o  cases,  and  the  smaller  diameters,  or  those  at 
same  distance  from  centres,  O  O,  being  used 
in  the  computations. 

A  tube,  angle  of  convergence,  O,  of  which 
is  5°  nearly,  with  a  head  of  from  i  to  10 
a  feet,  axial  length  of  which  is  3.5  ins.,  small 
diameter  i  inch,  and  large  diameter  1.3  ins., 
b  gives,  when  placed  as  at  Fig.  6,  .921  for  co- 
efficient ;  but  when  placed  as  at  Fig.  7,  co- 
efficient increases  up  to  .948.  Coefficient  of  velocity  is,  however,  larger  for 
Fig.  6  than  for  Fig.  7,  and  discharging  jet  has  greater  amplitude  in  falling. 
If  a  prismatic  tube  project  beyond  sides  into  a  vessel,  coefficient  will  be  re- 
duced to  .715  nearly. 

Form  of  tube  which  gives  greatest  discharge  is  that  of  a  truncated  cone, 
lesser  base  being  fitted  to  reservoir,  Fig.  7.  Venturi  concluded  from  his  ex- 


Fig.  7- 


HYDRAULICS.  537 

periments  that  tube  of  greatest  discharge  has  a  length  9  times  diameter  of 
lesser  opening  base,  and  a  diverging  angle  of  5°  6'— discharge  being  2.5 
greater  than  that  through  a  thin  plate,  1.9  times  greater  than  through  a 
short  cylindrical  tube,  and  1.46  greater  than  theoretic  discharge. 

Compound.   3V£ou.tli-pieees   and.   Ajutages. 
Fig.  8.  a  Fig.  9.  ^  Fig.  10. 


Coefficients    for   Month -pieces,   Short   Tntoes,  and.   Cyl- 
indrical  ^Prolongations. 

Computed  and  reduced  by  Mr.  Neville,  from  Venturis  Experiments. 


Description  of  Aperture,  Mouth-piece,  or  Tube. 


C.  for          C.  for 
Diam.  a  b.  Diain.  o  r, 


.622 
.823 

.611 
.607 

.561 
.928 

.823 
.823 
.911 

1.02 
I.2I5 

.895 


•974 
.823 

.956 
•934 

.948 
1.4*8 

.266 
.266 

:k 

•855 
1-377 


1.  An  aperture  i.  5  ins.  diameter,  in  a  thin  plate 

2.  Tube  i.  5  ins.  diameter,  and  4. 5  ins.  long,  Fig.  4 

3.  Tube,  Fig.  5,  having  junction  rounded  to  form  of  contracted 

vein 

4.  Short  conical  convergent  mouth  piece,  Fig.  6 

5.  Like  tube  divergent,  with  smaller  diameter  at  junction  with 

reservoir;  length  3.5  ins.,or  =  i  in., and  ab  =  i.$  ins.  ... 

6.  Double  conical  tube,  a  o,  8  T,  r  b.  Fig.  9,  when  a6  =  ST  =  i.5 

ins., or  =  1.21  ins.,  ao  =  -92  in.,  and  oS  =  4. i  ins 

7.  Like  tube  when,  as  in  Fig.  8,  aor6  =  oSTr,  and  a  o  S  = 

1.84  ins 

8.  Like  tube  when  S  T  =  1.46  ins. ,  and  o  S  =  2. 17  ins 

9.  Like  tube  when  ST=3  ins.,  and  08  =  9.5  ins 

10.  Like  tube  when  o  S  =  6.5  ins.,  and  S  T  =  1.92  ins 

ti.  Like  tube  when  ST  =  2.25  ins.,  and  08  =  12.125  ins 

12.  A  tube,  Fig.  10,  when  o  s  =  r  t  =  3  ins.,  o  r  =  s  t  =  i. 21  ins., 

and  tube  o  S  T  r,  as  in  No.  6,  S  T  =  i.  5  ins. ,  and  s  8=4.  i  ins. 

Mean  of  various  experiments  with  tubes  of  .5  to  3  ins.  in  diameter,  and 
with  a  head  of  fluid  of  from  3  to  20  feet,  gave  a  coefficient  of  .813 ;  and  as 
mean  for  circular  apertures  in  a  thin  plate  is  .63,  it  follows  that  under 
similar  circumstances,  .813  -f-  .63=  1.29  times  as  much  fluid  flows  through 
a  tube  as  through  a  like  aperture  hi  a  thin  plate. 

Preceding  Table  gives  coefficients  of  discharge  for  figures  given,  and  it 
will  be  found  of  great  value,  as  coefficients  are  calculated  for  large  as  well 
as  small  diameters,  and  the  necessity  for  taking  into  consideration  form  of 
junction  of  a  pipe  with  a  reservoir  will  be  understood  from  the  results. 

Circular  Sluices,  etc. 
To    Compute    Discharge. 

Height  measured  from  Surface  of  Head  of  Water  to  Centre  of  Opening. 

RULE.— Multiply  square  root  of  product  of  64.333  and  depth  of  centre  of 
opening  from  surface  of  water,  by  area  of  opening  in  square  feet,  and  this 
product  by  coefficient  for  the  opening,  and  whole  product  will  give  discharge 
in  cube  feet  per  second. 

Or,  -\/20d,  a  C  =  V.  a  representing  area  in  sq.  feet,  and  d  depth  of  surface  qf 
fluid  from  centre  of  opening  in  feet. 


538  HYDEAULICS. 

EXAMPLE.— Diameter  of  a  circular  sluice  is  i  foot,  and  its  centre  is  1.5  feet  below 
surface  of  the  water;  what  is  discharge  in  cube  feet  per  second? 


Area  of  i  foot  =  .7854;  C  =  .64,  and  ^64.333  X  1.5  X  -7854  X  .64  =  4.938  cube  feet. 

When  Circumference  reaches  Surface  of  Water.    Vz  gr,  .9604  a  C  =  V. 
r  representing  radius  of  circle  in  feet. 

ILLUSTRATION.  —  In  what  time  will  800  cube  feet  of  water  be  discharged  through  a 
circular  opening  of  .025  sq.  foot,  centre  of  which  is  8  feet  below  surface  of  water? 

800  Boo 

C  =  .  63.         __  -  =  —  ^—  -  —  —  =  2239.  58  =  37  min.  19.  6  sec. 
5X.63      22.68  X.  025  X.  63 


NOTE.—  For  circular  orifices,  the  formula  \/2  g  d  a  C  =  V  is  sufficiently  exact  for 
all  depths  exceeding  3  times  diameter;  the  finish  of  openings  being  of  more  effect 
than  extreme  accuracy  in  coefficient. 

Semicircular    Sluices. 

When  Diameter  is  either  Upward  or  Downward,  -\fzgd  a  C  =  V.  d  repre- 
senting depth  of  centre  of  gravity  of  figure  from  surface. 

When  Diameter  as  above  is  at  Depth  d,  beloiv  Surface.    V?  gd  i  .  188  a  C  =  V. 

Circular,    Semicircular,    Triangular,   Trapezoidal,    Pris- 
matic  Wedges,  Sluices,  Slits,  etc. 

See  Neville,  London,  1860,  pp.  51-63,  and  Weisbach,  vol.  \.p.  456. 
For  greater  number  of  apertures  at  any  depth  below  surface  of  water, 
product  of  area,  and  velocity  of  depth  of  centre,  or  centre  of  gravity, 
if  practicable  to  obtain  it,  will  give  discharge  with  sufficient  accuracy. 

Discharge   from   "Vessels    not    Receiving   any    Supply. 

For  prismatic  vessels  the  general  law  applies,  that  twice  as  much  would 
be  discharged  from  like  apertures  if  the  vessels  were  kept  full  during  the 
time  which  is  required  for  emptying  them. 

2  A  ^/h       2  A  h 
To    Compute   Time.     -  -  —  =  =  t. 

CaVsg         v 

ILLUSTRATION.  —  A  rectangular  cistern  has  a  transverse  horizontal  section  of  14 
feet,  a  depth  of  4  feet,  and  a  circular  opening  in  its  bottom  of  2  ins.  in  diameter;  in 
what  time  will  it  discharge  its  volume  of  water,  when  supply  to  it  is  cut  off  aud 
cistern  allowed  to  be  emptied  of  its  contents? 

h  —  4  feet,  a  —  22  X  -7854-:-  144  =  .0218,  €  =  .613,  and  Vz  gh  x  a  X  C  =  .2143 
cube  foot  per  second.  Then  -  —  —  =  522.  6  seconds. 

To   Compute   Time   and   Fall. 

Depression  or  subsidence  of  surface  of  water  in  a  vessel,  corresponding  to 
a  given  time  of  efflux,  is  h  —  h1.  h1  representing  lesser  depth. 


Inversely,  (Vh-CaV^  t\*  =  h'. 
\  2A/ 


ILLUSTRATION.  —  In  what  time  will  the  water  in  cistern,  as  given  in  preceding 
case,  subside  1.6  feet,  and  how  much  will  it  subside  in  that  time  ? 

A  =  14,    C  =  .6,    a  =  .0218,    -\/2  0  =  8.02,    ft  =  4,    h'  =r  4  —  1.6  =  2.4. 
2X14     —  X  (-y/4  —  V2-  4)  =  -^-  X  (2  —  i.  55)  =  120.  i  seconds. 


.6  X. 0218  X  8.02  ~  -.1049' 

/  .6  X  .0218  X  8.02  \2      

IV4 — Xi2o.il  =  2  —  .4s  =  2.4feet;  hence,  4  —  2.4  =  1.6  feet 

\  2  X  14  / 

When  Supply  is  maintained. — Divide  result  obtained  as  preceding  by  2. 


HYDRAULICS.  539 

Discharge,  -when.    Form.    and.    Dimensions    of   Vessel    of 
Efflvix    are   not   kno^wn. 

Volume  discharged  may  be  estimated  by  observing  heads  of  the  water  at 
equal  intervals  of  time  ;  and  at  end  of  half  time  of  discharge,  head  of  water 
will  be  .25  of  whole  height  from  surface  to  delivery. 

When  t  =  such  interval.    For  openings  in  bottom  or  side,  C  a  t  V^g  (  M 

=  V,/or  i  depth;       C  at^Tg  ^V*  +  4  V*i  +  V*a\  =  V  f&r  2  4^.       and 


NOTK.—  At  end  of  half  time  of  discharge,  head  of  water  will  be  .  25  of  whole  height 
from  surface  to  delivery. 

"Weirs   or   !N"otch.es. 

-  C  6  t  VTg  (V&3  +  4  VA3i  +  V^3z)  =  V.    6  representing  breadth  in  feet. 

9 

ILLUSTRATION.—  A  prismatic  reservoir  9  feet  in  depth  is  discharged  through  a 
notch  2.222  feet  wide,  surface  subsiding  6.75  feet  in  935  seconds;  what  is  volume 
discharged  ? 

C  =  .6,  h^  —  g  —  6.75  =  2.25/66*,  and  -  6X2.222X935X8.02  (V^  +  t 
V2.253-J-  Vo^)  =  2221.  6  x  40.  5  =  89  974.  8  cube  feet. 

When  there  is  an  Influx  and  Efflux. 

If  a  reservoir  during  an  efflux  from  it  has  an  influx  into  it,  determination 
of  time  in  which  surface  of  water  rises  or  falls  a  certain  height  becomes  so 
complicated  that  an  approximate  determination  is  here  alone  essayed. 

A  state  of  permanency  or  constant  height  occurs  whenever  head  of  water  is  in- 
creased or  decreased  by  —  (  —  J  =  k.  I  representing  influx  in  cube  feet  per  second. 


Time  (t)  in  which  variable  head  (x)  increases  by  volume  (v)  =  j  _  ^ 


and  time  in  which  it  sinks  height,  k,  by  --  —  —  -  .    Time  of  efflux,  in  which 

GaV?  gx  —  I 
subsiding  surface  falls  from  A  to  Ax,  etc.,  and  head  of  water  from  h  to  AI,  when 

k  is  represented  by  -  —  =  -^/Ar,  is 

Ca  Vs  g 
h  —  h4       /A  4Ai  2A2  4A3  A4       \ 

~ 


ILLUSTRATION.  —  In  what  time  will  surface  of  water  in  a  pond,  as  in  a  previous 
example,  fall  6  feet,  if  there  is  an  influx  into  it  of  3.0444  cube  feet  per  second? 


_  20  —  14  _       /   600000         4X495000  .2  X  410  OOP  i   4X325ooo 
12  X  -537X  .8836X  8.02       ^4.472—  s"1"  4.301—  .8  "^  4.  123  -.8  ~*~  3-937—  -8 

_|_  _?      —  -g  )  =  —  gg-  X  1  480  201  =  194  486  seconds  =  54  h.  ,  i  mm  ,  26  sec. 


If  vessel  has  a  uniform  transverse  section,  A. 

Then      2  V.  f  V*  -  V*x  +  V*  X  hyp.  log.  (^~V^)  ]  =  t  =  time  in  which 

CaVz^L  \v»x  —  v  fc/ J 

eati  of  water  flows  from  h  to  ht. 


54O  HYDRAULICS. 

ILLUSTRATION.—  A  reservoir  has  a  surface  of  500000  sq.  feet,  a  depth  of  20  feet;  it 
is  fed  by  a  stream  affording  a  supply  of  3.0444  cube  feet  per  second,  and  outlet  has 
an  area  of  .8836  sq.  foot;  in  what  time  will  it  subside  6  feet? 

•v/fc,  as  before,  =  .8,    C  =  .537,    and    2  X  SOOOOQ  x  [^20__^^+  8  x  nyp  log 


0/i°  —  87  X  2<3°3j  =  2384M  seconds  =  66  h.  13  min.  34  sec. 

To    Compute    Fall   in   a  given    Time. 

This  is  determining  head  hi  at  end  of  that  time,  and  it  should  be  sub- 
tracted from  head  h  at  commencement  of  discharge.  Put  into  preceding 
equation  several  values  of  hlt  until  one  is  found  to  meet  the  condition. 

ILLUSTRATION.  —  Take  a  prismatic  pond  having  a  surface  of  38  750  sq.  feet,  a  depth 
to  centre  of  opening  of  sluice  of  10.  5  feet,  a  supply  of  33.6  cube  feet,  and  a  discharge 
of  40  cube  feet  per  second. 

Vfc  =  .84. 

Putting  these  numerical  values  into  the  equation,  and  assuming  different  values 
for  hi,  a  value  which  nearly  satisfies  the  equation  is  4.  Consequently,  10.5  —  4  = 
6.  5  feet,  fall. 


. 

(  -  -  —  —  )*  =  &;  arc  (tang.  =  y,  arc  tangent  of  which  =  y,  and  I  as  preceding. 
\t  C  6  v  2  gf 

According  as  k  is  ^  h,  and  influx  of  water,  I  ^  £  C  I  V^gh^  there  is  a  rise  or  fall 

of  fluid  surface,  the  condition  of  permanency  occurring  when  hf  —  k,  and  time  cor- 
responding becomes  co. 

ILLUSTRATION.  —  In  what  time  will  water  in  a  rectangular  tank,  12  feet  in  length 
by  6  feet  in  breadth,  rise  from  sill  of  a  weir  or  notch,  6  inches  broad,  to  2  feet 
above  it,  when  5  cube  feet  of  water  flow  into  the  tank  per  second  ? 

ht  =  2,    h  =  o,     A  =  12X6  =  72,     1  =  5,     b  =  .5,     C  =  .6. 


)  J  = 


10.2423  X  0-961  —  (3-46i  X  arc,  tangent  of  which  =  .56497,  or  29°  28'  =  29.466, 
length  of  which  =  .5143)  =  1.781]  =x  10.2423  —  7.961  —  1.781  ==  10.2423  X  6.18  = 
63.297  seconds. 

Discharge   of*  Water  -under  "Variable  Pressures. 
To  Compute  Time,  Rise  and  Fall,  and  "Volume. 

•—  V*  gv  =  v.    »  representing  variable  head,  A  and  a  areas  of  transverse  horizon- 
tal section  of  vessel  and  discharge,  and  v  theoretical  velocity  of  efflux. 

To    Compute   "Volume. 

A  y  =  V.    y  representing  extent  of  fall,  and  V  volume  of  water  discharged,  as 

ILLUSTRATION. — Assume  elements  of  preceding  case. 

A  =  14.        y  =  4  feet.       Then  56  x  4  =  224  cube  feet 


HTDEAULICS. 


541 


Discharge   from  Vessels   of  Communication. 

When  Reservoir  of  Supply  is  maintained  at  a  uniform  Height.— Fig.  n. 

To   Compute   Time.        — ^-—  =  t. 

ILLUSTRATION  i.— In  what  time  will  level  of  water  in  a  receiving  vessel  having  a 
section  of  14  sq.  feet  attain  height  of  that  in  supply,  through  a  pipe  2  ins.  in  diam- 
eter/placed 4  feet  below  level  of  supply? 

C  =  .6x3.  2XHXV4       =  _56_  =  522.3  seconds, 

.613  X  .0218  X  8.02      . 1072 

2 Assume  C,  vessel,  Fig.  n,  to  be  a  cylinder  18 

ins.  in  diameter,  head  of  water  in  A  =  4  feet,  at  A' 
i  foot,  and  2  feet  below  outlet  o;  in  what  time  vrill 
water  in  vessel  run  out  and  over  at  o  through  a  pipe, 
a,  1.5  ins.  diameter? 


«  =  ^j='«- 


111611  .8X8.03 

When  Vessel  of  Supply  has  no  Influx,  and  is  not  indefinitely  great  compared 
with  Receiving  Vessel. 

^ =  t.    A'  representing  section  of  receiving  vessel,  t  time  in  which 

C  a  (A  -f-  A')  ^/Tg 

the  two  surfaces  of  water  attain  same  level;  and  —         *  —  =  t,  time  within 


which  level  falls  from  h  to  h'. 

ILLUSTRATION.  —  Section  of  a  cistern  from  which  water  is  to  be  drawn  is  10  sq. 
feet,  and  section  of  receiving  cistern  is  4  sq.  feet;  initial  difference  of  level  is  3  feet, 
and  diameter  of  communicating  pipe  is  i  inch;  in  what  time  will  surfaces  of  water 
in  both  vessels  attain  like  levels? 


.7854. 


2  X  io  X  4  V3 
. 82  X. 7854X^X8.02 


.502 


Discharge   from    a   Notch*   in    Side   of  a  Vessel. 

When  it  has  no  Influx.    ^^7=  (^r, Jr)  =  '•  *>  breadth  of  notch  in  feet. 

CbxVzg  W *      v */ 

ILLUSTRATION. — If  a  reservoir  of  water,  no  feet  in  length  by  40  in  breadth,  has  a 
notch  in  end  of  9  ins.  in  width;  in  what  time  will  head  of  water  of  15  ins.  fall  to  6? 

3X110X40        /i  i     \      13200 — 

.6  X. 75X8  os  ><  (71-7^)  =767  X  «-4.4-«94=  V»  "«•*. 
NOTE.— For  discharge  of  vessels  in  motion,  see  Weisbach,  vol.  i,  pp.  394-396. 
Reservoirs   or   Cisterns. 

To  Compxate  Time  of  Trilling  and  of  Emptying  a  Reser- 
voir under  Operation  of  "both   Supply  and  Discharge. 

V  V 

T,  and  — — ^  =.  t.    V  representing  volume  of  vessel,  S  supply  of  water, 


and  D  discharge  of  water,  both  per  minute,  and  in  cube  feet.    T  time  of  filling  wawJ, 
and  t  time  of  discharging  it,  both  in  minutes. 

*  When  the  notch  extends  to  the  bottom  of  the  reservoir,  etc.,  the  timt  for  the  water  to  mm  out  i* 
' 


542  HYDRAULICS. 

Irregular-Shaped.  Vessels,  as   a  Pond,  liaise,  etc. 

To  Compute  Time  and.  "Volume  Discharged. 
Operation. — Divide  whole  mass  of  water  into  four  or  six  strata  of  equal 
depths. 

h  —  h4          /a    ,  4  01  ,  2  ct2      4  a3  ,    a  4\ 

Then,  for  4  Strata, —  x  (_ + 1_  4.  -_  + 1— +  -nrj  =  ? ;  *,  h', 

i2GaV2ff      W*     v"       v/i2     v713     v^4/ 
etc.,  representing  depths  of  strata  at  a,  ai,  etc.,  commencing  at  surface;  a*,  as, 

etc.,  6ewp  area*  of  first,  second,  etc.,  transverse  sections  of  pond,  etc.  ;  and   -^- 


ILLUSTRATION.  —  In  what  time 
will  depth  of  water  in  a  lake, 
A  6  C,  Fig.  12,  subside  6  feet,  sur- 
faces of  its  strata  having  follow- 
ing areas,  outline  of  sluice  being 
a  semicircle,  18  ins.  wide,  9  deep, 
and  60  feet  in  length? 

a  at  20  feet  (h  )  depth  of  water  =  area  of  600000  sq.  feet 
ai  "18.5  "  (fti)        "        "        =      "      495000      " 
a2  "17      v  (h2)  =  410000 

03   "  15.5    "    (&3)  «          "          =  325000 

a4  "  14      "  (&4)  "       =      "     265000      " 

a  =  area  of  18  -=-  2  =  .8836  sq.  feet ;  C  =  .537. 

20  —  14 /6ooooo     4X495000      2X410000  .  4X325000 

*  12  X. 537  X. 8836X8.02       \4-472  4-30i  4-123  3.937 


,   26sooo\  6 

~~       = 


X  X  I94  431  =  I5   93    Se°'  =  43     '  35  mm>  38  5eC< 
And  discharge  =  —  x  (600000  +  4  X  495000-1-2X410000+4X3250004-265000) 
=  .  5  X  4  965  ooo  =  2  482  500  cube  feet. 

For  6  Strata,  put  2  a4,  instead  of  a4,  and  4  as  and  a6  additional,  and  divide  by 
1  8  instead  of  12. 

Flow   of  TVater  in.   Beds. 

Flow  of  water  in  beds  is  either  Uniform  or  Variable.  It  is  uniform  when 
mean  velocity  at  all  transverse  sections  is  the  same,  and  consequently  when 
areas  of  sections  are  equal  ;  it  is  variable  when  mean  velocities,  and  there- 
fore areas  of  sections,  vary. 

To   Compute    Fall   of  Flow. 

C  -?  x  —  =  h.    C  representing  coefficient  of  friction,  I  length  of  flow,  p  perimeter 

a       2  g 
of  sides  and  bottom  of  bed,  and  hfatt  in  feet. 

ILLUSTRATION.—  A  canal  2600  feet  in  length  has  breadths  of  3  and  7  feet,  a  depth 
of  3  feet,  with  a  flow  of  40  cube  feet  per  second;  what  is  its  fall  ? 

C  =  as  per  table  below  .007565;  p  =  -\/32  +  22x  2  +  3  =  10.2;  0=15;  and 
«  =  40-:-  15  =  2.66.  Hence  .007  565  X 


To  Compute  Velocity  of  Flow. 

ILLUSTRATION.  —A  canal  5800  feet  in  length  has  breadths  of  4  and  12  feet,  a  depth 
of  5,  and  a  fall  of  3  ;  what  is  velocity  and  volume  of  flow  ? 


Then       .ooyS65xT8ooX^8X64-33X3  =  V'°542  X  '93  =  3"3  ^     **" 
volume  =  40  x  3. 23  =  129. 2  cube  feet. 


HYDRAULICS. 


543 


Coefficients    of  Friction    of  ITlo-w    of  "Water   in    Beds,  as 
in    Ravers,  Canals,  Streams,  etc. 

In  Feet  per  Second. 


Velocity. 

c. 

Velocity. 

C. 

Velocity. 

C. 

Velocity. 

C. 

•3 
•4 

:I 

.00815 
.00797 
.00785 
.00778 

.1 

•9 

i 

•00773 
.00769 

.00766 

.00763 

i-5 

2 

2-5 

3 

.00759 
.00752 
•  00751 
.00749 

8 
10 

12 

•00745 
.00744 
.00743 
.00742 

Forms   of  Transverse    Sections   of  Canals,  etc. 

Resistance  or  friction  which  bed  of  a  stream,  etc.,  opposes  to  flow  of  water, 
in  consequence  of  its  adhesion  or  viscosity,  increases  with  surface  of  contact 
between  bed  and  water,  and  therefore  with  the  perimeter  of  water  profile,  or 
of  that  portion  of  transverse  section  which  comprises  the  bed. 

Friction  of  flow  of  water  in  a  bed  is  inversely  as  area  of  it. 

Of  all  regular  figures,  that  which  has  greatest  number  of  sides  has  for 
same  area  least  perimeter ;  hence,  for  enclosed  conduits,  nearer  its  trans- 
verse profile  approaches  to  a  regular  figure,  less  the  coefficient  of  its  friction ; 
consequently,  a  circle  has  the  profile  which  presents  minimum  of  friction. 

When  a  canal  is  cut  in  earth  or  sand  and  not  walled  up,  the  slope  of  its 
sides  should  not  exceed  45°. 

"Variable   Motion. 

Variable  motion  of  water  in  beds  of  rivers  or  streams  may  be  reduced  to 
rules  of  uniform  motion  when  resistance  of  friction  for  an  observed  length 
of  river  can  be  taken  as  constant. 

To   Compute  "Volume   of  "Water  flowing   in   a   River. 


-  =  V.  A  and  Ar  representing  areas  of  upper 
and  lower  transverse  sections  of 
flow. 

ILLUSTRATION  —A  stream  having  a  mean  perimeter  of  water  profile  of  40  feet  for 
a  length  of  300  feet  has  a  fall  of  9.6  ins. ;  area  of  its  upper  section  is  70  sq.  feet,  and 
of  its  lower  60;  what  is  volume  of  its  discharge? 


To  obtain  C  for  velocity  due  to  this  case,  92.35 
coefficient  for  which,  see  Table  above,  =.00744. 


V  - 


7o  +  6oX 


9.6 


40X300 


_        7-174        _ 


f~*          i~ 
V  ^~6o^ 


300X40 
~ 


V.ooo33o89 


=  394.6  cube  feet; 


and  mean  velocity  =  394'       2  =  6.07  feet,  C  for  which  is  .007  45. 

FRICTION  IN  PIPES  AND  SEWERS. 

Fi'iction  in  flow  of  water  through  pipes,  etc.,  of  a  uniform  diameter  is  in- 
dependent of  pressure,  and  increases  directly  as  length,  very  nearly  as  square 
of  velocity  of  flow,  and  inversely  as  diameter  of  pipe. 

With  wooden  pipes  friction  is  1.75  times  greater  than  in  metallic. 

Time  occupied  in  flowing  of  an  equal  quantity  of  water  through  Pipes  or 
Sewers  of  equal  lengths,  and  with  equal  heads,  is  proportionally  as  follows  : 

In  a  Right  Line  as  90,  in  a  True  Curve  as  100,  and  in  a  Right  Angle  as  140. 


544 


HYDRAULICS. 


To   Compute   Head  necessary   to  overcome   Friction   of 
Pipe.    (Weisbach.) 


.0144  4-  > 


)  X  -r  X  —  = 


h'  representing  head  to  overcome  friction  of 


-r 

a       5-4 

flow  in  pipe,  I  length  of  pipe,  and  v  velocity  of  water  per  second,  all  in  feet,  and  d 
internal  diameter  of  pipe  in  ins. 

ILLUSTRATION.—  Length  of  a  conduit-pipe  is  1000  feet,  its  diameter  3  ins.,  and  the 
required  velocity  of  its  discharge  4  feet  per  second;  what  is  required  head  of  water 
to  overcome  friction  of  flow  in  pipe? 

.0144  +  ' 


X  ~  =  .023  13  X  333-333  X  2.963  =  22.845^. 

Head  here  deduced  is  height  necessary  to  overcome  friction  of  water  in 
pipe  alone. 

Whole  or  entire  head  or  fall  includes,  in  addition  to  above,  height  between 
surface  of  supply  and  centre  of  opening  of  pipe  at  its  upper  end.  Conse- 
quently, it  is  whole  height  or  vertical  distance  between  supply  and  centre 
of  outlet. 


Xo    Compute   whole    Head,  or   Height   from    Surface    of 
Supply   to    Centre   of  Discharge. 


1.5  is  taken  as  a  mean,  and  is  coefficient  of  friction  for  interior  orifice,  or  that  of 
upper  portion  of  pipe. 

To  obtain  C  or  coefficient.      (.0144  -f  'OI™  j  =  C. 

For  facilitating  computation,  following  Table  of  coefficients  of  resistance 
is  introduced,  being  a  reduction  of  preceding  formula  : 


Coefficients   of  Friction    of  Water. 
In    I»ipes   at   Different  "Velocities. 


V. 

C. 

V. 

C. 

V. 

C. 

V. 

C. 

V. 

C 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

4 

•0443 

2  8 

025 

5 

.0221 

7  4 

.0208 

i  6 

.0195 

8 

.0356 

3 

0244 

5  4 

.O2I9 

7  8 

.0206 

2 

.0194 

X 

•OS'? 

3  4 

0239 

5  8 

.02I7 

8 

.0205 

2   6 

.0193 

*  4 

.0294 

3  8 

0234 

6 

.O2I5 

8  6 

0204 

3 

.0191 

i  8 

.0278 

4 

0231 

6  4 

.0213 

9 

0202 

4 

.0189 

2 

.0266 

4  4 

0227 

6  8 

.O2I  I 

10 

0199 

.0188 

2   4 

.0257 

4  8 

0224 

7 

.0209 

ii 

0196 

6 

.0187 

ILLUSTRATION  i. — Coefficient  due  to  a  velocity  of  4  feet  per  second  is  .0231. 
2.— Take  elements  of  preceding  case. 

(.0231  XIO°°3    I2  +  i-5)  X  g^  =  93.9  X  g^  =  23.35 /ee<. 

NOTE.— In  preceding  formula  I  was  taken  in  feet,  as  the  multiplier  of  12  for  ins. 
was  cancelled  by  taking  5.4  for  2  g,  but  jn  above  formula  it  is  necessary  to  restore 
this  multiplier. 

R-adii   of  Curvatures. 

When  Pipes  branch  off  from  Mains,  or  when  they  are  deflected  at  right 
angles,  radius  of  curvature  should  be  proportionate  to  their  diameter.  Thus, 


Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Diameter  .   . 

2  tO  7 

1  tO  A 

,. 

g 

Radius... 

83 

2O 

10 

A3 

60 

urim'AULICS. 


545 


Curves   and    Sends. 

Resistance  ox  toss  of  head  due  to  curves  and  bends,  alike  to  that  of  friction, 
increases  as  square  of  velocity ;  when,  however,  curves  have  a  long  radius 
and  bends  are  obtuse,  the  loss  is  small 

"  x  ^-  —  h. 

a  representing  angle  of  curve,  d  diameter  of  pipe,  r  radius  of  curve,  and  h  height 
due  tofi  iction  or  resistance  of  curve,  all  in  feet. 


Curved  Circular  Pipe.  (Wdsbach).     -^  x  [.131  + 1.847  (^) 

vmeter  of  pipe 
all  in  feet. 

For  ft«£ility  of  computations,  following  values  of  .131  -f- 1.847  ( — \*  are  intro- 


Coefficients    of  Resistance. 
In    Curved   JPipes   -with    Section   of  a   Circle. 


.138 


.158 
.178 


.206 


•44 
•54 

.661 


.806 
•977 


1.177 


•9 
•95 


1.978 


.6 

.244  II  .65 
.294  II  .7 

ILLUSTRATION.  —  If  in  a  pipe  18  ins.  in  diameter  and  i  mile  in  length  there  is  a 
right  angled  curve  of  5  feet  radius,  what  additional  head  of  flow  should  be  given  to 
attain  velocity  due  to  a  head  of  20  feet  ? 

a  =  90°,    v  for  such  a  pipe  and  head  =  4  feet  per  second;    18  =  1.5  and  —  — 


=  .  15,  and  ,  15  by  table  =  .  133. 


NOTE.—  If  angle  is  greater  than  90°,  head  should  be  proportionately  increased. 

Bent   or   Angular    Circular    Pipes. 

Coefficient  tor  angle  of  bend  =  .9457  sin.2  x  +  2.047  sin-4  *•    Hence, 


10° 

20° 

30° 

40° 

45° 

50° 

55° 

60° 

65° 

70° 

.046 

•139 

.364 

•74 

.984 

1.26 

1.556 

1.861 

2.158 

2-43J 

and  —  x  C  =  h.     x  representing  half  angle  of  bend. 

00° 

ILLUSTRATION.  —  Assume  v  =  4  feet,  and  angle  ==  90° ;    x  =  ?—-  •=.  45°. 

>2 

Then  ^ —  X  .984  =  .2447/00^  additional  head  required. 

In  "Valve   Q-ates   or   Slide  "Valves. 
In  Rectangular  Pipes. 


r 

x 

•9 

.8 

•7 

.6 

•5 

•4 

•3 

.2 

.x 

G 

.0 

.09 

•39 

•95 

2.08 

4.02 

8.12 

x7.8 

44-5 

193 

r  =  ratio  of  cross  section. 

In  Cylindrical  Pipes. 


h 

0 

.125 

•25 

•375 

•  5 

.625 

•75 

.875 

r 
C 

X 
.0 

.948 

.07 

.856 
.26 

;8 

•609 
2.06 

.466 
5-52 

•3i5 
17 

•159 

97-8 

h  =  relative  height  of  opening. 

In   a   Throttle  "Valve.    In  Cylindrical  Pipes. 


•913 
•24 


.826 
•52 


.741 


.658 
i-54 


25° 

30° 

35° 

40° 

45° 

50° 

60° 

•577 
2.51 

•5 
3-91 

.426 

6.22 

•357 
10.8 

•293 
18.7 

.234 
32.6 

.134 
118 

70° 
.06 

75* 


A  =  angle  of  position. 


Z    * 


546 


HYDRAULICS. 


In   a   Claelz   or  Trap  "Valve. 


Angle  of  opening 


20° 


60° 


In.  a  Cock.     In  Cylindrical  Pipes. 


.772 


.692 
1.56 


.613 


•535 


35° 


9.68 


40° 


'7-3 


45° 

50° 

55° 

60° 

•315 
31.2 

•25 

52.6 

•'9 

1  06 

•i37 
206 

70° 


65° 


In  a  Conical  "Valve.  (1.645  —,  —  i  J  =C.  a  and  a' '  =  areas  of  pipe 
and  opening. 

In  Imperfect  Contractions.  (~~;  —  *)  =  c-  c  =  a  factor,  rang- 
ing from  .624  for  —  =  .1  to  i  for  —  =  i,  being  greater  the  greater  the  ratio. 

ILLUSTRATION. — If  a  slide  valve  is  set  in  a  cylindrical  pipe  3  ins.  in  diameter  and 
500  feet  in  length,  is  opened  to  .375  of  diameter  of  pipe  (hence,  .625  diameter  closed), 
what  volume  of  water  will  it  discharge  under  a  head  of  100  feet,  coefficient  of  en- 
trance of  pipe  assumed  at  .5? 

C,  by  table,  p.  545,  pipe  being  .625  closed  =  $.52. 


C  —from,  table,  p.  544, for  an  assumed  velocity  of  u  feet  6  ins.  =  .0195. 
•V/64-33  X  -y/ioo 8.03  X  10     80. 3 


Then 


500  X  I2\ 




<J  ^i-5  +  5-  52  +  .0195 
Hence,  area  of  3  ins.  =  7.07,  and  7.07  X  12  X  11.85  =  1005.4  cube  feet  per  second. 

"Valves.     (Conical,  Spherical,  or  Flap.) 
Conical   or   Spherical    Valve   3?uppet. 

Height  due  to  resistance  or  loss  of  head  of  water  =  n  ^-.  v  representing 
velocity  of  water  in  full  diameter  of  pipe  or  vessel. 

(~ i )  —  C.    A  and  A'  representing  transverse  areas  of  vessel  and  of  valve 

\A'  C'        / 

opening,  and  (1.645  —,  —  i  j  =  C  of  contraction  in  general. 
ILLUSTRATION.— If  Ar=. 5  of  vessel,  C=  (1.645  X  —  —  i  J  =2.292  =  5.24. 

Clack  or  Trap  Valve. — C  decreases  with  diameter  of  vessel. 

ILLUSTRATION. — If  a  single-acting  force-pump,  6  ins.  in  diameter,  delivers  at  each 
stroke  5  cube  feet  of  water  in  4  seconds,  diameter  of  valve  seat  3.5  ins.,  and  of  valve 
4. 5 ;  what  resistance  has  water  in  its  passage,  and  what  is  loss  of  mechanical  effect  ? 

a  =  .196.    ( — )  =-34  ratio  of  transverse  area  of  opening,     i  —  ( ~  \  = .  44  ratio 
of  annular  contraction  to  transverse  area  of  vessel. 
Hence,  '34''44  =  .39  mean  ratio,  and  coefficient  of  resistance  corresponding 

thereto  =  (— ^  —  i)  =3.222=io.37.  — - — -  —  6.37  velocity  per  second. 

\  -39  /  4  X  .190 


HYDRAULICS. 


547 


_ 
64-33 


=  .63  height  due  to  velocity.    Consequently,  10.37  X  .63  =  6.53  height  due  to 


resistance  of  valve,  and  -  X  62.5  X  6.53  =  510.15  Ibs.  mechanical  effect  lost. 

Discharge   of  "Water  in   IPipes. 

For    any    Length    and.    Head,    and     for    Diameters    from. 
1    Inch,   to   1O   Feet.     In  Cube  Feet  per  Minute.     (Beardmore.) 


Diarn. 

Tab.  No. 

Diam. 

Tab.  No. 

Diam. 

Tab.  No. 

Diam. 

Tab.  No. 

Diam. 

Tab.  No. 

us. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

4.71 

9 

i  147.6 

ii 

11983 

3       i 

39329 

4     9 

"5854 

25 

8.48 

10 

M93-5 

I3328 

3       2 

42040 

5 

131  703 

5 
75 

13.02 

19-  1  5 
26.69 

ii 

I 

1894.9 
2876.7 

I 

2 

3 

14758 
16278 
17889 

3       3 
3       4 

3       5 

44863 
47794 
50835 

5     3 
5    6 
5    9 

148791 
167  139 
186786 

*-S 

46.67 

2 

3463-3 

4 

19592 

3      6 

53995 

6 

207  754 

3 

73-5 

3 

4  "5-9 

5 

21390 

3      7 

57265 

6    6 

25378i 

3-5 

108.  14 

4 

4836.9 

6 

23282 

3      8 

60648 

7 

305  437 

4 

151.02 

5 

5628.5 

7 

25270 

3      9 

64156 

7    6 

362  935 

4-5 

194.84 

6 

6  493-  1 

8 

27358 

3     10 

67782 

8 

426481 

5 

263.87 

7 

7433 

9 

29547 

3     " 

71526 

8    6 

496  275 

6 

416.54 

8 

8449 

10 

31834 

4 

75392 

9 

572  508 

7 

612.32 

9 

9544 

ii 

34228 

4      3 

87730 

9    6 

655  369 

8 

854.99 

10 

10722 

3 

36725 

4      6 

101  207 

10 

745038 

This  Table  is  applicable  to  Sewers  and  Drains  by  taking  same  proportion 
of  tabular  numbers  that  area  of  cross-section  of  water  in  sewer  or  drain 
bears  to  whole  area  of  sewer  or  drain. 

Formula  upon  which  the  table  is  constructed  is,    2356  ./—  x  d5  =  V  in 


—  X  d5  =  V  in  cube  feet  per  second,    h  represent- 


cube feel  per  minute,  and  39.  27 
ing  height  of  fall  of  water  and  d  diameter  of  pipe  and  I  length,  att  in  feet. 
To   Compute   Discharge. 


(Eytelwein.) 


4.71  =  V,  and 


ins.,  I  length  of  pipe  and  h  head  of  water,  both  in  feet. 


.  538  =  d.    d  =  diameter  of  pipe  in 


=  G.    G  =  number  of  Imperial 


(Hawksley.)      f /— =  d,  and 

V     A     15 

gallons  per  hour,  and  I  length  of  pipes  in  yards. 

(Neville.)  140 Vr  s  —  n  $'r  s=  v  in  feet  per  second,  r •=. hydraulic  mean  depth 
in  feet,  and  s  sine  of  the  inclination  or  total  fall  divided  by  total  length. 

v  47. 124  d'1  =  V,  and  v  293.7286  d2  =  Imperial  gallons  per  minute,  d  =  diameter 
of  pipe  in  feet. 

To    Compute   "Volume    discharged. 

When  Length  of  Pipe,  Height  or  Fall,  and  Diameter  are  given.  RULE. 
—  Divide  tabular  number,  opposite  to  diameter  of  tube,  by  square  root  of 
rate  of  inclination,  and  quotient  will  give  volume  required  in  cube  feet  per 
minute. 

EXAMPLE.— A  pipe  has  a  diameter  of  g  ins.,  and  a  length  of  4750  feet;  what  is 
its  discharge  per  minute  under  a  head  of  17.5  feet? 

__  ._    C-  __  ._    £- 

Tab.  No.  9  ins.  =  1147.6,  and 


548  HYDRAULICS. 

To    Compvite   Diameter. 

When  Length,  Head,  and  Volume  are  given.  RULE.  —  Multiply  discharge 
per  minute  by  square  root  of  ratio  of  inclination  ;  take  nearest  corresponding 
number  in  Table,  and  opposite  to  it  is  diameter  required. 

EXAMPLE.—  Take  elements  of  preceding  case. 

69.67  x  ^/^-  =  1147-61,  and  opposite  to  this  is  9  ins. 

Or,  Jy  —  —  r  =  d  in  feet,    v  representing  velocity  in  feet  per  second  and  I  length 

V  1542  h 
in  feet. 

To    Compute    Head. 

When  Length,  Discharge,  and  Diameter  are  given.  RULE.  —  Divide 
tabular  number  for  diameter  by  discharge  per  minute,  square  quotient,  and 
divide  length  of  pipe  by  it  ;  quotient  will  give  head  necessary  to  force  given 
volume  of  water  through  pipe  in  one  minute. 

EXAMPLE.—  Take  elements  of  preceding  cases. 

'         '  =  16.47;  i6.472  =  27i.3;  4750  -7-271.  2  =  17.  5  feet 


To  Compute  whole  Head  necessary  to  furnish  requisite 
Discharge. 

See  Formula  and  Illustration,  page  544. 

To   Compute   "Velocity. 

When  Volume  and  Diameter  alone  are  given.  RULE.  —  Divide  volume 
when  in  feet  per  minute  by  area  in  feet,  and  quotient,  divided  by  60,  will 
give  velocity  in  feet  per  second. 

EXAMPLE.—  Take  elements  of  preceding  case. 
69.67 


-•-6o:=s 

When  Volume  is  not  given.  RULE.  —  Multiply  square  root  of  product  of 
height  of  pipe  by  diameter  in  feet,  divided  by  length  in  feet,  by  50,  and 
product  will  give  velocity  in  feet  per  second.  (Beardmore.) 

To   Compute   Inclination   of  a   Pipe. 

When  Volume,  Diameter,  and  Length  are  given.     I— - j  ^  =  y  • 
ILLUSTRATION.— Take  elements  of  preceding  case. 

f^lY  x  — ^  =  .000874  X  4-214  =  .00368,  and  ^-5  =  .00368,  or  4750  X  .00368 
\23s6  /       .75*  4750 

=  17-  49  f^  head. 

To   Compute   Elements   of*  Long   IPipes. 

V2#  h 


5    I  y2 

and    .  4787  ./  (i.  505  X  d  -J-  c  1)  -^  =  d  in  ins. 

' 


•    This  latter  'formula  will  only  give  an  approximate  dimension  in  consequence  of 
unknown  element  d,  and  also  of  C,  as  v  = 


3.141 
For  Illustration,  see  Miscellaneous  Illustration,  page  556. 


HYDRAULICS.  549 

To  Compute  Vertical  Height  of  a  Stream  projected  from 
JPipe    of  a    Fire-engine    or    [Pump. 

RULE. — Ascertain  velocity  of  stream  by  computing  volume  of  water  run- 
ning or  forced  through  opening  in  a  second ;  then,  by  Rule  in  Gravitation, 
page  488,  ascertain  height  to  which  stream  would  be*elevated  if  wholly  un- 
obstructed, which  multiply  by  a  coefficient  for  particular  case. 

In  great  heights  and  with  small  apertures,  coefficients  should  be  reduced. 
In  consequence  of  the  varying  elements  and  conditions  of  operation  of  fire- 
engines,  it  is  difficult  to  assign  a  coefficient  for  them.  Difference  between 
actual  discharge  and  that  as  computed  by  capacity  and  stroke  of  cylinder, 
as  ascertained  by  Mr.  Larned,  1859,  was  *°*  per  cent.  =  a  coefficient  of  .82. 

A  steam  fire-engine  of  the  Portland  Company,  discharging  a  stream  1. 125  ins.  in 
diameter,  through  100  feet  2.5  inch  hose,  gave  a  theoretical  head,  computed  from 
actual  discharge,  of  225  feet,  and  stream  vertically  projected  was  200  feet;  hence 
coefficient  in  this  case  was  .88. 

EXAMPLE.  —  If  a  fire-engine  discharges  14  cube  feet  of  water  vertically  through  a 
P'Pe  i75  inch  in  diameter  in  one  minute,  how  high  will  the  water  be  projected? 

14  X  1728  -r-  .4417  area  of  pipe,  -r- 12  ins.  in  a  foot,  -r-  60  seconds  =  76.07  feet  le- 
locity;  and  as  coefficient  of  such  a  stream  =  at  .85,  then  114.1  x  .85  =  96.98 y«j6<. 

Or,  H  — ' — —  h.     H  representing  head  at  nozzle,  and  d  height  of  jet,  Loth  in 

feet,  and  d  diameter  of  nozzle  in  ins.     (R.  F.  Hartford.) 
ILLUSTRATION.— Assume  head  of  no  feet  and  diameter  of  nozzle  .75  inch. 

.0022  x  no2 
IIO =no  —  35.5  =  74.5  feet. 

NOTE.  —The  loss  of  head  is  greater  with  ring  than  with  smooth  nozzles.  E.  B. 
Weston,  Am.  Soc.  C.  E.,  puts  the  difference  at  .000  171  v2. 

The  loss  of  head  increases  with  the  absolute  height  of  the  jet,  and  is  less  with  an 
increase  of  its  diameter.  This  loss  increases  nearly  in  ratio  of  square  of  height  of 
jet,  and  varies  nearly  in  inverse  ratio  to  its  diameter 

Cylindrical  Ajutage. 

Mean  coefficient  as  determined  by  Mariotte  and  Bossut  =  .003  066  square 
of  effective  head  for  cylindrical  ajutages ;  hence,  for  conical,  alike  to  that  of 
an  engine  pipe,  coefficient  ranges  from  .72  to  .9,  or  a  mean  of  .81. 

By  formula  of  D'Aubuisson,  .003047  h2  —  h'. 

Effective  head,  or  h,  in  preceding  example  =  114.1.  Then  114.1 —.003047  X 
1 14.  i 2  =  1 14.  i  —  39. 67  =  74. 43  feet  height  of  jet. 

Hence,  for  a  conical  or  engine  pipe,  74. 43  X  .81  =  60.  zgfeet,  or  a  coefficient  of .  535. 

To  Compute  Distance  a  Jet  of  Water  -will  "be  projected 
from    a    Vessel   through    an    Opening   in    its    Side. 

B  C,  Fig.  13,  is  equal  to  twice  square  root  of  A  o  X  o  B. 
If  s  is  4  times  as  deep  below  A  as  a  is,  s  will  discharge 
twice  volume  of  water  that  will  flow  from  a  in  same  time, 
as  2  is  -y/  of  A  s  and  i  is  ^/  of  A  a. 

NOTE. — Water  will  spout  farthest  when  o  is  equidistant 
from  A  and  B;  and  if  vessel  is  raised  above  a  plane,  B  must 
be  taken  upon  plane. 
C  B  Volumes  of  water  passing  through  equal  apertures  in 

same  time  are  as  square  roots  of  their  depths  from  surface. 

RULE. — Multiply  square  root  of  product  of  distance  of  opening  from  sur- 
face of  water,  and  its  height  from  plane  upon  which  water  flows,  in  feet  by 
2,  and  product  will  give  distance  in  feet. 

EXAMPLE.— A  vessel  20  feet  deep  is  raised  5  feet  above  a  plane;  how  far  will  a  jet 
reach  that  is  5  feet  from  bottom  of  vessel  ? 

20  —  5  X  5  +  5  =  i5°>  and  VI5°  X  2  =  24-495/eet 


Fig.  13. 


550 


HYDRAULICS. 


Velocity  of  a  jet  of  water  flowing  from  a  cylindrical  tube  is  determined  to 
be  .974  to  .98  of  actual  to  theoretic  velocity,  or  =  .82  of  that  due  to  height 
of  reservoir.  Hence  volume  of  discharge  through  a  cylindrical  opening 
=s=  .82  a  Vzgh. 

Fig.  14.  Jets    d.'Eau.     (Fig.  14.) 

That  a  jet  may  ascend  to  greatest  practicable  height, 
communication  with  supply  should  be  perfectly  free. 

Short  tubes  shaped  alike  to  contracted  fluid  vein,  and 
conically  convergent  pipes,  are  those  which  give  greatest 
velocities  of  efflux.  Hence,  to  attain  greatest  effect,  as  in 
fire-engines,  long  and  slightly  conically  convergent  tubes 
or  pipes  should  be  applied. 

In  order  to  diminish  resistance  of  descending  water,  a 
jet  must  be  directed  with  a  slight  inclination  from  vertical. 

Effect  of  combined  causes  which  diminish  height  of  a  jet  from  that  due 
to  elevation  of  its  supply  can  only  be  determined  by  experiments.  Great 
jets  rise  higher  than  small  ones. 

With  cylindrical  tubes,  velocity  being  reduced  in  ratio  of  i  to  .82,  and  as 
heights  of  jets  are  as  squares  of  these  coefficients  or  ratios,  or  as  i  to  .67, 
height  of  a  jet  through  a  cylindrical  tube  is  two  thirds  that  of  head  of 
water  from  which  it  flows. 

H  C  =  h.  H  representing  head  of  water,  C  coefficient,  and  h  height  of  jet.  (Moles- 
worth.) 


When  d 


H  -r-  300,  C  =  .96. 

«-*-     450,   "  =  .93. 
"-:-    600,  '-  =  .0. 
"-5-    800,  "=.87. 
"  -r-  1000,  "  =  .85. 


When  d  =  H  -r-  1500,  C  =  .8. 


, 

2800,  "  =  .6. 
"=''-3500,  "  =  .5. 
"=  "  —  4500,  "  =  .25. 


FLOW  OF  WATER  IN  RIVERS,  CANALS,  AND  STREAMS. 

Running  Water. — Water  flows  either  in  a  natural  or  artificial  bed 
or  course.  In  first  case  it  forms  Streams,  Brooks,  and  Rivers  ;  in 
second,  Drains,  Cuts,  and  Canals. 

Bed  of  a  water-course  is  formed  of  a  Bottom  and  two  Banks  or  Shores. 

Transverse  Section  is  a  vertical  plane  at  right  angles  to  course  of  the 
flowing  water ;  Perimeter  is  length  of  this  section  in  its  bed. 

Longitudinal  Section  or  Profile  is  a  vertical  plane  in  the  course  or  thread 
of  current  of  flowing  water. 

Slope  or  Declivity  is  the  mean  angle  of  inclination  of  surface  of  the  water 
to  the  horizon. 

Fall  is  vertical  distance  of  the  two  extreme  points  of  a  denned  length  of 
the  flowing  course,  measured  upon  a  horizontal  plane,  and  this  fall  assigns 
angle  for  defined  length  of  the  course. 

Line  or  Thread  of  Current  is  the  point  where  flowing  water  attains  its 
maximum  velocity. 

Mid-channel  is  deepest  point  of  the  bed  in  thread  of  current.  Velocity  is 
greatest  at  surface  and  in  middle  of  current ;  and  surface  of  flowing  water 
is  highest  in  current,  and  lowest  at  banks  or  shore. 

A  River,  Canal,  etc.,  is  in  a  state  of  permanency  when  an  equal  quantity 
of  water  flows  through  each  of  its  transverse  sections  in  an  equal  time,  or 
when  V,  product  of  area  of  section,  and  mean  velocity  through  whole  extent 
of  the  stream,  is  a  constant  number. 


HYDRAULICS.  5  5  I 


To   Compute   Mean   Depth   of  Flowing   "Water. 

RULE.—  Set  off  breadth  of  the  stream,  etc.,  into  any  convenient  number  of 
divisions  ;  ascertain  mean  depths  of  these  divisions  ;  then  divide  their  sum 
by  number  of  divisions,  and  quotient  is  the  mean  depth. 

To    Compute    Mean   Area   of  Flowing   "Water. 

RULE  i.—  Multiply  breadth  or  breadths  of  the  stream,  etc.,  by  the  mean 
depth  or  depths,  and  product  is  the  area. 

2._Divide  the  volume  flowing  hi  cube  feet  per  second  by  mean  velocity 
in  feet  per  second,  and  quotient  is  area  in  sq.  feet. 

To    Compute    "Volume   of   Flo-wing   "Water. 

RULE.—  Multiply  area  of  the  stream,  etc.,  in  sq.  feet,  by  the  mean  velocity 
of  its  flow  in  feet,  and  product  is  volume  in  cube  feet. 

To    Compute    Mean   "Velocity    of  Flo^wing    "Water. 

RULE.  —  Divide  surface  velocity  of  flow  in  feet  per  second  by  area  of  the 
stream,  etc.,  and  quotient,  multiplied  by  coefficient  of  velocity,  will  give 
mean  velocity  in  feet. 

Mean  velocity  at  half  depth  of  a  stream  has  been  ascertained  to  be  as  .915  to  i, 
and  at  bottom  of  it  as  .83  to  i,  compared  with  velocity  at  surface.  Again,  the  ve- 
locity diminishes  from  line  of  current  toward  banks,  and,  to  obtain  mean  superficial 
velocity,  t>i4-  i>2-l-v* 

n          ='9ist*;  hence' 

To  Compute  Mean.  Velocity  in  whole  Profile  of  a  Navi- 
gable   River,  etc., 

V+i  —  2  Vv  =  velocity  at  bottom,  and  V-f  .5  —  Vv  =  mean  velocity. 
In  rivers  of  low  velocities  multiply  mean  velocity  by  .8. 

Obstruction   in    Rivers.     (Molesworth.) 


^1 — 1_  .05  X  f— V  —  x  =  R.    v  representing  velocity  in  ins.  per  second  previous 

58.6  \aj 

to  obstruction,  A  and  a  areas  of  river  unobstructed  and  at  obstruction  in  sq.feet,  and 
R  rise  in  feet 

ILLUSTRATION.— Velocity  of  obstructed  flow  of  a  river  is  6  feet  per  second,  and 
areas  of  section  before  and  after  obstruction  are  100  and  90  sq.  feet ;  what  would  be 
rise  in  feet  ?  

-|^  +  -05  X  f^V  —  i  =  -664  X  .234  =  .i55/eet 
ITlcrw  of  Water  in  Lined   Channels.    (Batin.) 

/5J?  _  v  ' =  C.     D  representing  mean  hydraulic  depth  in  feet,  F 

V    F  '       (     i  _L\  fall,  or  length  of  cftannel  to  fall  of  i,  x  and 

\y  *   D/  yas  per  table,  and  C  as  per  table  p.  543. 

x         I       y  *      I      v 

Plastered 0000045        10.16  Rubble  Masonry 00006        1.219 

Cut  Stone 000013      |     4-354         Earth .00035    |     .214 

For  Sections  of  Uniform  Area,  as  Canals,  Sewers,  etc.  y/^  2  D  =  v.  A  = 
area  of  flow  in  sq,  feet,  P  wet  perimeter  of  section,  and  D  fall  of  stream  per  mile 

"^ILLUSTRATION.—  Area  of  transverse  section  of  a  sewer  is  50  sq.  feet,  its  wet  perim- 
eter 20  feet,  and  its  fall  5  feet  per  mile. 

/  (¥.  x  2  x  5)  =  V25  =  5  feet    For  Sections  of  Rivers.     12  ^/D  p  =  v. 
ILLUSTRATION.— Assume  area  500  sq.  feet,  wet  perimeter  200,  and  fall  5  feet  per  milt, 


552  HYDRAULICS. 

Hydraulic  Radius  or  Mean  Depth  is  obtained  by  dividing  area  of  trans- 
verse section  by  wet  perimeter,  both  in  feet. 

To  Compute  Fall  per  Mile  for  a  required.  Mean  Velocity. 

r3- — iij  -=-  2  r  —  D.    r  representing  hydraulic  radius  in  feet. 

Upper  surface  of  flowing  water  is  not  exactly  horizontal,  as  water  at  its  surface 
flows  with  different  velocities  with  respect  to  each  other,  and  consequently  exert 
on  each  other  different  pressures. 

If  v  and  vi  are  velocities  at  line  of  current  and  bank  of  a  stream,  the  difference 
of  the  two  levels  is  V  ~~Vl  =  h. 


ILLUSTRATION. — If  v  =  5  feet  and  vx  .o  v ;  then ^ — -  =  ^^-  = .  0738  foot. 

20  64.33 

A  velocity  of  7  to  8  ins.  per  second  is  necessary  to  prevent  deposit  of  slime  and 
growth  of  grass,  and  15  ins.  is  necessary  to  prevent  deposit  of  sand. 

Maximum  velocity  of  water  in  a  canal  should  depend  on  character  of  bed  of  the 
channel. 

Thus,  Mean  Velocity  should  not  exceed  per  second  over 


Fine  clay 6  ins. 

A  slimy  bed 8   " 

Common  clay. 6    " 


River  sand i  ft.  I  Broken  stones 4  ft. 

Small  gravel iu     Stones 6U 

Large  shingle 3  "   |  Loose  rocks 10  " 


3.  W 
vide  vo 


To  Compute  "Velocity-  of  Flow  or  Discharge  of  \Vater  in 
Streaixis,  3?ipes,  Canals,  etc. 

i.  When  Volume  discharged  per  Minute  is  given  in  Cube  Feet,  and  Area  of 
Canal,  etc.,  in  Sq.  Feet.  RULE.  —  Divide  volume  by  area,  and  quotient,  di- 
vided by  60,  will  give  velocity  in  feet  per  second. 

a.  When  Volume  is  given  in  Cube  Feet,  and  A  rea  in  Sq.  Tns.    RULE.  —  Di- 
vide volume  by  area  ;  multiply  quotient  by  144,  and  divide  product  by  60. 
When  Volume  is  given  in  Cube  Ins.,  and  Area  in  Sq.  Ins.    RULE.  —  Di- 
volume by  area,  and  again  by  12  and  by  60. 

To   Compute   Flow  or  "Volume   of  Discharge. 

1.  When  Area  is  given  in  Sq.  Feet.    RULE.—  Multiply  area  of  flow  by  its 
velocity  in  feet  per  second,  and  product,  multiplied  by  60,  will  give  volume 
in  cube  feet  per  minute. 

2.  When  Area  is  given  in  Sq.  Tns.    RULE.—  Multiply  area  by  its  velocity, 
and  again  by  60,  and'divide  product  by  144. 

NOTE  i.  —  Velocities  and  discharges  here  deduced  are  theoretical,  actual  results  de- 
pending upon  coefficient  of  efflux  used.  Mean  velocity,  however,  as  before  given, 
page  529,  may  be  taken  at  VTg  .673  =  5.4  feet,  instead  of  8.02  feet. 

2.  —  As  a  rule,  with  large  bodies,  as  vessels,  etc.,  their  floating  velocity  is  some- 
what greater  than  that  of  flow  of  water,  not  only  because  in  floating  they  descend 
an  inclined  plane,  formed  by  surface  of  the  water,  but  because  they  are  but  slightly 
affected  by  the  irregular  intimate  motion  of  water:  the  variation  for  small  bodies 
is  BO  slight  that  it  may  be  neglected. 

To    Compute   Height   of  Head   of  Flowing   "Water. 

When  Volume  and  Area  of  Flow  are  given  in  Feet.  RULE.  —  Divide  vol- 
ume in  feet  per  second  by  product  of  area,  and  £  coefficient  for  opening,  and 
square  of  quotient,  divided  by  64.33,  will  give  height  in  feet. 

EXAMPLK.—  Assume  volume  266.48  cube  feet,  area  40  sq.  feet,  and  C  =  $23. 


HYDEAULICS.  553 

Submerged,   or   Dro^wned    Orifices   and.   "Weirs. 

When  wholly  submerged  (Fig.  15) Available  pressure  at  any  point  in  depth 

of  orifice  is  equal  to  difference  of  pressure  on 
F'g-  «5-  &  each  side. 

Whence,  C  Vzgh  =  v,  and   C  a  \fzgh  =  V. 
a  representing  area  of  sluice  in  sq.feet. 
ILLUSTRATION.— Assume  opening  3  feet  by  5, 
h  =  4  feet,  and  C  = .  5. 

^      Then,  5  X  3^5  V64.33  X  4  =  7-5  X  16.04  = 
120. 3  cube  feet  per  second. 

When  partly  submerged  (Fig.  16).     h'  —  h  =  d  =  submerged  depth,  and  h  — 
«:-  T<e  ft"  =  d'  —  remaining  portion  of  depth;  whence 

<T  +  d  =  entire  depth,  and_ 


ILLUSTRATION.  —  Assume  opening  as  above,  fc 
4  feet,  h'  =  6,  h"  =  3,  and  C  =  .5.     Then  d  =  6 


Then  .5  X  5  X  8.02  (2  V4  +  £  X  4  \/4  —  3  A 
=:  20.05  X  5.869  =  117.67  cube  feet  per  second. 

When  drowned  (Fig.  17). 


ILLUSTRATION.  —  Assume  opening  as  above, 
|g§   h=4feet,    d  =  2,  and    G  =  .$2. 


Fig.  17 


Then,  .52  X  5  X  V64.33  x  4  X  (2  -f  £  4)  =  2.6 
X  16.04  X  4.66  =  194.34  cube  feet  per  second. 

CANAL  LOCKS. 
Single    Locks. 

When  a  fluid  passes  from  one  level  or  reservoir  to  another,  through  an 
aperture  covered  by  the  fluid  in  the  latter,  effective  head  on  each  point  of 
aperture,  and  consequently  head  due  to  velocity  of  efflux  at  each  instant,  is 
the  difference  of  levels  of  the  two  reservoirs  at  that  instant. 


Hence  C  a 


&'  =  V  per  second,    h'  representing  difference  of  levels. 


To  Compute  Time  of  Filling   and.  Discharging  a  Single 
Look.—  Fig.  IS. 

When  Sluice  in  Uj)per  Gate  is  entirely  under  Water,  and  above  Lower  Level. 


Ah" 


—  =  time  of  filling  up  to  centre  of  sluice. 


Fig.  1 8. 


Ca  \ 

h  representing  height  of  centre  of  sluice  in  upper 
gate  from  surface  of  canal  or  reservoir,  and  h'  height 
of  centre  of  sluice  in  upper  gate  from  lower  sur- 
face, or  water  in  the  lock  or  river,  all  infect;  and 

— - — =  time  of  filling  the  remaining  space, 

C  a  Vz  g  h 

where  a  gradual  diminution  of  head  of  water  occurs. 

Consequently,  —  -  2         =  t  time  of  filling  a  single  lock. 
Ca V? gh 

When  Aperture  or  Sluice  in  Lower  Gate  is  entirely  under  Water,  and  above 

Lower  Level.     ^— — ^=r  =  time  of  emptying  or  discharging  it.    a' representing 

G' 
area  of  lower  sluice. 


554 


HYDRAULICS. 


ILLUSTRATION.—  Mean  dimensions  of  a  lock,  Fig.  18,  are  200  feet  in  length  by  24 
in  breadth;  height  of  centre  of  aperture  of  sluice  from  upper  and  lower  surfaces  is 
5  feet;  breadth  of  both  upper  and  lower  sluices  is  2.5  feet;  height  of  upper  is  4  feet, 
and  of  lower—  entirely  under  water—  5  feet;  required  the  times  of  filling  and  dis- 
charging. 
h  =  5,  h'  =  5,  A  =  200  X  24  =  4800,  C  =  .545,  a  =  4  X  2.  5  =  10,  a'  =  5  X  2.5 

4_oo  —  5 


•  545X  ioX 
sluice;  and 


12.  5. 
_  ^  ^  seconds  =  time  of  filling  lock  up  to  centre  of 


seconds  =  ftwe  of  fitting  remain- 


97-72 

tn0  space,  or  Jocfc  aoove  centre  of  sluice,  and  245.594-491.18  =  736.77  seconds, 
iime. 


Or  (5  +  2X5)X48oo 
.545  X  10  X  A/2  0/i 
30358.08 
54-7 


:720QQ_ 
"  97-72 

=  554-9  seconds  =  time  of  discharging. 


2  X  4800  A 


.  545  x  12.5  x 


When  Aperture  or  Sluice  in  Upper  Gate  is  entirely  under  Water  and  below 

Lower  Level.     -  -  —  —  =  time  of  filling  lock. 
CaVsg 

When  Sluice  in  the  Lower  Gate  is  in  part  above  Surface  of  Lower  Level 
and  in  part  below  it.     -  -  -          —  =  time  of  dis- 


\    »  / 

charging,    d  and  d'  representing  distances  of  part  of  aperture  above  and  of  below 
surface  of  lower  water,  b  breadth  of  aperture,  and  h  and  h'  as  before. 

ILLUSTRATION.— Assume  sluice  in  preceding  example  to  be  i  foot  above  lower 
level  of  water,  or  that  of  lower  canal;  what  is  time  of  discharge  of  lock,  distance 
of  part  of  aperture  i  foot  and  of  that  below  surface  of  water  4  feet? 

2  X  4800  (5  -|-  5) 


96000 


•545X2.5X8.02 

96000 
-2 —  558.3  seconds. 

Don"ble    Hioclt.    (J.  D.  Van  Buren,  Jr.) 

A  double  lock  is  not  a  duplication  of  a  single  lock  in  its  operation,  for  in 
lower  chamber  supply  of  water 
is  from  upper  one,  having  no 
influx,  instead  of  a  uniform  sup- 
ply flowing  directly  from  sur- 
face level  of  canal  or  feeder. 

Operation,    therefore,    of 
double  lock  is  complex,  addition 
to  formula  for  a  single  lock  be- 
ing that  of  discharging  of  water 
in  upper  lock  to  fill  lower,  the 

hi^i^  ^cCS^hkh^ 

closed  from  upper  reach  during  discharge  into  lower. 

To    Compute    Time    required,    for   "Water   to   Fall    from 
Upper   to    Uniform   "Water   Level. 

i. ~  (  V/~h  "V/2  h — Vz  h  —  2  d)  =  t.     A  representing  horizontal  area  of  lock, 

and  a  area  of  sluice  opening,  both  in  sq.  feet,  C  coefficient  of  discharge  =  .545  for 
Openings  urilh  square  arrises,  g  acceleration  of  gravity,  f  depth  of  centre  of  sluice 


HYDRAULICS.  555 

below  uniform  level,  h  dtpth  of  centre  sluice  opening  below  upper  water  level,  and  d 
height  of  centre  of  sluice  above  lower  water  level,  all  infect,  and  t  time  for  water  to 
fall  from  upper  to  uniform  water  level,  in  seconds. 

ILLUSTRATION.—  A  =  2000  sq.feet;  €  =  .545;  a  =  s;  /=6;  h  =  14;  andd  = 
2  feet.  (Fig.  19.) 

2000  2000  ,      ,  , 

Then = X  7>  74  —  4-  9  =  3"7- o  seconds. 

'.545X5X5-67      15-45 

2  Ifd  =  o-    — ^-  =  t\    = — ^^ —      ,    =  — =  366.34 seconds. 

'    Ca^g  .545X5X5-67      15-45 

NOTE  —/is  never  greater  than  I  (lift  in  feet);  it  is  equal  to  I  when  d  =  o;  /2  is 
equal  to  I  when/i  =  o,  never  greater.  In  each  case  it  is  the  unbalanced  head  abovt 
eluice,  however  far  below  the  lowest  water  level  the  sluice  is. 

To   Fill    Upper   Locli   or   Empty    Lower. 

To  fill  upper  lock  or  empty  lower,  when  the  sluice  is  below  the  lowest  water-line, 
in  either  case,  takes  the  same  time;  for  the  head  diminishes  at  the  same  rate,  one 
from  the  upper  surface,  the  other  from  the  bottom. 


z    _  t     Here^  j.  bcing  below  lowest  wai€r  faa  Of  fafr  _  g  yj^  as  d  =  o, 
C  a  yg 

and/=  whole  lift  =    2°°°  —  -  -  —  =  -  =  517.  8  seconds. 
•545X5X5.67      15-45 

To  Discharge  a  like  Volume  under  a  Constant  Head. 

AX//          A       //  2000         /    8 

4.  -  ^=  =  7r-v/—  =*•     =  -  v/z  -  =  258-9  seconds, 
CaVTg      CaVaflr  .545X5V64-33 

Or,  one  half  the  time  given  by  preceding  case. 

The  times  deduced  by  preceding  formulas  are  in  the  following  proportions  in 
order,  as  i  :  V2  :  ^^  ,  or  i  :  V2  :  ~r~  • 

If  sluice  of  upper  lock,  through  which  it  is  filled,  is  above  lowest  water  level, 
then,  by  combining  formulas  3  and  4.  the  time  is  thus  deduced. 

To  Jill  from  Lowest  Water  Level  of  said  Lock  to  Level  of  Centre  of  Sluice. 

A  \!  f 

5.  —  v        =  t'.    f  representing  height  of  centre  of  sluice  above  said  lowest  water 
C  a  vTg  level 

To  Jill  remaining  Portion  of  Lock  above  Sluice. 

6.  -  Z—z  =  t".    f"  representing  depth  below  upper  water  level  of  centre  of 
CaVzg 

sluice  or  remaining  portion  of  lift.    Hence,  £'  -f-  «"  =  -^  -  -.  —  (V/'  -f  2  \//")  =  *• 

C  a  x/2  g 

To  Jill  Lower  Lock  under  Constant  Head  from  Upper  Canal  Level, 


8.  If  both  lifts  are  the  same,  h  —f=  I,  and  -A^-  L  -}.  *  _  2    /1\  =  t 

CaVzg^       h        V  i2/ 


If  lower  lock  is  filled  from  upper  one  under  a  constant  head,  when  latter  is  drawn 
down  to  lowest  level,  formula  7  will  apply  by  making  h  =/,  and 


—  -  (2 
2r  V 


which  is  identical  with  7,  for/=/2  and  d=f,  the  cases 


being  the  same. 


556 


HYDRAULICS. 


MISCELLANEOUS  ILLUSTRATIONS. 


1.  If  external  height  of  fresh  water,  at  60°  above  injection  opening  in  condensei 
of  a  steam-engine,  is  3  feet,  and  the  indicated  vacuum  at  23  ins  ,  velocity  of  water 
flowing  into  condenser  is  thus  determined.     (Formula  page  532.) 

v  =  -\/2  g  (h  +  h').  h'  representing  height  of  a  column  of  water  equivalent  to  press- 
ure of  atmosphere  within  condenser. 

Assuming  mean  pressure  of  atmosphere  —  14.7  Ibs.per  sq.  inch,  height  of  a  column 
of  fresh  water  equivalent  thereto  =  33.95  feet. 

Then,  if  i  inch  =  .4912  Ibs.,  23  ins.  =  11.3  Ibs.;  and  if  14.7  Ibs.  =  33.95  feet,  11.3 
Ibs.  =  26.  i  feet. 

Hence  v  —  V*  g  (3  -f-  26.  i)  =  43.27  feet,  less  retardation  due  to  coefficient  of  both 
influx  and  efflux. 

2.  What  breadth  must  be  given  to  a  rectangular  weir,  to  admit  of  a  flow  of  6  cube 
feet  of  water,  under  a  head  of  8  ins.  ?    (Formula  page  533.) 

-  —  —  :  —  =  -  2  —  '  =  a.  21  feel. 
$X.625V20  66      -417X6.55 

3.  It  being  required  to  ascertain  volume  of  water  flowing  in  a  stream,  a  tem- 
porary dam  is  raised  across  it,  with  a  notch  in  it  2  feet  in  breadth  by  i  in  depth, 
which  so  arrests  flow  that  it  raises  to  a  head  of  1.75  feet  above  sill  of  notch;  what 
is  volume  of  flow  per  second.?    (Formula  page  533.  ) 

€  =  .635.      -  X-  635X2X1-75  Vzgx  1.75  =  1-481  X  10.6  —  15.7  cube  feet. 

4.  A  rectangular  sluice  6  feet  in  breadth  by  5  in  depth,  has  a  depth  of  9  feet  of 
water  over  its  sill,  and  discharges,  as  per  example  page  535,  380.95  cube  feet  per 
second  ;  what  is  velocity  of  flow  ?    (Formula  page  535.  ) 


If  volume  was  not  given:    — 
3 

Then  -  X  .625  X  8.02  X     7*9  ~~      4  =  3-  34'  X  3-8  =  12.7  feet. 

3  9  —  4 

5.  If  a  river  has  an  inclination  of  i.  5  feet  per  mile,  is  40  feet  in  breadth  with  nearly 

vertical  banks,  and  3  feet  depth  ;  what  is  volume  of  its  discharge  ?  (Formula  p.  542.  ) 

Perimeter  40  -\-z  X  3  =  46  feet;      hydraulic  mean  depth  —  —  2.61  feet; 

46 

a  =  120  feet;       Cper  table,  page  543,/or  assumed  velocity  of  2.5  feet  =.007  5. 
Then      ___-__  x  64.  33  x  i.  5  =  ^.0659X96.5  =  2.  52  feet  velocity. 


Hence  120  X  2.52  =  302.4  vubefeet. 

6.  What  is  head  of  water  necessary  to  give  a  discharge  of  25  cube  feet  of  water 
per  minute,  through  a  pipe  5  ins.  in  diam.  and  150  feet  in  length?  (Formula  p.  548.) 

Tabular  number  for  diameter  5  ins.,  page  547,  =  263.87. 

Then  263.  87  -=-25  =  111.3,  and  150  -5-111.3  =  1.35  /get 

If  this  pipe  had  2  rectangular  knees  or  bends,  what  then  would  be  head  of  water 
required?    (Formula  page  545.) 

C,  page  545,/or  ^-  =  .984,  area  of  5  ins.  =  .  136  feet,  and  -25-  -4-  60  =  3.06  feet 

2  .130 

velocity.    Then  j^  —  X  .984  X  2  =  .2863,  which,  added  to  1.35  =  1.64  feet. 

By  formulas  foot  of  page  548,  0^.024,  and  c  .505  velocity  =  3.06  feet  ;  head  = 
i.+gfeet,  and  volume  26.38  cube  feet. 

7.  If  a  stream  of  water  has  a  mean  velocity  of  2.25  feet  per  second  at  a  breadth 
of  560  feet,  and  a  mean  depth  of  9  feet,  what  will  be  its  mean  velocity  when  it  has 
a  breadth  of  320  feet,  and  a  mean  depth  of  7.  5  feet  ?    (Rule  page  548.  ) 

560X9X2.25^1134? 
320X7-5  2400 


HYDEAULICS.  557 

8.  What  volume  will  a  pipe  48  feet  in  length  and  2  ins.  in  diameter,  under  a  head 
of  5  feet,  deliver  per  second  ?    ( Formula  page  547. ) 
Tabular  number  for  diameter  2  ins.,  page  547,  =  26.69. 

I  —  =  3.  i.     Then         9  =  8.61,  which  -r-  60  = .  143  cube  feet. 

If  this  pipe  had  5  curves  of  90°,  with  radii  — =  -  =.5;  what  would  be  its  dis- 
charge per  second  ? 
V  =  .i43;  a  =  2-7- 144  =.0139;  Gper  tables  —  =  .294;    v  =  ^^-  =  10.29 feet. 

O  I0  2    2  2T  '°139 

Then  .294  X  ^5  X  *6'       — ,  147  X  1.64  =  241,  which  x  5  for  5  curves  =  1.2  = 
height  due  to  resistance  of  curves,     h  =  5  —  1.2=  3. 8. 
Hence,  if -v/20  5  =  .143;  Vzg  3.8  =  .125  cube  feet. 

g.  If  a  slide  stop  valve,  set  in  a  cylindrical  conduit  500  feet  in  length  and  3  ins.  in 
diameter,  is  raised  so  as  to  close  .625  of  conduit;  what  volume  will  it  discharge 
under  a  head  of  4  feet  ?  (Formula  page  546. ) 

Cfor  conduit  =  .5,  for  friction  .025,  and  for  slide  valve  .375  open,  table,  page  545, 
5.52,  d  =  .25,  and  a  =  7.07  sq.  ins. 

2  9  h                                  16.06 
Ihen —          =  -j- -r- — -  =  2.13  feet  velocity,  and 

*/  (  l  +  •  5  +  5-  52  +  »025  — -J 
2. 13  X  12  X  7-  07  =  180. 71  cw&e  ITIS. 

10.  If  a  single  lock  chamber  is  200  feet  in  length  by  24  in  breadth,  with  a  depth 
of  10  feet,  centre  of  upper  gate,  which  is  4  feet  in  depth  by  2.5  in  breadth,  is  at 
middle  of  depth  of  chamber,  lower  gate,  5  feet  in  depth  by  2.5  in  breadth  and  wholly 
immersed;  what  is  time  required  for  filling  and  discharging  it?    (Formula p.  553.) 

C  =  .6is,  h  =  $,  h'  =  $,  A  =  200 X  24  =  4800,  0  =  4X2.5  =  10,  and  a' =  5 
X  2.5  =  12.5 

=  652. 8  seconds  time  of  filling. 
t5-  =  6        =  49x-4  seconds  time  of  emptying. 

11.  In  a  moderately  direct  and  uniform  course  of  a  river,  the  depths  and  velocities 
are  as  follows ;  what  is  the  volume  of  its  flow  and  what  its  mean  velocity  ?    (p.  551. ) 

Feet.      Feet.       Feet.       Feet.      Feet. 

Distances 5         12         20         15         7       I     Area  of  profiles  =  5  x  3  + 

Depths 3  6         ir  8         4          12X6  +  20X11  +  15X8  + 

Mean  velocity 1.9        2.3        2.8        2.4      2.1    |  7  X  4  =  455  sq.feet. 

15  X  1.9  +  72X2.3  +  220X2.8  +  120X2.4  +  28X2.1  =  1156.9  cube  feet  volume, 
and  *156'9  =  2. 54  feet  velocity. 

Miner's    Inch. 

A  '  Miner's  inch  "  is  a  measure  for  flow  of  water,  and  is  an  opening  one 
i'lch  square  through  a  plank  two  inches  in  thickness,  under  a  head  of  six 
inches  of  water  to  upper  edge  of  opening. 

It  will  discharge  11.625  U.  S.  gallons  water  in  one  minute. 

Theoretical  H?  under  different  Reads. 


Heads  in  feet.|ioo 
Ins.  pe 


in  feet.  1 100     190     |8o      70 
jrB?... I    3.251  3.61 1  4.06   4.64 


1 60 
I  5-4i 


3° 


Water  Inch  (Pouce  d"eau). — Circular  opening  of  i  inch  in  a  thin  plate  is 
equal  to  a  discharge  of  19.1953  cube  meters  per  24  hours. 

A* 


558  HYDRODYNAMICS. 

HYDRODYNAMICS. 

Hydrodynamics  treats  of  the  force  of  action  of  Liquids  or  Inelastic 
Fluids,  and  it  embraces  Hydraulics  and  Hydrostatics:  the  former  of 
which  treats  of  liquids  in  motion,  as  flow  of  water  in  pipes,  etc.,  and 
latter  of  pressure,  weight,  and  equilibrium  of  liquids  in  a  state  of  rest. 

Fluids  are  of  two  kinds,  aeriform  and  liquid,  or  elastic  and  inelastic, 
and  they  press  equally  in  all  directions,  and  any  pressure  communicated 
to  a  fluid  at  rest  is  equally  transmitted  throughout  the  whole  fluid. 

Pressure  of  a  fluid  at  any  depth  is  as  depth  or  vertical  height,  and 
pressure  upon  bottom  of  a  containing  vessel  is  as  base  and  perpendicu- 
lar height,  whatever  may  be  the  figure  of  vessel.  Pressure,  therefore, 
of  a  fluid,  upon  any  surface,  whether  Vertical,  Oblique,  or  Horizontal,  is 
equal  to  weight  of  a  column  of  the  fluid,  base  of  which  is  equal  to  sur- 
face pressed,  and  height  equal  to  distance  of  centre  of  gravity  of  sur- 
face pressed,  below  surface  of  the  fluid. 

Side  of  any  vessel  sustains  a  pressure  equal  to  its  area,  multiplied  by 
half  depth  of  fluid,  and  whole  pressure  upon  bottom  and  against  sides 
of  a  cubical  vessel  is  equal  to  three  times  weight  of  fluid. 

Pressure  upon  a  number  of  surfaces  is  ascertained  by  multiplying 
sum  of  surfaces  into  depth  of  their  common  centre  of  gravity,  below 
surface  of  fluid. 

When  a  body  is  partly  or  wholly  immersed  in  a  fluid,  vertical  press- 
ure of  the  fluid  tends  to  raise  the  body  with  a  force  equal  to  weight  of 
fluid  displaced ;  hence  weight  of  any  quantity  of  a  fluid  displaced  by  a 
buoyant  body  equals  weight  of  that  body. 

Centre  of  Pressure  is  that  point  of  a  surface  against  which  any  fluid 
presses,  to  which,  if  a  force  equal  to  whole  pressure  were  applied,  it 
would  keep  surface  at  rest.  Hence  distance  of  centre  of  pressure  of 
any  given  surface  from  surface  of  fluid  is  same  as  Centre  of  Percussion. 

Centres   of  fressnre. 

Parallelogram,  Side,  Base,  Tangent,  or  Vertex  of  Figure  at  Surface  of  Fluid,  is  at 
.66  of  line  (measuring  downward)  that  joins  centres  of  two  horizontal  sides. 

Triangle,  Base  uppermost,  is  at  centre  of  a  line  raised  from  lower  apex,  and  join 
ing  it  with  centre  of  base;  and  Vertex  uppermost,  it  is  at  .75  of  a  line  let  fall  from 
vertex,  and  joining  it  with  centre  of  base. 

Right-angled  Triangle,  Base  uppermost,  is  at  intersection  of  a  line  extended  from 
centre  of  base  to  extremity  of  triangle  by  a  line  running  horizontally  from  centre 
of  side  of  triangle.  Vertex  or  Extremity  uppermost,  is  at  intersection  of  a  line  ex- 
tended from  the  centre  of  the  base  to  the  vertex,  by  a  line  running  horizontally  from 
.375  of  side  of  triangle,  measured  from  base. 


Trapezoid,  either  of  parallel  Sides  at  Surface,  ^"v3  6/  X  a  =  d.  b  and  &'  repre- 
senting breadths  of  figure,  d  distance  from  surface  of  fluid,  and  a  length  of  line  join- 
ing opposite  sides. 

Circle,  at  1.25  of  its  radius,  measured  from  upper  edge. 

Semicircle,  Diameter  at  Surface  of  Fluid,  ±*^L  —  d.r  representing  radius  of  circle 

ID 

and  p  =  3. 1416.    Diam.  downward,  ^fl     "       =  <*• 


HYDRODYNAMICS.  559 

Side,   Base,  or    Tangent    of  Figure    "belcrw    Surface    of* 
TTluid. 

Rectangle  or  Parallelog'm.  -  x  i^^2  =  d  ;  or,  3mo  +  m*  =  d ;  and  -  =  d". 
3       h'2  —  h2  30  30 

h  and  h'  representing  depths  of  upper  and  lower  surfaces  of  figure  and  d  depth, 
both  from  surface  of  fluid,  m  half  depth  of  figure,  o  depth  of  centre  of  gravity  of 
figure  from  surface  of  fluid,  d'  distance  from  upper  side  of  figure,  and  d"  distance 
from  centre  of  gravity. 

Triangle.  —  Vertex  Uppermost.       -^- =  d;     =  d'.     Base  Uppermost. 

1 8  o  18  o 

"*"'    °  =  d.    I  representing  depth  of  figure,  d  distance  from  surface  of  fluid  upon 

Io  O 

a  line  from  vertex  to  centre  of  base,  and  d"  distance  from  centre  of  gravity  of  figure. 

A  o2  -4-  r2  r2 

Circle. =  d,  or  —  =  distance  from  centre  of  circle. 

Semicircle. — Diam.  Horizontal  and  Upward  or  Downward. 1-  o  =  d\ 

40      9J>o^ 

~4    =  d' :     —  =  d",  and =  c.     d  representing  distance  from 

3P  3P  4  °      9P  ° 

surface  of  fluid,  d'  distance  of  centre  of  gravity  from  centre  of  arc,  d"  distance  of 
centre  of  gravity  from  diameter  when  it  is  uppermost,  and  c  centre  of  pressure. 

^Pressure. 
To    Compute    Pressure    of  a   Fluid,    upon    Bottom    of  its 

Containing   "Vessel. 

RULE.— Multiply  area  of  base  by  height  of  fluid  in  feet,  and  product  by 
weight  of  a  cube  foot  of  fluid. 

To    Compute   Pressure    of  a   Fluid,    upon    a  "Vertical,  In- 
clined, Curved,  or    any    Surface. 

RULE. — Multiply  area  of  surface  by  height  of  centre  of  gravity  of  fluid 
in  feet,  and  product  by  weight  of  a  cube  foot  of  fluid. 

EXAMPLE  i.— What  is  pressure  upon  a  sloping  side  of  a  pond  of  fresh  water  10  feet 
square  and  8  feet  in  depth  * 

Centre  of  gravity,  8  -=-  2  =  4/0  -tfrom  surface.    Then  io2  X  4  X  62. 5  =  25  ooo  Its. 

2. — What  is  pressure  upon  staves  of  a  cylindrical  reservoir  when  filled  with  fresh 
water,  depth  being  6  feet,  and  diameter  of  base  5  feet? 

5X3. 1416  =15. 708  feet  curved  surface  of  reservoir,  which  is  considered  as  a  plane. 
1 5. 708  X  6  X  6  -i-  2  =  282. 744,  which  X  62. 5  =  17  671. 5  Ibs. 

3. — A  rectangular  flood-gate  in  fresh  water  is  25  feet  in  length  by  12  feet  deep; 
what  is  pressure  upon  it? 

25  X  12  X  i2-r-  2  =  1800,  which  X  62.5  =  112500  Ibs. 

When  water  presses  against  both  sides  of  a  plane  surface,  there  arises  from 
resultant  forces,  corresponding  to  the  two  sides,  a  new  resultant,  which  is 
obtained  by  subtraction  of  former,  as  they  are  opposed  to  each  other. 

ILLUSTRATE-  -Depth  of  water  in  a  canal  is  7  feet;  in  its  adjoining  lock  it  is  4 
feet,  and  breadtn  of  gates  is  15  feet;  what  mean  pressure  have  they  to  sustain,  and 
what  is  depth  of  point  of  its  application  below  surface? 

7  x  15  =  105,  and  4  X  15=60  sq.feet.  (105  X  -  —60  X  2)  X  62.5  =  15468.75  Ibs., 
mean  pressure. 

Then  15468.754-62.5  =  247.5  =  cube  feet  pressing  upon  gates  upon  high  side,  and 
247. 5-1-15  X  7  =  2.35  feet  =  depth  of  centre  of  gravity  of  mean  pressure. 

To    Compute    Pressure    on    a    Sluice. 

A  iv  d  =  P,  and  C  P  =  I".  A  representing  area  of  sluice  in  sq.  feet,  w  weight  of 
water  per  cube  foot,  d  mean  depth  of  sluice  below  surface,  in  feet,  P  pressure  on  sluice, 
and  P'  power  required  to  operate  it,  both  in  Ibs. 

C  =  .68  when  sluice  is  of  wood,  and  .31  when  of  iron. 


560 


HYDRODYNAMICS. 


EXAMPLE — What  is  pressure  on  a  sluice-gate  3  feet  square,  its  centre  of  gravity 
being  30  feet  below  surface  of  a  pond  of  fresh  water? 

3  X  3  X  30=270,  which  x  62.5  =  16875  Ibs. 

To    Compute    IPressxire   of  a   Colxixxixi   of"  a    Fluid,    per 
Sq..  Inch. 

RULE. — Multiply  height  of  column  in  feet  by  weight  of  a  cube  foot  of 
fluid,  and  divide  product  by  144 ;  quotient  will  give  weight  or  pressure  per 
sq.  inch  in  Ibs. 

NOTB When  height  is  given  in  ins.,  omit  division  by  144. 

PIPES. 

To   Compute   required.   Thickness   of*  a   Pipe. 
RULE. — Multiply  pressure  in  Ibs.  per  sq.  inch  by  diameter  of  pipe  in  ins., 
and  divide  product  by  twice  assumed  tensile  resistance  or  value  of  a  sq. 
inch  of  material  of  which  pipe  is  constructed. 

By  experiment,  it  has  been  found  that  a  cast-iron  pipe  15  ins.  in  diameter,  and 
.75  of  an  inch  thick,  will  support  a  head  of  water  of  600  feet;  and  that  one  of  oak, 
of  same  diameter,  and  2  ins.  thick,  will  support  a  head  of  180  feet? 

EXAMPLE  i.— Pressure  upon  a  cast-iron  pipe  15  ins.  in  diameter  is  300  Ibs.  per  sq. 
inch;  what  is  required  thickness  of  metal? 

300  X  15  =  45oo>  which  -~  3000  X  2  = .  75  inch. 

NOTE.— Here  3000  is  taken  as  value  of  tensile  strength  of  cast  iron  in  ordinary 
small  water-pipes.  This  is  in  consequence  of  liability  of  such  castings  to  be  im- 
perfect from  honey-combs,  springing  of  core,  etc. 

2. Pressure  upon  a  lead  pipe  i  inch  in  diameter  is  150  Ibs.  per  sq.  inch;  what  is 

required  thickness  of  metal  ? 
Here  500  is  taken  as  value  of  tensile  strength. 

150  X  i  =  150,  which  -r-  500  x  2  =  .15  inch. 

Cast-iron   Pipes. 
To   Compute   Thickness,  etc.,  of*  Flanged    Pipes. 

For  75  Ibs.  Pressure.  For  100  Ibs.  Pressure. 


.0250+    .25  =T 

•03    D  +   .3  =< 

.05    D  +  I.IS  =1 

•03    P-f    -35  =/ 

1.05    D  -f-  4.25  d  4-  1-25  =  o 
1.05    D  +  z  X  d  4-i       =  o' 


.03    D4-    .3  =T 

.o35D4-    .45  =t 

.05    D-j-  1. 15  =1 

•04    D+    -6  =/ 

i.i      D  +  5  X  d+  1.5  =o 

i.i      D4-  2.5X^4-1.4  =o' 


.7D  4-2.2  =  *;        Ax*-n=q,    and       /   «    4-C  =  d. 

4000  y  -7854 

D  representing  diam.  of  pipe,  T  thickness  of  metal,  t  thickness  and  I  length  of  boss, 
f  thickness  of  flange,  o  diam.  of  flange,  o'  diam.  of  centres  at  bolt  holes,  and  d  diam. 
of  bolts,  all  in  ins.;  A  area  of  pipe  and  a  area  of  bolt  at  base  of  its  thread,  in  sq.  ins., 
p  pressure  in  Ibs.  per  sq.  inch,  and  C  a  coefficient  due  to  diam.  of  bolt. 

Thus,  diam.  .125 +  .032,  .25 4-. 064,  .5 4-- 107,  i  4~-i6,  1.54- -214,  and  24-. 285. 

ILLUSTRATION.— What  should  be  dimensions  of  a  flanged  pipe,  10  ins.  in  diameter, 
for  a  pressure  of  100  Ibs.  per  sq.  inch  ? 

.7  X  10 -f-  2.2  =  9.2  =  10  number  of  bolts,  and  diam.  10  ins.  =  78.54  ins.  area  =  A. 

Z!^i^±^  =  .I9635,andN/S  +  C  =  V.25  =  .s;  hence,. 5+  ,o7  = 

.  607  = .  625  Ibs.  diameter  of  bolls  ;  .03  X  10  -f- .  3  = .  6  =  thickness  of  metal ;  .  035  X  10 
4-. 45  =  .8  —  thickness  of  flange;  .05  X  10  + 1. 15  =  1.65  =  length  of  boss;  .04  X  10 
4-  .6  —  i  =  thickness  of  flange  ;   i.i  X  10 4~  5  X  .625  4- *•  5  =  15-625  =  diameter  of 
flange ;  and  1. 1  X  10  4-2.5  X -625  4- 1. 4  =  13. 9625  =  diameter  of  bolt  holes. 
For  Tables  of  Cast-iron  Pipes,  see  page  132. 


HYDRODYNAMICS. 

To   Compmte   Elements   of*  'Water-pipes. 

.0001245  Pd  +  C  =  t;  or,  .000054  H  d  +  C  —  t;  .4336  H^P;  and 

02  _  d2  x  2.  45  =  W.  P  representing  pressure  of  water  in  Ibs.  per  sq.  inch,  D  and  d 
external  and  internal  diameters  of  pipe,  and  t  thickness  of  metal,  all  in  ins.,  C  coeffi- 
cient for  diameter  of  pipe,  and  H  head  of  water  in  feet. 

C  =s  .  37  for  pipes  less  than  12  ins.  in  diameter,  .  5  from  12  to  30,  and  6  from  30  to  50 

To   Compnte   "Weight   of  Pipes. 

To  Diameter  add  thickness  of  metal,  multiply  sum  by  10  times  thickness, 
and  product  will  give  weight  in  Ibs.  per  foot  of  length. 
Weight  of  Faucet  end  is  equal  to  8  ins.  of  length  of  pipe. 

Hydrostatic   3?ress. 
To  Compute   Elements   of  a   Hydrostatic    Press. 


ure  applied,  W  weight  or  resistance  in  Ibs.,  I  and  If  lengths  of  lever  and  fulcrum  in 
ins.  or  feet,  and  A  and  a  areas  of  ram  and  piston  in  sq.  ins. 

ILLUSTRATION.—  Areas  of  a  ram  and  piston  are  86.6  and  i  sq.  ins.,  lengths  oflever 
and  fulcrum  4  feet  and  9  ins.,  end  power  applied  20  Ibs.  ;  what  is  weight  that  may 
be  sustained  ?  _ 

20  X  4X  12  X  86.6      83  136 

-!>n  —  =-V=  9*37.3  »». 

To   Compute   Thickness    of  Metal    to   Resist  a  given 
Pressure. 

RULE.  —  Multiply  pressure  per  sq.  inch  in  Ibs.  by  diameter  of  cylinder  in 
ins.,  and  divide  product  by  twice  estimated  tensile  resistance  or  value  of 
metal  in  Ibs.  per  sq.  inch,  and  quotient  will  give  thickness  of  metal  required. 

EXAMPLE.—  Pressure  required  is  9000  Ibs.  per  sq.  inch,  and  diameter  of  cylinder  is 
5.3  ins.  ;  what  is  required  thickness  of  metal  of  cast  iron? 


Value  of  metal  is  taken  at  6000.    9°°°X5'3  =  ^-^  =  3.  975  in*. 

6000X2  12000 

Values  of  Different  Metals  in  Tons.    (Molesworth.) 
Cast  iron  .......  41  1  Gun  metal  ......  22  |  Wrought  iron.  .  .14  |  Steel  ..........  06 

Hydraulic    Rarru 

Useful  effect  of  an  Hydraulic  Ram,  as  determined  by  Eytelwein,  varied 
from  .9  to  .18  of  power  expended.  When  height  to  which"  water  is  raised 
compared  to  fall  is  low,  effect  is  greater  than  with  any  other  machine  ;  but 
it  diminishes  as  height  increases. 

Length  of  supply  pipe  should  not  be  less  than  .75  of  height  to  which 
water  is  to  be  raised,  or  5  times  height  of  supply  ;  it  may  be  much  longer. 

To   Compute   Elements. 


efficiency.  V  and  v  representing  volumes  expended  and  raised,  in  cube  feet  per 
minute,  h  and  h'  heights  from  which  water  is  drawn  and  elevated  in  feet,  D  and  d 
diameters  of  supply  and  discharging  pipes  in  ins.,  and  IP  effective  horse-power. 

ILLUSTRATION.  —  Heights  of  a  fall  and  of  elevation  ar3  10  and  26.3  feet,  and  vol- 
umes expended  and  raised  per  minute  are  1.71  and  .543  cube  feet 

.OOH3X  1.71  X  10  =  .01933*;    -  r  iQ°I93  =  i.7i  cube  feet;    1.45-^1.71  =  1.89 

=  .6g6  efficiency. 


562 


HYDRODYNAMICS. 


Results  of  Operations   of  Hydraulic   Rams* 


Strokes 
perM. 

Fall. 

Eleva- 
tion. 

Wat 
Eipen'd. 

er 
Raised. 

Useful 
Effect. 

Strokes 
per  M. 

Fall. 

Eleva- 
tion. 

Wai 

Expen'd. 

er 
Raised. 

Useful 
Effect. 

No. 
66 

52 
36 

3' 

Feet. 
10.  06 

9-93 
6.05 
5-o6 

Feet. 
26.3 
38.6 
38.6 
38.6 

C.  Ft. 
1.71 
1-93 
J-43 
1.29 

C.  Ft. 

•543 
.421 
.169 
•"3 

:§5 
$ 

No. 
15 

10 

Feet. 
3.22 
1.97 
22.8 
8-5 

Feet. 
38.6 
38.6 
196.8 
52-7 

C.  Ft. 

1.98 
1.58 
•38 

2 

C.  Ft. 

.058 
.014 
.029 
.186 

1 
.67 
•57 

NOTE.  — Volume  of  air  vessel  =  volume  of  delivery  pipe.  One  seventh  of  water 
may  be  raised  to  about  4  times  head  of  fall,  or  one  fourteenth  8  times,  or  one  twenty- 
eighth  ib  times. 

WATER  POWER. 

Water  acts  as  a  moving  power,  either  by  its  weight  or  by  its  vis  viva,  and 
in  latter  case  it  acts  either  by  Pressure  or  by  Impact. 

Natural  Effect  or  Power  of  a  fall  of  water  is  equal  to  weight  of  its  volume 
and  vertical  height  of  its  fall. 

IfSvater  is  made  to  impinge  upon  a  machine,  the  velocity  with  which  it 
impinges  may  be  estimated  in  the  effect  of  the  machine.  Result  or  effect, 
however,  is  in  nowise  altered ;  for  in  first  case  P  =  V  w  h,  and  in  latter  = 

—  V  w.    V  representing  volume  in  cube  feet,  w  weight  in  Ibs.,  and  v  velocity 

of  flow  in  feet  per  second. 

62. 5  V  h  =  P,  and  3. 2  *  a  ^h  =  V.  P  representing  pressure  in  Ibs. ,  a  area  of  open- 
ing in  sq.feet,  and  h  height  of  flow  in  feet  per  second. 

To    Cotnpnte    Power   of  a    Fall    of  "Water. 
RULE.-— Multiply  volume  of  flowing  water  in  cube  feet  per  minute  by 
62.5,  and  this  product  by  vertical  height  of  fall  in  feet. 

NOTE.— When  Flow  is  over  a  Weir  or  Notch,  height  is  measured  from  surface  of 
tail  race  to  a  point  four  ninths  of  height  of  weir,  or  to  centre  of  velocity  or  pressure 
*f  opening  of  flow. 

When  Flow  is  through  a  Sluice  or  Horizontal  Slit,  height  is  measured  from  sur- 
face of  tail-race  to  centre  of  pressure  of  opening. 

EXAMPLE.— What  is  power  of  a  stream  of  water  when  flowing  over  a  weir  5  feet 
in  breadth  by  i  in  depth,  and  having  a  fall  of  20  feet  from  centre  of  pressure  of  flow? 

By  Rule,  page  533,  -5X1  Vzgi  X  .625  =  16.72  cube  feet  per  second. 

16.68  X  60  X  62. 5  X  20  =  i  251  ooo  Ibs. ,  which  -r-  33  ooo  =  37.91  horses'  power. 
Or, .  1135  V  h  =  theoretical  IP.    h  representing  height  from  race  in  feet. 
ILLUSTRATION. — If  flow  of  a  stream  is  17.9  cube  feet  per  second,  to  what  height 
and  area  of  flow  of  i  foot  in  depth  should  it  be  dammed  to  attain  a  power  of  10 
horses. 

33  ooo  X  10  _          ^  per  secona^  an$  5500  _  88  cube  jeei  ^  secona     _. 

60  62 . 5 


17.9 


2  feet  height.    Hence,  -  .6 


/2gx  1  =  3.2,  and  17.9-^-3.2  =  5.59  sg. 


Water  sometimes  acts  by  its  weight  and  vis  viva  simultaneously,  by  com- 
bining effect  of  an  acquired  velocity  with  fall  through  which  it  flows  upon 
wheel  or  instrument. 


In  this  case 


/ 

[h  -\ 
\       2 


)  V  X  62.5  =  mechanical  effect. 


*  As  determined  by  ~  C. 


HYDRODYNAMICS.  563 

WATER-WHEELS. 

WATER- WHEELS  are  divided  into  two  classes,  Vertical  and  Horizontal. 
Vertical  comprises  Overshot,  Breast,  and  Undershot ;  and  Horizontal, 
Turbine,  Impact,  or  Reaction  wheels. 

Vertical  wheels  are  limited  by  construction  to  falls  of  less  than  60  feet 
Turbines  are  applicable  to  falls  of  any  height  from  i  foot  upward. 

Vertical  wheels  applied  to  a  fall  of  from  20  to  40  feet  give  a  greater 
effect  than  a  Turbine,  and  for  very  low  falls  Turbines  give  a  greater  effect. 

Sluices. — Methods  of  admitting  water  to  an  Overshot  or  Breast 
Wheel  are  various,  consisting  of  Overfall,  Guide-bucket,  and  Penstock. 

An  Overfall  Sluice  is  a  saddle-beam  with  a  curved  surface,  so  as  to  direct  the 
current  of  water  tangentially  to  buckets;  a  Guide-bucket  is  an  apron  by  which 
water  is  guided  in  a  course  tangential  to  buckets;  and  a  Penstock  is  sluice-board  or 
gate,  placed  as  close  to  wheel  as  practicable,  and  of  such  thickness  at  its  lower  edge 
as  to  avoid  a  contraction  of  current.  Bottom  surface  of  penstock  is  formed  with  a 
parabolic  lip. 

Sh.rondlng  of  a  wheel  consists  of  plates  at  its  periphery,  which 
form  the  sides  of  the  bucket. 

Height  of  fall  of  a  water-wheel  is  measured  between  surfaces  of  water  in  penstock 
and  in  tail-race,  and,  ordinarily,  two  thirds  of  height  between  level  of  reservoir  and 
point  at  which  water  strikes  a  wheel  is  lost  for  all  effective  operation. 

Velocity  of  a  wheel  at  centre  of  percussion  of  fluid  should  be  from  .5  to  .6  that 
of  flow  of  the  water. 

Total  effect  in  a  fall  of  water  is  expressed  by  product  of  its  weight 
and  height  of  its  fall. 

Ratio   of   Effective    IPower   of  Water    IVlotors. 

I  fw™  *Q  *«  <   *,»       Undershot,  Poncelet's,  from. 6  to. 4  toi 
jfrom.68to.6   to  x    Undershot' ]    «     Lte.Jitoi 

Turbine "     .6   to  .8    to  i    * 

Breast "     . 45  to. 65  to  i 


Hydraulic  Ram "  .6    to  i 


Water-pressure  engine  "  .8  to  i 


Oversliot-\vlieel. 

OVERSHOT-WHEEL. — The  flow  of  water  acts  in  some  degree  by  impact, 
but  chiefly  by  its  weight. 

Lower  the  speed  of  wheel  at  its  circumference,  the  greater  will  be  mechan- 
ical effect  of  the  water,  in  some  cases  rising  to  80  per  cent. ;  with  velocities 
of  from  3  to  6.5  feet,  efficiency  ranges  from  70  to  75  per  cent.  Proper  ve- 
locity is  about  5  feet  per  second. 

Number  of  buckets  should  be  as  great,  and  should  retain  water  as  long,  as 
practicable.  Maximum  effect  is  attained  when  the  buckets  are  so  numerous 
and  close  that  water  surface  in  the  bucket  commencing  to  be  emptied  should 
come  in  contact  with  the  under  side  of  the  bucket  next  above  it.  Moles- 
worth  gives  12  ins.  apart. 

Curved  buckets  give  greatest  effect,  and  Radial  give  but  .78  of  effect  of 
Elbow  buckets.  Wheel  40  feet  in  diameter  should  have  152  buckets. 

Small  wheels  give  a  less  effect  than  large,  in  consequence  of  their  greater 
centrifugal  action,  and  discharging  water  from  the  buckets  at  an  earlier 
period  than  with  larger  wheels,  or  when  their  velocity  is  lower. 

When  head  of  water  bears  to  fall  or  height  of  wheel  a  proportion  as  great 
as  i  to  4  or  5,  ratio  of  effect  to  power  is  reduced.  The  general  law  there- 
fore is,  that  ratio  of  effect  to  power  decreases  as  proportion  of  head  to  total 
head  and  fall  increases. 


564 


HYDRODYNAMICS. 


Wheel  with  shallow  Shrouding  acts  more  efficiently  than  one  where  it  is 
deep,  and  depth  is  usually  made  10  or  12  ins.,  but  in  some  cases  it  has  been 
increased  to  15. 

Breadth  of  a  wheel  depends  upon  capacity  necessary  to  give  the  buckets 
to  receive  required  volume  of  water. 

Form  of  Buckets.—  Radial  buckets— that  is,  when  the  bottom  is  a  right  line— in- 
volve so  great  a  loss  of  mechanical  effect  as  to  render  their  use  incompatible  with 
economy;  and  when  a  bucket  is  formed  of  two  pieces,  lower  or  inner  piece  is 
termed  bottom  or  floor,  and  outer  piece  arm  or  wrist.  Former  is  usually  placed  in 
a  line  with  radius  of  wheel. 

Line  of  a  circle  passing  through  elbow,  made  by  junction  of  floor  and  arm,  is 
termed  division  circle,  or  bucket  pitch,  and  it  is  usual  to  put  this  at  one  half  depth 
of  shrouding. 

•560° 

When  arm  of  a  bucket  is  included  in  division  angle  of  buckets,  that  is,  - — ,  n 

representing  number  of  buckets,  the  cells  are  not  sufficiently  covered,  except  for  very 
shallow  shrouding;  hence  it  is  best  to  extend  arm  of  a  bucket  over  1.2  of  division 
angle,  so  as  to  cover  or  overlap  elbow  of  bucket  next  in  advance  of  it. 

Construction  of  Buckets  (Fig.  i).— Capacity  of  bucket  should  be  3  times  volume 
of  water. 

Fairbairn  gives  area  of  opening  of  a  bucket  in  a 
wheel  of  great  diameter,  compared  to  the  volume  of  it, 
as  5  to  24. 

Buckets  having  a  bottom  of  two  planes,  that  is,  with 
two  bottoms,  and  two  division  circles  or  bucket  pitches 
and  an  arm,  give  a  greater  effect  than  with  one  bottom. 

When  an  opening  is  made  in  base  of  buckets,  so  as 
to  afford  an  escape  of  air  contained  within,  without  a 
loss  of  water  admitted,  the  buckets  are  termed  ven- 
tilated, and  effective  power  of  wheel  is  much  greater 
than  with  closed  buckets. 

D  =  distance  apart  at  periphery  =  d,  d  depth  of 
shrouding,  s  length  of  radial  start—  .33  d,  I  length  of 
bucket  curve  =  1.25  d  in  large  wheels,  and  i  in  wheels 
under  25  feet,  a  angle  of  radius  of  curve  of  bucket, 
with  radial  line  of  wheel  at  points  of  bucket  =  15°. 
(Molesworth.) 

To  Compute    Radius   and.  Revolutions  of  an  Overshot- 
wheel,  and    Height   of  Fall    of  \Vater. 


When  whole  Fall  and  Velocity  of  Flow,  etc.,  are  given, 
he  v2  ,.  .  3.1416  nr 

h 


and 


i  -f  cos.  a 
=  c.    h  representing  height  of  whole 


3.1416  r 

fall,  h'  height  between  the  centre  of  gravity  of  discharge  and  half  depth  of  bucket 
upon  which  water  flows,  v  velocity  of  flow  in  feet  per  second,  a  angle  which  point  of 
entrance  of  water  into  a  bucket  makes  with  summit  of  wheel,  n  number  of  revolutions 
per  minute,  c  velocity  of  wheel  at  its  circumference  per  second,  and  r  its  radius. 

NOTE.— Height  of  whole  fall  is  distance  between  surface  of  water  in  flume  and 
point  at  which  lower  buckets  are  emptied  of  water,  and  as  a  proportion  of  velocity 
of  flow  is  lost,  it  is  proper  to  assume  height  h'  as  above  given. 

ILLUSTRATION.— A  fall  of  water  is  30  feet,  velocity  of  its  flow  is  16  feet  per  second, 
angle  of  its  impact  upon  buckets  is  12°,  and  required  velocity  of  wheel  is  8  feet  per 
--  number  of  revolutions,  and  height  of  fall  upon 


second;  what  is  required  radius, 
wheel? 


30X8 
3.1416  X  12.95" 


^ 


HYDRODYNAMICS.  565 

When  Number  of  Revolutions  and  Ratio  between  Velocities  ef  Flow  and  at 
Circumference  of  Wheel  are  given. 

y/.ooo 772  (x  n)2  h  +  (i  -f  cos,  a)2  —  i  -f-  cos,  a)  _        _v        ^  3.1416  nr 

~~   .000386  (3571)2      ~  -   C  '  io" 

ILLUSTRATION.— If  number  of  revolutions  are  5,  x  =  2,  and  fall,  etc. ,  as  in  previous 
case;  what  is  radius  of  wheel,  velocity  of  flow,  and  height  of  fall? 


-518 

3'4  J 


= 
.000386  (2  Xs)2  .0386 

3.1416x5X13-41  _  7.03/ee<     Hence  7>o3  x  2  =  14.06  velocity  of  flow,  and  I4'°62 
30  64.33 

X  i  i=3.38/eet 

To    Compute   Width.   of  an    Overshot-wlieel. 

C  V 

—  ?  =  w.    C  representing  a  coefficient  =  3,  when  buckets  are  jilted  to  an  excess,  and 

5  when  they  are  deficiently  filled,  V  volume  of  water  in  cube  feet  per  second,  s  depth 
of  shrouding,  w  width  of  buckets,  both  in  feet,  and  c'  velocity  of  wheel  at  centre  of 
shrouding,  in  feet  per  second. 

ILLUSTRATION  —  A  wheel  is  to  be  31  feet  in  diameter,  with  a  depth  of  shrouding  of 
i  foot,  and  is  required  to  make  5  revolutions  per  minute  under  a  discharge  of  10 
cube  feet  of  water  per  second;  what  should  be  width  of  buckets? 

Assume  C  =  4,  and  c'  =  3-I~I  X  j*  I4*6  X  5  =  7.854-    Then  -4-x-IO_  5.09/5* 
60  i  X  7-054 

To    Compute    Number   of  Buckets. 

7  (  i  +  -|-  J  -=-  12  =  d,  and     **S  =  n.     D  representing  diameter  of  wheel,  d  dis- 
tance between  centres  of  buckets,  in  feet,  and  n  number  of  buckets. 
ILLUSTRATION  —Take  elements  of  preceding  case. 

Then  7  (,  +  -|-)  =  7  X  2.2  +  12  ^  1.283,  and  3I  ~*  x  ^l6  X  '  =  73.4>  say  7. 

060° 
buckets,  hence  -  —  =  5°,  angle  of  subdivision  of  buckets. 

To    Compnte   Kffect  of  an   Oversliot-\vlieel. 


--  —  —  —  =  P.     w  representing  weight  of  cube  foot  of  water  in  Ibs., 

v"  velocity  of  it  discharged  at  tail  of  wheel,  in  feet  per  second,  V  volume  of  flow  in 
cube  feet,  and  f  friction  of  wheel  in  Ibs. 

ILLUSTRATION  —A  volume  of  12  cube  feet  per  second  has  *  fall  of  10  feet,  wheel 
using  but  8.5  feet  of  it,  and  velocity  of  water  discharged  is  o  feet  per  second;  what 
is  effect  of  fall  ? 

Friction  of  wheel  is  assumed  to  be  750  Ibs. 

=  6375_(r.26  x  75o+75o)  ^  .,680  = 

I2XIOX62.5  7500  7500 

.  624  =  ratio  of  effect  to  power  ;  and  4680  X  60  seconds  H-  33  ooo  —  8.  5  1  IP. 

To  Compute  3?ower  of  an.  Overshot--wh.eel. 
RULE.  —  Multiply  weight  of  water  in  Ibs.  discharged  upon  wheel  in  one 
minute  by  height  or  distance  in  feet  from  centre  of  opening  in  gate  to  sur- 
face of  tail-race  ;  divide  product  by  33  ooo,  and  multiply  quotient  by  as- 
sumed or  determined  ratio  of  effect  to  power.  Or,  for  general  purposes, 
divide  product  by  50  ooo,  and  quotient  is  H*. 

Or,  .0852  V  h  =  IP,  and  ^-^  =  V  per  second;  or,  ~-  —  =  V  per  minute. 
3B 


566  HYDRODYNAMICS. 

Mechanical  Effect  of  water  is  product  of  its  weight  into  height  from  which 
it  falls. 

EXAMPLE.— Volume  of  water  discharged  upon  an  overshot- wheel  is  640  cube  feet 
per  minute,  and  effective  height  of  fall  is  22  feet;  what  is  IP? 

— — —  =  26.67,  which,  x  .75  =  assumed  ratio  of  effect  to  power  —  20  IP. 

Useful    Effect    of  an    Oversliot-\vlieel. 

With  a  large  wheel  running  in  most  advantageous  manner,  .84  of  power 
may  be  taken  for  effect. 

Velocity  of  a  wheel  bears  a  constant  ratio,  for  maximum  effects,  to  that 
of  the  flowing  water,  and  this  ratio  is  at  a  mean  .55. 

Ratio  of  effect  to  power  with  radial-buckets  is  .78  that  of  elbow-buckets. 
Ratio  of  effect  decreases  as  proportion  of  head  to  total  head  and  fall  increases. 
Thus,  a  wheel  10  feet  in  diameter  gave,  with  heads  of  water  above  gate, 
ranging  from  .25  to  3.75  feet,  a  ratio  of  effect  decreasing  from  .82  to  .67  of 
power. 

Higher  an  overshot-wheel  is,  in  proportion  to  whole  descent  of  water, 
greater  will  be  its  effect.  Effect  is  as  product  of  volume  of  water  and  its 
perpendicular  height. 

Weight  of  arch  of  loaded  buckets  in  Ibs.  is  ascertained  by  multiplying 
.444  of  their  number  by  number  of  cube  feet  in  each,  and  that  product  by  40. 

TJnd.ersh.ot-wh.eel. 

UNDERSHOT-WHEEL  is  usually  set  in  a  curb,  with  as  little  clearance  for 
escape  of  water  as  practicable ;  hence  a  curb  concentric  to  this  wheel  is  more 
effective  than  one  set  straight  or  tangential  to  it. 

Computations  for  an  undershot-wheel  and  rules  for  construction  are  near- 
ly identical  with  those  for  a  breast-wheel. 

Buckets  are  usually  set  radially,  but  they  may  be  inclined  upward,  so  as 
to  be  more  effectively  relieved  of  water  upon  their  return  side,  and  they  are 
usually  filled  from  .5  to  .6  of  their  volume.  Depth  of  shrouding  should  be 
from  15  to  18  ins.,  in  order  to  prevent  overflow  of  water  within  the  wheel, 
which  would  retard  it. 

Velocity  of  periphery  should  equal  theoretical  velocity  due  to  head  of 
water  x  .57. 

NOTE.— When  constructed  without  shrouding,  as  in  a  current- wheel,  etc.,  buckets 
become  blades. 

Sluice-gate  should  be  set  at  an  inclination  to  plane  of  curb,  or  tangential 
to  wheel,  in  order  that  its  aperture  may  be  as  close  to  wheel  as  practicable ; 
and  in  order  to  prevent  partial  contraction  of  flow  of  water,  lower  edge  of 
sluice  should  be  rounded. 

Effect  of  an  undershot-wheel  is  less  than  that  of  a  breast-wheel,  as  the 
fall  available  as  weight  is  less  than  with  latter. 

To    Compute    I>o\ver   of*  an  Under  shot- wheel. 

Proceed  as  per  rule  for  an  overshot-wheel,  using  93  750  for  50000,  and  .4 
for  .75. 

Or,  V  h  .  ooo  66  =  IP ;    or,  ^| —  =  V.    V  representing  volume  of  water  in  cube 

n 
feet  per  minute,  and  h  head  of  water  in  feet. 


HYDRODYNAMICS.  567 

IPoncelet's    "Wheel. 

PONCELET'S  WHEEL. — Buckets  are  curved,  so  that  flow  of  water  is  in 
course  of  their  concave  side,  pressing  upon  them  without  impact;  and  effect 
is  greater  than  when  water  impinges  at  nearly  right  angles  to  a  plane  sur- 
face or  blade. 

This  wheel  is  advantageous  for  application  to  falls  under  6  feet,  as  its 
effect  is  greater  than  that  of  other  undershot  wheels  with  a  curb,  and  for 
falls  from  3  to  6  feet  its  effect  is  equal  to  that  of  a  Turbine. 

For  falls  of  4  feet  and  less,  efficiency  is  65  per  cent.,  for  4.25  to  5  feet,  60 
per  cent.,  and  from  6  to  6.5  feet,  55  to  50  per  cent. 

In  its  arrangement,  aperture  of  sluice  should  be  brought  close  to  face  of 
wheel.  First  part  of  course  should  be  inclined  from  4°  to  6° ;  remainder  of 
course,  which  should  cover  or  embrace  at  least  three  buckets,  should  be  car- 
ried concentric  to  wheel,  and  at  end  of  it  a  quick  fall  of  6  ins.  made,  t«  guard 
against  effect  of  back-water.  Sluice  should  not  be  opened  over  i  foot  in  any 
case,  and  6  ins.  is  a  suitable  height  for  falls  of  5  and  6  feet. 

Distance  between  two  buckets  should  not  exceed  8  or  10  ins.,  and  radius 
of  wheel  should  not  be  less  than  40  ins.,  or  more  than  8  feet. 

Plane  of  stream  or  head  of  water  should  meet  periphery  of  wheel  at  an 
angle  of  from  24°  to  30°,  Space  between  wheel  and  its  curb  should  not  ex- 
ceed .4  of  an  inch. 

Depth  of  shrouding  should  be  at  least  .25  depth  of  head  of  water,  or  such 
as  to  prevent  water  from  flowing  through  it  and  over  the  buckets,  and  width 
of  wheel  should  be  equal  to  that  of  stream  of  impinging  water. 

Effect  of  this  wheel  increases  with  depth  of  water  flow,  and,  therefore, 
other  elements  being  equal,  as  filling  of  buckets,  to  obtain  maximum  effect, 
water  should  flow  to  buckets  without  impact,  and  velocity  of  wheel  should 
be  only  a  little  less  than  half  that  of  velocity  of  water  flowing  upon  wheel. 

To    Compute    Proportions   of  a,  I*oiioelet   Wheel. 

NOTE.  —  As  it  is  impracticable  to  arrive  at  the  results  by  a  direct  formula,  they 
must  be  obtained  by  gradual  approximation. 

EXAMPLE.— Height  of  fall  is  4.5  feet;  volume  of  water  40  cube  feet  per  second; 
radius  of  wheel  =  2  h,  or  9  feet;  depth  of  the  stream  =.j  5  feet;  and  C  assumed  at.  9. 

V  representing  volume  of  water  in  cube  feet  per  second,  h  height  of  fall,  d  depth  of 
shrouding  ==  —  . 1-  d' ;  d'  opening  of  and  e  width  of  sluice,  r  radius  of  curva- 
ture of  buckets  — ,  and  a  of  wheel,  all  in  feet ;  n  number  of  revolutions  =  ^-? 

cos.  z  pa 

per  minute;  c  velocity  of  circumference  of  wheel  and  v  velocity  of  water,  both  in  feet 
per  second ;  C  coefficient  of  resistance  of  flow  of  water ;  x  angle  between  plant  of 

flowing  water  and  that  of  circumference  of  wheel  at  point  of  contact,  sin.  of  -  = 


Vcos.  z;  z  angle  made  by  circumference  of  wheel  with  end  of  buckets  =  2  tang,  y; 

and  y  angle  of  direction  of  water  from  circumference  of  wheel  =.  —       I . 

V*  +  £ 

I  .  J*. 

Then  v  =  -9^/2  g  ( h J  =  .9  x  16.29  =  J4-66  feet  :.  velocity  of  wheel,  being 


568  HYDRODYNAMICS. 

less  than  half  velocity  of  water  ;  c  =  '4-66  —  .66  _  ^  ^  ^          d  =  -  x 


>  x 


.25,  angle  corresponding  to  which  =  14°  30';  n  =  3°  7  =7.43  revolutions; 
z  =  2  tang,  i/  =  2  X-  258  62  =  .517  24  .-.  *  =  27°2o';  e  =  £°^66  =  3.63  feet; 
r  —  -  LJL  —  __  '-5  _  x  8  ^  .  x  _  gin  *  _  ^/QQS'Z  _  Vcos.  27°  20'  =  .943 

COS.  27°  20  .88835  2 

:=  sin.  of  70°  34'  /.  x  =  141°  8'.    Effect  is  a  maximum  when  c  =  .5  v  cos.  y. 

Fig.  2.  Construction  of  Buckets  (Fig.  2).     (Molesworth.) 

From  point  of  bucket,  a,  draw  a  line,  a  6,  at  an  angle  of  26° 
with  radial  line,  point  6,  where  this  line  cuts  an  imaginary  cir- 
cle, drawn  at  a  distance  of  s  x  1.17  from  periphery  of  wheel,  is 
centre  from  which  bucket  is  struck  with  radius,  6  a.  Radius  of 
wheel  should  not  be  less  than  7,  or  more  than  16  feet. 

Curb  should  fit  wheel  accurately  for  18  or  20  ins.,  measured 
back  from  perpendicular  line  which  passes  through  axis  of 
wheel,  the  breast  should  then  incline  i  in  10,  or  i  in  15  towards 
sluice. 

After  passing  axis  of  wheel  in  tail-race,  curb  should  make  a 
sudden  dip  of  6  ins. 

To   Compute    I?o~wer  of  a    IPoncelet   "Wheel. 

880  TP 
V  h  .001  13  =  EP,    and  —  -  —  =  V.    V  =  velocity  of  theoretical  periphery  =  .  55.  * 

Number  of  buckets  1.6  D-f-  1.6,  D  =  diameter  of  wheel  in  feet.  Shrouding  .33  to 
.5  depth  of  head  of  water,  and  D  •=  2  h,  and  not  less  than  7  or  more  than  16  feet. 

Breast-wheel. 

BREAST-WHEEL  is  designed  for  fails  of  water  varying  from  5  to  15  feet, 
and  for  flows  of  from  5  to  80  cube  feet  per  second.  "  It  is  constructed  with 
either  ordinary  buckets  or  with  blades  confined  by  a  Curb. 

Enclosure  within  which  water  flows  to  a  breast-wheel  as  it  leaves  the  sluice 
is  termed  a  Curb  or  Mantle. 

When  blades  are  enclosed  in  a  cur  J,  they  are  not  required  to  hold  water  ; 
hence  they  may  be  set  radial,  and  they  should  be  numerous,  as  the  loss  of 
water  escaping  between  the  wheel  and  the  curb  is  less  the  greater  their  num- 
ber ;  and  that  they  may  not  lift  or  carry  up  water  with  them  from  tail-race, 
it  is  proper  to  give  them  such  a  plane  that  it  may  leave  the  water  as  nearly 
vertical  as  may  be  practicable. 

Distance  between  two  buckets  or  blades  should  be  from  1.3  to  1.5  times 
head  over  gate  for  low  velocity  of  wheel  and  more  for  a  high  velocity,  or 
equal  to  depth  of  shrouding,  or  at  from  10  to  15  ins. 

It  is  essential  that  there  should  be  air-holes  in  floor  of  buckets,  to  prevent 
air  from  impeding  flow  of  water  into  them,  as  the  water  admitted  is  nearly 
as  deep  as  the  interval  between  them  ;  and  velocity  of  wheel  should  be  such 
that  buckets  should  be  filled  to  .5  or  .625  of  their  volume. 

.When  wheels  are  constructed  of  iron,  and  are  accurately  set  in  masonry, 
a  clearance  of  .5  of  an  inch  is  sufficient. 

*  -\/2gh  in  ft*  ftf  tecond. 


HYDRODYNAMICS.  569 

High  Breast-wheel  is  used  when  level  of  water  in  tall-race  and  penstock 
or  forebay  are  subject  to  variation  of  heights,  as  wheel  revolves  in  direction 
in  which  water  flows  from  blades,  and  back-water  is  therefore  less  disad- 
vantageous, added  to  which,  penstocks  can  be  so  constructed  as  to  admit  of 
an  adjustable  point  of  opening  for  the  water  to  flow  upon  the  wheel. 

Effect  of  this  wheel  is  equal  to  that  of  the  overshot,  and  in  some  instances, 
from  the  advantageous  manner  in  which  water  is  admitted  to  it,  it  is  greater 
when  both  wheels  have  same  general  proportions. 

Under  circumstances  of  a  variable  supply  of  water,  Breast-wheel  is  better 
designed  for  effective  duty  than  Overshot,  as  it  can  be  made  of  a  greater 
diameter;  whereby  it  affords  an  increased  facility  for  reception  of  water 
into  its  buckets,  also  for  its  discharge  at  bottom ;  and  further,  its  buckets 
more  easily  overcome  retardation  of  back-water,  enabling  it  to  be  worked 
for  a  longer  period  in  back-water  consequent  upon  a  flood. 

In  a  well-constructed  wheel  an  efficiency  of  93  per  cent,  was  observed  by  M. 
Morin,  and  Sir  Wm.  Fairbairn  gives,  at  a  velocity  of  circumference  of  wheel  of 
5  feet,  an  efficiency  of  75  per  cent.  Velocity  usually  adopted  by  him  was  from  4 
to  6  feet  per  second,  both  for  high  and  low  falls;  a  minimum  of  3.5  feet  for  a  fall  of 
40  and  a  maximum  of  7  feet  for  a  fall  of  5  to  6  feet. 

When  water  flows  at  from  10°  to  12°  above  horizontal  centre  of  wheel,  Fairbairn 
gives  area  of  opening  of  buckets,  compared  with  their  volume,  as  8  to  24. 

The  capacity  between  two  buckets  or  blades  should  be  very  nearly  double  that  of 
volume  of  water  expended. 

To  Compute  Proportions  and  Effect  of  a  Breast-awheel. 

ILLUSTRATION. — Flow  of  water  is  15  cube  feet  per  second ;  height  of  fall,  measured 
from  centre  of  pressure  of  opening  to  tail  race,  is  8.5  feet;  velocity  of  circumference 
of  wheel  5  feet  per  second ;  and  depth  of  buckets  or  blades  i  foot,  filled  to  .5  of  their 
volume. 

Width  of  wheel  =  — ,  d  representing  depth,  and  v  velocity  of  buckets ;  —  —  =  3 ; 

and  as  buckets  are  but .  5  filled,  3  -r- .  5  =  6  feet.  Assume  water  is  to  flow  wi  th  double 
velocity  of  circumference  of  wheel ;  v  =  5  X  2  =  10  feet ;  and  fall  required  to  gen- 
erate this  velocity  =  —  X  i.i  —  ^'  — 7 X  i-i  =  1.71  feet. 

Deducting  this  height  from  total  fall,  there  remains  for  height  of  curb  or  shroud- 
ing,  or  fall  during  which  weight  of  water  alone  acts,  h  —  h'  =  8. 5  — 1.71  =  6.79  feet. 

Making  radius  of  wheel  12  feet,  and  radius  of  bucket  circle  u  feet,  whole  mechan- 
ical effect  of  flow  of  water  =  15  x  62.5  X  8.5  =  7968.75  Ibs.,  from  which  is  to  be  de- 
ducted from  10  to  15  per  cent,  for  loss  of  water  by  escape. 

Theoretical  effect,  as  determined  by  M.  Morin,  velocity  of  circumference  about 
.5  of  that  of  water,  and  within  velocities  of  1.66  to  6  feet. 

((—  i_^_j_  ft"\  v  62.5.    a  representing  angle  of  direction  of  velocity  with 

which  water  flows  to  wheel  at  centre  of  thread  of  flow  and  direction  of  velocity  of 
wheel  at  this  line,  and  h"  h  —  h'  in  feet. 

a  is  here  assumed  at  20°.  See  Weisbach,  London,  1848,  vol.  ii.  page  197,  and  for 
the  necessarily  small  value  of  a,  its  cosine  may  be  taken  at  i.  Cos.  20°  =  .94. 

Then  ^IOX'94  —  5)  5_|_6     \  x  Ig  x  6a>5  _ ?  474 x  I$  x  62  5  _ 7006.9  ibs.,  which 

\         32.16  / 

is  to  be  reduced  by  a  coefficient  of  .77  for  a  penstock  cluice,  and  .8  for  an  overfall 
sluice. 

Theoretical  effect,  as  determined  by  Weisbach,  7273  Ibs.,  from  which  are 
to  be  deducted  losses,  which  he  computes  as  follows : 

Loss  by  escape  of  water  between  wheel  and  curb =    916 

Loss  by  escape  at  sides  of  wheel  and  curb =    180 

Friction  and  resistance  of  water  =  2. 5  per  cent =    160 

1256  »t. 
3B* 


HYDRODYNAMICS. 

Friction  of  wheel  as  per  formula,  page  571,  =  W  r  n  G  .0086;    a=  .048     I—  = 

.048     /  -  =  4.  36  ins.  :        and  n  =  -  -  --  -  =  4  revolutions.        C  =  .08. 

4   V      2  12X2X3-1416 

r=  4.  36-7-  2  =  2.18.     Then  i6sooX  2.18  X  4  X  .08  X  .0086  =  98.99  Ibs. 

Whence,  -  —   -  —  "  =  .72  efficiency,  upon  assumption  of  losses  as  com- 

puted by  Weisbach. 

To    Compute    IPower   of  a    Breast-wliee!. 

RULE.  —  Proceed  as  per  rule  for  an  overshot-wheel,  using  55  ooo  and  .65 
with  a  high  breast,  and  62  500  and  .6  for  a  low  breast. 

Or,  High  breast,  .0612  V  h  =  IP,    and  ^  —  =  V  ;    and  Low  breast  .0546  V  h  = 

a-, 


=   . 

ILLUSTRATION.—  Assume  elements  of  preceding  case.    Then  -5  ><62-5X  8.5X60 

33000 
=  14.49,  which  X  -7  =  10.  14  horses. 

7006.0  —  1256-1-102.6x60 
Or,  '-  -  -  -  -  —  —  =  10.27  horses. 

33000 

Openings  of  Buckets  or  Blades.  —  High  Breast,  .33  sq.  foot,  and  Low  Breast,  .2  sq. 
foot  for  each  cube  foot  of  their  volume,  or  generally  6  to  8  in  opening  in  a  high 
breast  and  9  to  12  in  a  low  breast. 

Forms  of  Buckets.—  Two  Part.  d  —  D,  s  =  .$  d,  I  1.25  d  in  large  wheels,  and  =d 
in  wheels  less  than  25  feet  in  diameter. 

Three  Part  Buckets.—  d  divided  into  3  equal  parts;  I  =  .25  d,  d  =  D,  s  =  .33  d,  I  = 
d  in  large  wheels,  and  .75  d  in  wheels  less  than  25  feet  in  diameter. 

Ventilating  Buckets  (Fairbairn's).     Spaces  are  about  i  inch  in  width. 

NOTES.  —  A  Committee  of  the  Franklin  Institute  ascertained  that,  with  a  high 
breast-  wheel  20  feet  in  diameter,  water  admitted  under  a  head  of  9  ins.,  and  at  17 
feet  above  bottom  of  wheel,  elbow-buckets  gave  a  ratio  of  effect  to  power  of  .731  at 
a  maximum,  and  radial-blades  .653.  With  water  admitted  at  a  height  of  33  feet 
8  ins.,  elbow-buckets  gave  .658,  and  radial  blades  .628. 

At  10.96  feet  above  bottom  of  wheel,  with  a  head  of  4.29  feet,  elbow-buckets  gave 
.  544,  and  blades  .  329. 

At  7  feet  above  bottom  of  wheel,  and  a  head  of  2  feet,  a  low  breast  gave  for 
elbow-buckets  .62,  and  for  blades  .531. 

At  3  feet  8  ins.  above  bottom  of  wheel,  and  a  head  of  i  foot,  elbow-buckets  gave 
.555,  and  blades  .533. 

Current-  wheel. 

CURRENT-WHEEL.—  D.  K.  Clark  assigns  the  most  suitable  ratio  of  veloc- 
ity of  blades  to  that  of  current  as  40  per  cent. 

Depth  of  blades  should  be  from  .25  to  .2  of  radius  ;  it  should  not  be  less 
than  12  or  14  ins.  Diameter  is  usually  from  13  to  16.5  feet,  with  12  blades  ; 
but  it  is  thought  that  there  might  be  an  advantage  in  applying  18  or  even 
24.  The  blades  should  be  completely  submerged  at  lower  side,  but  not  more 
than  2  ins.  under  water,  and  not  less  than  2  at  one  time. 

—  (v  —  s)2  =  H*-     a  representing  area  of  vertical  section  of  immersed  blades  in 
150 
tq.  feet,  s  velocity  of  wheel  at  circumference,  and  v  of  stream,  both  in  feet  per  second. 

Or,  .38  —  V  62.  5  =  useful  effect.    Hence,  efficiency  =  .  sa 


HYDRODYNAMICS.  5/1 

ITTu.tter-\vlieel. 

Flutter  or  Saw-mitt  Wheel— Is  a  small,  low  breast-wheel  operating  under 
a  high  head  of  water ;  the  design  of  its  construction,  water  being  plenty,  is 
the  attainment  of  a  simple  application  to  high-speed  connections,  as  a  gang 
or  circular  saw.  In  effect  it  is  from  .6  to  .7  that  of  an  overshot-wheel  of 
like  head  of  fall. 

V  s 

—  (v  —  s)  =  H?.     v  and  s  as  preceding. 

150 

Friction   of  Journals   or   Grudgeons. 

A  very  considerable  portion  of  mechanical  effect  of  a  wheel  is  lost  in  ef- 
fect absorbed  by  friction  of  its  gudgeons. 

To  Compute    Friction,   of*  Journals  or  Gudgeons   of  a 
\Vater-\vheel. 

W  rn  C  .0086  =/  W  representing  weight  of  wheel  in  Ibs.,  r  radius  of  gudgeon  in 
ins.,  and  n  number  of  revolutions  of  wheel  per  minute. 

For  well-turned  surfaces  and  good  bearings,  C  =  .o75  with  oil  or  tallow;  when 
best  of  oil  is  well  supplied  =  .054;  and,  as  in  ordinary  circumstances,  when  a  black- 
lead  unguent  is  alone  applied  = .  u. 

ILLUSTRATION.— A  wheel  weighing  25000  Ibs.  has  gudgeons  6  ins.  in  diameter,  and 
makes  6  revolutions  per  minute;  what  is  loss  of  effect? 

Assume  C  =  .08.    Then  25000  X  -  X  6  X  .08  X  .0086  =  309.6  Ibs. 

"Weights. — Iron  wheels  of  18  to  20  feet  in  diameter  will  weigh  from  800  to 
jooo  Ibs.  per  H?. 
Wood  wheels  of  30  feet  in  diameter,  2000  to  2500  Ibs.  per  H*. 

To  Compute  Diameter   and  Journals  of  a  Shaft,  Stress 
laid   uniformly   along   its    Length. 

tXwl  /IP 

Cast  Iron,  — —  =  d.     Wood,  6. 12   3/  —  =  d.    W  representing  weight  or  load  in 

Ibs.,  I  length  of  shaft  between  journals  in  feet,  and  d  diameter  of  shaft  in  its  body 
in  ins. 


Journals  or  Gudgeons. — Cast  Iron,  .048     / — =d. 


When  Shaft  has  to  resist  both  Lateral  and  Torsional  Stress.— Ascertain 
the  diameter  for  each  stress,  and  cube  root  of  sum  of  their  cubes  will  give 
diameter. 

To    Compute    Dimensions    of   Arms. 

Cast  Iron,  ^—  =  w.    d  representing  diameter  of  shaft,  and  w  width  of  arm,  both 
yn 

in  ins.,  n  number  of  arms,  —  =  t,  and  t  thickness  of  arm. 

When  Arm  is  )f  Oak,  w  should  be  1.4  times  that  of  iron,  and  thickness  .7  that 
of  width. 

Memoranda. 

A  volume  of  water  of  17. 5  cube  feet  per  second,  with  a  fall  of  25  feet,  applied  to  an 
undershot-wheel,  will  drive  a  hammer  of  1500  Ibs.  in  weight  from  100  to  120  blows 
per  minute,  with  a  lift  of  from  i  to  1.5  feet.* 

A  volume  of  water  of  21.5  cube  feet  per  second,  with  a  fall  of  12.5  feet,  applied  to 
a  wheel  having  a  great  height  of  water  above  its  summit,  being  7.75  feet  in  diame- 
ter, will  drive  a  bammer  of  500  Ibs.  in  weight  100  blows  per  minute,  with  a  lift  of  a 
feet  10  ins.  Estimate  of  power  31. 5  horses. 

*  Volume  of  water  required  for  a  hammer  increases  in  a  much  greater  ratio  than  velocity  to  be  given 
to  it.  it  being  nearly  as  cube  of  velocity. 


5  72  HYDRODYNAMICS. 

A  Stream  and  Overshot  Wheel  of  following  dimensions— viz.,  height  of  head  to 
centre  of  opening,  24.875  ins. ;  opening,  1.75  by  80  ins. ;  wheel,  22  feet  in  diameter 
by  8  feet  face;  52  buckets,  each  i  foot  in  depth,  making  3.5  revolutions  per  minute 
,— drove  3  run  of  4.5  feet  stones  130  revolutions  per  minute,  with  all  attendant  ma- 
chinery, and  ground  and  dressed  25  bushels  of  wheat  per  hour. 

4.5  bushels  Southern  and  5  bushels  Northern  wheat  are  required  to  make  i  bar- 
rel of  flour. 

A  Breast-wheel  and  Stream  of  following  dimensions — viz.,  head,  20  feet;  height 
of  water  upon  wheel,  16  feet;  opening,  18  feet  by  2  ins. ;  diameter  of  wheel,  26  feet 
4  ins. ;  face  of  wheel,  20  feet  9  ins. ;  depth  of  buckets,  15.75  ins. ;  number  of  buck- 
ets, 70;  revolutions,  4.5  per  minute  — drove  6144  self-acting  mule  spindles;  160 
looms,  weaving  printing-cloths  27  ins.  wide  of  No.  33  yarn  (33  hanks  to  a  Ib. ),  and 
producing  24000  hanks  in  a  day  of  n  hours. 

Horizontal   "Wheels. 

In  horizontal  water-wheels,  water  produces  its  effect  either  by  Impact, 
Pressure,  or  Reaction,  but  never  directly  by  its  weight. 

These  wheels  are  therefore  classed  as  Impact,  Pressure,  and  Reaction,  but 
are  now  designated  by  the  generic  term  of  Turbine. 

Tu.r~bin.es. 

TURBINES,  being  operated  at  a  higher  number  of  revolutions  than  Ver- 
tical Wheels,  are  more  generally  applicable  to  mechanical  purposes ;  but 
in  operations  requiring  low  velocities,  Vertical  Wheel  is  preferred. 

For  variable  resistances,  as  rolling-mills,  etc.,  Vertical  Wheel  is  far 
preferable,  as  its  mass  serves  to  regulate  motion  better  than  a  small 
wheel. 

In  economy  of  construction  there  is  no  essential  difference  between 
a  Vertical  Wheel  and  a  Turbine.  When,  however,  fall  of  water  and 
volume  of  it  are  great,  the  Turbine  is  teast  expensive.  Variations  in 
supply  of  water  affect  vertical  wheels  less  than  Turbines. 

Durability  of  a  Turbine  is  less  than  that  of  a  Vertical  Wheel ;  and  it  is 
indispensable  to  its  operation  that  the  water  should  be  free  from  sand,  silt, 
branches,  leaves,  etc. 

With  Overshot  and  Breast  Wheels,  when  only  a  small  quantity  of  water  is 
available,  or  when  it  is  required  or  becomes  necessary  to  produce  only  a  por- 
tion of  the  power  of  ihsfall,  their  efficiency  is  relatively  increased,  from  the 
blades  being  but  proportionately  filled ;  but  with  Turbines  the  effect  is  con- 
trary, as  when  the  sluice  is  lowered  or  supply  decreased  water  enters  the 
wheel  under  circumstances  involving  greater  loss  of  effect.  To  produce 
maximum  effect  of  a  stream  of  water  upon  a  wheel,  it  must  flow  without  im- 
pact upon  it,  and  leave  it  without  velocity ;  and  distance  between  point  at 
which  the  water  flows  upon  a  wheel  and  level  of  water  in  reservoir  should 
be  as  short  as  practicable. 

Small  wheels  give  less  effect  than  large,  in  consequence  of  their  making  a 
greater  number  of  revolutions  and  having  a  smaller  water  arc. 

In  High-pressure  Turbines  reservoir  (of  wheel)  is  enclosed  at  top,  and  water 
is  admitted  through  a  pipe  at  its  side.  In  Low-pressure,  water  flows  into  res- 
ervoir, which  is  open. 

In  Turbines  working  under  water,  height  is  measured  from  surface  of 
water  in  supply  to  surface  of  discharged  water  or  race ;  and  when  they  work 
in  air,  height  is  measured  from  surface  in  supply  to  centre  of  wheel. 

In  order  to  obtain  maximum  effect  from  water,  velocity  of  it,  when  leav- 
ing a  Turbine,  should  be  the  least  practicable. 


HYDRODYNAMICS.  573 

Efficiency  is  greater  when  sluice  or  supply  is  wide  open,  and  it  is  less  af- 
fected by  head  than  by  variations  in  supply  of  water.  It  varies  but  little 
with  velocity,  as  it  was  ascertained  by  experiment  that  when  35  revolutions 
gave  an  effect  of  .64,  55  gave  but  .66." 

When  Turbines  operate  under  water,  the  flow  is  always  full  through  them ; 
hence  they  become  Reaction-wheels,  which  are  the  most  efficient. 

Experiments  of  Morin  gave  efficiency  of  Turbines  as  high  as  .75  of  power. 

Angle  of  plane  of  water  entering  a  Turbine,  with  inner  periphery  of  it, 
should  be  greater  than  90°,  and  angle  which  plane  of  water  leaving  reservoir 
makes  with  inner  circumference  of  Turbine  should  be  less  than  90°. 

When  Turbines  are  constructed  without  a  guide  curve*,  angle  of  plane  of 
flowing  water  and  inner  circumference  of  wheel  =  90°. 

Great  curvature  involves  greater  resistance  to  efflux  of  water ;  and  hence 
it  is  advisable  to  make  angle  of  plane  of  entering  water  rather  obtuse  than 
acute,  say  100° ;  angle  of  plane  of  water  leaving,  then,  should  be  50°,  if  in- 
ternal pressure  is  to  balance  the  external ;  and  if  wheel  operates  free  of 
water,  it  may  be  reduced  to  25°  and  30°. 

If  blades  are  given  increased  length,  and  formed  to  such  a  hollow  curve 
that  the  water  leaves  wheel  in  nearly  a  horizontal  direction,  water  then  both 
impinges  on  blades  and  exerts  a  pressure  upon  them;  therefore  effect  is 
greater  than  with  an  impact-wheel  alone. 

Turbines  are  of  three  descriptions :  Outward,  Downward,  and  Inward  flow. 

Ou.tward.-fi.ow   Turbines. 

FOURNEYRON  TURBINE,  as  recently  constructed,  may  be  considered  as  one 
of  the  most  perfect  of  horizontal  wheels;  it  operates  both  in  and  out  of 
back-water,  is  applicable  to  high  or  low  falls,  and  is  either  a  high  or  low 
pressure  turbine. 

In  high-pressure,  the  reservoir  is  closed  at  top  and  the  water  is  led  to  it 
through  a  pipe.  In  low-pressure,  the  water  flows  directly  into  ,an  open  res- 
ervoir. Pressure  upon  the  step  is  confined  to  weight  of  wheel  alone. 

Foumeyron  makes  angle  of  plane  of  water  entering  =  90°,  and  angle  of 
plane  of  water  leaving  =  30°. 

Efficiency  is  reduced  in  proportion  as  sluice  is  lowered,  for  action  of  water 
on  wheel  is  less  favorably  exerted.  M.  Morin  tested  a  Fourneyron  turbine 
6.56  feet  in  diameter,  and  he  found  that  efficiency  varied  from  a  minimum 
of  24,  to  79  per  cent.,  when  supply  of  water  was  reduced  to  .25  of  full  supply. 
In  practice,  radial  length  of  blades  of  wheel  is  .25  of  radius,  for  falls  not  ex- 
ceeding 6.5  feet,  .3  for  falls  of  from  6.5  to  19  feet,  and  .66  for  higher  falls. 

To    Compnte    Elements   and.   Results. 

High  Pressure,  6.6  ^/h  =  v:      ^  =  A;      ^I>77  V=Dt;     12.6—  =  V:     and 

v  -\/h  h 

,079  V  h  =  IP.  h  representing  head  of  water,  v  velocity  of  turbine  at  periphery  per 
minute,  and  D  internal  diameter  of  turbine,  all  in  feet,  V  volume  of  water  in  cube  feet 
per  second,  A  sum  of  area  of  orifices  in  sq.feet,  aria  IP  effective  horse-power. 

1.2  D  =  external  diameter  of  turbine  in  feet,  when  it  is  more  than  6  feet,  and  1.4 
when  it  is  less  than  6  feet.  Number  of  guides  =  number  of  blades  J  when  less  than 
24,  and  number  -H  3  when  greater  than  24.  Area  of  section  of  supply-pipe  =  .4  V. 

For  construction  of  blades  and  guides,  see  Molesworth,  London,  1882,  page  540. 

*  Guide  curves  are  plates  upon  centre  body  of  a  Turbine,  which  give  direction  to  flowing  water, 
or  to  blades  of  wheel  wnich  surround  them. 

f  In  extreme  cases  of  very  high  falls  diameter  given  by  this  formula  may  be  increased. 
t  Fourneyron '»  rule  for  the  number  of  blades  it  constant  number  36,  irrespective  of  size  of  turbine. 


574  HYDRODYNAMICS. 


Operation   of  High-Pressure   Turbines. 


30 
4.2 

36 


60 


70 


80 

1.6 
59 


1.27 

66 


1.05 
73 


140 
78' 


•9 


1 80 


200 


.63 


94 


h  =  head  of  water  in  feet,  V  volume  of  water  in  cube  feet  required  for  each  10  IP, 
and  v  velocity  of  periphery  of  turbine  in  feet  per  second. 

BOYDEN  TURBINE.  —  Mr.  -  Boy  den,  of  Massachusetts,  designed  an 
outward-flow  turbine  of  75  IP,  which  realized  an  efficiency  of  88  per  cent. 
Peculiar  features,  as  compared  with  a  Fourneyron  turbine,  are,  ist,  and  most 
important,  the  conduction  of  the  water  to  turbine  through  a  vertical  trun- 
cated cone,  concentric  with  the  shaft.  The  water,  as  it  descends,  acquires  a 
gradually  increasing  velocity,  together  with  a  spiral  movement  in  direction 
of  motion  of  wheel.  The  spiral  movement  is,  in  fact,  a  continuation  of  the 
motion  of  the  water  as  it  enters  cone.  —  2d.  Guide-plates  at  base  are  inclined, 
so  as  to  meet  tangentially  the  approaching  water.  —  3d.  A  "  diffuser,"  or  annu- 
lar chamber  surrounding  wheel,  into  which  water  from  wheel  is  discharged. 
This  chamber  expands  outwardly,  and,  thus  escaping  velocity  of  water,  is 
eased  off  and  reduced  to  a  fourth  when  outside  of  diffuser  is  reached.  Effect 
of  diffuser  is  to  accelerate  velocity  of  water  through  machine  ;  and  gain  of 
efficiency  is  3  per  cent.  Diffuser  must  be  entirely  submerged.  (D.  K.  Clark.) 

PONCELET  TURBINE.  —  This  wheel  is  alike  to  one  of  his  undershot-wheels 
set  horizontally,  and  it  is  the  most  simple  of  all  horizontal  wheels. 

To    Compu.te    Elements    of    Q-eneral    Proportion    and. 
Results.    (Lt.F.A.Mahan,U.S.A.) 

J)>  -5l>2V^  =  V;  .1  D  =  H;  '4.49^=1); 


3(D+io)=N;       =  u;;          =  W;    D-        =  d 

V 

and  C  coefficient  for  V  in  terms  of  V=  —  -.    D  and  d  representing  exterior  and  in- 

terior diameters  of  wheel,  H  and  h  heights  of  orifices  of  discharge  at  outer  circum- 
ference and  of  fall  acting  on  wheel,  w  and  w  shortest  distances  between  two  adjacent 
blades  and  two  adjacent  guides,  all  in  feet,  V,  V,  and  v  velocities  due  to  fall  of  water 
passing  through  narrowest  section  of  wheel,  and  of  interior  circumference  of  wheel, 
all  in  feet  per  second,  N  and  n  numbers  of  blades  and  guides,  and  IP  actual  horse- 
power. 

For  falls  of  from  5  feet  to  40,  and  diameters  not  less  than  2  feet,  n  w  should  be 
equal  to  diameter  of  wheel.  H  equal  to  .  i  D,  n  w'  =  d,  and  4  w  =  width  of  crown. 
For  falls  exceeding  this,  H  should  be  smaller,  in  proportion  to  diameter  of  wheel. 

Downward-flow  Turbines. 

In  turbines  with  downward  flow,  wheel  is  placed  below  an  annular  series 
of  guide-blades,  by  which  water  is  conducted  to  wheel.  The  water  strikes 
curved  blades,  and  falls  vertically,  or  nearly  so,  into  tail-race  ;  consequently, 
centrifugal  action  is  avoided,  and  downward  flow  is  more  compact. 

FONTAINE  TURBINE  yields  an  efficiency  of  70  per  cent.,  when  fully 
charged.  When  supply  of  water  is  shut  off  to  .75,  by  sluice,  efficiency  is 
57  per  cent.  Best  velocity  at  mean  circumference  of  wheel  is  equal  to  55 
per  cent,  of  that  due  to  height  of  fall.  It  may  vary  .25  of  this  either  way, 
without  materially  affecting  efficiency. 

In  operation  the  water  in  race  is  in  immediate  contact  with  wheel,  and  its 
efficiency  is  greatest  when  sluice  is  fully  opened.  Its  efficiency,  also,  is  less 
affected  by  variations  of  head  of  flow  than  in  volume  of  water  supplied; 
hence  they  are  adapted  for  Tide-mills. 


HYDRODYNAMICS. 


575 


JONVAL  TURBINE.— This  wheel  is  essentially  alike  in  its  principal  propor- 
tions to  Fontaine's,  and  in  principle  of  operation  it  is  the  same.  Water  in 
race  must  be  at  a  certain  depth  below  wheel. 

For  convenience,  it  is  placed  at  some  height  above  level  of  tail-race,  within 
an  air-tight  cylinder,  or  "  draft-tube,"  so  that  a  partial  vacuum  or  reduction 
of  pressure  is  induced  under  wheel,  and  effect  of  wheel  is  by  so  much  in- 
creased. Resulting  efficiency  is  same  as  if  wheel  was  placed  at  level  of  tail- 
race  ;  and  thus,  while  it  may  be  placed  at  any  level,  advantage  is  taken  of 
whole  height  of  fall,  and  its  efficiency  decreases  as  volume  of  water  is  di- 
minished or  as  sluice  is  contracted. 

To   Compute   Elements   and.   Results. 

Low  Pressure.— For  falls  of  30  feet  and  less. 

V  Vi.77  V  IP 

6<Jh  =  v;    =  —  =  A;     -       { —  =  D*;    12.7  -r-  =  V;    and  .070  V  h  =  H*. 
v  Y/i  n> 

h  representing  head  of  water,  v  velocity  of  turbine  at  periphery  per  minute,  and  D 
internal  diameter  of  turbine,  all  in  feet,  V  volume  of  water  in  cube  feet  per  second, 
A  sum  of  area  of  orifices  in  sq.feet,  and  IP  effective  horse-power. 

1.2  Dm  external  diameter  of  turbine  in  feet,  when  it  is  more  than  6  feet,  and  1.4 
when  it  is  less  than  6  feet.  Number  of  guides  =  number  of  blades  t  when  less  than 
24,  and  number  -+-  3  when  greater  than  24.  Area  of  section  of  supply-pipe  =  .4  V. 

For  construction  of  blades  and  guides,  see  Molesworth,  London,  1882.  page  540. 

ire   Turbines.     (Molesworth.) 


•0 

5l 

P 

10 

HP 

IS 

rP 

20 

H? 

30 

IP 

40 

H> 

50 

ff 

1 

v 

V 

R 

V 

R 

V 

R 

V 

R 

V 

R 

V 

R 

V 

R 

n  *R 

IOO 

2-5 

5 

n.38 

12.5 

if 

25 

57 

38 

47 

50 

41 

75 

33 

IOO 

28 

126 

26 

7-5 

16.38 

8.S 

136 

17 

35 

79 

33 

68 

5i 

56 

68 

48 

8s 

43 

10 

18.96 

6.3 

1  80 

12.6 

128 

19 

105 

25 

go 

38 

75 

50 

64 

63 

58 

IS 

20 

23.22 
26.82 

4.2 

319 

8.4 
6.3 

226 
329 

12.6 

9-3 

i85 
273 

Hi 

232 

25 
18.9 

IQ4 

33 

2.S 

"3 
164 

42 
li 

100 

148 

25 

30 

— 

— 

7-5 

35« 

10 

310 

15 

253 

20 

220 

25 

196 

3° 

32.88 

— 

— 

— 

— 

— 

8.4 

380 

12.6 

310 

'7 

268 

21 

240 

v  representing  velocity  of  centre  of  blades  in  feet  and  V  volume  of  water,  in  cube 
feet,  both  per  second,  R  revolutions  per  minute,  and  IP  effective  horse-power. 

Vertical  Shaft.      3/-^n —  =  diameter  of  sJiqft  in  ins. 

Inward-flow   Turbine. 

INWARD-FLOW  TURBINE.  —  Inward-flow  or  vortex  wheel  is  made  with 
radiating  blades,  and  is  surrounded  by  an  annular  case,  closed  externally, 
and  open  internally  to  wheel,  having  its  inner  circumference  fitted  with  four 
curved  guide-passages.  The  water  is  admitted  by  one  or  more  pipes  to  the 
case,  and  it  issues  centripetally  through  the  guide-passages  upon  circum- 
ference of  wheel.  The  water  acting  against  the  curved  blades,  wheel  is 
driven  at  a  velocity  dependent  on  height  of  fall,  and  water  having  expended 
its  force,  passes  out  at  centre.  This  wheel  has  realized  an  efficiency  as  high 
as  77.5  per  cent.  It  was  originally  designed  by  Prof.  James  Thomson. 

SWAIN  TURBINE. — Combines  an  inward  and  a  downward  discharge.  Re- 
ceiving edges  of  buckets  of  wheel  are  vertical  opposite  guide-blades,  and 
lower  portions  of  the  edges  are  bent  into  form  of  a  quadrant.  Each  bucket 
thus  forms,  with  the  surface  of  adjoining  bucket,  an  outlet  which  combines 
an  inward  and  a  downward  discharge.  One,  72  ins.  in  diameter,  was  tested 

*  In  extreme  cases  of  very  high  falls  diameter  given  by  this  formula  may  be  increased. 

t  Pourneyron's  rule  for  the  number  of  blades  is  constant  nmmber  36,  irrespective  of  site  of  turbine. 


576 


HYDEODYNAMICS. 


by  Mr.  J.  B.  Francis,  for  several  heights  of  gate  or  sluice,  from  2  to  13.08 
ins.,  and  circumferential  velocities  of  wheel  ranging  from  60  to  80  per  cent, 
of  respective  velocities  due  to  heads  acting  on  wheel. 

For  a  velocity  of  60  per  cent.,  and  for  heights  of  gate  varying  within  limits  al- 
ready stated,  efficiency  ranged  from  47.5  to  76.5  per  cent.,  and  for  a  velocity  of  80 
per  cent,  it  ranged  from  37.5  to  83  per  cent.  Maximum  efficiency  attained  was  84 
per  cent,  with  a  i2-inch  gate  and  a  velocity-ratio  of  76  per  cent. ;  but  from  g-inch 
to  i3-inch  gate,  or  from  .66  gate  to  full  gate,  maximum  efficiency  varied  within 
very  narrow  limits— from  83  to  84  per  cent ,— velocity-ratios  being  72  per  cent,  for 
9-inch  gate,  and  76.5  per  cent,  for  full  gate.  At  half-gate,  maximum  efficiency  was 
78  per  cent.,  when  velocity-ratio  was  68  per  cent.  At  quarter-gate,  maximum  effi- 
ciency was  61  per  cent.,  and  velocity-ratio  66  per  cent 

TREMONT  TURBINE,  as  observed  by  Mr.  Francis,  in  his  experiments  at 
Lowell,  Mass.,  gave  a  ratio  of  effect  to  power  as  .793  to  i. 

VICTOR  TURBINE  is  alleged  to  have  given  an  effect  of  .88  per  cent,  under 
a  head  of  18.34  feet,  with  a  discharge  of  977  cube  feet  of  water  per  minute, 
and  with  343.5  revolutions. 

Tangential  Wheel. 

Wheels  to  which  water  is  applied  at  a  portion  only  of  the  circumference 
are  termed  tangential.  They  are  suited  for  very  high  falls,  where  diameter 
and  high  tangential  velocity  may  be  combined  with  moderate  revolutions. 
The  Girard  turbine  belongs  to  this  class.  It  is  employed  at  Goeschenen 
station  for  St.  Gothard  tunnel ,  it  operates  under  a  head  of  279  feet.  The 
wheels  are  7  feet  10.5  ins.  in  diam.,  having  80  blades,  and  their  speed  is  160 
revolutions  per  minute,  with  a  maximum  charge  of  water  of  67  gallons  per 
second.  An  efficiency  of  87  per  cent,  is  claimed  for  them  at  the  Paris 
water-works ;  ordinarily  it  is  from  75  to  80  per  cent.  (D.  K.  Clark.) 

Impact   and.   Reaction  Wheel. 

IMPACT-WHEEL. — Impact  Turbine  is  most  simple  but  least  efficient  form 
of  impact-wheel.  It  consists  of  a  series  of  rectangular  buckets  or  blades, 
set  upon  a  wheel  at  an  angle  of  50°  to  70°  to  horizon ;  the  water  flows  to 
blades  through  a  pyramidal  trough  set  at  an  angle  of  20°  to  40°,  so  that 
the  water  impinges  nearly  at  right  angles  to  blades.  Effect  is  .5  entire  me- 
chanical effect,  which  is  increased  by  enclosing  blades  in  a  border  or  frame. 

If  buckets  are  given  increased  length,  and  formed  to  such  a  hollow  curve 
that  the  water  leaves  wheel  in  nearly  a  horizontal  direction,  the  water  then 
impinges  on  buckets  and  exerts  a  pressure  upon  them ;  effect  therefore  is 
greater  than  with  the  force  of  impact  alone. 

By  deductions  of  Weisbach  it  appears  that  effect  of  impact  is  only  half 
available  effect  under  most  favorable  circumstances. 

REACTION-WHEEL.— Reaction  of  water  issuing  from  an  orifice  of  less 
capacity  than  section  of  vessel  of  supply,  is  equal  to  weight  of  a  column  of 
water,  basis  of  which  is  area  of  orifice  or  of  stream,  and  height  of  which  is 
twice  height  due  to  velocity  of  water  discharged. 

Hence,  the  expression  is  2.  —  a  w  =  R.  to  representing  weight  of  a  cube  foot  of 
water  in  Z6*.,  and  a  area  of  opening  in  sq.feet 

WHITELAW'S  is  a  modification  of  Barker's ;  the  arms  taper  from  centre 
towards  circumference  and  are  curved  in  such  a  manner  as  to  enable  the 
water  to  pass  from  central  openings  to  orifices  in  a  line  nearly  right  and 
radial,  when  instrument  is  operating  at  a  proper  velocity ;  in  order  that  very 
little  centrifugal  force  may  be  imparted  to  the  water  by  the  revolution  of 
the  arms,  and  consequently  a  minimum  of  frictional  resistance  is  opposed 
to  course  of  the  water. 


HYDRODYNAMICS.  577 

A  Turbine  9.55  feet  in  diameter,  with  orifices  4.944  ins.  in  diameter,  oper- 
ated  by  a  fall  of  25  feet,  gave  an  efficiency  of  75  per  cent ,  including  friction 
of  gearing  of  an  inclined  plane. 

When  a  reaction  wheel  is  loaded,  so  that  height  due  to  velocity,  corresponding  to 
velocity  of  rotation  v,  is  equal  to  fall,  or  —  •=  ft,  or  v  =  Vzgh,  there  is  a  loss  of  17 

v2 
per  cent,  of  available  effect;  and  when  —  =  2  h,  there  is  a  loss  of  but  10  per  cent. ; 

v2 
and  when  — t-  =  4  h,  there  is  a  loss  of  but  6  per  cent.    Consequently,  for  moderate 

falls,  and  when  a  velocity  of  rotation  exceeding  velocity  due  to  height  of  fall  may 
be  adopted,  this  wheel  works  very  effectively. 

Efficiency  of  wheel  is  but  one  half  that  of  an  undershot- wheel. 

When  sluice  is  lowered,  so  that  only  a  portion  of  wheel  is  opened,  efficiency 
of  a  Reaction-wheel  is  less  than  that  of  a  Pressure  Turbine. 

Ratio  of  Effect  to  Power  of  several  Turbines  is  as  follows: 

Poncelet 65  to  75  to  i  I  Jonval „ , 6  to  7  to  i 

Fourneyron 6    to  75  to  i  J  Fontaine 6  to  7  to  i 

BARKER'S  MILL. — Effect  of  this  mill  is  considerably  greater  than  that 
which  same  quantity  of  water  would  produce  if  applied  to  an  undershot- 
wheel,  but  less  than*  that  which  it  would  produce  if  properly  applied  to  an 
overshot-wheel. 

For  a  description  of  it,  see  Grier's  Mechanics'  Calculator,  page  234;  and  for  ita 
formulas,  see  London  Artisan,  1845,  page  229. 

IMPULSE   AND   RESISTANCE  OP  FLUIDS. 

Impulse  and  Resistance  oi"Water. — Water  or  any  other  fluid, 
when  flowing  against  a  body,  imparts  a  force  to  it  by  which  its  condition  of 
motion  is  altered.  Resistance  which  a  fluid  opposes  to  motion  of  a  body 
does  not  essentially  differ  from  Impulse. 

Impulse  of  one  and  same  mass  of  fluid  under  otherwise  similar  circum- 
stances is  proportional  to  relative  velocities  c  =p  v  of  fluid. 

For  an  equal  transverse  section  of  a  stream,  the  impulse  against  a  surface 
at  rest  increases  as  square  of  velocity  of  water. 

Impulse  against  Plane  Surfaces.— The  impulse  of  a  stream  of  water  de- 
pends principally  upon  angle  under  which,  after  impulse,  it  leaves  the  water ; 
it  is  nothing  if  the  angle  is  o,  and  a  maximum  if  it  is  deflected  back  in  a 

line  parallel  to  that  of  its  flow,  or  180°,  2  -  — -  V  w  =  P*. 

When  Surface  of  Resistance  is  a  Plane,  and  =  90°,  then  — ^  V  w  =  P,  and 
for  a  surface  at  res/,  2  a  h  w  =  P.  a  representing  area  of  opening  in  sq.feet. 

P  =  2  A  h  w ;  c  and  v  representing  velocities  of  water  and  of  surface  upon  which  it 
impinges  in  feet  per  second,  w  weight  of  fluid  per  cube  foot  in  Ibs.,  A  transverse  section 
of  stream  in  sq.  ins. ,  and  c=p  v  relative  motions  of  water  and  surface. 

Normal  impulse  of  water  against  a  plane  surface  is  equivalent  to  weight 
of  a  column  which  has  for  its  base  transverse  section  of  stream,  and  for 

c2 
altitude  twice  height  due  to  its  velocity,  2^  =  2  —  . 

Resistance  of  a  fluid  to  a  body  in  motion  is  same  as  impulse  of  a  fluid 
moving  with  same  velocity  against  a  body  at  rest. 

*  Weiabach,  New  York,  1870,  vol.  i.  page  1008. 


57* 


HYDRODYNAMICS. 


Maximum  Effect  of  Impulge.  —  Effect  of  impulse  depends  principally  on 
velocity  v  of  impinged  surface.  It  is,  for  example,  p,  both  when  v  =.  c  and 
v  =  o ;  hence  there  is  a  velocity  for  which  effect  of  impulse  is  a  maximum 

e=  (c  — 1>)  v;  that  is,  v=  — ,  and  maximum  effect  of  impulse  of  water  is  ob- 
tained when  surface  impinged  moves  from  it  with  half  velocity  of  water. 

ILLUSTRATION. — A  stream  of  water  having  a  transverse  section  of  40  sq.  ins.,  dis 
charges  5  cube  feet  per  second  against  a  plane  surface,  and  flows  off  with  a  velocity 

of  12  feet  per  second ;  effect  of  its  impulse,  then,  is  ^^  V  w  =  P;  c=  - — —  =  i8j 

9  4° 

0  =  32.16;    10  =  62.5;    -  ^-^  X  5X62. 5=  58.28  Ibs. 
Hence  mechanical  effect  upon  surface  =  P  v  =  58.28  x  12  =  699.36  Ibs. 
Maximum  effect  would  be  v  =  ^  =  i  x  5  XQ*44  =  9  feet,  and  -  x  —  X  5  X  62. 5 

=  -  X  5-036  x  312.5  =  786.87  Ibs.;  and  hydraulic  pressures- — —  =  87.44  Ibs. 

When  Surface  is  a  Plane  and  at  an  Angle,  then  (i  —  cos.  a)  —  V  «>=  P. 

ILLUSTRATION.— A  stream  of  water,  having  a  transverse  section  of  64  sq.  ins.,  dis- 
charges 17.778  cube  feet  per  second  against  a  fixed  cone,  having  an  angle  of  con- 
vergence from  flow  of  stream  of  50°,  hydraulic  pressure  in  direction  of  stream ; 

thenc  =  ~^—  =  40;  cos.  50°  =  . 64279.      (i  — .64279)  —^  x  17.778  X  62.5  = 
.357  21  X  1382.2  =  494.26  Ibs. 

When  Surface  of  Resistance  is  a  Plane  at  90°,  and  has  Borders  added  to 
its  Perimeter,  effect  will  be  greater,  depending  upon  height  of  border  and 
ratio  of  transverse  section  between  stream  and  part  confined. 

Oblique  Impulse. — In  oblique  impulse  against  a  plane,  the  stream  may  flow 
in  one,  two,  or  in  all  directions  over  plane. 

When  Stream  is  confined  at  Three  Sides,  (i  cos.  a)  -^-  V  w  =  P. 
When  Stream  is  confined  at  Two  Sides,  —  —  sin.  a2  V  w  =  P. 

Normal  impulse  of  a  stream  increases  as  sine  of  angle  of  incidence ;  par- 
allel impulse  as  square  of  sine  of  angle ;  and  lateral  impulse  as  double  the 
angle. 

When  an  Inclined  Surface  is  not  Bm^dered,  then  stream  can  spread  over 
it  in  all  directions,  and  impulse  is  greater,  because  of  all  the  angles  by 
which  the  water  is  deflected,  a  is  least ;  hence  each  particle  that  does  not 
move  in  normal  plane  exerts  a  greater  pressure  than  particle  in  that  plane, 

,    2  sin.  a2       c — v  ~,          ^ 
and     .    .      ~X-—  Vwj  =  P. 
i  -f-  sin.  a2        g 

Impulse  and  Resistance  against  Surfaces. 

Coefficient  of  resistance,  C,  or  number  with  which  height  due  to  velocity  is  to  be 
multiplied,  to  obtain  height  of  a  column  of  water  measuring  this  hydraulic  press- 
ure, varies  for  bodies  of  different  figures,  and  only  for  surfaces  which  are  at  right 
angles  to  direction  of  motion  is  it  nearly  a  definite  quantity. 

According  to  experiments  of  Du  Buat  and  Thibault,  C  =  1.85  for  impulse  of  air 
or  water  against  a  plane  surface  at  rest,  and  for  resistance  of  air  or  water  against  a 
surface  in  motion,  C  =•  1.4.  In  each  case  about  .66  of  effect  is  expended  upon  front 
surface,  and  .34  upon  near. 


HYDRODYNAMICS.  579 

Comparison  between  Turbines  and.  other  "Water-wheels. 

Turbines  are  applicable  to  falls  of  water  at  any  height,  from  i  to  500  feet. 

Their  efficiency  for  very  high  falls  is  less  than  for  smaller,  in  consequence 
of  the  hydraulic  resistances  involved,  and  which  increase  as  the  square  of 
the  velocity  of  the  water.  They  can  only  be  operated  in  clear  water. 

With  Fourneyron's,  the  stress  and  pressure  on  the  step  is  that  of  the  wheel 
in  motion  ;  with  Fontaine's,  the  whole  weight  of  the  water  is  added  to  that 
of  the  wheel  ;  they  are  well  adapted,  however,  for  tide-mills.  Experiments 
on  Jouval's  gave  equal  results  with  Fontaine's. 

Vertical  Water-wheels  are  limited  in  their  application  to  falls  under  60 
feet  in  height. 

For  falls  of  from  40  to  20  feet  they  give  a  greater  effect  than  any  turbine  ; 
for  falls  of  from  20  to  10  feet,  they  are  equal  to  them  ;  and  for  very  low 
falls,  they  have  much  less  efficiency. 

Variations  in  the  supply  of  water  effect  them  less  than  turbines. 

"Water-pressure    Engine. 

By  experiments  of  M.  Jordan,  he  ascertained  that  a  mean  useful  effect  of 
.84  was  attainable. 
Weisbach,  London,  1848,  vol.  ii.  page  349. 

PERCUSSION  OF  FLUIDS. 

When  a  stream  strikes  a  plane  perpendicular  to  its  action,  force  with 
which  it  strikes  is  estimated  by  product  of  area  of  plane,  density  of  fluid, 
and  square  of  its  velocity. 

Or,  A  d  v2  =  P.  A  representing  area  in  sq.  feet,  d  weight  of  fluid  in  Ibs.,  and  v 
velocity  in  feet  per  second. 

If  plane  is  itself  in  motion,  then  force  becomes  A  d  (v  —  v')2  =  P.  t>'  representing 
velocity  of  plane. 

If  C  represent  a  coefficient  to  be  determined  by  experiment,  and  h  height 
due  to  velocity  v.  then  v2  =  2  q  h.  and  expression  for  force  becomes 


CENTRIFUGAL  PUMPS.     (D.  K.  Clark.) 
Appold.  3?uinp,  made  with  curved  receding  blades,  is  the  form  of 

centrifugal  pump  most  widely  known  and  accepted.    M.  Morin  tested  three 

kinds  of  centrifugal  or  revolving  pumps  : 

ist,  on  model  of  Appold  pump;    2d,  one  having  straight  receding  blades 

inclined  at  an  angle  of  45°  with  the  radius,  and  30!,  one  having  radial  blades. 

They  were  12  ins.  in  diameter  and  3.125  ins.  in  length,  and  had  central  open- 

ings of  6  ins.    Their  efficiencies  were  as  follows  : 

i.  Curved  blades.  .  48  to  68  per  cent.     |    2.  Inclined  blades.  .  40  to  43  per  cent 
3.  Radial  blades  ........  24  per  cent. 

Height  to  which  water  ascends  in  a  pipe,  by  action  of  a  centrifugal  pump, 
would,  if  there  were  no  other  resistances,  be  that  due  to  velocity  of  circum- 

ference of  revolving  wheel,  or  to  ~  .    Results  of  experiments  made  by  the 

author  on  two  pumps,  in  1862,  yielded  following  data,  showing  height  to 
which  water  was  raised,  without  any  discharge  : 


Diameter  of  pump-  wheel  .....  ..........        4  feet.  4  feet  7  ins. 

Revolutions  per  minute  ................  177  95.4 

Velocity  of  circumference  per  second.  .  .  37.05  feet.  22.9     feet 

Head  due  to  the  velocity  ...............  21.45    "  8.194    " 

Actual  head  ...........................      18.21    "  5.833    " 

Do.  do.  in  parts  of  head  due  to  velocity,  85  per  cent.  71.2  per  cent 


580         HYDKODYNAMICS. IMPACT    OE    COLLISION. 

Mr.  David  Thomson  made  similar  experiments  with  Appold  pumps  of  from  1.25 
to  1.71  feet  in  diameter,  the  results  of  which  showed  that  the  actual  head  was  about 
90  per  cent,  of  the  head  due  to  the  velocity. 

M.  Tresca,  in  1861,  tested  two  centrifugal  pumps,  18  ins.  in  diameter,  with  a  cen- 
tral opening  of  9  ins.  at  each  side.  The  blades  were  six  in  number,  of  which  three 
sprung  from  centre,  where  they  were  .5  inch  thick;  the  alternate  three  only  sprung 
at  a  distance  equal  to  radius  of  opening  from  centre.  They  were  radial,  except  at 
ends,  where  they  were  curved  backward,  to  a  radius  of  about  2.25  ins. ;  and  they 
joined  the  circumference  nearly  at  a  tangent.  Width  of  blades  was  taper,  and  they 
were  5.75  ins.  wide  at  nave,  and  only  2.625  ins.  at  ends:  so  designed  that  section  of 
outflowing  water  should  be  nearly  constant. 

M.  Tresca  deduced  from  his  experiments  that,  in  making  from  630  to  700  revolu- 
tions per  minute,  efficiency  of  the  pump,  or  actual  duty  in  raising  water,  through  a 
height  of  31.16  feet,  amounted  to  from  34  to  54  per  cent,  of  work  applied  to  shaft; 
or  that,  in  the  conditions  of  the  experiment,  the  pump  could  raise  upward  of  16  200 
cube  feet  of  water  per  hour,  through  a  height  of  33  feet,  with  about  30  HP  applied 
to  shaft,  and  an  efficiency  of  45  per  cent. 

According  to  Mr.  Thomson,  maximum  duty  of  a  centrifugal  pump  worked  by  a 
steam-engine  varies  from  55  per  cent,  for  smaller  pumps  to  70  per  cent,  for  larger 
pumps.  They  may  be  most  effectively  used  for  low  or  for  moderately  high  lifts,  of 
from  15  to  20  feet;  and,  in  such  conditions,  they  are  as  efficient  as  any  pumps  that 
can  be  made.  For  lifts  of  4  or  5  feet  they  are  even  more  efficient  than  others. 

At  same  time,  larger  the  pump  higher  lift  it  may  work  against.  Thus,  an  1 8- inch 
pump  works  well  at  20- feet  lift,  and  a  3-feet  pump  at  3o-feet  lift.  A  21  inch  wheel 
at  4o-feet  lift  has  not  given  good  results:  high  lifts  demand  very  high  velocities. 

Efficiency  is  influenced  by  form  of  casing  of  pump.  Hon.  R.  C.  Parsons  made  exper- 
iments with  two  i4-inch  wheels  on  Appold's  and  on  Rankine's  forms.  In  Rankiue's 
wheel  blades  are  curved  backwards,  like  those  of  Appold's,  for  half  their  length ; 
and  curved  forwards,  reversely,  for  outer  half  of  their  length.  Deducing  results  of 
performance  arrived  at,  following  are  the  several  amounts  of  work  done  per  Ib.  of 
water  evaporated  from  boiler  : 

Work  done  per  Ib.  of 
water  evaporated. 

Foot-lbs.  Ratio. 

Appold  wheel,   in  concentric  circular  casing u  385  1.06 

"  "        in  spiral  casing 15996  1.5 

Rankine  wheel,  in  concentric  circular  casing 10  748  i 

"        in  spiral  casing 12954  '-2 

These  data  prove:— ist,  that  spiral  casing  was  better  than  concentric  casing;  2d 
that  Appold's  wheel  was  more  efficient  than  Rankine's  wheel. 


IMPACT  OR  COLLISION. 

IMPACT  is  Direct  or  Oblique.  Bodies  are  Elastic  or  Inelastic.  The 
division  of  them  into  hard  and  elastic  is  wholly  at  variance  with  these 
properties ;  as,  for  instance,  glass  and  steel,  which  are  among  hardest 
of  bodies,  are  most  elastic  of  all. 

Product  of  mass  and  velocity  of  a  body  is  the  Momentum  of  the  body. 

Principle  upon  which  motions  of  bodies  from  percussion  or  collision  are 
determined  belongs  both  to  elastic  and  inelastic  bodies ;  thus  there  exists  in 
bodies  the  same  momentum  or  quantity  of  motion,  estimated  in  any  one  and 
same  direction,  both  before  collision  and  after  it. 

A  ctlon  and  reaction  are  always  equal  and  contrary.  If  a  body  impinge 
obliquely  upon  a  plane,  force  of  blow  is  as  the  sine  of  angle  of  incidence. 

When  a  body  impinges  upon  a  plane  surface,  it  rebounds  at  an  angle  equal 
to  that  at  which  it  impinged  the  plane,  that  is,  angle  of  reflection  is  equal  to 
that  of  incidence. 

Effect  of  a  blow  of  an  elastic  body  upon  a  plane  is  double  that  of  an  in- 
elastic one,  velocity  and  mass  being  equal  in  each^  for  the  force  of  blow 


IMPACT   OR   COLLISION.  jgl 

from  inelastic  body  is  as  its  mass  and  velocity,  which  is  only  destroyed  by 
resistance  of  the  plane  ;  but  in  an  elastic  body  that  force  is  not  only  destroyed, 
being  sustained  by  plane,  but  another,  also  equal  to  it,  is  sustained  by  plane, 
in  consequence  of  the  restoring  force,  and  by  which  the  body  is  repelled  with 
an  equal  velocity  ;  hence  intensity  of  the  blow  is  doubled. 

If  two  perfectly  elastic  bodies  impinge  on  one  another,  their  relative  ve- 
locities will  be  same,  both  before  and  after  impact  ;  that  is,  they  will  recede 
from  each  other  with  same  velocity  with  which  they  approached  and  met. 

If  two  bodies  are  imperfectly  elastic,  sum  of  their  moments  will  be  same, 
both  before  arid  after  collision,  but  velocities  after  will  be  less  than  in  case 
of  perfect  elasticity,  in  ratio  of  imperfection. 

Effect  of  collision  of  two  bodies,  as  B  and  b,  velocities  of  which  are  differ- 
ent, as  v  and  v',  is  given  in  f  ollowing  formulas,  in  which  B  is  assumed  to 
have  greatest  momentum  before  impact. 

If  bodies  move  in  same  direction  before  and  after  impact,  sum  of  their 
moments  before  impact  will  be  equal  to  their  sum  after. 

If  bodies  move  in  same  direction  before,  and  in  opposite  direction  after 
impact,  sum  of  their  moments  before  impact  will  be  equal  to  difference  of  their 
sums  after. 

If  bodies  move  in  opposite  directions  before,  and  in  same  direction  after 
impact,  difference  of  their  moments  before  impact  will  be  equal  to  their 
sum  after. 

If  bodies  move  in  opposite  directions  before,  and  in  opposite  directions 
after  impact,  difference  of  their  moments  before  impact  will  be  equal  to  their 
difference  after. 

To  Compute  "Velocities  of  Inelastic  Bodies  after  Impact. 

When  Impelled  in  Same  Direction.     —    "^    v  =  r.    B  and  b  representing 

weights  of  the  two  bodies,  V  and  v  their  velocities  before  impact,  and  r  velocity  of  bodies 
after  impact,  all  in  feet. 

Consequently,  —  ^|  x  b  =  velocity  lost  by  B,  and  —^  X  B  =  velocity  gained  by  b. 

--  -- 


NOTE.—  In  these  formulas  it  is  assumed  that  V>r.  If  V<t>  the  result  will  be 
negative,  but  may  be  read  as  positive  if  lost  and  gained  are  reversed  in  places. 

ILLUSTRATION.—  An  inelastic  body,  b,  weighing  30  Ibs.,  having  a  velocity  of  3  feet, 
is  struck  by  another  body,  B,  of  50  Ibs.,  having  a  velocity  of  7  feet;  the  velocity  of 
b  after  impact  will  be  ,  - 

50  X  7  +  30  X  3      440 
__F-  —  =  — 

When  Impelled  ir,  Opposite  Directions.     -  —  —  ~=r. 

B-}-  b 

ILLUSTRATION.—  Assume  elements  of  preceding  case. 

*b 


50  +  30  80 

B  V 

When  One  Body  is  at  Rest.     ^—^  =  r. 


ILLUSTRATION.—  Assume  elements  as  preceding. 
50X7      35o 


When  Bodies  are  inelastic,  their  velocities  after  impact  will  be  alike. 

30* 


5  82  IMPACT   OK   COLLISION. 

To  Compxite  Velocities  of  Elastic   JBodios  after  Impact 

„   ,  .     -.       _^.      A.         B— b  V  +  a&v                ,,26  V—  B  —  bv 
When  Impelled  in  One  Direction.    =  R,  and =  r. 

ILLUSTRATION.— Assume  elements  as  preceding. 

=g=4^^ax5oxr-r55x^fe=8^ 


50+30  80  50+30  80 

Or>  v  —  S^-TT  V  — v  =  velocity  of  ri,  and  v  +  -^r  V^v  =  velocity  of  r. 

o  +  0  i>  +  0          / 

TF%en  Impelled  in  Opposite  Directions. 
B  — 6  V<x>2  fe  v  2  BV+  B— 6  v 

- 


B  +  6 
ILLUSTRATION.— Assume  elements  as  preceding. 

50  —  30  X  7^2X  30  X  3_i4Q/vi8o_  ,2X50X7  +  50  —  3oX3_^ 


Rest.          7   =  R,  and  =r. 

ILLUSTRATION.  —  Assume  elements  as  preceding. 


=^o=  pd  «Z== 


50+30          80  50+30 

To  Compute  "Velocities  of*  Imperfect  Elastic  Bodies  after 
Impact. 

Effect  of  Collision  is  increased  over  that  of  perfectly  inelastic  bodies,  but 
not  doubled,  as  in  case  of  perfectly  elastic  bodies  ;  it  must  be  multiplied  by 

i  +  ^  or  mj^n  ,  when  £  represents  degree  of  elasticity  relative  to  both  per- 
fect inelasticity  and  elasticity. 

Moving  in  same  Direction.    V  --  3l_  x  (  V  —  v)  =  R  :  and  v  -j  —  -£_ 

in         D-\-  o  m 

X  p  ,  .  (V  —  v)  =  r.    m  and  n  representing  ratio  of  perfect  to  imperfect  elasticity. 

D  +  O 

ILLUSTRATION.—  Assume  elements  as  preceding.       m  and  n  =  2  and  i. 

,  and  3  + 


When  Moving  in  Opposite  Directions. 


T-=±5  x  =  B  and  x 


When  One  Body  is  at  Rest.         •       *  '  =  R,  and       3+6 
ILLUSTRATION.—  Assume  elements  of  preceding  case. 


50+30 

3SQ  Xr  5 

-    -  =  6.56*5  /««*. 


LIGHT. 


LIGHT. 

LIGHT  is  similar  to  Heat  in  many  of  its  qualities,  being  emitted  in 
form  of  rays,  and  subject  to  same  laws  of  reflection. 

It  is  of  two  kinds,  Natural  and  Artificial ;  one  proceeding  from  Sun 
and  Stars,  the  other  from  heated  bodies. 

Solids  shine  in  dark  only  at  a  temperature  from  600°  to  700°,  and  in 
daylight  at  1000°. 

Intensity  of  Light  is  inversely  as  square  of  distance  from  luminous 
body. 

Velocity  of  Light  of  Sun  is  185  ooo  miles  per  second. 

Standard  of  Intensity  or  of  comparison  of  light  between  different  methods 
of  Illumination  is  a  Sperm  Candle  "short  6,"  burning  120  grains  per  hour. 

Candles. 

A  Spermaceti  candle  .85  of  a  inch  in  diameter  consumes  an  inch  in  length 
in  i  hour. 


Decomposition   of  Light. 

Maximum 

Contrasts. 

Combinations. 

COLORS. 

Ray. 

Primary. 

Second'y. 

Tertiary. 

Primary. 

Secondary. 

Tertiary. 

Violet.. 
Indigo.  . 

Chemical. 

- 

- 

Brown. 

Blue...) 
Yellow.  } 

Green.  .  i 

Dark. 

Blue  .  .  . 

Electrical. 

Blue. 



— 

Blue...) 

Purple.  ) 

Green. 

Green  .  . 

— 



Green. 

Green. 

Red.  .  .  .  J 

Orange.  ) 

_ 

Yellow. 

Light. 

Yellow. 

— 

— 

— 

Green.  .  J 

i*ray. 

Orange  . 
Red.... 

Heat. 

Red. 

Orange. 
Purple. 

Broken. 
Green. 

Yellow.  ) 
Red.  .  .  .  } 

Purple.  ) 
Orange.  ) 

Brown. 

All  colors  of  spectrum,  when  combined,  are  white. 

Consumption    and.    Comparative    Intensity    of  Light 
of  Candles. 


CANDLK. 

No.  in  a 
Lb. 

Diameter. 

Length. 

Consumption 
per  Hour. 

Light  comp'd 
with  Carcel. 

Wax  

Inch. 

Ins. 
12 

Grains. 

3 

.875 

}         135 

.09 

Q 

I  e 

) 

:§ 

13.5 

}         156 

u 

6 

84 

) 

wy 

Tallow  

t 

3 

•  0 

15 

204 

4 

.07 

« 

4 

.8 

13.7=5 

Compared  with  1000  Cube  Feet  of  Gas. 


CANDLB. 

Gas=i. 

Con- 
sump- 
tion. 

Light. 

Con-    1 
sumption 
for  equal 
Light.  | 

CANDLE. 

Gaa=i. 

Con- 
sump- 
tion. 

Light. 

Con- 
sumption 
for  equal 
Light. 

Paraffine. 
Sperm  .  .  . 

.098 
.005 

Lbs. 
3-5 
3.Q 

Lbs. 
35-5 
41.  1 

.03 

120 

Adamantine. 
Tallow  

.108 

.074 

Lbs. 
5-i 
5-  1 

Lbs. 
47.2 

53-8 

137 
155 

In  combustion  of  oil  in  an  ordinary  lamp,  a  straight  or  horizontally  cut  wick 
gives  great  economy  over  one  irregularly  cut. 


584 


LIGHT. 


Relative     Intensity-,  Consumption,    Illumination,   and. 
Cost    of  various    Ivlotles    of  Illumination. 

Oil  at  ii  cents,  Tallow  at  14  cents,  Wax  at  52  cents,  and  Stearine  at  32  cents  pei 
Ib.     ioo  cube  feet  coal  gas  at  14  cents,  and  100  cube  feet  of  oil  gas  at  52  cents. 


ILLUMINATOR. 

Illumi- 
nation. 
Carcel 
Lamp 

=  100. 

Actual 
Cost 

Hour. 

Cost  for 
equal 
Inten- 
sity. 

ILLUMINATOR. 

Illumi- 

Carcel' 
Lamp 

=  IOO. 

Actual 
Cost 

deSir. 

Costfoi 

ssi 

•ity. 

Carcel  Lamp  
Lamp  with    in-) 
verted  reserv'r.  j 
Astral  Lamp  

IOO 

57-8 
48  7 

Cents. 
.87 

.89 
.56 

PerH'r. 
.87 

•99 
1.78 

Stearine  Candle  5  to  Ib. 
Tallow        u      6    " 
Sperm        «      6    " 
Coal  Gas  

66.6 
3.5 

Cents. 
•59 

% 

1.22 

PerH'r. 
4-13 
2-34 

5-7 
.06 

Wax  Candle  6  to  Ib. 

61.6 

.02 

1.70 
6.^1 

Oil  Gas... 



1.25 

5 

1000  cube  feet  of  13- candle  coal  gas  is  equal  to  7.5  gallons  sperm  oil,  52.9  Ibs.  mold, 
and  44.6  Ibs.  sperm  candles. 

Candles,  .Lamps,  Fluids,  and    GJ-as. 

Comparison  of  several  Varieties  of  Candles,  Lamps,  and  Fluids,  with  Coal*  Gas,  de- 
duced from  Reports  of  Com.  of  Franklin  Institute,  and  of  A.  Frye,  M.D.,etc. 


CANDLB. 

B 

:i 

.58 

t  Coi 
consum 

en-     Li 

rof      F' 
ht.      g 

If! 

Cost  com- 
pared with 
Gas  for 

CANDLB. 

3  s 

|I| 

"3 

1! 

Diaphane  

•  5 
•5 
.8 

upar 
It* 

f 
ual 

St. 

4 

Bdl 

cub 

Tin 
Bu 
i 
of 

15 

16 

7 

vitha 
e  foot 

lie  of 

nnf 

Oil. 

I 

2 

5 

fisl 
per 

Tallow,  short  6's,  ) 
double  wick  .  .  ] 
Wax,  short  6's.... 
Palm  oil  

i 
.8 

is,  conta 

Inten- 
sity of 
Light. 

i 
.61 
•77 
ning  12 

Light 
at 
Equal 

Cost. 

7-1 
14.4 
10.5 

per  cent. 

Time  of 
Burning 
i  Pint 
of  Oil. 

Spermaceti,  short 
Tallow,  short  6's,  1 

single  wick  .  .  .  j 

*CityofPhiladelph 
of  condensable  matter 

LAMP  AND  FLUID. 

5»a 

id. 
and 

Int 

Bit) 

Li? 

-tail  jet  of  Edinburgh  g 
hour. 

LAMP  AND  FLUID. 

Carcel. 
Sperm  oil,  max'm 
'  '       mean. 
'  '       wuw'wi 
Lard  oil.  .  . 

2.15 
1.22 
.69 
.77 

1.8 
i-35 

1.2 

.07 

Hours 

9.87 
14.6 

Ga 
Se 
So 
Ca 

S  

I 

l'*76 
1.75 

•93 
i-55 
1.08 

Hours. 
8^2 

mi-solar,  Sperm  oil 
lar,  Sperm  oil  
mphene  .  .  . 

Loss  of  Light  by  Use  of  Glass  Globes. 
Clear  Glass,  12  per  cent.    |   Half  ground,  35  per  cent.    |   Full  ground,  40  per  cent. 

Refraction. 

Relative  Index  of  Refraction—  Is,  Ratio  of  sine  of  angle  of  incidence  to  sine  of 
angle  of  refraction,  when  a  ray  of  light  passes  from  one  medium  into  another. 

Absolute  Index  or  Index  of  Refraction  —  Is,  When  a  ray  passes  from  a  vacuum  into 
any  medium,  the  ratio  is  greater  than  unity. 

Relative  index  of  refraction  from  any  medium,  as  A,  into  another,  as  B,  is  always 
equal  to  absolute  index  of  B,  divided  by  absolute  index  of  A. 

Absolute  index  of  air  is  so  small,  that  it  may  be  neglected  when  compared  with 
liquids  or  solids;  strictly,  however,  relative  index  for  a  ray  passing  from  air  into  a 
given  substance  must  be  multiplied  by  absolute  index  for  air,  in  order  to  obtain 
like  index  of  refraction  for  the  substance. 


Mean    Indices   of  Refraction. 


Air  at  32° i 

Alcohol i.  37 

Canada  balsam ......  i.  54 

Crystalline  lens x.^4 


Glass,  fluid J  J'j?8 

"  crown }i:ll 


Humors  of  eye.  . 
Salt,  rock 
Water,  fresh 


1.34 
1.55 
1.34 
1-34— 


LIGHT. 


585 


GJ-as. 

Retort. — A  retort  produces  about  600  cube  feet  of  gas  in  5  hours  with  a 
charge  of  about  1.5  cwt.  of  coal,  or  2800  cube  feet  in  24  hours. 

In  estimating  number  of  retorts  required,  one  fourth  should  be  added  for 
being  under  repairs,  etc. 

Pressure  with  which  gas  is  forced  through  pipes  should  seldom  exceed  2.5 
ins.  of  water  at  the  Works,  or  leakage  will  exceed  advantages  to  be  obtained 
from  increased  pressure. 

The  a\  erage  mean  pressure  in  street  mains  is  equal  to  that  of  i.  inch  of 
water. 

When  pipes  are  laid  at  an  inclination  either  above  or  below  horizon,  a  cor- 
rection will  have  to  be  made  in  estimating  supply,  by  adding  or  deducting 
.01  inch  from  initial  pressure  for  every  foot  of  rise  or  fall  in  the  length  of  pipe. 

It  is  customary  to  locate  a  governor  at  each  change  of  level  of  30  feet. 

Illuminating  power  of  coal-gas  varies  from  1.6  to  4.4  times  that  of  a  tallow 
candle  6  to  a  Ib. ;  consumption  being  from  1.5  to  2.3  cube  feet  per  hour,  and 
specific  gravity  from  .42  to  .58. 

Higher  the  flame  from  a  burner  greater  the  intensity  of  the  light,  the 
most  effective  height  being  5  ins. 

Standard  of  gas  burning  is  a  i5-hole  Argand  lamp,  internal  diameter  .44 
inch,  chimney  7  ins.  in  height,  and  consumption  5  cube  feet  per  hour,  giving 
a  light  from  ordinary  coal-gas  of  from  10  to  12  candles,  with  Cannel  coal 
from  20  to  24  candles,  and  with  rich  coals  of  Virginia  and  Pennsylvania  of 
from  14  to  1 6  candles. 

In  Philadelphia,  with  a  fish-tail  burner,  consuming  4.26  cube  feet  per  hour, 
illuminating  power  was  equal  to  17.9  candles,  and  with  an  Argand  burner, 
consuming  5.28  cube  feet  per  hour,  illuminating  power  was  20.4  candles. 

Gas,  which  at  level  of  sea  would  have  a  Value,  of  100,  would  have  but  60 
in  city  of  Mexico. 

Internal  lights  require  4  cube  feet,  and  external  lights  about  5  per  hour. 
When  large  or  Argand  burners  are  used,  from  6  to  10  are  required. 

An  ordinary  single-jet  house  burner  consumes  5  to  6  cube  feet  per  hour. 

Street-lamps  in  city  of  New  York  consume  3  cube  feet  per  hour.  In  some 
cities  4  and  5  cube  feet  are  consumed.  Fish-tail  burners  for  ordinary  coal 
gas  consume  from  4  to  5  cube  feet  of  gas  per  hour. 

A  cube  foot  of  good  gas,  from  a  jet  .033  inch  in  diameter  and  height  of 
flame  of  4  ins.,  will  burn  for  65  minutes. 

Resin  Gas.— Jet  .033,  flame  5  ins.,  1.25  cube  feet  per  hour. 

Purifiers. — Wet  purifiers  require  i  bushel  of  lime  mixed  with  48  bushels 
of  water  for  10000  cube  feet  of  gas. 

Dry  purifiers  require  i  bushel  of  lime  to  10000  cube  feet  of  gas,  and  i 
superficial  foot  for  every  400  cube  feet  of  gas. 

Intensity   of*  Light   -with    Kqnal   "Volumes    of   Gras   from 
cUfferent    Burners. 

Equal  to  Spermaceti  Candle  burning  120  Grains  per  Hour. 


BUBNIRS. 

**! 

enditi 
eet  pe 

re  in 
r  Hou 
3 

2ube 
r. 
4 

BURNERS. 

Exp 

z 

enditii 

>etp€ 

2 

re  in  < 
r  Hou 
3 

?ubt 
r. 
4 

Single-jet,  i  foot  
Fish-tail  No.  3  
Bat's  wine.  .  ." 

2.6 

3-5 
q 

4 
4-1 

4.2 
4-3 

4.5 

Argand,  16  holes.  .  .  . 
Argand,  24  holes.  .  .  . 
Argand,  28  holes.  .  . 

•32 
•33 

.34 

1.9 

2.2 
2.3 

3-3 
3-4 
3.* 

3-8 

5:8 

586 


LIGHT. 


Volume  of  G^as  obtained  from  a  Ton  of  Coal,  Resin*  etc. 


Material. 

Cube 
Feet. 

Material. 

Cube 
Feet. 

Material. 

Cube 
Feet. 

Boghead  Cannel 

I  -I   -3-2  A 

Cumberland  

o  8bo 

Pittsburgh 

O  S2O 

Wigan  Cannel  

15  42O 

Resin  

15600 

Cfinncl                   < 

8960 

Newcastle               < 

9500 

Scotch                  | 

10300 

Cape  Breton,     ) 

15000 

Oil  and  Grease  

IOOOO 

23  ooo 

I5OOO 
8960 

"Cow  Bay,"}  .. 
etc  ) 

9500 

Pictou  and  Sidney.  . 
Pine  wood  .  .  , 

8000 

11800 

44       West'n.. 
Walls-end  .  .  . 

9500 
12  OOO 

i  Chaldron  Newcastle  coal,  3136  Ibs.,  will  furnish  8600  cube  feet  of  gas  at 
a  specific  gravity  of  .4,  1454  Ibs.  coke,  14.1  gallons  tar,  and  15  gallons  am- 
moniacal  liquor. 

Australian  coal  is  superior  to  Welsh  in  producing  of  gas. 

Wigan  Cannel,  i  ton,  has  produced  coke,  1326  Ibs. ;  gas,  338  Ibs, ;  tar, 
250  Ibs. ;  loss,  326  Ibs. 

Peat,  i  Ib.  will  produce  gas  for  a  light  of  one  hour. 

Fuel,  required  for  a  retort  18  Ibs.  per  100  Ibs.  of  coal. 

In  distilling  56  Ibs.  of  coal,  volume  of  gas  produced  in  cube  feet  when 
distillation  was  effected  in  3  hours  was  41.3,  in  7,  37.5,  in  20,  33.5,  and  in 
25, 31.7- 

Flow   of  Q-as   in    IPipes. 

Flow  of  Gas  is  determined  by  same  rules  as  govern  that  of  flow  of  water. 
Pressure  applied  is  indicated  and  estimated  in  inches  of  water,  usually  from 
.5  to  i  inch. 

Volumes  of  gases  of  like  specific  gravities  discharged  in  equal  times  by  a 
horizontal  pipe,  under  same  pressure  and  for  different  lengths,  are  inversely 
as  square  roots  of  lengths. 

Velocity  of  gases  of  different  specific  gravities,  under  like  pressure,  are  in- 
versely as  square  roots  of  their  gravities. 

By  experiment,  30  ooo  cube  feet  of  gas,  specific  gravity  of  .42,  were  dis- 
charged in  an  hour  through  a  main  6  ins.  in  diameter  and  22.5  feet  in  length. 

Loss  of  volume  of  discharge  by  friction,  in  a  pipe  6  ins.  in  diameter  and  i 
mile  in  length,  is  estimated  at  95  per  cent. 

Diameter    and.    Length    of  Q-as-pipes   to   transmit  given 
Volumes   of  <3-as   to    Branch-pipes.    (Dr.  Ure.) 


Volume 
per  Hour. 

Diameter. 

Length. 

Volume 
per  Hour. 

Diameter. 

Length. 

Volume 
per  Hour. 

Diameter. 

Length. 

Cube  Feet. 
50 
250 
500 
700 

Ins. 
•4 

1.97 
2.65 

Feet. 

100 
200 
600 
IOOO 

Cube  Feet. 
IOOO 

1500 

2000 
2000 

Ins. 
3.16 
3-87 
5.-  32 
6-33 

Feet. 

IOOO 
IOOO 
2OOO 

4000 

Cube  Feet. 

2OOO 
6000 
6oOO 
8000 

Ins. 
7 
7-75 
9.21 

8-95 

Feet. 
6000 
IOOO 
2000 
IOOO 

Regulation    of  Diameter  and    Extreme   .Length   of  Tub- 
ing, and    Number   of  Burners   permitted. 


Diameter 

Capacity 

Diameter 

Capacity 

of 
Tubing. 

Length. 

of 
Meters. 

Burners. 

Tubing. 

Length. 

of 
Meters. 

Burners. 

Ins. 

Feet. 

Light. 

No. 

Ins. 

Feet. 

Light. 

No. 

•25 

6 

3 

9 

•75 

50 

30 

90 

•375 
•5 

20 
30 

5 
10 

15 

30 

I 
1.25 

70 

IOO 

£ 

'M 

.625 

40 

20 

60 

i-5 

«50 

IOO 

300 

LIGHT.  587 

Temperature  of  Gases. — Combustion  of  a  cube  foot  of  common  gas  will 
heat  650  Ibs.  of  water  i°. 

Services    for    Lamps 


Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

No. 

2 

i 

Feet. 
40 
40 
50 

Ins. 
•375 

^625 

No. 
10 
15 

20 

Feet. 
100 

130 
150 

Ins. 
•75 
I 
1.25 

No. 
25 
30 

Feet. 
1  80 
200 

Ins. 
1-75 

Volumes  of  Q-as  Discharged  per  Hour  under  a  Pressure 
of  Half  an    Inch   of  Water. 


Diam.  of 
Opening. 

Volume. 

Diam.  of 
Opening. 

Specific  Gr 
Volume. 

avity  .42. 
Diam.  of 
Opening. 

Volume. 

Diam.  of 
Opening. 

Volume. 

Ins. 
•25 
•5 

Cube  Feet. 
80 
321 

Ins. 
•75 

i 

Cube  Feet. 
723 
1287 

Ins. 
1.125 
1.25 

Cub«  Feet. 
1625 

2OIO 

Ins. 
i-5 
5 

Cube  Feet. 
2885 
46150 

To  Compute  Volume  of  G-as  Discharged  through  a  IPipe. 

yd*  h  /Va  G  I 

—  —  =  V,  and   063  5  /  —  -  —  —  d.    d  representing  diameter  of  pipe,  and 
(r  I  \       n, 

h,  height  of  water  in  ins.  ,  denoting  pressure  upon  gas,  I  length  of  pipe  in  yards,  G 
specific  gravity  of  gas,  arid  V  volume  in  cube  feet  per  hour. 
G  may  be  assumed  for  ordinary  computation  at  .42,  and  h  .5  to  i  inch. 
ILLUSTRATION.  —  Assume  diameter  of  pipe  i  inch,  pressure  1.68  ins.,  and  length 
of  pipe  i  yard. 

tooo  x    /r  —  :;  —  —  I00°  X  .  /-  —  =  2000  cube  feet, 


and  .063  X 


—  :; 

40oooooX  -42X  i 
--  -- 


=  -OS  «•* 


NOTE.—  For  tables  deduced  by  above  formulas  see  Molesworth,  1878,  page  226. 
Dimensions   of  Mains,  \vith  "Weight   of  One   Length. 


Diameter  in  ins  ..... 

Length  in  feet  ...... 

Thickness  in  ins.  .  .  . 

Weight  in  Ibs  ....... 


•375 


6 
9 
•375 


400 


9 
9 

•5 
454 


489 


'4 

9 

868 


.625 


•75 


1316 


t484 


•75 


GAS  ENGINES. 


In  the  Lenoir  engine,  the  best  proportions  of  air  and  gas  are,  for  common 
gas,  8  volumes  of  air  to  i  of  gas,  and  for  cannel  gas,  n  of  air  to  i  of  gas. 

The  time  of  explosion  is  about  the  27th  part  of  a  second. 

An  engine,  having  a  cylinder  4.625  ins.  in  diameter  and  8.75  ins.  stroke  of 
piston,  making  185  revolutions  per  minute,  develops  a  half  horse-power. 

Distribution  of  Heat  Generated  in  the  Cylinder.    (M.  Tresca.) 

Per  cent.  Per  cent. 

Dissipated  by  the  water  and  prod-         I  Losses 27 

ucts  of  combustion 69  ~^ 

Converted  into  work 4  | 

Hence  efficiency  as  determined  by  the  brake  —  4  per  cent. 

Atmospheric  Gas  Engine. 

A  single-acting  cylinder  6  ins.  in  diameter,  making  81  strokes  per  minute,  devel- 
oped .456  IP,  and  the  gas  consumed  per  minute  for  cylinder  20  cube  feet  and  for  in- 
flaming 2  cube  feet.  (M.  Tresca.) 


588      LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 

LIMES/CEMENTS,  MORTARS,  AND  CONCRETES. 

Essentially  from  a  Treatise  by  Brig.-Geril  Q.  A.  Gillmore,  U.S.A.* 

Lime. 

Calcination  of  marble  or  any  pure  limestone  produces  lime  (quick" 
lime).  Pure  limestones  burn  white,  and  give  richest  limes. 

Finest  calcareous  minerals  are  rhombohedral  prisms  of  calcareous 
spar,  the  transparent  double-reflecting  Iceland  spar,  and  white  or  statu- 
ary marble. 

Property  of  hardening  under  water,  or  when  excluded  from  air,  con- 
ferred upon  a  paste  of  lime,  is  effected  by  presence  of  foreign  sub- 
stances— as  silicum,  alumina,  iron,  etc. — when  their  aggregate  presence 
amounts  to  .  I  of  whole. 

Limes  are  classed :  i.  Common  or  Fat  limes,  which  do  not  set  in  water. 

2.  Poor  or  Meagre,  mixed  with  sand,  which  does  not  alter  its  condition. 

3.  Hydraulic  Lime,  containing  8  to  12  per  cent,  of  silica,  alumina,  iron, 
etc.,  set  slowly  in  water.    4.  Hydraulic,  containing  12  to  20  per  cent,  of 
similar  ingredients,  sets  in  water  in  6  or  8  days.    5.  Eminently  Hydraulic, 
containing  20  to  30  per  cent,  of  similar  ingredients,  sets  in  water  in  2  to  4 
days.    6.  Hydraulic  Cement,  containing  30  to  50  per  cent,  of  argil,  sets  in  a 
few  minutes,  and  attains  the  hardness  of  stone  in  a  few  months.    7.  Natural 
Pozzuolanas,  including  pozzuolana  properly  so  called,  Trass  or  Terras,  Arenes, 
Ochreous  earths,  Basaltic  sands,  and  a  variety  of  similar  substances. 

Indications  of  Limestones.  They  dissolve  wholly  or  partly  in  weak  acids 
with  brisk  effervescence,  and  are  nearly  insoluble  in  water. 

Rich  Limes  are  fully  dissolved  in  water  frequently  renewed,  and  they 
remain  a  long  time  without  hardening ;  they  also  increase  greatly  in  vol- 
ume, from  2  to  3.5  times  their  original  bulks,  and  will  not  harden  without 
the  action  of  air.  They  are  rendered  Hydraulic  by  admixture  of  pozzuolana 
or  trass. 

Rich,  fat,  or  common  Limes  usually  contain  less  than  10  per  cent,  of  im- 
purities. 

Hydraulic  Limestones  are  those  which  contain  iron  and  clay,  so  as  to  en- 
able them  to  produce  cements  which  become  solid  when  under  water. 

Poor  Limes  have  all  the  defects  of  rich  limes,  and  increase  but  slightly  in 
bulk,  the  poorer  limes  are  invariably  basis  of  the  most  rapidly  -  setting 
and  most  durable  cements  and  mortars,  and  they  are  also  the  only  limes 
which  have  the  property,  when  in  combination  with  silica,  etc.,  of  indurating 
under  water,  and  are  therefore  applicable  for  admixture  of  hydraulic  cements 
or  mortars.  Alike  to  rich  limes,  they  will  not  harden  if  in  a  state  of  paste 
under  water  or  in  wet  soil,  or  if  excluded  from  contact  with  the  atmosphere 
or  carbonic  acid  gas.  They  should  be  employed  for  mortar  only  when  it  is 
impracticable  to  procure  common  or  hydraulic  lime  or  cement,  In  which  case 
it  is  recommended  to  reduce  them  to  powder  by  grinding. 

Hydraulic  Limes  are  those  which  readily  harden  under  water.  The  most 
valuable  or  eminently  hydraulic  set  from  the  2d  to  the  4th  day  after  immer- 
sion ;  at  end  of  a  month  they  become  hard  and  insoluble,  and  at  end  of  six 
months  they  are  capable  of  being  worked  like  the  hard,  natural  limestones. 
They  absorb  less  water  than  pure  limes,  and  only  increase  in  bulk  from  1.75 
to  2.5  times  their  original  volume. 

*  See  also  his  Treatises  on  Limes,  Hydraulic  Cements,  and  Mortars,  in  Papers  on  Practical  Engineer- 
ing, Engineer  Department,  U.  S.  A. 


LIMES,  CEMENTS,  MORTARS,  AND   CONCRETES.      589 

Inferior  grades,  or  moderately  hydraulic,  require  a  period  of  from  15 
to  20  days'  immersion,  and  continue  to  harden  for  a  period  of  6  months. 

Resistance  of  hydraulic  limes  increase  if  sand  is  mixed  in  proportion 
of  50  to  1 80  per  cent,  of  the  part  in  volume ;  from  thence  it  decreases. 

M.  Vicat  declares  that  lime  is  rendered  hydraulic  by  admixture  with  it  of  from 
33  to  40  per  cent,  of  clay  and  silica,  and  that  a  lime  is  obtained  which  does  not 
slake,  and  which  quickly  sets  under  water. 

Artificial  Hydraulic  Limes  do  not  attain,  even  under  favorable  circum- 
stances, the  same  degree  of  hardness  and  power  of  resistance  to  compression 
as  natural  limes  of  same  class. 

Close-grained  and  densest  limestones  furnish  best  limes. 

Hydraulic  limes  lose  or  depreciate  in  value  by  exposure  to  the  air. 

Pastes  of  fat  limes  shrink,  in  hardening,  to  such  a  degree  that  they  can- 
not be  used  as  mortar  without  a  large  proportion  of  sand. 

Arenes  is  a  species  of  ochreous  sand.  It  is  found  in  France.  On  account 
of  the  large  proportion  of  clay  it  contains,  sometimes  as  great  as  .7,  it  can  be 
made  into  a  paste  with  water  without  any  addition  of  lime ;  hence  it  is  some- 
times used  in  that  state  for  walls  constructed  en  pise,  as  well  as  for  mortar. 
Mixed  with  rich  lime  it  gives  excellent  mortar,  which  attains  great  hardness 
under  water,  and  possesses  great  hydraulic  energy. 

Pozzuolana  is  of  volcanic  origin.  It  comprises  Trass  or  Terras,  the  Arenes, 
some  of  the  ochreous  earths,  and  the  sand  of  certain  graywackes,  granites, 
schists,  and  basalts;  their  principal  elements  are  silica  and  alumina,  the 
former  preponderating.  None  contain  more  than  10  per  cent,  of  lime. 

When  finely  pulverized,  without  previous  calcination,  and  combined  with  paste 
of  fat  lime  in  proportions  suitable  to  supply  its  deficiency  in  that  element,  it  pos- 
sesses hydraulic  energy  to  a  valuable  degree.  It  is  used  in  combination  with  rich 
lime,  and  may  be  made  by  slightly  calcining  clay  and  driving  off  the  water  of  com- 
bination at  a  temperature  of  1200°. 

Brick  or  Tile  Dust  combined  with  rich  lime  possesses  hydraulic  energy. 

Trass  or  Terras  is  a  blue-black  trap,  and  is  also  of  volcanic  origin.  It 
requires  to  be  pulverized  and  combined  with  rich  lime  to  render  it  fit  for 
use,  and  to  develop  any  of  its  hydraulic  properties. 

General  Gillmore  designates  the  varieties  of  hydraulic  limes  as  follows:  If,  after 
being  slaked,  they  harden  under  water  in  periods  varying  from  15  to  20  days  after 
immersion,  slightly  hydraulic ;  if  from  6  to  8  days,  hydraulic;  and  if  from  i  to  4 
days,  eminently  hydraulic. 

Pulverized  silica  burned  with  rich  lime  produces  hydraulic  lime  of  ex- 
cellent quality.  Hydraulic  limes  are  injured  by  air-slaking  in  a  ratio  vary- 
ing directly  with  their  hydraulicity,  and  they  deteriorate  by  age. 

For  foundations  in  a  damp  soil  or  exposure,  hydraulic  limes  must  be  ex- 
clusively employed. 

Hydraulic  Lime  of  Teil  is  a  silicious  hydraulic  lime ;  it  is  slow  in  setting, 
requiring  a  period  of  from  18  to  24  hours. 

Cements. 

Hydraulic  Cements  contain  a  larger  proportion  of  silica,  alumina,  magnesia, 
etc.,  than  any  of  preceding  varieties  of  lime ;  they  do  not  slake  after  calcina- 
tion, and  are  superior  to  the  very  best  of  hydraulic  limes,  as  some  of  them 
set  under  water  at  a  moderate  temperature  (65°)  in  from  3  to  4  minutes ; 
others  require  as  many  hours.  They  do  not  shrink  in  hardening,  and  make 
an  excellent  mortar  without  any  admixture  of  sand. 

7  D 


59O      LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 

When  exposed  to  air,  they  absorb  moisture  and  carbonic  acid  gas,  and  are 
rapidly  deteriorated  thereby. 

Roman  Cement  is  made  from  a  lime  of  a  peculiar  character,  found  in  Eng- 
land and  France,  derived  from  argillo-calcareous  kidney-shaped  stones  termed 
Septaria. 

It  is  about  .33  strength  of  Portland,  and  is  not  adapted  for  use  with  sand. 

Rosendale  Cement  is  from  Rosendale,  New  York. 

Portland  Cement  is  made  in  England,  Germany,  France,  and  the  United 
States.  It  requires  less  water  (cement  i,  water  .29)  than  Roman  cement, 
sets  slowly,  and  can  be  remixed  with  additional  water  after  an  interval  of  12 
or  even  24  hours  from  its  first  mixture. 

Property  of  setting  slow  may  be  an  obstacle  to  use  of  some  designations  of  this 
cement,  as  the  Boulogne,  when  required  for  localities  having  to  contend  against 
immediate  causes  of  destruction,  as  in  sea  constructions,  having  to  be  executed  un- 
der water  and  between  tides.  On  the  other  hand,  a  quick-setting  cement  is  always 
difficult  of  use  ;  it  requires  special  workmen  and  an  active  supervision.  A  slow- 
setting  cement,  however,  like  natural  Portland,  possesses  the  advantage  of  being 
managed  by  ordinary  workmen,  and  it  can  also  be  remixed  with  additional  water 
after  an  interval  of 12  or  even  24  hours  from  its  first  mixing. 

Conclusions  derived  from  Mr.  Grant's  Experiments. 

1.  Portland  cement  improves  by  age,  if  kept  from  moisture. 

2.  Longer  it  is  in  setting,  stronger  it  will  be. 

3.  At  end  of  a  year,  i  of  cement  to  i  sand  is  about  .75  strength  of  neat  cement; 
i  to  2,  .5  strength;  i  to  3,  .33;  i  to  4,  .25;  i  to  5,  .16. 

4.  Cleaner  and  sharper  the  sand,  greater  the  strength. 

5.  Strong  cement  is  heavy;  blue  gray,  slow-setting.     Quick-setting  has  generally 
too  much  clay  in  its  composition — is  brownish  and  weak. 

6.  Less  water  used  in  mixing  cement  the  better. 

7.  Bricks,  stones,  etc.,  used  with  cement  should  be  well  wetted  before  use. 

8.  Cement  setting  under  stitt  water  will  be  stronger  than  if  kept  dry. 

9.  Bricks  of  neat  Portland  cement  in  a  few  months  are  equal  to  Blue  bricks, 
Bramley-Fall  stone,  or  Yorkshire  landings. 

10.  Bricks  of  i  cement  to  4  or  5  of  sand  are  equal  to  picked  stock  bricks. 

11.  When  concrete  is  being  used,  a  current  of  water  will  wash  away  the  cement. 

A  rtificial  Cement  is  made  by  a  combination  of  slaked  lime  with  unburned 
clay  in  suitable  proportions. 

Artificial  Pozzuolana  is  made  by  subjecting  clay  to  a  slight  calcination. 

Salt  water  has  a  tendency  to  decompose  cements  of  all  kinds,  and  their 
strength  is  considerably  impaired  by  their  mixture  with  it. 

Mortar. 

Lime  or  Cement  paste  is  the  cementing  substance  in  mortar,  and  its  pro- 
portion should  be  determined  by  the  rule  that  Volume  of  cementing  substance 
should  be  somewhat  in  excess  of  volume  of  voids  or  spaces  in  sand  or  coarse 
material  to  be  united,  the  excess  being  added  to  meet  imperfect  manipulation 
of  the  mass. 

Hydraulic  Mortar,  if  re-pulverized  and  formed  into  a  paste  after  having 
once  set,  immediately  loses  a  great  portion  of  its  hydraulicity,  and  descends 
to  the  level  of  moderate  hydraulic  limes. 

The  retarding  influence  of  sea-water  upon  initial  hydraulic  induration  is 
not  very  great,  if  the  cement  is  mixed  with  fresh  water.  The  strength  of 
mortars,  however,  is  considerably  impaired  by  being  mixed  with  sea-water. 

Pointing  Mortar  is  composed  of  a  paste  of  finely-ground  cement  and  clean 
sharp  siliceous  sand,  in  such  proportions  that  the  volume  of  cement  paste  is 
slightly  in  excess  of  the  volume  of  voids  or  spaces  in  the  sand.  The  volume 


LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES.       59! 

of  sand  varies  from  2.5  to  2.75  that  of  the  cement  paste,  or  by  weight,  i  ol 
cement  powder  to  3  to  3.33  of  sand.  The  mixture  should  be  made  under 
shelter,  and  in  small  quantities. 

All  mortars  are  much  improved  by  being  worked  or  manipulated;  and  as  rich 
limes  gain  somewhat  by  exposure  to  the  air,  it  is  advisable  to  work  mortar  in 
large  quantities,  and  then  render  it  fit  for  use  by  a  second  manipulation. 

White  lime  will  take  a  larger  proportion  of  sand  than  brown  lime. 

Use  of  salt-water  in  the  composition  of  mortar  injures  adhesion  of  it 

When  a  small  quantity  of  water  is  mixed  with  slaked  lime,  a  stiff  paste 
is  made,  which,  upon  becoming  dry  or  hard,  has  but  very  little  tenacity,  but, 
by  being  mixed  with  sand  or  like  substance,  it  acquires  the  properties  of  a 
cement  or  mortar. 

Proportion  of  sand  that  can  be  incorporated  with  mortar  depends  partly 
upon  the  degree  of  fineness  of  the  sand  itself,  and  partly  upon  character  of 
the  lime.  For  rich  limes,  the  resistance  is  increased  if  the  sand  is  in  pro- 
portions varying  from  50  to  240  per  cent,  of  the  paste  hi  volume ;  beyond 
this  proportion  the  resistance  decreases. 

Lime,  i,  clean  sharp  sand,  2.5.  An  excess  of  water  in  slaking  the  lime 
swells  the  mortar,  which  remains  light  and  porous,  or  shrinks  in.  drying ;  an 
excess  of  sand  destroys  the  cohesive  properties  of  the  mass. 

It  is  indispensable  that  the  sand  should  be  sharp  and  clean. 

Stone  Mortar.— %  parts  cement,  3  parts  lime,  and  31  parts  of  sand ;  or  i 
cask  cement,  325  Ibs.,  .5  cask  of  lime,  120  Ibs.,  and  14.7  cube  feet  of  sand= 
18.5  cube  feet  of  mortar. 

Brick  Mortar.— %  parts  cement,  3  parts  lime,  and  27  parts  of  sand ;  or  i 
cask  cement,  325  Ibs.,  .5  cask  of  lime,  120  Ibs.,  and  12  cube  feet  of  sand= 
16  cube  feet  of  mortar. 

Brown  Mortar. — Lime  i  part,  sand  2  parts,  and  a  small  quantity  of  hair. 
Lime  and  sand,  and  cement  and  sand,  lessen  about  .33  in  volume  when  mixed 
together. 

Calcareous  Mortar,  being  composed  of  one  or  more  of  the  varieties  of  lime 
or  cement,  natural  or  artificial,  mixed  with  sand,  will  vary  in  its  properties 
with  quality  of  the  lime  or  cement  used,  the  nature  and  quality  of  sand,  and 
method  of  manipulation. 

Tuirlzisli    3?laster,   or    Hydraulic    Cement. 

100  Ibs.  fresh  lime  reduced  to  powder,  10  quarts  linseed-oil,  and  i  to  2 
ounces  cotton.  Manipulate  the  lime,  gradually  mixing  the  oil  and  cotton,  in 
a  wooden  vessel,  until  mixture  becomes  of  the  consistency  of  bread-dough. 

Dry,  and  when  required  for  use,  mix  with  linseed-oil  to  the  consistency  of  paste, 
and  then  lay  on  in  coats.  Water-pipes  of  clay  or  metal,  joined  or  coated  with  it, 
resist  the  effect  of  humidity  for  very  long  periods. 

Stucco. 

Stucco  or  Exterior  Plaster  is  term  given  to  a  certain  mortar  designed  for 
exterior  plastering;  it  is  sometimes  manipulated  to  resemble  variegated 
marble,  and  consists  of  i  volume  of  cement  powder  to  2  volumes  of  dry  sand. 

In  India,  to  water  for  mixing  the  plaster  is  added  i  Ib.  of  sugar  or  molas- 
ses to  8  Imperial  gallons  of  water,  for  the  first  coat ;  and  for  second  or  finish- 
ing, i  Ib.  sugar  to  2  gallons  of  water. 

Powdered  slaked  lime  and  Smith's  forge  scales,  mixed  with  blood  in  suit- 
able proportions,  make  a  moderate  hydraulic  mortar,  which  adheres  well  to 
masonry  previously  coated  with  boiled  oil. 


5Q2     LIMES,  CEMENTS,  MORTARS,  AND   CONCRETES. 

Plaster  should  be  applied  in  two  coats  laid  on  in  one  operation,  first  coat  being 
thinner  than  second.  Second  coat  is  applied  upon  first  while  latter  is  yet  soft. 

The  two  coats  should  form  one  of  about  1.5  inches  in  thickness,  and  when  fin- 
ished it  should  be  kept  moist  for  several  days. 

When  the  cement  is  of  too  dark  a  color  for  desired  shade,  it  may  be  mixed  w*tti 
white  sand  in  whole  or  in  part,  or  lime  paste  may  be  added  until  its  volume  equals 
that  of  the  cement  paste. 

Klliorassar,    or   Tvirlzish.   Miortar, 

Used  for  the  construction  of  buildings  requiring  great  solidity,  .33  pow- 
dered brick  and  tiles,  .66  fine  sifted  lime.  Mix  with  water  to  required  con- 
sistency, and  lay  between  the  courses  of  brick  or  stones. 

Mortars. 

Mortars  used  for  inside  plastering  are  termed  Coarse,  Fine,  Gauge  or  hard 
finish,  and  Stucco. 

Plastering. — i  bushel,  or  1.25  cube  feet  of  cement,  mortar,  etc.,  will  cover  1.5 
square  yards  .75  inch  thick.  75  volumes  are  required  upon  brick  work  for  70  upon 
laths. 

When  full  time  for  hardening  cannot  be  allowed,  substitute  from  15  to  20  per 
cent,  of  the  lime  by  an  equal  proportion  of  hydraulic  cement. 

For  the  second  or  brown  coat  the  proportion  of  hair  may  be  slightly  diminished. 

Coarse  Stu.fr.  —  Common  lime  mortar,  as  made  for  brick  masonry, 
with  a  small  quantity  of  hair ;  or  by  volumes,  lime  paste  (30  Ibs.  lime)  i 
partj  sand  2  to  2.25  parts,  hair  .16  part. 

Fine  Stuff  (lime  putty). — Lump  lime  slaked  to  a  paste  with  a  mod- 
erate volume  of  water,  and  afterwards  diluted  to  consistency  of  cream,  and 
then  to  harden  by  evaporation  to  required  consistency  for  working. 

In  this  state  it  is  used  for  a  slipped  coat,  and  when  mixed  with  sand  or  plaster  of 
Paris,  it  is  used  for  finishing  coat. 

Q-auge,  or  Hard  Finish,  is  composed  of  from  3  to  4  volumes  fine 
stuff  and  i  volume  plaster  of  Paris,  in  proportions  regulated,  by  rapidity  re- 
quired in  hardening;  for  cornices,  etc.,  proportions  are  equal  volumes  of 
each,  fine  stuff  and  plaster. 

Scratch  Coat. — First  of  three  coats  when  laid  upon  laths,  and  is  from  .25  to 
•375  °f  an  incn  m  thickness. 

One-coat  Work. — Plastering  in  one  coat  without  finish,  either  on  masonry 
or  laths — that  is,  rendered  or  laid. 

Two-coat  Work.— Plastering  in  two  coats  is  done  either  in  a  laid  coat 
and  set,  or  in  a  screed  coat  and  set. 

Screed  coat  is  also  termed  a  Floated  coat.  Laid  first  coat  in  two-coat 
work  is  resorted  to  in  common  work  instead  of  screeding,  when  finished  sur- 
face is  not  required  to  be  exact  to  a  straight-edge.  It  is  laid  in  a  coat  of 
about  .5  inch  in  thickness. 

Laid  coat,  except  for  very  common  work,  should  be  hand-Jloated. 

Firmness  and  tenacity  of  plastering  is  very  much  increased  by  hand-floating. 

Screeds  are  strips  of  mortar  6  to  8  inches  in  width,  and  of  required  thick 
ness  of  first  coat,  applied  to  the  angles  of  a  room,  or  edge  of  a  wall  and  paral- 
lelly,  at  intervals  of  3  to  5  feet  over  surface  to  be  covered.  When  these  have 
become  sufficiently  hard  to  withstand  pressure  of  a  straight-edge,  the  inter- 
spaces between  the  screeds  are  filled  out  flush  with  them. 

Slipped  Coat  is  the  smoothing  off  of  a  brown  coat  with  a  small  quantity 
of  lime  putty,  mixed  with  3  per  cent,  of  white  sand,  so  as  to  make  a  compar- 
atively even  surface. 

this  finish  answers  when  the  surface  is  to  be  finished  in  distemper,  or  paper. 


LIMES,  CEMENTS,  MORTARS,  AND    CONCRETES.       593 

Concrete    or    Beton 

fs  a  mixture  of  mortar  (generally  hydraulic)  with  coarse  materials,  as 
gravel,  pebbles,  stones,  shells,  broken  bricks,  etc.  Two  or  more  of  these 
materials,  or  all  of  them,  may  be  used  together.  As  lime  or  cement  paste  is 
the  cementing  substance  in  mortar,  so  is  mortar  the  cementing  substance  in 
concrete  or  beton.  The  original  distinction  between  cement  and  beton  was, 
that  latter  possessed  hydraulic  energy,  while  former  did  not. 

Hydraulic.  —  1.5  parts  unslaked  hydraulic  lime,  1.5  parts  sand,  i  part 
gravel,  and  2  parts  of  a  hard  broken  limestone. 

This  mass  contracts  one  fifth  in  volume.  Fat  lime  may  be  mixed  with  concrete, 
without  serious  prejudice  to  its  hydraulic  energy. 

"Various    Compositions   of  Concrete. 

Hydrauli c.— 308  Ibs.  cement  =  3.65  to  3.7  cube  feet  of  stiff  paste.  12  cube 
feet  of  loose  sand  =  9.75  cube  feet  of  dense. 

For  Superstructure.— 11.75  cube  feet  of  mortar  as  above,  and  16  cube  feet 
of  stone  fragments. 

Sea  Wall. — Boston  Harbor. — Hydraulic. — 308  Ibs.  cement,  8  cube  feet  of 
sand,  and  30  cube  feet  of  gravel.  Whole  producing  32.3  cube  feet. 

Superstructure. — 308  Ibs.  cement,  80  Ibs.  lime,  and  14.6  cube  feet  dense 
sands.  Whole  producing  12.825  cube  feet. 

I?ise  fs  made  of  clay  or  earth  rammed  in  layers  of  from  3  to  4  ins.  in  depth.  In 
moist  climates,  it  is  necessary  to  protect  the  external  surface  of  a  wall  constructed 
in  this  manner  with  a  coat  of  mortar. 

A.splialt   Composition. 

Asphaltum  3  parts,  residuum  oil  or  soft  bitumen  i  part,  powdered  stone  or  fine 
sand  12  parts. 

Ashes  2  parts,  powdered  clay  3  parts,  sand  i  part.  Mixed  with  soft  bitumen 
makes  a  very  fine  and  durable  cement,  suitable  for  external  use. 

Flooring. — 8  Ibs.  of  composition  will  cover  i  sup.  foot,  .75  inch  thick.  Asphaltic 
limestone  55  Ibs.  and  gravel  28.7  Ibs.  will  cover  10.75  sq.  feet,  .75  inch  thick. 

Asphaltic  Mastic. — Mix  hot  asphaltic  limestone  8  parts,  asphaltum  i  part;  add 
sufficient  sand  for  density  needed  for  floor,  roof,  or  walk. 

Waterproofing. — Asphaltum  4  parts,  linseed  oil  2  parts,  sand  14  parts,  pulverized 
limestone  14  parts,  by  weight.  Materials  to  be  well  dried,  hot,  and  apply  to  dry 
surface. 

For  Roads. — Asphaltum  12.5  parts,  soft  bitumen  or  maltha  2.5  parts,  powdered 
limestone  5  parts,  sand  80  parts,  mixed  at  temperature  of  300°.  Thickness,  2  ina 

Artificial  Mastic.  —Composition  of  i  square  yard  .9  inch  thick: 

Mineral  tar. 205  cube  ins.    I    Gravel 275  cube  ins. 

Pitch 165    "  Slakedlime 55    "      « 

Sand 549    "  1249  cube  ins. 

M:  viral    "Efflorescence.— White  alkaline  efflorescence  upon  the  surface 
of  brick  walls  laid  in  mortar,  of  which  natural  hydraulic  lime  or  cement  is  the  basis. 
Mortar  mixed  with  animal  fat  in  the  proportion  of  .025  of  its  weight  will  prevent 
its  formation. 

Crystallization  of  these  salts  within  the  pores  of  bricks,  into  which  they  have 
been  absorbed  from  the  mortar,  causes  disintegration. 

Distemper  is  term  for  all  coloring  mixed  with  water  and  size. 

Gr0H#w<7.— Mortar  composed  of  lime  and  fine  sand,  in  a  semi-fluid  state, 
poured  into  the  upper  beds  and  internal  joints  of  masonry. 

Laitance  is  the  pulpy  and  gelatinous  fluid,  of  a  milky  hue,  that  is  washed 
from  cement  upon  its  being  deposited  in  water.  It  is  produced  more  abun- 
dantly in  sea  water  than  in  fresh ;  it  sets  very  imperfectly,  and  has  a  ten- 
dency to  lessen  the  strength  of  the  concrete. 


594      LIMES>  CEMENTS,  MORTARS,  AND    CONCRETES. 

Slaking. 

Slaked  Lime  is  a  hydrate  of  lime,  and  it  absorbs  a  mean  of  2.5  times  its 
volume,  and  2.25  times  its  weight  of  water. 

Lime  (quicklime)  must  be  slaked  before  it  can  be  used  as  a  matrix  for 
mortar. 

Ordinary  method  of  slaking  is  by  submitting  the  lime  to  its  full  propor- 
tion of  water  (previously  known  or  attained  by  trial)  in  order  to  reduce  it  to 
the  consistency  of  a  thick  pulp.  The  volume  of  water  required  for  this  pur- 
pose will  vary  with  different  limes,  and  will  range  from  2.5  to  3  volumes 
that  of  the  lime,  and  it  is  imperative  that  it  should  all  be  poured  upon  it  so 
nearly  at  one  time  as  to  be  in  advance  of  the  elevation  of  the  temperature 
consequent  upon  its  reduction. 

This  process,  when  the  water  used  is  in  an  excessive  quantity,  is  termed 
"  drowning,"  and  when  the  volume  of  lime  has  increased  by  the  absorption 
of  water  it  is  termed  its  "  growth." 

If  too  much  water  is  used,  the  binding  qualities  of  the  lime  is  injured  by 
its  semi-fluidity ;  and  if  too  little,  it  is  injurious  to  add  after  the  reduction  of 
the  lime  has  commenced,  as  it  reduces  its  temperature  and  renders  it  granu- 
lar and  lumpy. 

While  lime  is  in  progress  of  slaking  it  should  be  covered  with  a  tarpaulin 
or  canvas  (a  layer  of  sand  will  suffice),  in  order  to  concentrate  its  evolved 
heat. 

The  essential  point  in  slaking  is  to  attain  the  complete  reduction  of  the 
lime,  and  the  greater  the  hydraulic  energy  of  a  lime,  the  more  difficult  it  be- 
comes to  effect  it. 

Whitewash  or  Grouting.— When  lime  is  required  for  a  whitewash  or  for 
grouting,  it  should  be  thoroughly  "  drowned,"  and  then  run  off  into  tight  ves- 
sels and  closed. 

Slaking  by  Immersion  is  the  method  of  suspending  lime  in  a  suitable  ves' 
sel  in  water  for  a  very  brief  period,  and  withdrawing  it  before  reduction 
commences.  The  lime  is  then  transferred  to  casks  or  like  suitable  receptacles, 
and  tightly  enclosed,  until  it  is  reduced  to  a  fine  powder,  in  which  condition, 
if  secured  from  absorption  of  air,  it  may  be  preserved  for  several  months 
without  essential  deterioration. 

Spontaneous  or  Air  Slaking. — When  lime  is  not  wholly  secured  from  ex- 
posure to  the  air,  it  absorbs  moisture  therefrom,  slakes,  and  falls  into  a  powder. 

Limes  and  Cements. — A  Cask  of  Lime  =  240  Ibs.,  will  make  from  7.8  to 
8.15  cube  feet  of  stiff  paste. 

A  Cask  of  Cement  =  300*  Ibs.,  will  make  from  3.7  to  3.75  cube  feet  of 
stiff  paste. 

A  Cask  of  Portland  Cement  =  4  bushels  or  5  cube  feet  =  420  Ibs. 

A  Cask  of  Roman  Cement  =  3  bushels  or  3.75  cube  feet  =  364  Ibs. 

.5  inch.  .75  inch.  i  inch. 

A  Bushel  of  cement  will  cover 2.25  yards  1.5  yards  1.14  yards. 

From  experiments  of  General  Totten,  it  appeared  that 
i  volume  of  lime  slaked  with  .33  its  volume  of  water  gave  2.27  volumes  of  powder. 
i      "  "  "         .66  "  "  1.74        "  " 

i      «  »  "       i  «  »  2.o6        " 

One  cube  foot  of  dry  cement,  mixed  with  .33  cube  foot  of  water,  will  make  63  to 
.635  cube  foot  of  stiff  paste. 

Lime  should  be  slaked  at  least  one  day  before  it  is  incorporated  with  the 
sand,  and  when  they  are  thoroughly  mixed,  the  mortar  should  be  heaped  into 
one  volume  or  mass,  for  use  as  required. 

*  300  Ibs.  net  is  standard ;  it  usually  overruns  8  Ibs. 


LIMES,  CEMENTS,  MORTARS,  AND    CONCRETES.        595 


Mortar,  Cement,  &c.     (Molesworth.) 

Mortar. — i  of  lime  to  2  to  3  of  sharp  river  sand. 

Or,  i  of  lime  to  2  sand  and  i  blacksmith's  ashes,  or  coarsely  ground  coke. 

Coarse  Mortar. — i  of  lime  to  4  of  coarse  gravelly  sand. 

Concrete.^i  of  lime  to  4  of  gravel  and  2  of  sand. 

Hydraulic  Mortar.— i  of  blue  lias  lime  to  2.5  of  burnt  clay,  ground  to- 
gether. 

Or,  i  of  blue  lias  lime  to  6  of  sharp  sand,  i  of  pozzuolana  and  i  of  calcined 
ironstone. 

Beton. — i  of  hydraulic  mortar  to  1.5  of  angular  stones. 

Cement. — i  of  sand  to  i  of  cement. — If  great  tenacity  is  required,  the  ce- 
ment should  be  used  without  sand. 

Portland    Cement 

Is  composed  of  clayey  mud  and  chalk  ground  together,  and  afterwards  cal- 
cined at  a  high  temperature — after  calcining  it  is  ground  to  a  fine  powder. 

Strength,   of  Mortars,  Cements,  and   Concretes. 

Deduced  from  Experiments  of  Vicat,  Paisley,  Treussart,  and  Voisin. 

Tensile 

Weight  or  Power  required  to  Tear  asunder  One  Sq.  Inch. 

Cement    Mortar.    (42  days  old.) 

Proportion  of  Sand  to  i  of  Cement. 


0 

i 

t 

2 

roport 
3 

Roman  
Portland  

284 
142 

284 
142 

199 

"3 

166 
92 

5 

6 

7 

8 

9 

xo 

128 
67 

116 

57 

106 
42 

99 
35 

92 
25 

95lbs. 

79 
Brick,  Stone,  and.    Q-ranite   Masonry.     (320  days  old.) 


Experiments  of  General  Gillmore,  U.  S.  A. 


Cement  on  Bricks. 


Cement  on  Granite. 


Lba. 

Pure,  average  30.8 

Pure  

Lb6. 

Lbs. 
Sand  i      ) 

Sand  i      ) 
Cement  i  J  *5'7 
Sand  i      J 

Sand  i 
Cement  i 
Sand  i 

....    20.8 

.     .       12  6 

Cement4j  7'9 

^ateri    i  .          ..205 

Cement  2) 
Water  .42)              ..3725 

Cement  2  J  I2'3 
£and  '      1                     68 

Cement  2     '  ' 
Sand  i 

Cement  i  J  " 
Water  .33) 

Cements) 

Delafield  and  Baxter.  Lbs. 
Pure  cement  68 

Cement  3     *  ' 

James  River. 
Pure  cement.  . 
Cement  4  ) 

Lbs. 
87 

..    rfv> 

Cement  i  J   '  ' 

Lbs. 
Neivark  and  Rosendale. 
Cement  i  ) 

Cement  4)                   6g 

Sand  i      J  '  ' 
Cement  8  )                   ~ 

Sand  i      f  '  ' 
Newark  Lime  and  Cement 
Co. 
Pure  cement  93 

Sand  3      )  *  ' 
Pure,  without) 
mortar,  mean  j  ** 

Mortar. 
Lime  paste  i,  sand  2.5,    6 

"            "       I,       "       2            4 

I!        "     i,     "     3        6 
"        "     «,     "     3, 
cement  paste  5  it 

Siftings  i  j  '  ' 
Cement  i  )                   « 

Siftings  i  J  *  * 
Cement  i  ) 

Cement  i  ) 
Sand  2      }  4° 
Newark  and  Rosendale. 
Pure  cement       *»e 

Siftings  2  J  '  ' 
Lawrence  Cement  Co. 
Pure  cement  87 

Cement  i  ) 

j6 

"       54 

Sand  i      j  '  '  * 

596       LIMES,  CEMENTS,  MOKTAKS,  AND    CONCRETES. 


Pure  Cement. 


Boulogne  100,  water  50 112 

Portland,  natural,  i  year 675 

"         artificial,  Eiig.,  i  year...  462 

"         English,  320  days 1152 

"              "        i  month 393 

Newark  and  Rosendale 339 


Portland,  in  sea- water,  45  days 266 

' '         English,  6  months 424 

Roman  "Septaria,"  i  year 191 

"      masonry,  5  months „     77 

Rosendale,  o  months 700 

Lawrence  Cement  Co 1210 


Transverse. 

Reduced  to  a  uniform  Measure  of  One  Inch  Square  and  One  Foot  in  Length. 
Supported  at  Both  Ends. 

Experiments    of  Greneral    Grillmore. 

Formed  in  molds  under  a  pressure  of  32  Ibs.  per  sq.  inch,  applied  until  mortar 
had  set.     Exposed  to  moisture  for  24  hours,  and  then  immersed  in  sea- water. 
Prisms  2  by  2  by  8  ins.  between  supports. 


Reduced  by  Formula  - 

W        a 

:  C.     C  coefficient  of  rupture,  and  a  weight  of 

bd2      2  ~ 

portion  of  prism  I. 

Cement. 

Mortar. 

MATERIAL. 

t 

Z 

MATERIAL. 

1 

"S    M 
1"° 

"a  ** 

S* 

3l 

SI 

Days. 

Lbs. 

Days. 

Lbs. 

T,h». 

James  River. 

Portland,  Eng.,  stiff  paste 

320 

13 

10 

Thick  cream               .  .  > 

•3    Q 

Roman        "       "       " 

2   C 

Thin  paste  

320 

5.8 

6 

Stiff  paste 

6  Q 

Cumberland  Md 

12  8 

8 

Rosendale  "Hoffman  " 

Akron,  N  Y  

320 

8  8 

8  4 

Thin  paste 

James  River  Va 

8  6 

8  8 

Stiff  paste 

320 

ii 

Pulverized    and    re-  ) 

* 

"  Delafield  and  Baxter." 

mixed  after  set  i 

3 

3-6 

— 

Thin  paste               . 

320 

8.5 

Fresh    

Stiff  paste  

020 

12 

Kingston  and  Rosendale. 

76 

6.6 

English. 

tf\ 

High  Falls,  UM 

95 

3-2 

Stiff  paste  

320 
320 

13 

Fresh  water  to  a  stiff  ) 

Cumberland,  Md.,  pure  
High  Falls,  UM 

320 
95 

13-2 
8.4 

paste  ) 

95 
95 

— 

4.4 

2.6 

Sea-water  to  a  stiff  paste 
Lawrence  Cement  Co. 

sterCo.,N.Y.  j  

Comolete  calcination.  .  . 

Q< 

4.2 

Fresh.  .  . 

32O 

IO.2 



Crnsliing. 

Cements,  Stones,  etc.    (Crystal  Palace,  London.) 
Reduced  to  a  uniform  Measure  of  One  Sq.  Inch. 


MATERIAL. 

Destructive 
Pressure. 

MATERIAL. 

Destructi 
Pressure 

Portl'd  cem't,  area  i,  height  i. 

Lbs. 
1680 

Portland  coment  i  ) 

Lbs. 
1244 

"     cement  ) 

1244 

"      sand.  .  .  )  " 

"       cement  i  ) 

692 

Roman  cement,  pure.  .  .  , 

14.2 

G-eneral   Deductions. 

i.  Particles  of  unground  cement  exceeding  .0125  of  an  inch  in  diameter  may  be 
allowed  in  cement  paste  without  sand,  to  extent  of  50  per  cent,  of  whole,  without 
detriment  to  its  properties,  while  a  corresponding  proportion  of  sand  injures  the 
strength  of  mortar  about  40  per  cent. 


LIMES,  CEMENTS,  MORTARS,  ETC. — MASONRY.       597 

2  When  these  unground  particles  exist  in  cement  paste  to  extent  of  66  per  cent, 
of  whole,  adhesive  strength  is  diminished  about  28  per  cent.  For  a  corresponding 
proportion  of  sand  the  diminution  is  68  per  cent. 

3.  Addition  of  siftings  exercises  a  less  injurious  effect  upon  the  cohesive  than  upon 
the  adhesive  property  of  cement.    The  converse  is  true  when  sand,  instead  of  sift- 
ings,  is  used. 

4.  In  all  mixtures  with  siftings,  even  when  the  latter  amounted  to  66  per  cent,  of 
whole,  cohesive  strength  of  mortars  exceeded  their  adhesion  to  bricks.     Same  re- 
sults appear  to  exist  when  siftings  are  replaced  by  sand,  until  volume  of  the  latter 
exceeds  20  per  cent,  of  who^  after  which  adhesion  exceeds  cohesion. 

5.  At  age  of  320  days  (and  perhaps  considerably  within  that  period)  cohesive 
strength  of  pure  cement  mortar  exceeds  that  of  Croton  front  bricks.    The  converse 
is  true  when  the  mortar  contains  50  per  cent,  or  more  of  sand. 

6.  When  cement  is  to  be  used  without  sand,  as  may  be  the  case  when  grouting  is 
resorted  to,  or  when  old  walls  are  to  be  repaired  by  injections  of  thin  paste,  there  is 
no  advantage  in  having  it  ground  to  an  impalpable  powder. 

7.  For  economy  it  is  customary  to  add  lime  to  cement  mortars,  and  this  may  be 
done  to  a  considerable  extent  when  in  positions  where  hydraulic  activity  and 
strength  are  not  required  in  an  eminent  degree. 

8.  Ramming  of  concrete  under  water  is  held  to  be  injurious. 

9.  Mortars  of  common  lime,  when  suitably  made,  set  in  a  very  few  days,  and  with 
such  rapidity  that  there  is  no  need  of  awaiting  its  hardening  in  the  prosecution  of 
work. 

Fire  Clay.— The  fusibility  of  clay  arises  from  the  presence  of  impurities, 
such  as  lime,  iron,  and  manganese.  These  may  be  removed  by  steeping  the  clay  in 
hot  muriatic  acid,  then  washing  it  with  water.  Crucibles  from  common  clay  may 
be  made  in  this  manner. 

Notes  by  General  Gillmore,  U.  S.  A.—  Recent  experiments  have  developed  that 
most  American  cements  will  sustain,  without  any  great  loss  of  strength,  a  dose  of 
lime  paste  equal  to  that  of  the  cement  paste,  while  a  dose  equal  to  5  to  75  the  vol- 
ume of  cement  paste  may  be  safely  added  to  any  Rosendale  cement  without  pro- 
ducing any  essential  deterioration  of  the  quality  of  the  mortar.  Neither  is  the 
hydraulic  activity  of  the  mortars  so  far  impaired  by  this  limited  addition  of  lime 
paste  as  to  render  them  unsuited  for  concrete  under  water,  or  other  submarine 
masonry.  By  the  use  of  lime  is  secured  the  double  advantages  of  slow  setting  and 
economy 

Notes  by  General  Totten,  V.  S.  A— 240  Ibs.  lime  =  i  cask,  will  make  from  7.8  to 
8. 15  cube  feet  of  stiff  paste. 

i  cube  foot  of  dry  cement  powder,  measured  when  loose,  will  measure  .  78  to  8 
cube  foot  when  packed,  as  at  a  manufactory. 

For  composition  of  Concretes,  at  Toulon,  Marseilles,  Cherbourg,  Dover,  Alderney, 
etc.,  see  Treatise  of  General  Gillmore,  pp.  253-256. 


MASONRY. 
Brickwork. 

Bond  is  an  arrangement  of  bricks  or  stones,  laid  aside  of  and  above 
each  other,  so  that  the  vertical  joint  between  any  two  bricks  or  stones 
does  not  coincide  with  that  between  any  other  two. 

This  is  termed  "breaking  joints." 

Header  is  a  brick  or  stone  laid  with  an  end  to  face  of  wall. 

Stretcher  is  a  brick  or  stone  laid  parallel  to  face  of  wall. 

Header  Course  or  Bond  is  a  course  or  courses  of  headers  alone. 

Stretcher  Course  or  Bond  is  a  course  or  courses  of  stretchers  alone. 

Closers  are  pieces  of  bricks  inserted  in  alternate  courses,  in  order  to  obtain 
a  bond  by  preventing  two  headers  from  being  exactly  over  a  stretcher. 

English  Bond  is  laying  of  headers  and  stretchers  in  alternates  courses. 


598  MASONRY. 

Flemish  Bond  is  laying  of  headers  and  stretchers  alternately  in  each  course, 

Gauged  Work. — Bricks  cut  and  rubbed  to  exact  shape  required. 

String  Course  is  a  horizontal  and  projecting  course  around  a  building. 

Corbelling  is  projection  of  some  courses  of  a  wall  beyond  its  face,  in  order 
to  support  wall-plates  or  floor-beams,  etc. 

Wood  Bricks,  Pallets,  Plugs,  or  Slips  are  pieces  of  wood  laid  in  a  wall  in 
order  the  better  to  secure  any  woodwork  that  it  may  be  necessary  to  fasten 
to  it. 

Reveals  are  portions  of  sides  of  an  opening  in  a  wall  in  front  of  the  recesses 
for  a  door  or  window  frame. 

Brick  Ashlar. — Walls  with  ashlar-facing  backed  with  brick. 

Grouting  is  pouring  liquid  mortar  over  last  course  for  the  purpose  of  filling 
all  vacuities. 

Larrying  is  filling  in  of  interior  of  thick  walls  or  piers,  after  exterior  faces 
are  laid,  with  a  bed  of  soft  mortar  and  floating  bricks  or  spawls  in  it. 

Rendering  (Eng.)  is  application  of  first  coat  on  masonry,  Laying  if  one 
or  two  coats  on  laths,  and  "  Pricking  up  "  if  three-coat  work  on  laths. 

Bricks  should  be  well  wetted  before  use.  Sea  sand  should  not  be  used  in  the 
composition  of  mortar,  as  it  contains  salt  and  its  grains  are  round,  being  worn  by 
attrition,  and  consequently  having  less  tenacity  than  sharp-edged  grains. 

A  common  burned  brick  will  absorb  i  pint  or  about  one  sixth  of  its  weight  of 
water  to  saturate  it.  The  volume  of  water  a  brick  will  absorb  is  inversely  a  test  of 
its  quality. 

A  good  brick  should  not  absorb  to  exceed  .067  of  its  weight  of  water. 

The  courses  of  brick  walls  should  be  of  same  height  in  front  and  rear,  whether 
front  is  laid  with  stretchers  and  thin  joints  or  not 

In  ashlar- facing  the  stones  should  have  a  width  or  depth  of  bed  at  least  equal  to 
height  of  stone. 

Hard  bricks  set  in  cement  and  3  months  set  will  sustain  a  pressure  of  40  tons 
per  sq.  foot. 

The  compression  to  which  a  stone  should  be  subjected  should  not  exceed  .  i  of  its 
crushing  resistance. 

The  extreme  stress  upon  any  part  of  the  masonry  of  St.  Peter's  at  Rome  is  com- 
puted at  15.5  tons  per  sq.  foot ;  of  St.  Paul's,  London,  14  tons  ;  and  of  piers  of  New 
York  and  Brooklyn  Bridge,  5. 5  tons. 

The  absorption  of  water  in  24  hours  by  granites,  sandstones,  and  limestones  of  a 
durable  description  is  i,  8,  and  12  per  cent,  of  volume  of  the  stone. 

Color  of  Bricks  depends  upon  composition  of  the  clay,  the  molding  sand,  tem- 
perature, of  burning,  and  volume  of  air  admitted  to  kiln. 

Pure  clay  free  of  iron  will  burn  white,  and  mixing  of  chalk  with  the  clay  will 
produce  a  like  effect. 

Presence  of  iron  produces  a  tint  ranging  from  red  and  orange  to  light  yellow, 
according  to  proportion  of  iron. 

A  large  proportion  of  oxide  of  iron,  mixed  with  a  pure  clay,  will  produce  a  bright 
red,  and  when  there  is  from  8  to  10  per  cent.,  and  the  brick  is  exposed  to  an  intense 
heat,  the  oxide  fuses  and  produces  a  dark  blue  or  purple,  and  with  a  small  volume 
of  manganese  and  an  increased  proportion  of  the  oxide  the  color  is  darkened,  even 
to  a  black. 

Small  volume  of  lime  and  iron  produces  a  cream  color,  an  increase  of  iron  pro- 
duces red,  and  an  increase  oflime  brown. 

Magnesia  in  presence  of  iron  produces  yellow. 

Clay  containing  alkalies  and  burned  at  a  high  temperature  produces  a  bluish  green. 

For  other  notes  on  materials  of  masonry,  their  manipulation,  etc.,  see  "Limes, 
Cements,  Mortars,  and  Concretes,"  pp.  588-597. 

Pointing. — Before  pointing,  the  joints  should  be  reamed,  and  in  close  ma- 
sonry they  must  be  open  to  2  of  an  inch,  then  thoroughly  saturated  with  water, 
and  maintained  in  a  condition  that  they  will  neither  absorb  water  from  the  mortar 
or  impart  any  to  it.  Masonry  should  not  be  allowed  to  dry  rapidly  after  pointing, 
but  it  should  be  well  driven  in  by  the  aid  of  a  calking  iron  and  hammer. 

In  pointing  of  rubble  masonry  the  same  general  directions  are  to  be  observed. 


MASONEY. 


599 


Sand  is  Argillaceous,  Siliceous,  or  Calcareous,  according  to  its  composition. 
Its  use  is  to  prevent  excessive  shrinking,  and  to  save  cost  of  lime  or  cement  Or- 
dinarily it  is  not  acted  upon  by  lime,  its  presence  in  mortar  being  mechanical,  and 
with  hydraulic  limes  and  cements  it  weakens  the  mortar.  Rich  lime  adheres  better 
to  the  surface  of  sand  than  to  its  own  particles;  hence  the  sand  strengthens  the 
mortar. 

It  is  imperative  that  sand  should  be  perfectly  clean,  freed  from  all  impurities, 
and  of  a  sharp  or  angular  structure.  Within  moderate  limits  size  of  grain  does 
not  affect  the  strength  of  mortar;  preference,  however,  should  be  given  to  coarse. 

Calcareous  sand  is  preferable  to  siliceous. 

Sea  and  River  sand  are  suitable  for  plastering,  but  are  deficient  in  the  sharpness 
required  for  mortar,  from  the  attrition  they  are  exposed  to. 

Clean  sand  will  not  soil  the  hands  when  rubbed  upon  them,  and  the  presence  of 
gait  can  be  detected  by  its  taste. 

Scoriae,  Slag,  Clinker,  and  Cinder,  when  properly  crushed  and  used,  make  good 
substitutes  for  sand. 

Concrete. — In  the  mixing  of  concrete,  slake  lime  first,  mix  with  cement,  and  then 
with  the  chips,  etc.,  deposit  in  layers  of  6  ins.,  and  hammer  down. 

Bricks. 

Variations  in  dimensions  by  various  manufacturers,  and  different  degrees 
of  intensity  of  their  burning,  render  a  table  of  exact  dimensions  of  different 
manufactures  and  classes  of  bricks  altogether  impracticable. 

As  an  exponent,  however,  of  the  ranges  of  their  dimensions,  following 
averages  are  given : 


DESCRIPTION. 

Ins. 

DESCRIPTION. 

In§. 

Baltimore     front 

Maine    . 

7  e        y  o  •571:  V.  2  77C 

Philadelphia    " 

8.25  X  4-125  X  2.375 

Milwaukee  

8.  c      X  4  125  X  2  375 

Wilmington     " 
Croton 

8.5    X4        X2.25 

North  River  
Ordinary  

8         X3-5      X2.25 
(7-75    X  3-625X2.25 

Eng.  ordinary... 
"     Lond.  stock 
Dutch  Clinker.  .  . 

8.25  X  3-625  X  2.375 
9       X4-5      X2.5 
8.75X4-25    X2.5 
6.25X3        Xi.5 

Stourbridge       ) 
fire-brick.  .  .  .  j 
Amer.  do.,N.  Y.. 

(8         X  4-  125X2.5 
9.125X4-625X2.375 
8-875X4-5      Xa.625 

In  consequence  of  the  variations  hi  dimensions  of  bricks,  and  thickness  of 
the  layer  of  mortar  or  cement  in  which  they  may  be  laid,  it  is  also  impracti- 
cable to  give  any  rule  of  general  application  for  volume  of  laid  brick-work. 
It  becomes  necessary,  therefore,  when  it  is  required  to  ascertain  the  volume 
of  bricks  in  masonry,  to  proceed  as  follows : 

To  Compute  Volume  of  Bricks,  and.  Number  in  a  Cube 
Foot   of  3VIason.ry. 

RULE.— To  face  dimensions  of  particular  bricks  used,  add  one  half  thick- 
ness of  the  mortar  or  cement  in  which  they  are  laid,  and  compute  the  area ; 
divide  width  of  wall  by  number  of  bricks  of  which  it  is  composed  ;  multiply 
this  area  by  quotient  thus  obtained,  and  product  will  give  volume  of  the 
mass  of  a  brick  and  its  mortar  in  ins. 

Divide  1728  by  this  volume,  and  quotient  will  give  number  of  bricks  in  a 
cube  foot. 

EXAMPLE. — Width  of  a  wall  is  to  be  12.75  ins.,  and  front  of  it  laid  with  Philadel- 
phia bricks  in  courses  .25  of  an  inch  in  depth;  how  many  bricks  will  there  be  in 
face  and  backing  in  a  cube  foot? 
Philadelphia  front  brick,  8.25  x  2.375  ins.  face. 

8. 25  + .  25  X  2-7-2  =  8. 25  + .  25  =  8. 5     =  length  of  brick  and  joint ; 
2-375  -f-  -25  X  2  -r-  2  =  2.375  -f-  .25  =  2.625  =  width  of  brick  and  joint. 
Then  8.5  x  2.625  =  22.3125  ins.  —area  of  face;   12.75  -5-3  (number  of  bricks  in 
width  of  wall)  =  4. 25  ins. 
Hence  22.3125  x  4.25  =  94.83  cube  ins.  ;  and  1728 -7-94. 83  =  18.22  brickt. 


600 


MASONRY. 


One  rod  of  brick  masonry  (Eng.)r^  11.33  cube  yards  and  weighs  15  tons,  or  272 
superficial  feet  by  13.5  thick,  averaging  4300  bricks,  requiring  3  cube  yards  mortar 
and  120  gallons  water. 

Bricklayers'  hod  will  contain  16  bricks  or  .7  cube  feet  mortar. 


Fire-clay  contains  Silica,  Alumina,  Oxide  of  Iron,  and  a  small  proportion 
of  Lime,  Magnesia,  Potash,  and  Soda.  Its  fire-resisting  properties  depend- 
ing upon  the  relative  proportions  of  these  constituents  and  character  of  its 
grain. 

A  good  clay  should  be  of  a  uniform  structure,  a  coarse  open  grain,  greasy 
to  the  hand,  and  free  from  any  alkaline  earths. 

The  Stourbridge  clay  is  black  and  is  composed  as  follows  : 
Silica  .....  63.3   |   Alumina  .....  23.3   |   Lime  ......  73   |   Protoxide  of  iron....  1.8 

Water  and  organic  matter  ......  .  .  10.3 

Newcastle  clay  is  very  similar. 

Thickness   of  Brick   \Valls   for  \Varehcmses  in   Feet. 

(Molesworth.) 


Height  in  Feet  

IOO 

9° 

80 

7° 

60 

5° 

40 

30 

Length  Unlimited  
Thickness  in  Ins  

34 

34 

30 

26 

26 

26 

21.5 

'7-5 

Length  in  Feet  ..... 

70 

7° 

60 

45 

50 

70 

60 

5° 

Thickness  in  Ins  

30 

30 

26 

21.5 

21-5 

21.5 

17.5 

Length  in  Feet  

55 

60 

45 

3° 

35 

4° 

3° 

45 

Thickness  in  Ins  

26 

26 

21.5 

i7-5 

i7-5 

'7-5 

13 

13 

Stone    Masonry. 

Masonry  is  classed  as  Ashlar  or  Rubble. 

Ashlar  consists  of  blocks  dressed  square  and  laid  with  close  joints. 
Coursed  Ashlar  consists  of  blocks  of  same  height  throughout  each  course. 
Rubble  is  composed  of  small  stones  irregular  in  form,  and  rough. 
Rubble  Ashlar  is  ashlar  faced  stone  with  rubble  backing. 
Ashlar. 
Fig.  2. 


Fig.  i. 


Fig.  i.—  Coursed,  with  chamfered  and 
rusticated  quoins  and  plinth. 


Fig.  3- 


Fig.  2.— Coursed,  with  rock  face  and 
draft  edges. 

Fig.  4. 


?:..*.•*>.'•[•%:  •'*-;\w.,w'jt:f\ 


1 

1          1] 

1    \ 

1 

1        l\ 

1 

1 

1 

!           I 

Fig.  3.— Coursed,  with  rock  face. 


Fig.  4.— Regular  Courted. 


MASONRY. 


6O I 


2 
1  1  1 

landomed  Ashlar. 
-n              Fig-* 

—  m    h-M 

—  1  —  '  —  i  —  '  —  r—  '  — 

i  i  i 

—  H 

3 

-^vw 

—  i  *  1  1   i    i  \ 

i  '  i  '  i  ' 

3 

MH  HH 

Fig-  5- 


Fig.  5.— Irregular  Coursed.  Fig.  6.— Random  Coursed. 

Fig.  7.     | 1 1 T~l V  FiS  8. 


ffl 


Fig.  7.—  Ranged  Random,  level,  and 
broken  coursea 

Fig.  9. 


Fig.  8.—  Random,  level,  and  broken. 

le. 

Fig.  io. 


Fig.  9.  Block  Coursed.— f^arge  blocks 
in  courses  (regular  or  irregular),  Beds 
and  Joints  roughly  dressed. 

Fig.  xx. 


Fig.   io.— Coursed  and  Ranged 
Random. 


Fig.  12. 


Fig.  ii.  Ranged  Random.— Squared 
rubble    laid    in    level    and   broken 


courses. 

Dry    Tl\ 
is  a  wall  laid  without  cement  or  mortar. 


Fig-  13-     (TT 


Fig.  12.  Coursed  Random.—  Stones  laid 
in  courses  at  intervals  of  from  12  to  18 
ins.  in  height. 


Fig.  13.    Dry  Rubble.—  Without  mor- 
tar or  cement. 


Fig.  15. 


Tig.  14.  Rustic  or  .Rap.  — Stones  of 
irregular  form,  and  dressed  to  make 
close  joints. 

Fig.  16. 


Fig.  15.  ITncoursed  or  Random.  — 
Beds  and  Joints  undressed,  projections 
knocked  off,  and  laid  at  random.  In- 
terstices filled  with  spalls  and  mortar. 

NOTE  —Rustic  or  Rag  work  is  frequently  laid  in  mortar 
3E 


Fig.  16.  Laced  Coursed.— Horizontal 
bands  of  stone  or  bricks,  interposed  to 
give  stability. 


6O2 


MASONEY. 
Terra   Cotta. 


Terra  Cotta  in  blocks  should  not  exceed  4  cube  feet  in  volume.    When 
properly  burned,  it  is  unaffected  by  the  atmosphere  or  by  fumes  of  any  acid. 


and.   AValls. 

Singing.— Point  a,  Fig.  15,  on  each  side,  Fig.  15. 

from  which  arch  springs. 

Crown. — Highest  point  of  arch. 

Haunches. — Sides  of  arch,  from  springing 
half-way  up  to  crown. 

Spandrel. — Space  between  extrados,a  hor- 
izontal line  drawn  through  crown  and  a  ver- 
tical line  through  upper  end  of  skewback. 

Skewback  is  upper  surface  of  an  abut- 
ment or  pier  from  which  an  arch  springs, 
and  its  face  is  on  a  line  radiating  from  centre  of  arch. 

Abutment  is  outer  body  that  supports  arch  and  from  which  it  springs. 

Pier  is  the  intermediate  support  for  two  or  more  arches. 

Jambs  are  sides  of  abutments  or  piers. 

Voussoirs  are  the  blocks  forming  an  arch. 

Key-stone  is  centre  voussoir  at  crown. 

Span  is  horizontal  distance  from  springing  to  springing  of  arch. 

Rise. — Height  from  springing  line  to  under  side  of  arch  at  key-stone. 

length  is  that  of  springing  line  or  span. 

Ring-course  of  a  wall  or  arch  is  parallel  to  face  of  it,  and  in  direction  of 
its  span. 

String  and  Collar  courses  are  projecting  ashlar  dressed  broad  stones  at 
right  angles  to  face  of  a  wall  or  arch,  and  in  direction  of  its  length. 

Camber  is  a  slight  rise  of  an  arch  as  .125  to  .25  of  an  inch  per  foot  of 
span. 

Quoin  is  the  external  angle  or  course  of  a  wall. 

Plinth  is  a  projecting  base  to  a  wall. 

Footing  is  projecting  course  at  bottom  of  a  wall,  in  order  to  distribute  its 
weight  over  an  increased  area.  Its  width  should  be  double  that  of  base  of 
wall,  diminishing  in  regular  offsets  .5  width  of  their  height. 

Blocking  Course. — A  course  placed  on  top  of  a  cornice. 

Parapet  is  a  low  wall,  over  edge  of  a  roof  or  terrace. 

Extrados. — Back  or  upper  and  outer  surface  of  an  arch. 

Intrados  or  Soffit  is  underside  of  lower  surface  of  arch  or  an  opening. 

Groined  is  when  arches  intersect  one  another. 

Invert. — An  inverted  arch,  an  arch  with  its  intrados  below  axis  or  spring- 
ing line. 

Ashlar  masonry  requires  .125  of  its  volume  of  mortar.  Rubble,  1.2  cube 
yards  stone  and  .25  cube  yard  mortar  for  each  cube  yard. 

Rubble  masonry  in  cement,  160  feet  in  height,  will  stand  and  bear  20  ooo 
\bs.  per  sq.  inch. 

Stones  should  be  laid  with  their  strata  horizontal. 

When  "  through  "  or  "  thorough  bonds  "  are  not  introduced,  headers  should 
overlap  one  another  from  opposite  sides,  known  as  dogs'  tooth  bond. 

Aggregate  surface  of  ends  of  bond  stones  should  be  from  .125  to  .25  of 
area  of  each  face  of  wall. 

Weak  stones,  as  sandstone  and  granular  limestone,  should  not  have  a 
length  over  3  times  their  depth.  Strong  or  hard  stones  may  have  a  length 
from  4  to  5  times  their  depth. 


MASONRY. 


603 


Gdllets  are  small  and  sharp  pieces  of  stone  stuck  into  mortar  joints,  in 
which  case  the  work  is  termed  galleted. 

Snapped  work  is  when  stones  are  split  and  roughly  squared. 
Quarry  or  Rock-faced.— Quarried  stones  with  their  faces  undressed. 

Pitch-faced. — Stones  on  which  the  arris  or  angles  of  their  face,  with  their 
sides  and  ends,  is  defined  by  a  chisel,  in  order  to  show  a  right-lined  edge. 

Drafted  or  Drafted  Margin  is  a  narrow  border  chiselled  around  edges  of 
faces  of  a  block  of  rough  stone. 

Diamond-faced  is  when  planes  are  either  sunk  or  raised  from  each  edge 
and  meet  in  the  centre. 

Squared  Stones. — Stones  roughly  squared  and  dressed. 

Rubble. — Unsquared  stones,  as  taken  from  a  quarry  or  elsewhere,  in  their 
natural  form,  or  their  extreme  projections  removed. 

Cut  Stones.— Stones  squared  and  with  dressed  sides  and  ends. 


Dressed.    Stones. 

The  following  are  the  modes  of  dressing  the  faces  of  ashlar  in  engineering: 

Rough  Pointed. — Rough  dressing  with  a  pick  or  heavy  point. 

Fine  Pointed. — Rough  dressing,  followed  by  dressing  with  a  fine  point. 

Crandalled. — Fine  pointing  in  right  lines  with  a  hammer,  the  face  of 
which  is  close  serried  with  sharp  edges. 

Cross  Crandalled. — When  the  operation  of  crandalling  is  right  angled. 

Hammered. — The  surface  of  stone  may  be  finished  or  smooth  dressed  by 
being  Axed  or  Bushed;  the  former  is  a  finish  by  a  heavy  hammer  alike  to  a 
crandall,  the  latter  is  a  final  finish  by  a  heavy  hammer  with  a  face  serried 
with  sharp  points  at  right  angles. 

Thickness   of*  Brick:  \Valls  for  "Warehouses.    (Molesworth.) 


Length. 

Height. 

Thickness. 

Length. 

Height. 

Thickness. 

Length. 

Height. 

Thickne 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Unlimited. 

25 

'3 

Unlimit'd. 

100 

34 

45 

3° 

13 

do. 

30 

17-5 

60 

40 

'7-5 

30 

40 

13 

do. 

40 

21.5 

70 

21.5 

40 

So 

17-5 

do. 

50 

26 

50 

60 

21.5 

35 

60 

17-5 

do. 
do. 

60 
70 

26 
26 

i 

70 
80 

21.5 
26 

45 

70 

80 

'7-5 
21.5 

do. 

80 

30 

70 

90 

30 

60 

9° 

26 

do. 

90 

34 

70 

100 

30 

55 

100 

26 

For  drawings  and  a  description  of  stone-dressing  tools,  see  a  paper  by  J.  R.  Cross, 
W.  E.  Merrill,  and  E.  B.  Van  Winkle,  UA.  S. Civil  Engineer  Transactions,"  Nov.  1877. 

Walls  not  exceeding  30  feet  in  height,  upper  story  walls  may  be  8.5  ins.  thick. 

From  16  feet  below  top  of  wall  to  base  of  it,  it  should  not  be  less  than  the  space 
defined  by  two  right  lines  drawn  from  each  side  of  wall  at  its  base  to  16  feet  from 
top. 

Thickness  not  to  be  less  in  any  case  than  one  fourteenth  of  height  of  story. 

Laths. 

Laths  are  1.25  to  1.5  ins.  by  4  feet  in  length,  are  usually  set  .25  of  an  inch 
apart,  and  a  bundle  contains  100. 


604 


MASONRY. 

Plastering. 

Volumes  required  for  Various  Thickness. 


MATKEIAL. 

Sq 
•5 

uare  Yar 
•75 

ds. 

i 

MATERIAL. 

Sq 

uare  Yar 

•75 

di. 

Cube  Feet. 

2.25 
4-5 
6-75 

Ins. 
i-5 
3 
4-5 

Ins. 
I-I5 
2.25 
3-33 

Cube  Feet. 

Lime  i,  sand  2,  ) 
hair  3  75  i  "" 

Ins. 

75  ya 
dered 
brick 

rds,  sup 
and  s 
or  7001 

Ins. 
'1  ren- 
et  on 
i  lath. 

Cement  i,sand  i... 
Cement  i,  sand  2... 

Kstiinate  of  Materials  and  Labor  for  1OO  Sq..  Yards  of 
Lath,   and    blaster. 


Materials 
and  Labor. 

Three  Coats 
Hard  Finish. 

Two  Coats 
Slipped. 

Materials 
and  Labor. 

Three  Coats 
Hard  Finish. 

Two  Coats 
Slipped. 

Lime  

4    casks. 

3.  5  casks 

White  sand 

2  5  bushels 

Lump  lime  

.66  " 

Nails  

13  Ibs 

13  Ibs 

Plaster  of  Paris.. 

•5     " 

Masons  ....... 

4  days. 

3.  5  days. 

Hair  

4  bushels 

-3  bushels 

Laborer  

3     u 

2           " 

Sand  .  .  , 

7  loads. 

6  loads. 

Cartage... 

i     " 

.?«;  " 

Rough   Cast  is  washed  gravel  mixed  with  hot  hydraulic  lime  and 
water  and  applied  in  a  semi-fluid  condition. 


-A^rclies   and   .AJbutments. 

To  Compute  IDepth  of  Keystone  of  Circular  or  Elliptic 
Arch. 


f-  .25  —  d.    R  representing  radius,  s  span,  and  d  depth,  all  in  feet. 

This  is  for  a  rise  of  about  .25  of  span ;  when  it  is  reduced,  as  to  .  125,  add  .5  instead 
of  .25. 

ILLUSTRATION.— Arch  of  Washington  aqueduct  at  "Cabin  John"  has  a  span  of  220 
feet,  a  rise  of  57.25,  and  a  radius  of  134.25;  what  should  be  depth  of  its  keystone? 

4  4 

Vtaducts  of  several  arches  increase  results  as  determined  above  by  add- 
ing .125  to  .15  to  depth. 

For  arches  of  2d  class  materials  and  work,  and  for  spans  exceeding  10 
feet,  add  .125  to  depth  of  keystone,  and  for  good  rubble  or  brick- work 
add  .25. 

NOTE.— It  is  customary  to  make  the  keystones  of  elliptic  arches  of  greater  depth 
than  that  obtained  by  above  formula.  Trautwine,  however,  who  is  high  authority 
in  this  case,  declares  it  is  unnecessary. 


To   Compute    Radius  of  an   Arch,  Circular    or  Ellipse, 
f— j  -|-r2-:-2r  =  R.    r  representing  rise. 

Rail-way   Arches. 

For  Spans  between  25  and  70  feet.    Rise  .2  of  span.    Depth  of  arch  .055  of  span. 
Thickness  of  abutments  .  2  to  .  25  of  span,  and  of  pier .  14  to  .  16  of  span. 

Altmtments. 

When  height  does  not  exceed  i.  5  times  base.     R-t-$-}-.ir-}-2  =  thickness  at  spring 
qf  arch  in  feet.     (Trautwine.) 

Batter.— From  .5  to  1.5  ins.  per  foot  of  height  of  wall. 


MASONRY. — MECHANICAL   CENTRES. — GRAVITY.      605 

To   Compute   Depth   of*  Arch.     (Hurst.) 

c  ^R  =  D.     c  =  Stone  (block)  .3.    Brick  =  .4.    Rubble  =  .45. 
When  there  are  a  series  of  arches,  put  .3  =  .35,  .4  =  .45,  and  .45  — .5. 

Mininaviixi    Thickness   of*  AJtmtments   for   Bridge   and 
similar   Arches    of  ISO0.     (Hurst.) 

When  depth  of  crown  does  not  exceed  3  feet.    Computed  from  formula 

v  /6  R  4-  ( ^-=  )  —  2—  =  T.    H  representing  height  of  abutment  to  springing  in  feet 
V  \2  H/         z  U 


Radius 
of  Arch. 

Heif 
5 

rht  of  At 
7-5 

utment 

10 

to  Spring 
20 

ing. 
30 

Radius 
of  Arch. 

Heig 
5 

ht  of  Ab 
7-5 

utment  t 
10 

o  Spring 
20 

in*. 
30 

Feet. 
4 

Feet. 
3-7 

Feet. 
4.2 

Feet. 
4-3 

Feet. 
4.6 

Feet. 
4-7 

12 

5-6 

6.4 

6.9 

7.6 

Feet. 
7-9 

4-5 

3-9 

4.4 

4.6 

4-9 

5 

IS 

6 

7 

7-5 

8.4 

8.8 

5 

4.2 

4.6 

4.8 

5-1 

5-2 

20 

6-5 

7-7 

8.4 

9.6 

10 

6 

4-5 

4-7 

5-2 

5-6 

5-7 

25 

6.9 

8.2 

9.1 

0-5 

ii.  i 

7 

4-7 

5-2 

5-5 

6 

6.1 

30 

7-2 

8.7 

9-7 

1.4 

12 

8 

4.9 

5-5 

5-8 

6.4 

6-5 

35 

7-4 

9.1 

10.2 

1.8 

12.9 

9 

5-i 

5.8 

6.1 

6.7 

6.9 

40 

7.6 

9.4 

10.6 

2.8 

13-6 

10 

5-3 

6 

6.4 

7-i 

7-3 

45 

7.8 

9-7 

ii 

3-4 

14-3 

ii 

5-5 

6.2 

6.6 

7-3 

7.6 

So 

7-9 

10 

11.4 

4 

15 

NOTE.— Abutments  in  Table  are  assumed  to  be  without  counterforts  or  wing- 
walls.  A  sufficient  margin  of  safety  must  be  allowed  beyond  dimensions  here 
given. 

Culverts  for  a  road  having  double  tracks  are  not  necessarily  twice  the 
length  tor  a  single  track. 
For  other  and  full  notes,  tables,  etc.,  see  Trautwine's  Pocket  Book,  pp.  693-710. 


MECHANICAL  CENTRES. 

There  are  four  Mechanical  centres  of  force  in  bodies,  namely,  Centre 
of  Gravity,  Centre  of  Gyration,  Centre  of  Oscillation,  and  Centre  of 
Percussion. 

Centre   of  GS-ravity. 

CENTRE  OP  GRAVITY  of  a  body,  or  any  system  of  bodies  rigidly  con- 
nected together,  is  point  about  which,  if  suspended,  all  parts  will  be  in 
equilibrium. 

A  body  or  system  of  bodies,  suspended  at  a  point  out  of  centre  of  gravity, 
will  rest  with  its  centre  of  gravity  vertical  under  point  of  suspension. 

A  body  or  system  of  bodies,  suspended  at  a  point  out  of  centre  of  gravity, 
and  successively  suspended  at  two  or  more  such  points,  the  vertical  lines 
through  these  points  of  suspension  will  intersect  each  other  at  centre  of 
gravity  of  body  or  bodies. 

Centre  of  gravity  of  a  body  is  not  always  within  the  body  itself. 

If  centres  of  gravity  of  two  bodies,  as  B  C,  be  connected  by  a  line,  dis- 
tances of  B  and  C  from  their  common  centre  of  gravity,  c,  is  inversely  as 
the  weights  of  the  bodies.  Thus,  B  :  C : :  C  c :  c  B. 

To  Ascertain  Centre  of  Gravity  of  any  Plane  Figure  Mechanically. 

Suspend  the  figure  by  any  point  near  its  edge,  and  mark  on  it  direction 
of  a  plumb-line  hung  from  that  point ;  then  suspend  it  from  some  other 
point,  and  again  mark  direction  of  plumb-line.  Then  centre  of  gravity  of 
surface  will  be  at  point  of  intersection  of  the  two  marks  of  plumb-line. 

3E* 


606  MECHANICAL   CENTRES. — GRAVITY. 

Centre  of  gravity  of  parallel-sided  objects  may  readily  be  found  in  this 
way.  For  instance,  to  ascertain  centre  of  gravity  of  an  arch  of  a  bridge, 
draw  elevation  upon  paper  to  a  scale,  cut  out  figure,  and  proceed  with  it  as 
above  directed,  in  order  to  find  position  of  centre  of  gravity  in  elevation  of 
the  model.  In  actual  arch,  centre  of  gravity  will  have  same  relative  position 
as  in  paper  model. 

In  regular  figures  or  solids,  centre  of  gravity  is  same  as  their  geometrical 
centres. 

Line. 

Circular  Arc.     —=-  =.  distance  from  centre,  r  representing  radius,  c  chord,  and  I 

length  of  arc. 

Surfaces. 

Square,  Rectangle,  Rhombus,  Rhoinboid,  Gnomon,  Cube,  Regular  Polygon, 
Circle,  Sphere,  Spheroid  or  Ellipsoid,  Spheroidal  Zone,  Cylinder,  Circular 
Ring,  Cylindrical  Ring,  Link,  Helix,  Plain  Spiral,  Spindle,  all  Regular  Fig- 
ures, and  Middle  Frusta  of  all  Spheroids,  Spindles,  etc. 

The  centre  of  gravity  of  the  surfaces  of  these  figures  is  in  their  geometri- 
cal centre. 

Triangle. — On  a  line  drawn  from  any  angle  to  the  middle  of  opposite  side, 
at  two  thirds  of  the  distance  from  angle. 

Trapezium. — Draw  two  diagonals,  and  ascertain  centres  of  gravity  of  each 
of  four  triangles  thus  formed  >  join  each  opposite  pair  of  these  centres,  and  it 
is  at  intersection  of  the  lines. 

Trapezoid.      I  rT_L^  )  X  —  =  distance  from  B  on  a  line  joining  middle  of  two 
parallel  sides  Eb,m  representing  middle  line. 
Circular  Arc.     -y-  =  distance  from  centre  of  circle. 

Sector  of  a  Circle.    ,4244  r  =  distance  from  centre  of  circle,  c  representing  chord. 
Semicircle.    .4244  r  =  distance  from  centre. 

Semi-semicircle.  .4244  r  =  distance  from  both  base  and  height  and  at  their  inter- 
section. 

Segment  of  a  Circle.     =  distance  from  centre,  a  representing  area  of  segment. 

Sector  of  a  Circular  Ring.  —  X  "  ^  X  2_  t,2  =  distance  from  centre  of 
arcs,  r  and  r'  representing  the  radii. 

ILLUSTRATION.— Radii  of  surfaces  of  a  dome  are  5  and  3.5  feet,  and  angle  «)  at 
centre  =  130°. 

4       sin.  65°       125-42.875  _  4       .9063       82.125 

—  X  ~  X —  =  —  X  — 2o~  X ~ —  =  3-437eec- 

3      arc  130°        25  — 12.25         3      2.2609        I2-75 

Hemisphere,  Spherical  Segment,  and  Spherical  Zone,  At  centre  of  their 
heights. 

Circular  Zone.— Ascertain  centres  of  gravity  of  trapezoid  and  segments 
comprising  zone ;  draw  a  line  (equally  dividing  zone)  perpendicular  to 
chords;  connect  centres  of  segments  by  a  line  cutting  perpendicular  to 
chords. 

Then  centre  of  gravity  of  figure  will  be  on  perpendicular,  toward  lesser 
chord,  at  such  proportionate  distance  of  difference  between  centres  of  gravity 
of  trapezoid  and  line  connecting  centres  of  segments,  as  area  of  segments 
bears  to  area  of  trapezoid. 


MECHANICAL    CENTRES.  -  GRAVITY.  6O/ 

Prism  and  Wedge.—  When  end  is  a  Parallelogram,  in  their  geometrical 
centres  ;  when  the  end  is  a  Triangle,  Trapezium,  etc.,  it  is  in  middle  of  its 
length,  at  same  distance  from  base,  as  that  of  triangle  or  trapezoid  of  which 
it  is  a  section. 

Parabola  in  its  axis  =  .6  distance  from  vertex. 

Prismoid.—At  same  distance  from  its  base  as  that  of  the  trapezoid  or 
trapezium,  which  is  a  section  of  it. 

Lum.—On  a  line  connecting  centres  of  gravity  of  arcs  at  a  proportionate 
point  to  respective  areas  of  arcs. 

Co-ordinates.      \r-\-r'  --  ~r~>)  ~  —  z> 

* 


Solids. 

Cube,  Parallelopipedon,  Hexahedron,  Octahedron,  Dodecahedron,  Icosahe-* 
dron,  Cylinder,  Sphere,  Right  Spherical  Zone,  Spheroid  or  Ellipsoid,  Cylin- 
drical Ring,  Link,  Spindle,  all  Regular  Bodies,  and  Middle  Frusta  of  all 
Spheroids  and  Spindles,  etc.  Centre  of  gravity  of  these  figures  is  in  their 
geometrical  centre. 

Tetrahedron.—  In  common  centre  of  centres  of  gravity  of  the  triangles  made  by  a 
section  through  centre  of  each  side  of  the  figures. 

Cone  and  Pyramid.  .  25  of  line  joining  vertex  and  centre  of  gravity  of  base  =  dis- 
tance from  base. 

/r_j_r/\2_j_2  r2  t 

Frustum  of  a  Cone  or  Pyramid.  '  -  ,  X  -  h  =  distance  from  centre 

of  lesser  end,  r  and  r',  in  a  cone  representing  radii,  and  in  a  pyramid  sides,  and  A 
height. 

Cone,  Frustum  of  a  Cone,  Pyramid,  Frustum  of  a  Pyramid,  and  Ungula.  — 
At  same  distance  from  base  as  in  that  of  triangle,  parallelogram,  or  semicir- 
cle, which  is  a  right  section  of  them. 

Hemisphere.    .  375  r  =  distance  from  centre. 

Spherical  Segment.    3.  1416  vs2  (**——)  -*-  v  =  distance  from  centre,  vs  repre- 

senting versed  sine,  and  v  volume  of  segment.     I  —  r~3  .]  X  h  =  distance  from 

\i2r-  4  hf 

vertex. 

Spherical  Sector.    .75  (r  —  .  5  h)  =  d  istance  from  centre.     £jjt3_  —  distance 

8 
from  vertex. 

Spirals.  —  Plane,  in  its  geometrical  centre.  Conical,  at  a  distance  from  the 
base,  .25  of  line  joining  vertex  and  centre  of  gravity  of  base. 

r2  _  r'2 

Frustum  of  a  Circular  Spindle.     —  j-  —  =p-  =  distance  from  centre  of  spindle, 

h  representing  distance  between  two  bases,  D  distance  of  centre  of  spindle  from  centre 
of  circle,  and  z  generating  arc,  expressed  in  units  of  radius. 

r2 

Segment  of  a  Circular  Spindle.     —  -  -  —  —  =  distance  from  centre  of  spindle. 
2  (h  —  U.  z) 

Semi-spheroids.  —  Prolate.    .375  a.  —  Oblate.    .375  a  =  distance  from  centre. 
Semi-spheroid  or  Ellipsoid  and  its  Segment.—  See  HaswelVs  Mensuration,  pages 
281  and  282. 

Frusta  of  Spheroids  or  Ellipsoids.    Prolate.    .75  —       -^—^  =  distance  from 

centre  of  spheroid,  a  representing  semi-transverse  diameter  in  a  prolate  frustum,  and 
semi-conjugate  in  an  oblate  frustum. 


608  MECHANICAL    CENTRES.  —  GRAVITY. 


Segments  of  Spheroids.  —Prolate.    .  75  •  —Oblate.    ,  75  —  dietanct 

from  centre  of  spheroid,  d  and  d'  representing  distances  of  base  of  segments  froih 
centre  of  spheroid. 


Any  Frustum.     .  75     "*"  ^^-d2      =  distance  ^rom  centre  °f  *Phe~ 

roid,  d  and  d'  representing  distances  of  base  and  end  of  segments  from  centre  of  the 
spheroid. 

Segment  of  an  Elliptic  Spindle  at  two  thirds  of  height  from  vertex. 
Paraboloid  of  Revolution,  at  two  thirds  of  height  from  vertex. 
Segment  of  a  Hyperbolic  Spindle,  at  75  of  height  from  vertex. 

2  r2  -\-  r        h 
Frustum  of  Paraboloid  of  Revolution.     —  2_T  ,  X  -  =  distance  from  base^  r  and 

r'  representing  radii  of  base  and  vertex. 
Segment  of  Paraboloid  of  Revolution,  at  two  thirds  of  height  from  vertex. 

Segments  of  a  Circular  and  a  Parabolic  Spindle.—  See  HaswelVs  Mensuration, 
pages  192  and  199. 

Parabola.    .4  of  height  =  distance  from  base. 

Hyperboloid  of  Revolution.     \     ,        X  h  =  distance  from  vertex,  b  representing 

o  b  -j-  4  h> 
diameter  of  base. 


Frustum  of  Hyperboloid  of  Revolution.     .  75  _  =  distance 

from  centre  of  base,  a  representing  semi-transverse  axis,  or  distance  from  centre  of 
curve  to  vertex  of  figure  ;  d  and  d'  distances  from  centre  of  curve  to  centre  of  lesser 
and  greater  diameter  of  frustum. 

Segment  of  Hyperboloid  of  Revolution.     ^    J]3     X  h  •=.  distance  from  vertex. 

Of  Two  Bodies.  =  distance  from  V  or  volume  or  area  of  larger  body,  d  rep- 

resenting distance  between  centres  of  gravity  of  bodies,  and  v  volume  or  area  of  less 
body. 

Cycloid.  —  .833  of  radius  of  generating  circle  =  distance  from  centre  of 
chord  of  curve. 

Any  Plane  Figure.  —  Divide  it  into  triangles,  and  ascertain  .centre  of  grav- 
ity of  each  ;  connect  two  centres  together,  and  ascertain  their  common  cen- 
tre ;  then  connect  this  common  centre  and  centre  of  a  third,  and  ascertain 
the  common  centre,  and  so  on,  connecting  the  last-ascertained  common  centre 
to  another  centre  till  whole  are  included,  and  last  common  centre  will  give 
centre  required. 

Of  an  Irregular  Body  of  Rotation. 

Divide  figure  into  four  or  six  equidistant  divisions  ;  ascertain  volume  of 
each,  their  moments  with  reference  to  first  horizontal  plane  or  base,  and 
then  connect  them  thus  : 

(A  +  4  Ai-f-2  A2-f-4  A3  +  A4)  —  =  V,  A  AI,  etc.,  representing  volume  of  divis- 
ions, and  h  height  of  body  from  base; 

and  (°  A+  .  X  4  A.  +  .  X  .  A,  +  3  X  4  A3  +  4  A4)       ft  =-^  ^ 

A  +  4  Ai-f-2  A2  +  4  A3  +  A4  4 

gravity  from  base. 


MECHANICAL   CENTRES. — GYRATION.  OOQ 

Centre   of   Gfyratioru 

CENTRE  OP  GYRATION  is  that  point  in  any  revolving  body  or  system 
of  bodies  in  which,  if  the  whole  quantity  of  matter  were  collected,  the 
Angular  velocity  would  be  the  same ;  that  is,  the  Momentum  of  the  body 
or  system  of  bodies  is  centred  at  this  point,  and  the  position  of  it  is  a 
mean  proportional  between  the  centres  of  Oscillation  and  Gravity, 

If  a  straight  bar  of  uniform  dimensions  was  struck  at  this  point,  the 
stroke  would  communicate  the  same  angular  velocity  to  the  bar  as  if  the 
whole  bar  was  collected  at  that  point. 

The  A  ngular  velocity  of  a  body  or  system  of  bodies  is  the  motion  of  a  line 
connecting  any  point  and  the  centre  or  axis  of  motion :  it  is  the  same  in  all 
parts  of  the  same  revolving  body. 

In  different  unconnected  bodies,  each  oscillating  about  a  common  centre, 
their  angular  velocity  is  as  tho  velocity  directly,  and  as  the  distance  from 
the  centre  inversely.  Hence,  if  their  velocities  are  as  their  radii,  or  distances 
from  the  axis  of  motion,  their  angular  velocities  will  be  equal. 

When  a  bodv  revolves  on  an  axis,  and  a  force  is  impressed  upon  it  suffi- 
cient to  cause  "it  to  revolve  on  another,  it  will  revolve  on  neither,  but  on  a 
line  in  the  plane  of  the  axes,  dividing  the  angle  which  they  contain ;  so  that 
the  sine  of  each  part  will  be  in  the  inverse  ratio  of  the  angular  velocities 
with  which  the  bodies  would  have  revolved  about  these  axes  separately. 

Weight  of  revolving  body,  multiplied  into  height  due  to  the  velocity  with 
which  centre  of  gyration  moves  in  its  circle,  is  energy  of  body,  or  mechani- 
cal power,  which  must  be  communicated  to  it  to  give  it  that  motion. 

Distance  of  centre  of  gyration  from  axis  ot  motion  is  termed  the  Radius 
of  gyration ;  and  the  moment  of  inertia  is  equal  to  product  of  square  of 
radius  of  gyration  and  mass  or  weight  of  body. 

The  moment  of  inertia  of  a  revolving  body  is  ascertained  exactly  by  as- 
certaining the  moments  of  inertia  of  every  particle  separately,  and  adding 
them  together ;  or,  approximately,  by  adding  together  the  moments  of  the 
small  parts  arrived  at  by  a  subdivision  of  the  body. 

To  Compute  Moment  of  Inertia  of  a  Revolving  Body. 

RULE.— Divide  body  into  small  parts  of  regular  figure.  Multiply  mass 
or  weight  of  each  part  by  square  of  distance  of  its  centre  of  gravity  from 
axis  of  revolution.  The  sum  of  products  is  moment  of  inertia  of  body. 

NOTE.—  The  value  of  moment  of  inertia  obtained  by  this  process  will  be  more 
exact,  the  smaller  and  more  numerous  the  parts  into  which  body  is  divided. 

To  Compute  Radius  of*  GJ-yration  of  a  Revolving  Body 
at>out  its  Axis  of  Revolxition. 

RULE. — Divide  moment  ol  inertia  of  body  by  its  mass,  or  its  weight,  and 
square  root  of  quotient  is  length  of  radius  of  gyration. 

NOTE.— When  the  parts  into  which  body  is  divided  are  equal,  radius  of  gyration 
may  be  determined  by  taking  mean  of  all  squares  of  distances  of  parts  from  axis 
of  revolution,  and  taking  square  root  of  their  sum. 

Or,  VR2  -f-  r*  -4-  2  =  G.     R  and  r  representing  radii. 

EXAMPLE.— A  straight  rod  of  uniform  diameter  and  4  feet  in  length,  weighs  4  Ibs.  • 
what  is  its  inertia,  and  where  is  its  radius  or  centre  of  gyration? 

Each  foot  of  length  weighs  i  lb.,  and  if  divided  into  4  parts,  centre  of  gyration  of 
each  is  respectively  .5,  1.5,  2.5,  and  3.5  feet.  Hence, 

.  =  inertia,  which  -4-  4  =  5. 25,  and  ^/$.  25  =  2. 201 
feet  radius. 


6lO      MECHANICAL  CENTRES. — GYRATION. 

Following  are  distances  of  centres  of  gyration  from  centre  of  motion  in 
various  revolving  bodies : 

Straight,  uniform  Rod  or  Cylinder  or  thin  Rectangular  Plate  revolving  about  one 
end;  length  x  -5773>  and  revolving  about  their  centre;  length  X  .2886. 
The  general  expression  is,  when  revolving  at  any  point  of  its  length, 

/"       )  .     I  and  I'  representing  length  of  the  two  arms. 


Circular  Plane,  revolving  on  its  centre;  radius  of  circle  X  -7071  ;  Circle  Plane,  as 
a  Wheel  or  Disc  of  uniform  Thickness,  revolving  about  one  of  its  diameters  as  an 
axis  ;  radius  X  •  5- 

Solid  Cylinder,  revolving  about  its  axis;  radius  x  -7071. 

Solid  Sphere,  revolving  about  its  diameter  as  an  axis;  radius  X  .6325. 

Thin,  hollow  Sphere,  revolving  about  one  of  its  diameters  as  an  axis;  radius 
X  •  8  164.  Surface  of  sphere  .  86  1  5  r. 

Sphere  and  Solid  Cylinder  (vertical;,  at  a  distance  from  axis  of  revolution  = 
Vl>2  -J-  •  4  r2  for  sphere,  and  V^2-{-  -5  r2  for  cylinder,  I  representing  length  of  connec- 
tion to  centre  of  sphere  and  cylinder. 

Cone,  revolving  about  its  axis;  radius  of  base  X  .5447;  revolving  about  its  ver- 
tex =  Vi2  /i2  4-3  r2-r-2o,  h  representing  height,  and  r  radius  of  base,  revolving 
about  its  base=  -\/2  h2-t-$  r2-r-2o. 

Circular  Ring,  as  Rim  of  a  Fly-wheel  or  Hollow  Cylinder,  revolving  about  its 
diameter  =  VR2  +  r2-:-2,  R  representing  radius  of  periphery,  and  r  of  inner  circle 
of  ring. 

Fly-wheel  =     /  -  -  —       wTi^  4  7  •  -  —  -  ,  W  and  w  representing  weights  of 


rim  and  of  arms  and  hub,  and  I  length  of  arms  from  axis  of  wheel. 

Section  of  Rim.    *  --  \-r2  -\-r  d.    d  representing  depth  and  c  periphery 

of  rim. 

Parallelepiped,  revolving  about  one  end,  distance  from  end=r  */-  —  —  ,  b  rep- 
resenting breadth. 

ILLUSTRATION.  —  In  a  solid  sphere  revolving  about  its  diameter,  diameter  being 
2  feet,  distance  of  centre  of  gyration  is  12  X  -6325  =  7.59  ins. 

To    Compxvte    Elements    of  GS-yration. 

GWjy_  Prtg_n  GWv_  ?rtg_  GWv_ 

rig  ~  Wv    ~  Ptg~  G  t>    ~  Prg~ 

••  ?        =  v.     G  representing  distance  of  centre  of  gyration  from  axis  of  rotation, 

W  weight  of  body,  t  time  power  acts  in  seconds,  v  velocity  in  feet  per  second  acquired 
by  revolving  body  in  that  time,  and  r  distance  of  point  of  application  of  power  from 
axis  of  body,  as  length  of  crank,  etc. 

ILLUSTRATION  r.  —  What  is  distance  of  centre  of  gyration  in  a  fly-wheel,  power 
224  Ibs.,  length  of  crank  7  feet,  time  of  rotation  10  seconds,  weight  of  wheel  5600 
IDS.,  and  velocity  of  it  8  feet  per  second? 

224  X  7  X  io  X  32.  166  _  504  373 


2.—  What  should  be  weight  of  a  fly  wheel  making  12  revolutions  per  minute,  its 
diameter  8  feet,  power  applied  at  2  feet  from  its  axis  84  Ibs.,  time  of  rotation  6  sec 
onds,  and  distance  of  centre  of  gyration  of  wheel  3.5  feet? 

8X3.  1416  X  12  84  X  2  X  6  X  32.  166 

-  —  5.0265  feet  —  velocity.     Then  --  ---    —  -  -  =  1843.2  Ib3 
60  3-5X5-0265 


MECHANICAL    CENTRES.  —  GYRATION.  6  1  I 

When  the  Body  is  a  Compound  one.  RULE.—  Multiply  weight  of  several 
particles  or  bodies  by  squares  of  their  distances  in  feet  from  centre  of  mo- 
tion or  rotation,  and  divide  sum  of  their  products  by  weight  of  entire  mass  ; 
the  square  root  of  quotient  will  give  distance  of  centre  of  gyration  from 
centre  of  motion  or  rotation. 

EXAMPLE.  —  If  two  weights,  of  3  and  4  IDS.  respectively,  be  laid  upon  a  lever  (which 
is  here  assumed  to  be  without  weight)  at  the  respective  distances  of  i  and  2  feet, 
what  is  distance  of  centre  of  gyration  from  centre  of  motion  (the  fulcrum)  ? 


=      =  2.71,  and 

That  is,  a  single  weight  of  7  IDS.,  placed  at  1.64  feet  from  centre  of  motion,  and  re- 
volving in  same  time,  would  have  same  momentum  as  the  two  weights  in  their 
respective  places. 

When  Centre  of  Gravity  is  given.  RULE.  —  Multiply  distance  of  centre  of 
oscillation  from  centre  or  point  of  suspension,  by  distance  of  centre  of  grav- 
ity from  same  point,  and  square  root  of  product  will  give  distance  of  centre 
of  gyration. 

EXAMPLE.  —  Centre  of  oscillation  of  a  body  is  9  feet,  and  that  of  its  gravity  4  feet 
from  centre  ef  rotation  or  point  of  suspension;  at  what  distance  from  this  point  is 
centre  of  gyration  ? 

9  x  4  =  36,  and  ^36  =  6  feet. 

To    Compute    Centre    of  Gryration.   of  a  "Water-  -wheel. 

RULE.  —  Multiply  severally  twice  weight  of  rim,  as  composed  of  buckets, 
shrouding,  etc.,  and  twice  that  of  arms  and  that  of  water  in  the  buckets 
(when  wheel  is  in  operation)  by  square  of  radius  of  wheel  in  feet  ;  divide 
sum  by  twice  sum  of  these  several  weights,  and  square  root  of  quotient  will 
give  distance  in  feet. 

EXAMPLE.  —In  a  wheel  20  feet  in  diameter,  weight  of  rim  is  3  tons,  weight  of 
arms  2  tons,  and  weight  of  water  in  buckets  i  ton;  what  is  distance  of  centre  of 
gyration  from  centre  of  wheel  ? 

Rim        =3  tonsx  io2  X  2=  600  3-{-2-f-i  X  2  =  12  sum  of  weights. 

Buckets  =  2  tons  X  102  x  2  =  400 

Water    =  iton   X  io2        =  100  TT 

—  Hence 


GENERAL  FORMULAS.—?  representing  power,  H  Worses'  power,  F  force  applied  to 
rotate  body  in  Ibs.,  M  mass  of  revolving  body  in  Ibs.,  r  radius  upon  which  F  acts  in 
feet,  d  distance  from  axis  of  motion  to  centre  of  gyration  in  feet,  t  time  force  is  ap- 
plied in  seconds,  n  number  of  revolutions  in  time  t,  x  angular  velocity,  or  number  of 

revolutions  per  minute  at  end  of  time  t,  and  G  =  32 ''        — . 

/4prn_f        2pr*x  Mod2  _  M^n d*  _  2.56<2Fr_ 

V      G        =f|         6oG    ~    '       153.5  tr~     '      2.56<2F~  Md2 

~~Wd2~~=X]         ~x^~d2~==^'t  244*    ":?;          i34ioot  =  H< 

ILLUSTRATION.— Rim  of  a  fly-wheel  weighing  7000  Ibs.  has  radii  of  6.5  and  5.75 
feet;  what  is  its  centre  of  gyration,  and  what  force  must  be  applied  to  it  2  feet 
from  axis  of  motion  to  give  it  an  angular  velocity  of  130  revolutions  per  minute  in 
40  seconds?  how  many  revolutions  will  it  make  in  40  seconds?  and  what  is  its 
power  ? 

i3Q2  X  7000  X  6.  i42  _  4  459  862  680 
134  zoo  X  40  5  364  ooo 


612       MECHANICAL   CENTRES. — OSCILLATION,  ETC. 

Centres   of  Oscillation,  and   3?ercnssion. 

CENTRE  OF  OSCILLATION  of  a  body,  or  a  system  of  bodies,  is  that  point 
in  axis  of  vibration  of  a  vibrating  body  in  which,  if,  as  an  equivalent 
condition,  the  whole  matter  of  vibrating  body  was  concentrated,  it  would 
continue  to  vibrate  in  same  time.  It  is  resultant  point  of  whole  vibrat- 
ing energy,  or  of  action  of  gravity  in  producing  oscillation. 

As  particles  of  a  body  further  from  centre  of  its  suspension  have  greater 
velocity  of  vibration  than  those  nearer  to  it,  it  is  apparent  that  centre  of 
oscillation  is  further  from  its  centre  than  centre  of  gravity  is  from  axis  of 
suspension,  but  it  is  situated  in  centre  of  a  line  drawn  from  axis  of  a  body 
through  its  centre  of  gravity.  It  further  differs  from  centre  of  gyration 
in  this,  that  while  motion  of  oscillation  is  produced  by  gravity  of  a  body, 
that  of  gyration  is  caused  by  some  other  force  acting  at  one  place  only. 

Radius  of  oscillation,  or  distance  of  centre  of  oscillation  from  axis  of  sus- 
pension, is  a  third  proportional,  to  distance  of  centre  of  gravity  from  axis 
of  suspension  and  radius  of  gyration. 

CENTRE  OF  PERCUSSION  of  a  body,  or  a  system  of  bodies,  revolving 
about  a  point  or  axis,  is  that  point  at  which,  if  resisted  by  an  immov- 
able obstacle,  all  the  motion  of  the  body,  or  system  of  bodies,  would  be 
destroyed,  and  without  impulse  on  the  point  of  suspension.  It  is  also 
that  point  which  would  strike  any  obstacle  with  greatest  effect,  and 
from  this  property  it  has  been  termed  percussion. 

Centres  of  Oscillation  and  Percussion  are  in  same  point. — If  a  blow  is 
struck  by  a  body  oscillating  or  revolving  about  a  fixed  centre,  percussive 
action  is  same  as  if  its  entire  mass  was  concentrated  at  centre  of  oscillation. 
That  is,  centre  of  percussion  is  identical  with  centre  of  oscillation,  and  its 
position  is  ascertained  by  same  rules  as  for  centre  of  oscillation.  If  an  ex- 
ternal body  is  struck  so  that  the  mean  line  of  its  resistance  passes  through 
centre  of  percussion,  then  entire  force  of  percussion  is  transmitted  directly 
to  the  external  body ;  on  the  contrary,  if  a  revolving  body  is  struck  at  its 
centre  of  percussion,  its  motion  will  be  absolutely  destroyed,  so  that  the  body 
will  not  incline  either  way. 

As  in  bodies  at  rest,  the  entire  weight  may  be  considered  as  collected  in 
centre  of  gravity ;  so  in  bodies  in  vibration,  the  entire  force  may  be  consid- 
ered as  concentrated  in  centre  of  oscillation ;  and  in  bodies  in  motion,  the 
whole  force  may  be  considered  as  concentrated  in  centre  of  percussion. 

If  centre  of  oscillation  is  made  point  of  suspension,  point  of  suspension 
will  become  centre  of  oscillation. 

Angle  of  Oscillation  or  Percussion  is  determined  by  angle  delineated  by 
vertical  plane  of  body  in  vibration,  in  plane  of  motion  of  body. 

Velocity  of  a  Body  in  Oscillation  or  Percussion  through  its  vertical  plane. 
is  equal  to  that  acquired  by  a  body  freely  falling  through  a  vertical  line 
equal  in  height  to  versed  sine  of  the  arc. 

To    Compute   Centre   of*  Oscillation   or   Percussion   of*  a 
Body   of*  Uniform. Density   and   Figure. 

RULE.— Multiply  weight  of  body  by  distance  of  its  centre  of  gravity  from 
point  of  suspension ;  multiply  also  weight  of  body  by  square  of  its  length, 
and  divide  product  by  3. 

Divide  this  last  quotient  by  product  of  weight  of  body  and  distance  of 
its  centre  of  gravity,  and  quotient  is  distance  of  centre  from  point  of  sus- 
pension. 


MECHANICAL   CENTRES.  —  OSCILLATION,  ETC.        613 

Or,  --  •-  W  x  9  =  distance  from  axis.  Or,  square  radius  of  gyration  of  body 
and  divide  by  distance  of  centre  of  gravity  from  axis  of  suspension. 

EXAMPLE.—  Where  is  centre  of  oscillation  in  a  rod  9  feet  in  length  from  its  point 
Of  suspension,  and  weighing  9  Ibs.  ? 

9  X  -  =  40.  5  =  product  of  weight  and  its  centre  of  gravity  ;  2  —  ^-  =  243  =  quo- 
tient of  product  of  weight  of  body  and  square  of  its  length  -r-  3  ;  ^—^-  =  6  feet. 

When  Point  of  Suspension  is  not  at  End  of  Rod.  RULE.  —  To  cube  of 
distance  of  point  of  suspension  from  top  of  rod  or  bar,  add  cube  of  its  dis- 
tance from  lower  end,  and  multiply  sum  by  2. 

Divide  product  by  three  times  difference  of  squares  of  these  distances,  and 
quotient  is  distance  of  point  of  oscillation  from  point  of  suspension. 

EXAMPLE.—  A  homogeneous  rod  of  uniform  dimensions,  6  feet  In  length,  is  sus- 
pended 1.5  feet  from  its  upper  end;  what  is  distance  of  point  of  oscillation  from 
that  of  suspension  ? 


Centres    of  Oscillation,    and.    Fercnssion    in    Bodies    of* 
"Various   Figures. 

When  Axis  of  Motion  is  in  Vertex  of  Figure,  and  when  Oscillation  or  Motion 
is  Facewise. 

Right  Line,  or  any  figure  of  uniform  shape  and  density  =  .661 
Isosceles  Triangle  =  .75  h.  Circle  =  1.25  r. 

Parabola  =  .  714  h.  Cone  =  .8h. 

When  Axis  of  Motion  is  in  Centre  of  Body.     Wheel  =  .75  radius. 

When  Oscillation  or  Motion  is  Sidemse.  Right  Line,  or  any  figure  of  uni- 
form shape  and  density  =  66  I  Rectangle,  suspended  at  one  angle  =  .66  of  di- 
agonal 

Parabola,  if  suspended  by  its  vertex  =  .7  14  of  axis  -{-.33  parameter;  if  suspended 
by  middle  of  its  base  =  .  57  of  axis  -|-  .  5  parameter. 

Sector  of  a  Circle  =  -  -  -  ,  c  representing  chord  of  arc,  and  r  radius  of  base. 


2  r2 
Sphere  =  -{-  r  -{-  c,  c  representing  length  of  cord  by  which  it  is  suspended. 

To   Ascertain   Centres   of  Oscillation   and  IPercussion 
experimentally. 

Suspend  body  very  freely  from  a  fixed  point,  and  make  it  vibrate  in  small  arcs, 
.noting  number  of  vibrations  it  makes  in  a  minute,  and  let  number  made  in  a  min- 
ute be  represented  by  n;  then  will  distance  of  centre  of  oscillation  from  point  of 

140850 
suspension  be  =       2     =  ins. 

For  length  of  a  pendulum  vibrating  seconds,  or  60  times  in  a  minute,  being 
39.125  ins.,  and  lengths  of  pendulums  being  reciprocally  as  the  squares  of  number 

of  vibrations  made  in  same  time,  therefore  n2  :  6o2  :  :  39.  125  :  --  39^5  __  **°  ^°f 

being  length  of  pendulum  which  vibrates  n  times  in  a  minute,  or  distance  of  centre 
of  oscillation  below  axis  of  motion. 

3F 


6  14      MECHANICAL  CENTRES.  —  MECHANICS. 

To   Compxite   Centres   of*  Oscillation    or    3?ercu.ssion    of  a 

System    of*  Particles    or    Bodies. 

RULE.  —  Multiply  weight  of  each  particle  or  body  by  square  of  its  distance 
from  point  of  suspension,  and  divide  sum  of  their  products  by  sum  of  weights, 
multiplied  by  distance  of  centre  of  gravity  from  point  of  suspension,  and 
quotient  will  give  centre  required,  measured  from  point  of  suspension. 

W  d2  -f  W  d'z 

Or>    w    _LW    ~  =  distance  of  centre. 
w  9  -f-  W  g 

EXAMPLE  i.  —  Length  of  a  suspended  rod  being  20  feet,  and  weight  of  a  foot  in  length 
of  it  equal  100  pz.,  has  a  ball  attached  at  under  end  weighing  100  oz.  ;  at  what  point 
of  rod  from  point  of  suspension  is  centre  of  percussion  ? 

100  x  20  =  2000  =  weight  of  rod  ;  2000  X  —  =  20000  —  momentum  of  rod,  or  prod- 
uct of  its  weight,  and  distance  of  its  centre  of  gravity  ;  25^2  —  £°_  =266666.66  = 
force  of  rod  ;  1000  X  2o2  =  400  ooo  =  force  of  ball. 

266  666.  66  -4-  400  ooo 
Then  —  -  j-^4  —  —  =  *6.66feet. 

20  000  -f-  20  000 

2.—  Assume  a  rod  12  feet  in  length,  and  weighing  2  Ibs.  for  each  foot  of  its  length, 
with  2  balls  of  3  Ibs.  each—  one  fixed  6  feet  from  the  point  of  suspension,  and  the 
other  at  the  end  of  the  rod;  what  is  the  distance  between  the  points  of  suspension 
and  percussion  ? 

^X2X^-  =  ^  =  momentumofrod  24  X  »»'      g45g 

3X6  =   18=         "         ofistball 


3X12         =  36=         "         of  zd  ball.       3X   62  =  3X36  =  io8  —  "ofittbalL 
~i98~*tm  of  moments.  3X  i22=  3  X  144  =432  =  "  ofzd  ball. 

Then  1692  -f-  198  =  8.  545  feet.  1692  sum  of  forces. 


MECHANICS. 

MECHANICS  is  the  science  which  treats  of  and  investigates  effects  of 
forces,  motion  and  resistance  of  material  bodies,  and  of  equilibrium : 
it  is  divided  into  two  parts — STATICS  and  DYNAMICS. 

STATICS  treats  of  equilibrium  of  forces  or  bodies  at  rest.  DYNAMICS 
of  forces  that  produce  motion,  or  bodies  in  motion. 

These  bodies  are  further  divided  into  Mechanics  of  Solid,  Fluid,  and  Aeri- 
form bodies ;  hence  the  following  combinations : 

1.  Statics  of  Solid  Bodies,  or  Geostatics. 

2.  Dynamics  of  Solid  Bodies,  or  Geodynamics. 

3.  Statics  of  Fluids,  or  Hydrostatics. 

4.  Dynamics  of  Fluids,  or  Hydrodynamics. 

5.  Statics  of  A  eriform  Bodies,  or  A  erostatics. 

6.  Dynamics  of  Aeriform  Bodies,  Pneumatics  or  Aerodynamics. 
Forces  are  various,  and  are  divided  into  moving  forces  or  resistances ;  as 

Gravity,  Heat  or  Caloric,         Inertia, 

Muscular,  Magnetism,  Cohesion, 

Elasticity  and  Contractility,        Percussion,  A  dhesion, 

Central,  Expansion,  and  Explosion. 

Couple. — Two  forces  of  equal  magnitude  applied  to  or  operating  upon 
same  body  in  parallel  and  opposite  directions,  but  not  in  same  line  of  action, 
constitute  a  couple,  and  its  force  is  sum  or  magnitude  of  the  two  equal  forces. 
Moment. — Quantity  of  motion  in  a  moving  body,  which  is  always  equal 
to  product  of  quantity  of  matter  and  its  velocity. 

When  velocities  of  two  moving  bodies  are  inversely  as  their  quantities  of 
matter,  their  momenta  are  equal. 


MECHANICS. — STATICS. 


6i5 


Fig.  i.  o 


STATICS. 
Composition   and.    Resolution   of  Forces. 

When  two  forces  act  upon  a  body  in  same  or  in  an  opposite  direc- 
tion, effect  is  same  as  if  only  one  force  acted  upon  it,  being  sum  or 
difference  of  the  forces.  Hence,  when  a  body  is  drawn  or  projected  in 
directions  immediately  opposite,  by  two  or  more  unequal  forces,  it  is  affected 
as  if  it  were  drawn  or  projected  by  a  single  force  equal  to  difference  between 
the  two  or  more  forces,  and  acting  in  direction  of  greater  force. 

This  single  force,  derived  from  the  combined  action  of  two  or  more  forces, 
is  their  Resultant. 

The  process  by  which  the  resultant  of  two  or  more  forces,  or  a  single 
force  equivalent  in  its  effect  to  two  or  more  forces,  is  determined,  is  termed 
the  Composition  of  Forces,  and  the  inverse  operation ;  or,  when  combined 
effects  of  two  or  more  forces  are  equivalent  to  that  of  a  single  given  force, 
the  process  by  which  they  are  determined  is  termed  the  Decomposition  or 
Resolution  of  Forces.  Two  or  more  forces  which  are  equivalent  to  a  single 
force  are  termed  Components. 

When  two  forces  act  on  same  point  their  intensities  are  represented  by  sides 
of  a  parallelogram,  and  their  combined  effect  will  be  equivalent  to  that  of  a 
single  force  acting  on  point  in  direction  of  diagonal  of  parallelogram,  the 
intensity  of  which  is  proportional  to  diagonal. 

ILLUSTRATION.— Attach  three  cords  to  a  fixed  point,  c,  Fig.  i ;  let  c  a  and  c  6  pass 
over  fixed  rollers,  and  suspend  weights  A  and  B  therefrom. 

Point  c  will  be  drawn  by  the  forces  A  and  B  in  directions  a  c 
and  6  c.  Now,  in  order  to  ascertain  which  single  force,  P,  would 
produce  the  same  effect  upon  it,  set  off  the  distances  c  w  and 
c  n  on  the  cords  in  the  same  proportion  of  length  as  weights 
of  A  and  B ;  that  is,  so  that  cm:  en::  A  :  B ;  then  draw  par- 
allelogram cm  on  and  diagonal  o  c,  and  it  will  represent  a  sin- 
gle  force,  P,  acting  in  its  direction,  and  having  same  ratio  to 
weights  A  or  B  as  it  has  to  sides  c  m  or  c  n  of  parallelogram. 
Consequently,  it  will  produce  same  effect  on  point  c  as  com- 
bined actions  of  A  and  B. 

A  parallelogram,  constructed  from  lateral  forces,  and  diagonal  of  which  is 
jg  2       a  mean  force,  is  termed  a  Parallelogram  of  Forces. 

ILLUSTRATION.  — Assume  a  weight,  W,  Fig.  2,  to  be 
suspended  from  a;  then,  if  any  distance,  a  o,  is  set 
/  ^^^v.          off  m  numerical  value  upon  the  vertical  line,  aW, 
and  the  parallelogram,  o  r  a  s,  is  completed,  a  s  and 
•  o  IT    °  ri  measured  upon  the  scale,  a  o,  will  represent 

strain  upon  a  c  and  a  e  in  same  proportion  that  a  o 
W  bears  to  weight  W. 

If  several  forces  act  upon  same  point,  and  their  intensities  taken  in  order 
are  represented  by  sides  of  a  polygon,  except  one,  a  single  force  proportioned 
to  and  acting  in  direction  of  that  one  side  will  be  their  resultant. 

To  Resolve  a  Single  Force  into  a  Pair  of  Forces.— Figs.  3  and  4. 
The  ends  of  a  cord,  Fig.  3,  are  led  over  two  points,  a  and  6,  and  in  centre  of 
cord  at  c  a  weight  of  4  Ibs.  is  suspended.    If  distances  a  c,  b  c,  are  each  i  foot,  dis- 
tance a  b  should  be  18  ins.  Fig.  4. 
When  cord  is  in  this  posi- 
tion, weight  at  c  draws  upon 
c  a  and  c  6  with  a  force  of 
3  Ibs. ;  hence  c  of  4  Ibs.  is 
equal  to  two  forces  of  3  Ibs. 
each  in  direction  of  a  c  and  b  c. 
Apply  ends  of  cord  to  «/,  Fig.  4,  distance  being  22  ins.,  then  the  strain  on  ce,  c£ 
are  each  5  Ibs. ;  hence  one  force  of  4  Ibs.  is  equal  to  two  of  5  Ibs.  each- 


B 


6 1 6  MECHANICS. — STATICS. — DYNAMICS. 

Equilibrium    of  ITorees. 

Two  bodies  which  act  directly  against  each  other  in  same  line  are  in  equi- 
librium when  their  quantities  of  motion  are  equal;  that  is,  when  product  of 
mass  of  one,  into  velocity  with  which  it  moves  or  tends  to  move,  is  equal  to 
product  of  mass  of  other,  into  its  actual  or  virtual  *  velocity. 

When  the  velocities  with  which  bodies  are  moved  are  same,  their  forces 
are  proportional  to  their  masses  or  quantities  of  matter.  Hence,  when  equal 
masses  are  in  motion,  their  forces  are  proportional  to  their  velocities. 

Relative  magnitudes  and  directions  of  any  two  forces  may  be  represented 
by  two  right  lines,  which  shall  bear  to  each  other  the  relations  of  the  forces, 
PI  and  which  shall  be  inclined  to  each  other  in  an  angle 

-     <S       equal  to  that  made  by  direction  of  the  forces. 

ILLUSTRATION.— Assume  a  body,  W,  to  weigh  150  Ibs.,  and 
resting  upon  a  smooth  surface,  to  be  drawn  by  two  forces,  a 
and  6,  Fig.  5.  =  24  and  30  Ibs.,  which  make  with  each  other 
an  angle,  a  W  b  —  105°,  in  which  direction  and  with  what 
acceleration  will  motion  occur? 

<s_  a  W  &  =  105°,  and  cos.  180°  —  105°  =  cos.  75°,  mean 
force. 


P = Vso2 -f  24  2  —  2Xjo X  24  cos.  75°  =  VQOO -|- 576  -  1440  cos.  7^ 
=  ^1476 — (1440  X  258  82j  =  Vuo3.3  =  33.21  Z&«. 

The  acceleration  is  *g  =  33^  X  32-166  =  7.^5  feet. 
w  150 

Angle  of  Repose  is  greatest  inclination  of  a  plane  to  horizon  at  which  a 
body  will  remain  in  equilibrium  upon  it. 

Hence  greatest  angle  of  obliquity  of  pressure  between  two  planes,  consist- 
ent with  stability,  is  the  angle  tangent  of  which  is  equal  to  coefficient  of 
friction  of  the  two  planes. 

Inertia  is  resistance  which  a  body  at  rest  offers  to  an  external  power  to 
be  put  in  motion  or  to  change  its  velocity  or  direction  when  in  motion. 

To   Compute   Tiiertia  of  a   Revolving   Body. 
Divide  ft  into  small  parts  of  a  regular  figure,  multiply  weight  of  each  part 
by  square  of  its  distance  of  its  centre  of  gravity  from  axis  of  revolution, 
and  sum  of  products  will  give  moment  of  inertia  of  body. 

DYNAMICS. 

DYNAMICS  is  the  investigation  of  the  laws  of  Motion  of  Solid  Bodies, 
or  of  Matter,  Force,  Velocity,  Space,  and  Time.^ 

Mass  of  a  body  is  the  quantity  of  matter  of  which  it  is  composed. 

Force  is  divided  into  Motive,  Accelerative,  or  Retardative. 

Motive  Force,  or  Momentum,  of  a  body,  is  the  product  of  its  mass  and 
its  velocity,  and  is  its  quantity  of  motion.  This  force  can,  therefore,  be 
ascertained  and  compared  in  any  number  of  bodies  when  these  two 
quantities  are  known,  f 

Accelerative  or  Retardative  Force  is  that  which  respects  velocity  of 
motion  only,  accelerating  or  retarding  it ;  and  it  is  denoted  by  quotient 
of  motive  force,  divided  by  mass  or  weight  of  body.  Thus,  if  a  body 

•  Virtual  velocity  is  the  velocity  which  a  body  in  equilibrium  would  acquire  were  the  equilibrium 
to  be  disturbed. 

t  It  is  compared,  because  it  is  not  referable  to  any  standard,  as  a  ton,  pound,  etc.  Thus,  suppose 
a  cannon-ball  weighing  15  Ibs.,  projected  with  a  velocity  of  1500  feet  per  second,  strike  a  resisting 
body,  its  momentum,  according  to  the  above  rule,  would  be  15  X  1500  =  22  500  j  not  pounds,  for  weight 
is  a  pressure  with  which  it  cannot  be  compared. 


MECHANICS.  —  DYNAMICS.  6  1  J 

of  5  Ibs.  is  impelled  by  a  force  of  40  Ibs.,  accelerating  force  is  8  Ibs.  5 
but  if  a  force  of  40  Ibs.  act  upon  a  body  of  10  Ibs.,  accelerating  force 
is  only  4  Ibs.,  or  half  former,  and  will  produce  only  half  velocity. 

With  equal  masses,  velocities  are  proportional  to  their  forces. 

With  equal  forces,  velocities  are  inversely  as  the  masses. 

With  equal  velocities,  forces  are  proportional  to  the  masses. 

Work  is  product  of  force,  velocity,  and  time. 

Motion.  —  The  succession  of  positions  which  a  body  in  its  motion  pro- 
gressively occupies  forms  a  line  which  is  termed  the  trajectory,  or  path 
of  the  moving  body. 

A  motion  is  Uniform  when  equal  spaces  are  described  by  it  in  equal 
times,  and  Variable  when  this  equality  does  not  occur.  When  spaces 
described  in  equal  times  increase  continuously  with  the  time,  a  variable 
motion  is  termed  accelerated,  when  spaces  decrease,  retarded,  and  when 
equal  spaces  are  described  within  certain  intervals  only,  the  motion  is 
termed  periodic,  and  intervals  periods.  Uniform  motion  is  illustrated 
in  progressive  motion  of  hands  of  a  watch  ;  variable  in  progressive  ve- 
locity of  falling  and  upwardly  projected  bodies  ;  and  periodic  by  oscil- 
lation of  a  pendulum  or  strokes  of  a  piston  of  a  steam-engine. 

Uniform   Motion. 
L     * 


/»,        ,   H55o,  and       =  P;          , 


and 


7, 


P      H  550        .    W  ,     P  t     W  H  550  1  sf     t       W 

7'  —  iand  71—  ;     "'  7'  7'and  -,-=•'      T-  7-  75. 


-  =  H.    P  representing  power  in  effect,  body,  or  momentum,/  force  in  Ibs..  v  and 
55°  ' 

*  velocity  and  space  in  feet  per  second,  t  time  in  seconds,  H  horse-power,  and  W  work 
in  foot-lbs. 

If  two  or  more  bodies,  etc.,  are  compared,  two  or  more  corresponding  letters, 
as  P,  /?,/>',  V,  v,  v',  etc.,  are  employed. 

ILLUSTRATION  i.—  Two  bodies,  one  of  20,  the  other  of  10  Ibs.,  are  impelled  by  same 
momentum,  say  60.  They  move  uniformly,  first  for  8  seconds,  second  for  6;  what 
are  the  spaces  described  by  both? 

60  -i-  20  =  3  =  V,  and  60  -f-  10  =  6  =  v. 

Then  TV  =  3X8  =  24  =  S,  and  £v  =  6x6  =  36  =  *,  spaces  respectively. 

2.  —If  a  power  of  12  800  effects  has  a  velocity  of*  10  feet  per  second,  what  is  its 
force  ?  12  800  -f-  10  =  1280  Ibs. 

"Uniform.   Variable    Motion. 

Space  described  by  a  body  having  uniform  variable  motion  is  represented 
by  sum  or  difference  of  velocity,  and  product  of  acceleration  and  time,  ac- 
cording as  the  motion  is  accelerated  or  retarded. 

ILLUSTRATION  i.  —  A  sphere  rolling  down  an  inclined  plane  with  an  initial  velocity 
of  25  feet,  acquires  in  its  course  an  additional  velocity  at  each  second  of  time  oi  5 
feet;  what  will  be  its  velocity  after  3  seconds? 


2  —A.  locomotive  having  an  initial  velocity  of  30  feet  per  second  is  so  retarded 
that  in  each  second  it  loses  4  feet;  what  is  its  velocity  after  6  seconds? 

30  —  4X6  =  6  feet. 
3** 


6  1  8  MECHANICS.  —  DYNAMICS. 

"Uniform    Motion.   Accelerated.. 

In  this  motion,  velocity  acquired  at  end  of  any  time  whatever  is  equal  to  _ 
uct  of  accelerating  force  into  time,  and  space  described  is  equal  to  product  of  half 
accelerating  force  into  square  of  time,  or  half  product  of  velocity  and  time  of  ac- 
quiring the  velocity. 

Spaces  described  in  successive  seconds  of  time  are  as  the  odd  numbers,  i,  3,  5,  7, 
o,  etc. 

Gravity  is  a  constant  force,  and  its  effect  upon  a  body  falling  freely  in  a  vertical 
line  is  represented  by  g,  and  the  motion  of  such  body  is  uniformly  accelerated. 

The  following  theorems  are  applicable  to  aii  cases  of  motion  uniformly  acceler- 
ated by  any  constant  force,  F  : 


When  gravity  acts  alone,  as  when  a  body  falls  in  a  vertical  line,  F  it  omit- 
ted. Thus, 

V2  V  /2  3  V          2  S         V* 

.„«•  =  _=*        ,!  =  ,/.,.=«        j=VT  T  =  7?  =  J7  =  !'- 

t  representing  time  in  seconds,  and  s  velocity  in  feet  per  second. 

If,  instead  of  a  heavy  body  falling  freely,  it  be  projected  vertically  upward 
or  downward  with  a  given  velocity,  p,  then  s  =  tv  qp  .5  g  t2  ;  an  expression 
in  which  —  must  be  taken  when  the  projection  is  upward,  and  +  when  it  is 
downward. 

ILLUSTRATION  i.  —  If  a  body  in  10  seconds  has  acquired  a  velocity  by  uniformly 
accelerated  motion  of  26  feet,  what  is  accelerating  force,  and  what  space  described, 
in  that  time? 

26  -=-10  =  2.6  =  accelerating  force  ;     -—  X  io2  =  1  30  feet  =  space  described. 

2.—  A  body  moving  with  an  acceleration  of  15.625  feet  describes  in  1.5  seconds  a 
8pace  =  '5.625  X(i.S)a  =  I7.578/e^ 

3.—  A  body  propelled  with  an  initial  velocity  of  3  feet,  and  with  an  acceleration 
of  5  feet,  describes  in  7  seconds  a  space  =  3X7  +  5X  —  =  143.  5  feet. 

4.  —  A  body  which  in  180  seconds  changes  its  velocity  from  2.5  to  7.5  feet,  trav- 
erses in  that  time  a  distance  of  2'5*~7'5  x  180  =  900  feet. 

5.—  A  body  which  rolls  up  an  inclined  plane  with  an  initial  velocity  of  40  feet  per 
second,  by  which  it  suffers  a  retardation  of  8  feet,  ascends  only  —  =  5  seconds,  and 

4o2-f-2  X  8=  ioo  feet  in  height,  then  rolls  back,  and  returns,  after  io  seconds,  with 
a  velocity  of  40  feet,  to  its  initial  point;  and  after  12  seconds  arrives  at  a  distance 
of  40  X  12  —  4  X  i22  =  g6feet  below  point,  assuming  plane  to  be  extended  backward. 

Circular   Motion. 


_  ____=  _2  = 

60  t  rn         sprn'     J'  5500  ~~  550  X  60  ~~      ' 

fzprn'  —      2foTn  =  W.    r  representing  radiiis  in  feet,  n  number  of  revolutions 

of  circle  per  minute,  n'  total  revolutions,  f  force  in  Ibs.,  t  time  in  seconds,  and  BP 
horse-power. 


MECHANICS.  —  DYNAMICS.  lQ 

Motion   on   an    Inclined.   Plane. 

To  Ascertain  Conditions  of  Motion  by  Gravity. 
Fig.  6.  b  Assume  A  B,  Fig.  6,  an  inclined  plane,  B  C  its  base, 

~  A  C  its  height,  and  b  a  body  descending  the  plane  ;  from 

dot,  centre  of  gravity  of  body,  draw  b  a  perpendicular 
to  B  C,  representing  pressure  of  b  by  gravity  ;  draw  6  o 
\\  \r         parallel  and  6  r  perpendicular  to  A  B,  and  complete 
_  <**•''       I      parallelogram  ;  then  force  &  a  is  equal  to  both  6  o,  6  r, 
c    of  which  6  r  is  sustained  by  reaction  of  plane,  and 
force  6  o  is  wholly  effective  in  accelerating  motion  of  body. 

Let  this  force  be  represented  byf  and  ba,byg  or  force  of  gravity,  then  by  similar 
triangle,/:  g::bo  :  ba:  AC  :  A  B.  Hence,  ^^L-f. 

Put  A  B  =  ?,  A  C  =  fc  and  ^_  A  B  C  =  a,  then  force  which  produces  motion  on  the 
plane  on/  becomes  g  y  ,  and  g  sin.  a. 

Therefore,  accelerating  force  on  an  inclined  plane  is  constant,  and  equations  of 
motion  will  be  obtained  by  substituting  its  value  of  /for  g  in  equations  i3  2,  and 
3,  page  618. 


If,     111,    J2JL±1,    9  'sin.  a, 


, 
g  sin.  a  ' 


/_i^-  =  «.    a  representing  L  A  B  C. 
V  0  sm-  a 


VFAero  a  Body  is  projected  down  or  up  an  Inclined  Plane,  with  a  given  Ve- 
locity. —  The  distance  which  it  will  be  from  point  of  projection  hi  a  given 
time  will  be  ^ght*  t 


t 
r-  ,  and  —  (2  I  v  ±  g  h  t)  =  s. 

26  21 

ILLUSTRATION  i.—  Length  of  an  inclined  plane  is  100  feet,  and  its  angle  of  inclina- 
tion 60°;  what  is  time  of  a  body  rolling  down  it,  and  velocity  acquired  ? 
sin.  60°  =  .866. 


=  ^'  l8  =  2'68  *eMndS>  and  32'  l6  X  2'68  X  866  =  74.64  fed" 


2.—  If  a  body  is  projected  up  an  inclined  plane,  which  rises  i  in  6,  with  a  velocity 
of  50  feet  per  second,  what  will  be  its  place  and  velocity  at  end  of  6  seconds? 

6  x  50  -^6_^-Xj>!=       2      ^^  bo       and  SQ  _  /       x  6  x  i  \  _ 

2X0  \  o/ 

50  —  32.  16  =  1  7.  84  feet. 

To  effect  an  ascent  up  an  inclined  plane  in  least  time,  its  length,  to  its  height, 
must  be  as  twice  weight  to  power. 

"Work   Accoimnlateci   in    Gloving    Codies. 

Quantity  of  work  stored  in  a  body  in  motion  is  same  as  that  which  would 
be  accumulated  in  it  by  gravity  if  it  fell  from  the  height  due  to  the  velocity. 
Accumulated  work  expressed  in  foot-lbs.  is  equal  to  product  of  height  so 
found  in  feet,  and  weight  of  body  in  Ibs.  Height  due  to  velocity  is  equal 
to  square  of  velocity  divided  by  64.4,  and  work  and  velocity  may  be  de- 
duced directly  from  each  other  by  following  rules  : 

To   Compute    Accumulated.   "Work. 

RULE.  —  Multiply  weight  in  Ibs.  by  square  of  velocity  in  feet  per  second, 
and  divide  by  64.4,  and  quotient  is  accumulated  work  in  foot-lbs. 

Or,  W  =  ^  -  —,     or,  =wxh.      W  representing  work,  w  weight  in  Ibs.,  and 

64.4 
h  height  due  to  velocity  in  feet  per  second. 


62O 


MECHANICS.  —  DYNAMICS. 


"by   ^Percussive    Force. 

If  a  wedge  is  driven  by  strokes  of  a  hammer  or  other  heavy  mass,  effect 
of  percussive  force  is  measured  by  quantity  of  work  accumulated  in  stricken 
body.  This  work  is  computed  by  preceding  rules,  from  weight  of  body 
and  velocity  with  which  a  stroke  is  delivered,  or  directly  from  height  of 
fall,  if  gravity  be  percussive  power. 

Useful  work  done  through  a  wedge  is  equal  to  work  expended  upon  it, 
assuming  that  there  is  no  elastic  or  vibrating  reaction  from  the  stroke,  as  if 
the  work  had  been  exerted  by  a  constant  pressure  equal  to  weight  of  strik- 
ing body,  exerted  through  a  space  equal  to  height  of  fall,  or  height  due  to 
its  final  velocity. 

If  elastic  action  intervenes,  a  portion  of  work  exerted  is  absorbed  in  an 
elastic  stress  to  resisting  body  ;  and  the  elastic  action  may  be,  in  some  cases, 
so  great  as  to  absorb  the  work  expended. 

The  principle  of  action  of  a  blow  on  a  wedge  is  alike  applicable  to  action 
of  the  stroke  of  a  monkey  of  a  pile-driver  upon  a  pile. 

If  there  be  no  elastic  action,  the  work  expended  being  product  of  weight 
of  monkey  by  height  of  its  fall,  is  equal  to  work  performed  in  driving  the 
pile:  that  is,  to  product  of  resistance  to  its  descent  by  depth  through  which 
it  is  driven  by  each  blow  of  monkey. 

ILLUSTRATION  —  If  a  horse  draws  200  Ibs.  out  of  a  mine,  at  a  speed  of  2  miles  per 
hour,  how  many  units  of  work  does  he  perform  in  a  minute,  coefficient  of  friction  .05  ? 


Fig-  7- 


03        '"ft-  ~~ 


=  176  feet  per  minute.    Hence,  176  X  200  +  .05  x  200  =  35  210  units. 

Decomposition   of  Force. 

*  By  parallelogram  of  force  it  is  il- 

lustrated how  a  vessel  is  enabled  to 
T  be  sailed  with  a  free  wind  and  against 

one. 

Assume  wind  to  be  free  or  in  direction 
of  arrows,  Fig.  7,  and  perpendicular  to 
line  A  B,  the  course  of  vessel. 

Let  line  m  o  represent  direction  and 
B   force  of  wind,  and  r  s  plane  of  sail  ;  from 
o  draw  o  u  perpendicular  to  r  s,  and 
from  m  perpendicular,  m  v  on  r  s,  and 

/m  u  on  o  u. 
By  principle  of  parallelogram  offerees, 
force  m  o  may  be  decomposed  into  o  v 

and  OM,  since  they  are  the  sides  of  parallelogram  of  which  m  o,  representing  force 
of  wind,  is  diagonal.  Force  of  wind,  therefore,  is  measured  by  ow,  both  in  magni- 
tude and  direction,  and  represents  actual  pressure  on  sail. 

Draw  un  and  u  x  parallel  to  oA  and  om,  thus  forming  parallelogram  unox. 

Hence  force  o  M  is  equal  to  the  two,  o  n 
and  o  x.  Force  o  n  acts  in  a  direction 
perpendicular  to  vessel's  course  and  that 
of  o  a;  is  to  drive  vessel  onward. 

It  can  ^us  be  shown  that  when  di- 
rection  of  sail  bisects  angle  m  o  B,  the 
effect  of  o  a;  is  greater  than  when  sail  is 
in  any  other  position. 

Assume  wind  to  be  ahead  as  in  direc- 
tion of  arrows,  Fig.  8.  Let  o  m  repre- 
sent direction  and  force  of  wind,  and  r  s 
direction  of  sail;  from  o  draw  ow,  and 
proceed  as  before,  and  o  u  represents  the 
effective  force  that  acts  upon  the  sail, 
on  that  which  drives  her  to  leeward,  and 
o  x  that  which  drives  her  on  her  course. 

For  full  treatises  on  this  subject,  see  John  C.  Trautwine's  Engineer's  Pocket-book,  1872  ;  Bull's  Ex- 
perimental Mechanics,  London,  1871  ;  and  Dynamic*,  Construction  of  Machinery,  etc.,  by  G.  Finden 
Warr,  London,  1851. 


MECHANICS. — MOMENTS  OF  STRESS  ON  GIRDERS,  ETC.  621 


MOMENTS  OP  STRESS. 

To  Describe  and  Compute  Moments  of  Stress  on  <3-ird« 
era   or    Beams. 

Supported  at  Both  Ends. 


Fig.  i. 


Loaded  in  Centre,  Fig.  i.  —  Assume 
A  B,  a  beam.    At  centre  erect  We  = 

— .    Connect  A  c  and  c  B,  and  any 

vertical  distance  between  them  and 
A  B  will  give  moment  required  at  that 
point. 

Wa/ 

=  M  at  any  point.    W  represent- 
ing weight  or  load,  I  length  of  span,  x  horizontal  distance  from  nearest  support  at 
which  M,  the  moment  of  stress,  is  required. 
ILLUSTRATION*.  — Assume  I  =  10  feet,  W  =  10  Ibs. ,  and  x  =  3  feet. 

Then,  W  c  =  I0  X  I0  =  25  Ibs.  at  centre  of  span ;  and  ^-X~3-  =  15  Ibs.  at  x. 

Fig.  2.  c  Loaded  at  Any  Point,  Fig.  2.— 

Proceed  as  for  previous  figure. 


™ 


=  M  between  W  and  B. 


a  representing  least  distance  of  W  to  swpportf, 
and  6  greatest  distance. 
ILLUSTRATION.—  Take  elements  as  before  with  a  =  3  feet,  a?  1.5,  and  *'  3.5  feet. 


Wafr 
2 

Wxb 
__ 

Wxa 
T 


or  W  c  =  maximum  load. 


=  M  between  A  and  W. 


I0.5  U>s.  at  x 


Then,Wc  =  1-^    -*-?  =  21  Ibs.  at  point  of  stress;     IoX*-5X7  = 
between  A  and  W,  and  IoX  3-5X3  =         ^ 

10 

NOTE.— »  and  x'  must  be  taken  from  the  pier,  which  is  on  the  same  side  of  W  as 
that  of  the  stress  desired. 

Loaded  with  Two  Equal  Weights  at  Equal  Distances  from  Supports,  alike  to  a 
Transverse  Girder  in  a  Single  Line  of  Railway.—  Fig.  3. 

Fig.  3.  q £ d  At  point  of  stress  of  weights 

erect  W  c  and  W  d,  each  =  W  a. 
Connect  A  cd  and  B,  and  vertical 
distances  between  them  and  AB 
will  give  moments  required. 


Fig.  4. 


any  point  between  weights. 

Loaded  with  Four  Equal  Weights,  symmetrically  bearing  from  Centre,  alike  to  a 
Transverse  Girder  in  a  Double  Line  of  Railway.— Fig.  4. 

At  W  and  w"  erect  We,  and 
w"  i  =  2  W  a,  and  at  w  and  w/ 
erect  w  d,  w'  e,  each  =  W  (2  a-f-  a'). 
Connect  Acdei  and  B,  and  or- 
dinates  from  them  to  A  B  will  give 
moments  required. 

W  (2  a  -\-  a')  =  M  at  w  and  w'\ 
2  W  a  =  M  at  W  and  w". 

ILLUSTRATION.— Assume  W  each 
10  Ibs.  2  feet  apart,  and  1 10  feet. 
Then,  10  (2  x  2  -f-  2)  =  60  at  w  or  w',  and  2  x  10  x  2  ^  40  at  W  or  to". 


622  MECHANICS.  —  MOMENTS  OF  STRESS  ON  GIRDERS,  ETC. 


Loaded  at  Different  Points.—  Fig.  5. 

Locate  three  weights,  W,  w,  and 
tt>',  as  at  a  6,  at  6^  a2  bm. 

Draw  A  c  B,  A  d  B,  and  A  «  B,  as 
three  separate  cases,  by  formula, 
Wa& 


I 


-,  Fig.  -2. 


f 


Produce  We  until  Wo  =  Wr,Ws, 
*  and  We;  Wd  until  tow  — tow,  wv 
and  w;  d,  and  w'  e  to  w/  m  in  like 
manner. 

Connect  A  owm  and  B,  and  an  or- 
dinate  therefrom,  to  A  B  will  give 
moment  of  stress  at  the  point  taken. 
ILLUSTRATION.  —Take  a  =  2  feet,  a  =4,  a2  — 6,  6  =  8,  6X=:6,  62  =  4,  W,w,  and 
w'  each  10  Ibs.,  and  I  — 10  feet,  carefully  observing  Note  to  Fig.  2. 


Then  j  (W  6  x  +  w  bx  x  -f  w"  b2  x)  —  M  at  x. 


Take  x  =  2.    Then  —  (10x8x2  +  10x6x2-1-10X4X2)=:^—  =  36  Z&* 


360 


*'  =  4-      —  (io  X  2  X  6  +  I0  X  6  X  4  +  io  X  4  X  4)  =  ^-  =  52  «*. 
*"  =  6.      j-  (10  X  2  X  4  +  10  X  4  X  4  +  10  X  4  X  6)  =  ^  =  48  Ibs. 

Loaded  with  a  Rolling  Weight.  — 

Fig.  6. 

Define  parabola  A  c  B  as  deter- 
mined by  —  =  the  ordinate  at  c, 

B  and  vertical  distances  between  A  B 

<m"  ~VV"  Tl^-    will  giye  moments. 

$L~Z~«4 JT  W  x  (l-x) 

—  M  at  any  point. 

Loaded  Uniformly  its  Entire  Length. — Define  parabola  as  at  Fig.  6,  ordinate  of 
which  at  c  = .     L  representing  stationary  or  dead  load  per  unit  of  length. 


--- 
8 


w  I2 


—  (I— x)  =  M  at  any  point,  and —  M  at  centre. 

2  8 

Loaded  with  Two  Connected  Weights,  moving  in  either  Direction,  alike  to  a  Locomo- 
tive or  Car  on  a  Railway. — Fig.  7. 

Fig.  7.  ^  Define  parabola  A  c  B  as  deter- 

(W-4-w>)  I 

mined  by — —  =  c. 

4 

At  A  and  B  erect  A  e,  B  i  =  w  d, 
connect  A  i  and  B  e,  and  vertical 
i-o    distances  between  A  o  B  and  A  c  B 
|f    will  give  moments. 


Or  if  W  and  w  are 


at  any  point. 
Position  ofW  at  greatest  moment,  when  x=-  ± 

2          2 

equal,  when  x  =  —  it  —  . 


ILLUSTRATION.  —  Assume  x=  3,  d  =  4,  and  W  w  each  10  Ibs.  ,  and  1  10  feet 
Then  —  (io-f-io  X  10  —  3  —  10  X  4)  =  M  at  any  point,  as  at  W  r,  w  r. 


MECHANICS.  —  MOMENTS  OF  STRESS  ON  GIKDERS^ETC.   623 

Shearing    Stress. 

To  Determine  Shearing  Stress   at  any  3?art  of*  a  Girder 
or    13  earn    and.    -under   any    Distribution    of  Load. 

Fig.  8.  Required  to  determine  stress  of  a 

-  —  £  —  ~"VB    beam  at  any  P°int  as  c,  Fig.  8. 

I  j^        Assume  W  =  load  between  A  and 

-^  p>       Cj  and  w  t^t,  between  B  and  c. 

Then  S  x  at  c  =  P  —  W,  or  P'  —  w. 
The  greater  of  the  two  values  to  be  taken. 

S  x  representing  shearing  stress  at  any  point  x,  P  and  P'  the  reaction  on  supports 
due  to  total  load  on  beam  between  supports,  W  and  w  loads  or  stress  concentrated  at 
iny  point. 

Desoritoe    and    Ascertain    Shearing    Stress    in    a 
GHrder   or    Beam. 


To 


Fig.  9. 


Supported  or  Fixed  at  Both  Ends. 

Loaded  Uniformly. 


Fig.  9. 
At  A  and  B,  erect  A  c,  B  e.  each 

W  I 
equal  to  — .     Connect  c  and  e  at 

middle  of  span  as  at  w,  and  vertical 
distances  between  A  B  and  cue  will 
i-,,    give  shearing  stresses  as  determined 
rwl.      by  the  ordi  nates  tocne. 

— ,j >%       L  n  _  \  _  s  Si    of  result  to 

\2  / 

It  representing  distributed  load  per  unit  of  length. 


l^k  X, 

'  ^JL£  L 
>^Ww>^<w    ! 


be  disregarded. 
ILLUSTRATION.—  Assume  L  =  10  Iba  per  foot,  I  =  10,  and  x  =  2.5  feet. 


Then  10  (  --  2.5  j  =  25  /6s. 


NOTE.—  The  moment  of  rupture  at  any  point,  produced  by  several  loads  acting 
simultaneously  on  a  beam,  is  equal  to  the  sum  of  the  moments  produced  by  tha 
several  loads  acting  separately. 

For  other  Formulas  and  Diagrams  see  Strains  in  Girders,  by  William  Humber, 
A.I.C.E.,  London,  1872. 

Operation  deduced  by  Graphic  Delineation  of  Greatest  Stress,  with  a 

Uniformly  Distributed  Load  of  14  ooo  Lbs.  —  Fig.  10. 
Fig.  10.  Determine  moment  of  weights  by 

formulas  5-»,  =£f  ,  and  =?. 


Assume  W  =  7ooo  Iba,  »  =  4ooo, 
and  w'  =  3oc»,  m  =  7  feet,  n  =  13, 
r  =  13.  5=7,  0=3,  v=i7,  and  1=20. 


_  3000X3X17  _  7g50j  and  Jet  fal,  perpendic. 


w  =  4000X13X7  _ 

ulars  thereto,  as  3  d,  2  c,  and  i  b. 

Connect  d,  c,  and  6  with  A  B,  and  sum  of  distances  of  intersections  of  these  lines 
upon  perpendiculars,  from  3,  2,  and  i  respectively,  will  give  stress  upon  A  B  at 
these  points. 

To  determine  Greatest  Stress  at  Greatest  Load. 

Stress  at  3  d  =31  850     I     Stress  at  i  6  =  17  :  7650  :  3  =  i  350 

"      "20=13:18200:7=9800     |  ~^^> 

43  ooo  -f  7XI3X400QX.5  _  $2  IQQ  m  ^  concentrate(j  ioad  at  w>  ^^  proportion 

20 
of  uniformly  distributed  load  of  4000  Ibs. 


624  MECHANICAL    POWERS. — LEVEE. 

MECHANICAL  POWERS. 

MECHANICAL  POWER  is  a  compound  of  Weight,  or  Force  and  Velocity: 
it  cannot  be  increased  by  mechanical  means. 

The  Powers  are  three  in  number—viz.,  LEVER,  INCLINED  PLANE,  and 
PULLEY. 

NOTE.— A  Wheel  and  Axle  is  a  continuous  or  revolving  lever,  a  Wedge  a  double  in 
clined  plane,  and  a  Screw  a  revolving  inclined  plane. 

LEVER. 
Levers  are  straight,  bent,  curved,  single,  or  compound. 

To    Compute    Length,   of  a   Lever. 

When  Weight  and  Power  are  given.     RULE.-— Divide  weight  by  power, 
and  quotient  is  leverage,  or  distance  from  fulcrum  at  which  power  supports 
weight, 
w 

Or,  ~  =  p.  W  representing  weight,  P  power,  and  p  distance  of  power  from  fulcrum. 

EXAMPLE. — A  weight  of  1600  Ibs.  is  to  be  raised  by  a  power  or  force  of  80;  re- 
quired length  of  longest  arm  of  lever,  shortest  being  i  foot. 
1600  -i-  80  =  20  feet. 

To  Compute  \Veight  that  can  "be   raised,  "by  a   Lever. 

When  its  Length,  Power,  and  Position  of  its  Fulcrum  are  given.  RULE. — 
Multiply  power  by  its  distance  from  fulcrum,  and  divide  product  by  dis- 
tance of  weight  from  fulcrum. 

Or,  -^  =  W.    w  representing  distance  of  weight  from  fulcrum, 
w 

EXAMPLE. — What  weight  can  be  raised  by  375  Ibs.  suspended  from  end  of  a  lever 
8  feet  from  fulcrum,  distance  of  weight  from  fulcrum  being  2  feet? 
375X8-7-2  =  1500^5. 

To    Compute   Position   of*  Fulcrum. 

When  Weight  and  Power  and  Length  of  Lever  are  given,  and  when  Ful- 
crum is  between  Weight  and  Power.  RULE. — Divide  weight  by  power,  add 
i  to  quotient,  and  divide  length  by  sum  thus  obtained. 

Or,  L-f-(-p-f-i):=w>.    L  representing  entire  length  of  lever. 

EXAMPLE.— A  weight  of  2460  Ibs.  is  to  be  raised  with  a  lever  7  feet  long  and  a 
power  of  300;  at  what  part  of  lever  must  fulcrum  be  placed  ? 

2460-7-300  =  8.2,  and  8. 2-}- 1  =9-2.     Then  7  X  12 -1-9.2  =  9.13  irw. 

When  Weight  is  between  Fulcrum  and  Power.  RULE. — Divide  length 
by  quotient  of  weight,  divided  by  power. 

Or,  LH-^  =  M>. 

To    Compiate    Length    of*   Arm    of*    Lever    to    \vhich 
"Weight   is    attached. 

When  Weight,  Power,  and  Length  of  Arm  of  Lever  to  which  Power  is  ap- 
plied are  given.  RULE.  —  Multiply  power  by  length  of  arm  to  which  it  is 
applied,  and  divide  product  by  weight. 


MECHANICAL    POWERS. — LEVEK. 


625 


EXAMPLE.  —A  weight  of  1600  Ibs. ,  suspended  from  a  lever,  is  supported  by  a  power 
of  80,  applied  at  other  end  of  arm,  20  feet  in  length ;  what  is  length  of  arm  ? 

80  X  20 -r-  l6<X)  =  I  foot. 

NOTE.— These  rules  apply  equally  When  fulcrum,  (or  support)  of  lever  is  between 
weight  and  power  ;*  when  fulcrum  is  at  one  extremity  of  lever,  and  power,  or  weight, 
*,t  the  other  ;t  and  when  arms  of  lever  are  equally  or  unequally  bent  or  curved. 

To  Compute  I?ower  Required,  to  Raise  a  given  'Weight. 

When  Length  of  Lever  and  Position  of  Fulcrum  are  given.  RULE. — MuU 
tiply  weight  to  be  raised  by  its  distance  from  fulcrum,  and  divide  product 
by  distance  of  power  from  fulcrum. 

W  w 
Or,  =  P. 

-v.AV  V  p 

EXAMPLE Length  of  a  lever  is  10  feet,  weight  to  be  raised  is  3000  Ibs.,  and  its 

distance  from  fulcrum  is  2  feet;  what  is  power  required? 

a  =  6o~  , 

IO  —  0 

To  Compute  Length,  of  Arm  of  Lever  to  which.  IPower 
is    applied. 

When  Weight,  Power,  and  Distance  of  Fulcrum  are  given.  RULE.— Mul- 
tiply weight  by  its  distance  from  fulcrum,  and  divide  product  by  power. 

Ww 
Or,  -j-  =p. 

EXAMPLE.— A  weight  of  400  Ibs.,  suspended  15  ins.  from  fulcrum,  is  supported  by 
a  power  of  50,  applied  at  other;  what  is  length  of  the  arm  ? 

400  X  15  -i-  50  =  120  ins. 

When  Arms  of  a  Lever  are  bent  or  curved, 
Distances  taken  from  perpendiculars,  drawn 
from  lines  of  direction  of  weight  and  power, 
must  be  measured  on  a  line  running  horizon - 
G  tally  through  fulcrum,  as  a  b  c,  Figs,  i  and  2. 

When  A  rms  of  a  Lever  are  at  Right  A  ngles, 
and  Power  and  Weight  are  applied  at  a  Right 

Fig  3 


Fig.  i. 


- . 


Angle  to  each    other, 
Fig.  3,  The  moments 
are  computed  directly  as  abtob  c. 

Thrust,  or  press- 
ure on  fulcrum, 
is  in  this  case  less 
than  sum  of  pow- 
c  er  and  weight; 
'    and  it  may  be 
determined  by  a 
drawing  a  paral- 
lelogram   upon 


\ 


O 

p 


the  two  arms  of 
s  lever,  arms  repre-  | 
senting  inverse- 
ly their  respec- 
tive forces.  That  is,  a  b  represents  magnitude  and  direction  of  weight  W, 
and  6  c  of  power  P.  Diagonal  o  b  of  parallelogram  represents  magnitude 
and  direction  of  third  force,  or  thrust  upon  fulcrum. 

*  Pressure  upon  fulcrum  is  equal  to  sum  of  weight  and  power. 

t  Pressure  upon  fulcrum  is  equal  to  difference  of  weight  and  power. 

3  G 


626         MECHANICAL    POWEKS. — LEVEE. — WHEEL. 


Fig.  4.  When  same,  Lever  is  borne  into  an  Oblique 

Position,  Power  continuing  to  act  Horizontally, 
Fig.  4,  Draw  vertical  a  v  through  end  o  of 
lever,  and  produce  the  power  line  p  c  to  meet 
it  at  a.  Complete  parallelogram  avbr;  then 
sides  r  b  and  b  v  are  perpendiculars  to  direc- 
tions to  power  and  weight,  on  which  moments 
are  computed. 

Consequently,  moment  P  X  r  b  =  moment 
W  X  6  v,  and  a  diagonal,  b  a,  is  resultant  thrust 
at  fulcrum. 

When  Power  does  not  act  Horizon- 
tally, Fig.  5,  but  in  some  other  direc- 
tion, a  p,  produce  the  power  -  line  p  a 
and  draw  b  c  perpendicular  to  it ;  draw 
p  b  o,  then  moments  are  computed  on 
perpendiculars  b  c,  b  0,  and  P  x  c  b  = 
W  xbo. 

If  several  weights  or   powers   act 
upon  one  or  both  ends  of  a  lever,  con- 
/k   dition  of  equilibrium  is 

P        Pp  +  P'p'4-  P"j>",  etc.,  =  Ww  + 
W  w>',  etc. 

In  a  system  of  levers,  either  of  similar,  compound,  or  mixed 
kinds,  condition  is       p  p  p'  p" 
Vf  w  w'  w"  =  W" 

ILLUSTRATION.— Let  P  =  i  lb.,  p  and  p'  each  10  feet,  p"  i  foot;  and  if  w  and  w' 
be  each  i  foot,  and  w"  i  inch,  then 


=  1200;  that  is,  i  lb.  will  support  1200,  with  levers 


I  X  120  X  120  X  12  172  800 

12  X  12  X  I       "   144 

of  the  lengths  above  given. 

NOTE.— Weights  of  levers  in  above  formulas  are  not  considered,  centre  of  gravity 
being  assumed  to  be  over  fulcrum s. 


GENERAL  RULE,  therefore,  for  ascertaining  relation  of  POWER  to 
WEIGHT  jn  a  iever?  whether  straight  or  curved,  is,  Power  multiplied  by  its 
distance  from  fulcrum  is  equal  to  weight  multiplied  by  its  distance  from 
fulcrum.  Or,P:W::w:p,orPp  = 


;  and 


W  w 


:P.  e.  -^  =  T 


W  w 


Pp_ 


WHEEL  AND  AXLE. 

A.  "Wh.ee!  and   Axle  is  a  revolving  lever. 

Power,  multiplied  by  radius  of  wheel,  is  equal  to  weight,  multiplied  by 
radius  of  axle. 

As  radius  of  wheel  is  to  radius  of  axle,  so  is  effect  to  power. 

R  R  P 

Or,PR  =  Wr.     Or,  PV  =  Wv.     Or,  R:r::W:P.      Or,  P-^W;     ^f  =  r't 
-—  :=  R.    R  and  r  representing  radw,  and  V  and  v  velocities  of  wheel  and  axle. 


MECHANICAL   POWERS. — WHEEL    AND    AXLE.       627 

When  a  series  of  wheels  and  axles  act  upon  each  other,  either  by  belts  or 
teeth,  weight  or  velocity  will  be  to  power  or  unity  as  product  of  radii,  or 
circumferences  of  wheels,  to  product  of  radii,  or  circumferences  of  axles. 

ILLUSTRATION.— If  radii  of  a  series  of  wheels  are  9,  6,  9,  10,  and  12,  and  their  pin- 
ions have  each  a  radius  of  6  ins.,  and  power  applied  is  10  Ibs.,  what  weight  will 
they  raise? 

io  X  9  X  6  X  9  X  io  X  12  _  583200  _       lb 

6X6X6X6X6         ~    7776    ~75 
Or,  if  ist  wheel  make  io  revolutions,  last  will  make  75  in  same  time. 

Xo  Coxnprite  3?o\ver  of  a  Com"bination  of  "Wheels  and  an 
Axle   or   .A.xles,  as    in    Cranes,  etc. 

RULE. — Divide  product  of  driven  teeth  by  product  of  drivers,  and  quo- 
tient is  their  relative  velocity ;  which,  multiplied  by  length  of  lever  or  arm 
and  power  applied  to  it  in  pounds,  and  divided  by  radius  of  barrel,  will  give 
weight  that  can  be  raised. 

Or.  ^—  =  W ;  Or,  W  r  =  v  I  P;  Or,  -^  ==  P.  I  representing  length  of  lever  or 
arm,  r  radius  of  barrel,  P  power,  v  velocity,  and  W  weight. 

EXAMPLE  i.— A  power  of  18  Ibs.  is  applied  to  lever  or  winch  of  a  crane,  length  of 
it  being  8  ins.,  pinion  having  6  teeth,  driving-wheel  72,  and  barrel  6  ins.  diameter. 

^  =  12,  and  12  X  8  X  18  =  1728,  which,  -f-  3,  radius  of  barrel,  =  576  Ibs. 
o 

2.— A  weight  of  94  tons  is  to  be  raised  360  feet  in  15  minutes,  by  a  power,  velocity 
of  which  is  220  feet  per  minute;  what  is  power  required? 

360  -r- 1 5  =  24  feet  per  minute.    Hence  —  — —  =.  io.  2545  tons. 

Compound.  .A.xle,  or   Chinese  "Windlass. 

Axle  or  drum  of  windlass  consists  of  two  parts,  diameter  of  one 
being  less  than  that  of  the  other. 

The  operation  is  thus :  At  a  revolution  of  axle  or  drum,  a  portion  of  sus- 
taining rope  or  chain  equal  to  circumference  of  larger  axle  is  wound  up,  and 
at  same  time  a  portion  equal  to  circumference  of  lesser  axle  is  unwound. 
Effect,  therefore,  is  to  wind  up  or  shorten  rope  or  chain,  by  which  a  weight 
or  stress  is  borne,  by  a  length  equal  to  difference  between  circumferences  of 
the  two  axles.  Consequently,  half  that  portion  of  the  rope  or  chain  will  be 
shortened  by  half  difference  between  circumferences. 

To    Compnte    Elements    of   a   "Wheel    and    Conapou.net 
Axle,  or    Chinese    "Windlass.— Fig.  G. 

RULE.— Multiply  power  by  radius  of  wheel,  arm,  or      Fig.  6.  r 

bar  to  which  it  is  applied,  and  divide  product  by  half 
difference  of  radii  of  axle,  and  -quotient  is  weight  that  a' 
can  be  sustained.  ** 

P  R 
Or,  7-  =  W.    R  representing  radius  of  wteel,  etc. ,  and  r  and  r' 

radii  of  axle  at  its  greatest  and  least  diameters. 

EXAMPLE.— What  weight  can  be  raised  by  a  capstan,  radius  of  its  bar,  a, 
5  feet,  power  applied  50  Ibs.,  and  radii,  r  r',  of  axle  or  drum  6  and  5  ins.  ? 

50  X  5  X  12  _: 
•5(6-5) 


628         MECHANICAL   POWERS.  —  INCLINED    PLANE. 

"Wh.ee!    and.    IPinion    Combinations,  or    Complex 
"Wheel-  work. 

Power,  multiplied  by  product  of  radii  or  circumferences,  or  number  of 
teeth  of  wheels,  is  equal  to  weight,  multiplied  by  product  of  radii  or  circum- 
ferences, or  number  of  teeth  or  leaves  of  pinions. 

Or,  P  R  R'  R",  etc.,  =  W  r  r'  r",  etc. 

NOTE.—  Cogs  on  face  of  wheel  are  termed  teeth,  and  those  on  surface  of  axle  are 
termed  leaves  ;  the  axle  itself  in  this  case  is  termed  a  pinion. 

Tfcaok   and.   Opinion. 
To   Compute    3?o\ver   of  a   Rack   and.    Pinion. 

RULE.  —  Multiply  weight  to  be  sustained  by  quotient  of  radius  of  pinion, 
divided  by  radius  of  crank,  and  product  is  power  required. 

Or,  W  -£•  =  P- 

Jtv 

When  Pinion  on  Crank  Axle  communicates  with  a  Wheel  and  Pinion. 
RULE.  —  Multiply  weight  to  be  sustained  by  quotient  of  product  of  radii  of 
pinions,  divided  by  radii  of  crank  and  wheel,  and  product  is  power  required. 


EXAMPLE.—  If  radii  of  pinions  of  a  jack-screw  are  each  one  inch;  of  crank  and 
wheel  10  and  5  ins.  ;  what  power  will  sustain  a  weight  of  750  Ibs.  ? 


INCLINED  PLANE. 

To  Compute   Length   of  Base,  Height,  or   Length. 
When  any  Two  of  them  are  given,  and  when  Line  of  Direction  of  Power 
or  Traction  is  Parallel  to  Face  of  Plane. — Proceed  as  in  Mensuration  or 
Trigonometry  to  determine  side  of  a  right-angled  triangle,  any  two  of  thre« 
being  given. 

To  Compute  Power  necessary  to  Support  a  "Weight  on 
an.    Inclined    Plane. 

When  Height  and  Length  are  given.    RULE. — Multiply  weight  by  height 
of  plane,  and  divide  product  by  length. 

©r,  — —  =  P-    h  and  I  representing  height  and  length  of  plane. 

EXAMPLE.— What  is  power  necessary  to  support  1000  Ibs.  on  an  inclined  plane 
4  feet  in  height  and  6  feet  in  length  ? 

looo  X  4  -r-  6  =  666.67  lbs- 

To  Compute  "Weight  that  may  "be  Sustained  by  a  given 

Po*wer   on.    an    Inclined    Plane. 

When  Height  and  Length  of  Plane  are  given.    RULE.— Multiply  power 
by  length  of  plane,  and  divide  product  by  height. 


EXAMPLE.—  What  is  weight  that  can  be  sustained  on  an  inclined  plane  5  feet  in 
height  and  7  feet  in  length  by  a  power  of  700  Ibs.  ? 
700  x  7  -T-  5  =  980  Ibs. 

NOTE.—  In  estimating  power  required  to  overcome  resistance  of  a  body  being 
drawn  up  or  supported  upon  an  inclined  plane,  and  contrariwise,  if  body  is  de- 
scending; weight  of  body,  in  proportion  of  power  of  plane  (i.  e.,  as  its  length  to  its 
height),  must  be  added  to  resistance,  if  being  drawn  up  or  supported,  or  to  the  wo- 
ment  if  descending. 


MECHANICAL    POWERS.  —  INCLINED    PLANE.         62Q 

To   Compute   Heiglxt   or    Length    of  arx    Inclined    Plane. 

When  Weight  and  Power  and  one  of  required  Elements  are  given,  and 
when  Height  is  required.  RULE.—  Multiply  power  by  length,  and  divide 
product  by  weight. 

When  Length  is  required.  RULE.—  Multiply  weight  by  height,  and  divide 
product  by  power. 


To   Compute   Pressure  on   an   Inclined   Plane. 

RULE.—  Multiply  weight  by  length  of  base  of  plane,  and  divide  product 
by  length  of  face. 

w  b 
Or,  —j-  =  pressure,    b  representing  length  of  base  of  plane. 

EXAMPLE.—  Weight  on  an  inclined  plane  is  100  IDS.,  base  of  plane  is  4  feet,  and 
length  of  it  5  ;  required  pressure  on  plane. 


When  Two  Bodies  on  Two  Inclined  Planes  sustain  each  other,  as  by  Connection 
of  a  Cord  over  a  Pulley,  their  Weights  are  directly  as  Lengths  of  Planes. 
ILLUSTRATION.  —  If  a  weight  of  50  Ibs.  upon  an  inclined  plane,  of  10  feet  rise  in  100 

of  an  inclination,  is  sustained  by  a  weight  on  another  plane  of  10  feet  rise  in  90, 

what  is  the  weight  of*the  latter  ? 

ioo  :  90  ::  50  :  45  =  weight  that  on  shortest  plane  would  sustain  that  on  largest. 

When  a  Body  is  Supported  by  Two  Planes,  as  Fig.  7,  pressure  upon  them 
p.  will  be  reciprocally  as  sines  of  inclinations  of  planes. 

-i  Thus,  weight  is  as  sin.  A  B  D. 

Pressure  on  A  B  as  sin.  D  B  i. 
Pressure  on  B  D  as  sin.  A  B  h. 

Assume  angle  A  B  D  to  be  90°,  and  D  B  t,  60°  ;  then  angle 
A  B  h  will  be  30°;  and  as  sines  of  90°,  60°,  and  30°  are  respec- 
tively .1,  .866,  and  .5,  if  weight  =  ioo  Ibs.,  then  pressures  on 
A  B  and  B  D  will  be  86.6  and  50  Ibs.,  centre  of  gravity  of  weight  assumed  to  be  in  its 
centre. 

When  Line  of  Direction  of  Power  is  parallel  to  Base  of  Plane,  power  is 
to  weight  as  height  of  plane  to  length  of  its  base. 
Or,  P:  W.:h:b. 


When  ZAne  of  Direction  of  Poiner  is  neither  parallel  to  Face  of  Plane  nor 
to  its  Base,  but  in  some  other  Direction,  as  P',  Fig.  8,  power  is  to  weight  as 
sine  of  angle  of  plane's  elevation  to  cosine  of  angle  which  line  of  power  or 
traction  describes  with  face  of  plane. 
Fig.  8.  «\  Thus,  P'  :  W  ::  sin.  A  :  cos.  P*  e  c. 

Sin.  A:  cos.  P'ec::P':  W. 

Cos.  P'  ec:  sin.  A  :  :  W  :  P'. 


ILLUSTRATION.— A  weight  of  500  Ibs.  is  required  to  be 
sustained  on  a  plane,  angle  of  elevation  of  which, 

c  A  B,  is  10°;  line  of  direction  of  power  or  traction, 

A  n  B  P'e  c,  is  5°;  what  is  sustaining  power  required? 

Cos.  P'  e  c  (5°)  =  .996 19  :  sin.  A  (10°)  =  .173  65  : :  500  :  87. 16  Ibs. 
Or,  draw  a  line,  B  s,  perpendicular  to  direction  of  power's  action  from  end 
of  base  line  (at  back  of  plane),  and  intersection. of  this  line  on  length,  Ac, 
will  determine  length  and  height  (n  r)  of  the  plane. 

3  G* 


630          MECHANICAL    POWERS.  -  WEDGE.  -  SCREW. 

ILLUSTRATION.—  By  Trigonometry  (page  385),  A  B,  assumed  to  be  i,  A  r  and  n  r  are 
=  .985  and.  171. 

Hence  5°°X'171  =  86.8  Ibs.  ^product  of  weight  X  height  of  plane  -r-  length  of  it. 

•985 
NOTE.—  When  line  of  direction  of  power  is  parallel  to  plane,  power  is  least. 

Wedge. 
A  WEDGE  is  a  double  inclined  plane. 

To    Compnte   Power. 

1.  When  One  Body  is  to  be  Forced  or  Sustained.    RULE.  —  Multiply  weight 
or  resistance  to  be  sustained  by  depth  of  back  of  wedge,  and  divide  product 
by  length  of  its  base. 

EXAMPLE.  —  What  power,  applied  to  the  back  of  a  wedge  6  ins.  deep,  will  raise  a 
weight  of  15000  Ibs.,  the  wedge  being  100  ins.  long  on  its  base? 
15000X6^90  ooo=         bs 

IOO  IOO 

2.  When  Two  Bodies  or  Two  Parts  of  a  Body  are  Forced  or  Sustained  in  a 
Direction  Parallel  to  Back  of  Wedge.    RULE.  —  Multiply  weight  or  resist- 
ance to  be  sustained  by  half  depth  of  back  of  wedge,  and  divide  product  by 
length  of  wedge. 

Or,  Wd~2  =  P.    d  representing  depth  of  back,  and  I  length. 
I 

NOTE.—  The  length  of  a  single  wedge  is  measured  on  its  base,  and  of  a  double 
wedge,  from  centre  of  its  head  to  its  point. 

EXAMPLE.—  The  depth  of  the  back  of  a  double-faced  wedge  is  6  ins.,  and  the 
length  of  it  through  the  middle  10;  what  power  applied  to  it  is  necessary  to  sus- 
tain or  overcome  a  resistance  of  150  Ibs.  ? 

=  45° 


To   Compute   Elements   of  a  "Wedge. 


...  = 

NOTE.—  As  power  of  wedge  in  practice  depends  upon  split  or  rift  in  wood  to  be 
cleft,  or  in  rise  of  body  to  be  raised,  the  above  rules  as  regards  length  of  wedge  are 
only  theoretical  when  a  rift  or  rise  exists. 

SCREW. 

A  SCREW  is  a  revolving  inclined  plane. 
To  Compvite   Length   and    Height  of  Plane  of  a  Screw. 

As  a  screw  is  an  inclined  plane  wound  around  a  cylinder,  length  of  plane 
is  ascertained  by  adding  square  of  circumference  of  screw  to  square  of  dis- 
tance between  threads,  and  taking  square  root  of  sum. 

The  Pitch  or  height  of  a  screw  is  distance  between  its  consecutive  threads. 
To  Compute  IPower. 

RULE.  —  Multiply  weight  or  resistance,  to  be  sustained  by  pitch  of  threads, 
and  divide  product  by  circumference  described  by  power. 

W  n 

Or,  —  -  =  P.    p  representing  pitch,  and  c  circumference. 

EXAMPLE.—  What  is  power  requisite  to  raise  a  weight  of  8000  Ibs.  by  a  screw  of  12 
ins.  circumference  and  i  inch  pitch  ?  8000  X  i  -5-  ia  =  666.66  Ibt. 


MECHANICAL    POWERS.  —  SCREW.  63  I 

To   Compute  "Weight. 

RULE.  —  Multiply  power  by  circumference  described  by  it,  and  divide 
product  by  pitch  of  threads. 

Or,^=W. 
P 

To    Compute   Pitch.. 

RULE.—  Multiply  power  by  circumference  described  by  it,  and  divide 
product  by  weight. 


To  "Compute    Circumference. 

RULB.—  Multiply  weight  by  pitch,  and  divide  product  by  power. 

Or,  -—  -  =  c.    Or,       p     =  r.    r  representing  radius. 

r  0.20  r 

When  Power  is  applied  by  a  Lever  or  Wheelr  substitute  radius  of  power 
for  circumference. 

ILLUSTRATION.—  If  a  lever  30  ins.  in  length  was  added  to  circumference  of  screw 
in  preceding  example, 

Then,  12-7-3.416  =  3.819,  and  ^-^-|-3o=3i.9095  =  radtM»  of  power. 


Compound.    Screw. 

Fig.  9.         &  When  a  Lever  and  Endless  Screw  or  a  Series  of 

Wheels  are  applied  to  a  Screw,  as  Fig.  9.  RULE. 
—  Ascertain  result  of  each  application,  and  take 
their  continued  product. 

NOTE.—  If  there  is  more  than  one  thread  to  a  screw, 
pitch  must  be  increased  as  many  times  as  there  are 
threads. 

ILLUSTRATION.—  What  weight  can  be  raised  with  a  power  of 
10  Ibe.,  applied  to  a  crank,  c,  Fig,  9,  32  ins.  long,  turning  an  end- 
less screw,  6,  of  3.5  ins.  diameter  and  i  inch  pitch,  applied  to  a 
wheel,  d,  of  20  ins.  diameter,  upon  an  axle,  a,  of  5  ins.  ? 

ioX32X   .2   _  20Q9  6  _  quotient  of  product  of  power  and 

iV  Ifr'W 

circumference  described  by  it,  and  pitch,  and  2.°°9^X20  =  8038.4  Ibs.  =  quotient  of 
power  applied  to  wheel,  divided  by  its  axle. 

When  a  Series  of  Wheels  and  Axles  are  in  Connection  with  each  other, 
Weight  is  to  power,  as  continued  product  of  radii  of  wheels  is  to  continued 
product  of  radii  of  axles. 

W  :  P  :  :  R  n  :  r  n. 

Or,  rn  :  Rn  ::  P  :  W.  n  representing  continued  product  of  number  of  wheels  or 
axles. 

ILLUSTRATION.—  If  a  power  of  150  Ibs.  is  applied  to  a  crank  of  20  ins.  radius,  turn- 
ing  an  endless  screw  with  a  pitch  of  half  an  inch,  geared  to  a  wheel,  pinion  of 
which  is  geared  to  another  wheel,  and  pinion  of  second  wheel  is  geared  to  a  third 
wheel,  to  axle  or  barrel  of  which  is  suspended  a  weight;  it  is  required  to  know 
what  weight  can  be  sustained  in  that  position,  diameter  of  wheels  being  18,  and 
pinions  and  axle  2  ins. 

150X20X2X3-1416  =  3?699  2  ifa  =  power  applied  to  face  of  first  wheel. 

Diameters  of  wheels  and  pinions  being  18  and  2,  their  radii  are  9  and  i. 
Hence,  i  XxXx:9X9X9  ::  37  699-2  :  27482716.8  Ibs. 


632        MECHANICAL    POWERS. — SCREWS. — PULLEY. 


Differential    Screw. 

When  a  hollow  screw  revolves  upon  one  of  less  diameter  and  pitch  (as 
designed  by  Mr.  Hunter),  effect  is  same  as  that  of  a  single  screw,  in  which 
the  distance  between  threads  is  equal  to  difference  of  distances  between 
threads  of  the  two  screws. 

Therefore  power,  to  effect  or  weight  sustained,  is  as  difference  between 
distances  of  threads  of  the  two  screws  to  circumference  described  by  power. 

ILLUSTRATION.  —  If  external  screw  has  20  threads,  and  internal  one  21  threads  in 
pitch  of  j  inch,  and  power  applied  describes  a  circumference  of  35  ins.,  the  result  or 

35  =.4706. 


power  is  as  —  co  —  =  — ,  or .  002  38.     Hence 
21      20     420 


.00238" 


600 


PULLEY. 

PULLEYS  are  designated  as  Fixed  and  Movable,  according  as  cord  is  passed 
ovfer  a  fixed  or  a  movable  pulley.  A  movable  pulley  is  when  cord  passes 
through  a  second  pulley  or  block  in  suspension ;  a  single  movable  pulley  is 
termed  a  runner;  and  a  combination  of  pulleys  is  termed  a  system  of  pulleys. 

A  Whip  is  a  single  cord  over  a  fixed  pulley. 

To  Compxite  JPower  Required  to  Raise  a  given  "Weight. 

When  Number  of  Parts  of  Cord  supporting  Lower  Block  are  given,  and 
when  only  one  Cord  or  Hope  is  used.  RULE. — Divide  weight  to  be  raised  by 
number  of  parts  of  cord  supporting  lower  or  movable  block. 

Or,  W  -r-  n  =  P.  Or,  n  P  =  W.  n  representing  number  of  parts  of  cord  sustain- 
*ng  lower  block. 

EXAMPLE.— What  power  is  required  to  raise  600  Ibs.  when  lower  block  contains 
jix  sheaves  ? 

When  Cord  is  attached  to  Upper  or  Fixed  Block. 
-  =  50  Ibs.  =  weight-?- number  of  parts  of  rope  sustaining  lower  block. 

When  Cord  is  attached  to  Lower  or  Movable  Block, 
— — —  =  46.15  Ibs.  =  weight  -r-  number  of  parts  of  rope  sustaining  lower  block. 

To   Compute   "\Veiglit  a,  given   Power   "will   Raise. 

When  Number  of  Parts  of  Cord  supporting  Lower  Block  are  given.    RULE. 
—Multiply  power  by  number  of  parts  of  cord  supporting  lower  block. 
Or,  P  n  =  W. 

To   Compute   Numtoer  of  Cords   necessary  to   Sustain 
Lower   Block. 

When  Weight,  and  Power  are  given.    RULE. — Divide  weight  by  power. 

When  more  than  one  Cord  is  used. 

In  a  Spanish  Burton,  Fig.  10,  where  ends  of 
one  cord,  a  P,  are  fastened  to  support  and  power, 
and  ends  of  the  other,  c  o,  to  lower  and  upper 
blocks,  weight  is  to  power  as  4  to  i. 

In  another,  Fig.  n,  where  there  are  two  cords, 
a  and  o,  two  movable  pulleys,  and  one  fixed 
pulley,  with  ends  of  one  rope  fastened  to  sup- 
port and  upper  movable  pulley,  and  ends  of 
other  fastened  to  lower  block  and  power,  weight 
is  to  power  as  5  to  i. 


Fig.  to. 


MECHANICAL    POWERS.  —  PULLEY. 


633 


Compound    or    Fast    and    Loose    IPvilleys. 

When  Cord  is  attached  to  Fixed  Block,  Fig.  12.  RULE.— 
Multiply  power  by  the  power  of  2,  of  which  the  index  is 
number  of  movable  pulleys. 

Or,  P  2W  =  W. 

Or,  Multiply  power  successively  by  2  for  each  pulley. 
EXAMPLE  i.— What  weight  will  one  pound  support  in  a  system 
of  three  movable  pulleys,  the  cords  being  connected  to  a  fixed 
block  on  Fig.  12.  i  x  23  =  8  Ibs 

EXAMPLE  2.— What  would  a  like  power  support,  fixed  block  be- 
ing made  movable  and  cord  attached  thereto? 

If  Qxed  pulleys  were  substituted  for  hooks  abc,  Fig.  12,  power 
would  be  increased  threefold;  hence  i  x  33  =  27. 

In  a  System  of  Pulleys,  Figs.  13  and  14,  with  any  Number  of  Cords,  oo,ee, 
Ends  being  fastened  to  Support. 

W  F'g-  '4- 

W-r-2»=P;   2«XP  =  W;  —  =  2"     nrep- 

resenting  number  of  distinct  cords. 

ILLUSTRATION.  —  What  weight  will  a  power 
of  i  Ib.  sustain  in  a  system  of  two  movable  pul-       j  i 
leys  and  two  cords  ? 

i  x  2  X  2  —  4  Ibs. 

When  fixed  Pulleys,  e  e,  are  used  in  Place      p 
of  Hooks,  to  Attach  Ends  of  Rope  to  Sup- 
port.—Fig.  14.  £ 

ILLUSTRATION.— What  weight  will  a  power  of  5  Ibs.  sustain  with  two  movable  and 
three  fixed  pulleys,  and  two  cords?        5  x  3  x  3  _  45  fos 

When  Ends  of  Cord  or  Fixed  Pulleys  are  fastened  to  Weic/ht,  as  by  an  Inver- 
sion of  the  last  Figures,  putting  Supports  for  Weights,  and  contrariwise. — 
Figs.  13  and  14. 


Fig.  13. 


Fig.  13- 
Fig.  14. 


ILLUSTRATION What  weight  will  a  power  of  i  Ib.  sustain  in  a  system  of  two  mov- 
able pulleys  and  two  cords,  and  one  of  two  movable  and  two  fixed  pulleys  and  two 
3ords9  1X2X2  —  1  =  3/65.  1X3X3  —  i  =  8/6*. 

When  Cords  sustaining  Pulleys  are  not  in  a  Vertical  Direction. — Fig.  15. 
Fig.  15.  eo,  Fig.  15,  is  vertical  line  through  which  weight  bears,  and 

^-  ^    from  o  draw  or,os  parallel  to  De  and  A  e. 

Forces  acting  at  e  are  represented  by  lines  es,  er,  and  eo; 
and  as  tension  of  every  part  of  cord  is  same,  and  equal  to 
power  P,  sides  o  s  and  or  of  parallelogram  must  be  equal,  and 
therefore  diagonal  e  o  divides  the  angle  r  o  s  into  two  equal 
portions.  Hence  the  weight  will  always  fall  into  the  position 
in  which  the  two  parts  of  cord  A  e  and  e  D  will  be  equally 
inclined  to  vertical  line,  and  it  will  bear  to  power  same  ratio 
as  eo  to  es. 

Therefore  W  :  P  : :  2  cos. .  5  e  :  i.    e  representing  angle  A  e  D. 
Or,  2  P  X  cos. .  5  e  —  W.    That  is,  twice  pewer,  multiplied  by  cosine  of  half  angle 
of  cord,  at  point  of  suspension  of  weight,  is  equal  to  weight. 


634 


METALS. — ALLOYS    AND    COMPOSITIONS. 


ILLUSTRATION.— What  weight  will  be  sustained  by  a  power  of  5  Ibs.,  with  an  ob- 
lique movable  pulley,  Fig.  15,  having  an  angle,  A  c  D,  of  30°  ? 

5  X  2  X  .965  93  =  9.6593  Ibs.  =  twice  power  X  cos.  15°. 

When  Direction  of  Cord  is  Irregular,  Weight  not  resting  in  Centre  of  it. 
P  sin.  a  P  sin.  (a-f-6)  W  sin.  a 


W      sin.  (a  -f-  b) '          sin.  a 
greater  and  lesser  angles  of  cord  at  e. 


sin.  (a +  6) 


=  P.    a  and  b  representing 


METALS. 
ALLOYS  AND   COMPOSITIONS. 

Alloy  is  the  proportion  of  a  baser  metal  mixed  with  a  finer  or  purer, 
as  copper  is  mixed  with  gold,  etc. 

Amalgam  is  a  compound  of  Mercury  and  a  metal — a  soft  alloy. 

Compositions  of  copper  contract  in  admixture,  and  all  Amalgams  ex- 
pand. 

In  manufacture  of  Alloys  and  Compositions,  the  less  fusible  metals 
should  be  melted  first. 

In  Compositions  of  Brass,  as  proportion  of  Zinc  is  increased,  so  is 
malleability  decreased. 

Tenacity  of  Brass  is  impaired  by  addition  of  Lead  or  Tin. 

Steel  alloyed  with  one  five-hundredth  part  of  Platinum,  or  Silver,  is 
rendered  harder,  more  malleable,  and  better  adapted  for  cutting  instru- 
ments. 

Specific  gravity  of  alloys*  does  not  follow  the  ratios  of  those  of  their 
components ;  it  is  sometimes  greater  and  sometimes  less  than  the  mean. 

Composition    for   'Welding    Cast    Steel. 

Borax,  91  parts;  Sal-ammoniac,  9  parts.  Grind  or  pound  them  roughly  together; 
fuse  them  in  a  metal-pot  over  a  clear  fire,  continuing  heat  until  all  spume  has  dis- 
appeared from  surface.  When  liquid  is  clear,  pour  composition  out  to  cool  and  con- 
crete, and  grind  to  a  fine  powder;  then  it  is  ready  for  use. 

To  use  this  composition,  the  steel  to  be  welded  should  be  raised  to  a  bright  yellow 
heat;  then  dip  it  in  the  welding  powder,  and  again  raise  it  to  a  like  heat  as  before; 
it  is  then  ready  to  be  submitted  to  the  hammer. 

F'usi'ble    Compounds. 


COMPOUNDS. 

Zinc. 

Tin. 

Lead. 

Bismuth. 

Cadmium. 

25 

2C 

5® 

33.3 

33.  3 

33-  4 

Newton's,  fusing  at  less  than  212°. 
Fusing  at  iso°  to  160°  .  .  , 

19 

3' 

33 

50 

50 

13 

Solders. 

Solder  is  an  alloy  used  to  make  joints  between  metals,  and  it  must  be 
more  fusible  than  the  metals  it  is  designed  to  unite,  and  it  is  distinguished 
as  hard  and  soft,  according  to  the  temperature  of  its  fusing. 

The  addition  of  a  small  portion  of  Bismuth  increases  its  fusibility. 


*  For  a  table  of  Alloys,  having  densities  different  from  » 
Manual,  London,  1877,  page  201, 


D  of  their  components,  see  D.  K.  Clark'a 


METALS. — ALLOYS   AND    COMPOSITIONS. 


635 


Alloys   and.  Compositions. 


Copper. 

Zinc. 

Tin. 

Nickel. 

Lead. 

Anti- 
mony. 

Bis- 
muth. 

Ala- 

minqm. 

55 
95 
3-7 
84-3 
75 
79-3 
92.2 
90 
80 
88.8 
74-3 
5° 
88.9 
90 

10 

3 

4 
66 

s7 

86 
67.2 
90 
93 
95 
80 

93 
91.4 
58.1 
40.4 
80 
69 

87.5 
72 

33-3 

40.4 

49-5 
81.6 

g 

87-5 
77-4 

60 

50 
66.6 

33-4 

& 

73 

24 

5-2 
25 
6.4 

}-*•"- 

20 
II.  2 
22.3 

3;.8 

80 
90 

33 
34 

13 
ii.  i 

31-2 

5-5 
17.2 
25-4 
5-6 

33-4 
25.4 

24 

7 

40 

45 

21 

5 

12.3 

89~ 
10.5 

14-3 
7-8 
9 

3-4 

8^3 
10 

10 

25 

2.9 
1.6 

10 

7 

5 

20 

7 
1.4 

2.6 
10.  1 

31 
12.5 

26.5 

iZ4 
23 

20 

12.5 

15-6 

86~ 
80 

22 
29 

m 

28~4 
4.4 
(Magn 
iSaHu 

21 

X9 

n.6 
31-6 

46 

7-3 

7 
47 

25 
25 

25 

| 
55 

2 

««.|  ?  (Cobalt  of  Iron.  ,  ,  ,  ,  ,  |  ,  ,  j,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  «  , 

Babbitt's  metal  *  

"           "       hard  
"      instruments  
"      locomot.  bearings. 
"      Pinchbeck  

"      rolled  

"      Tutenag  

44      very  tenacious... 
««     wheels,  valves.... 
**      white  

«             u 

«            It 

"      yellow,  fine  

When  fused  add  ...... 

i-7 
4-3 

"       yellow    

**       Gun  metal,  large 
"               "         small 
"          soft. 

"       Medals  

"       Statuary  ....... 

Chinese  silver  

"       white  copper... 
Church  bells  

Clocks,  Musical  bells.  .  .  . 
Clock  bells   ,  ... 

33-3 
31.6 

24 

75 

16.7 

— 

i-5 

2.6 
2-5 

12    ' 

ar.6.5 
..  .1.3 

(t                       41 

1 

s 

8.3 

of  tart 
irae.  .  . 

"          "    fine  
Gongs  

House  bells  

Machinery  bearings  
hard. 
Metal  that  expands  in) 
cooling  f 

Muntz  metal...  .  ....... 

Pewter,  best  

esia... 

nmonia 

20 

£ 

..4.4 
c.2.5 

14 

25 
12.5 
56.8 

Cream 
Quickl 

Sheathing  metal  

Speculum     "     . 

F     u                u 

Telescopic  mirrors  
Temper  t  

Type  metal  and  stereo-  ) 

White  metal  

"       "     bard  

Orelde  

*  See  page  636  for  direction*. 


i  For  adding  small  quantities  of  copper. 


636 


METALS. — ALLOYS    AND    COMPOSITIONS. 


Solders. 


Copper. 

Tin. 

Lead. 

Zinc. 

Silver. 

Bis- 
muth. 

Gold. 

Cad- 
mium. 

Anti- 
mony. 

Tin  

•je 

g 

16 

16 

"   coarse,  melts  ) 
at  500°  .  .  .  J 
"   ordi'y,  melts) 
at  360°...} 
Spelter,  soft  

— 

33 
67 

67 
33 

— 

— 

— 

_ 

_ 

"      hard  

Lead  

' 

67 

Steel  

' 

82 

Brass  or  Copper... 
Fine  brass  

50 

- 

- 

50 

- 

- 

- 

- 

Pewterers'  or  Soft. 

Plumbers'     pot-  ) 
metal..} 
"      coarse  .  .  . 
"      fine  

33 
50 

33 

25 
67 

45 
25 
67 

75 

- 

22 
25 

- 

- 

- 

"      fusible... 
"  very  "    ... 
Gold  

— 

So 
25 

50 
25 

— 

— 

50 

— 

— 

— 

"  hard  

66 

9 

"   soft  

66 

Silver,  hard  

20 

80 

«     soft  

Pewter  

' 

21 

66 

Copper  .  .  , 

c;^ 

J.7 

T 

A  Plastic  Metallic  Alloy.— See  Journal  of  Franklin  Institute,  vol.  xxxix., 
for  its  composition  and  manufacture. 


Page  55, 


Soldering  Fluid  for  use  with  Soft  Solder. 

To  2  fluid  oz.  of  Muriatic  acid  add  small  pieces  of  Zinc  until  bubbles  cease  to  rise. 
Add  .  5  a  teaspoonful  of  Sal-ammoniac  and  two  fluid  oz.  of  Water. 

By  the  application  of  this  to  Iron  or  Steel,  they  may  be  soldered  without  their  sur- 
faces being  previously  tinned. 

Fluxes  for  Soldering  or  Welding. 


Iron Borax. 

Tinned  iron Resin. 

Copper  and  Brass Sal-ammoniac. 


Zinc Chloride  of  zinc. 

Lead Tallow  or  resin. 

Lead  and  tin Resin  and  sweet  oil. 


Babbitt's    Anti-attrition    3VEetal. 

Melt  4  Ibs.  copper  ;  add  12  Ibs.  Banca  tin,  8  Ibs.  Regulus  of  antimony,  and  12 
Ibs.  more  of  Tin.  After  4  or  5  Ibs.  Tin  have  been  added,  reduce  heat  to  a  dull 
red,  then  add  remainder  of  metal  as  above. 

This  composition  is  termed  hardening  ;  for  lining,  melt  i  Ib.  of  this  hardening 
with  2  Ibs.  tin,  which  produces  the  lining  metal  for  use. 

As  this  metal  was  introduced  in  1839,  it  is  now  maintained  by  engineers  that 
the  increased  weight  of  machines  and  the  velocity  of  engines  and  dynamos 
require  an  appropriate  alloy  ;  and  it  is  claimed  by  engineers  that  Phoenix  Metal 
meets  existing  requirements. 

Brass. 

Brass  is  an  alloy  of  copper  and  zinc,  in  proportions  varying  with  purpose 
of  metal  required,  its  color  depending  upon  the  proportions. 

It  is  rendered  brittle  by  continued  impacts ;  more  malleable  than  copper 
when  cold,  but  is  impracticable  of  being  forged,  as  its  zinc  melts  at  a  low 
temperature.  Its  fusibility  is  governed  by  the  proportion  of  zinc  in  it;  a 
small  quantity  of  phosphorus  gives  it  fluidity. 


METALS. — ALLOYS   AND    COMPOSITIONS. — IRON.      637 

Bronze. 

Bronze  is  an  alloy  of  copper  and  tin ;  it  is  harder,  more  fusible,  and 
stronger  than  copper.  It  is  usually  known  as  Gun-metal. 

Aluminum  Bronze  contains  90  to  95  per  cent,  of  copper,  and  5  to  10  per 
cent,  aluminum. 

Phosphor  Bronze  contains  copper  and  tin  and  a  small  proportion  of  phos- 
phorus. It  wears  better  than  bronze. 

IRON. 

Foreign  substances  which  iron  contains  modify  its  essential  proper, 
ties.  Carbon  adds  to  its  hardness,  but  destroys  some  of  its  qualities, 
and  produces  Cast  Iron  or  Steel,  according  to  proportion  it  contains. 
Thus,  .25  per  cent,  renders  it  malleable,  .5  steel,  1.75  is  limit  of  weld- 
ing steel,  and  2  is  lowest  limit  of  cast  iron.  Sulphur  renders  it  fusible, 
difficult  to  weld,  and  brittle  when  heated,  or  "hot  short."  Phosphorus 
renders  it  " cold  short"  but  may  be  present  in  proportion  of  .002  to 
.003,  without  affecting  injuriously  its  tenacity.  Antimony,  Arsenic,  and 
Copper  have  same  effect  as  sulphur,  the  last  in  a  greater  degree.  Sili- 
con renders  it  hard  and  brittle.  Manganese,  in  proportion  of  .02,  ren- 
ders it  "  cold  short"  and  Vanadium  adds  to  its  ductility. 

Cast   Iron. 

Process  of  making  Cast  Iron  depends  much  upon  description  of  fuel  used ; 
whether  charcoal,  coke,  bituminous,  or  anthracite  coals.  A  larger  yield  from 
same  furnace,  and  a  great  economy  in  fuel,  are  effected  by  use  of  a  hot  blast. 
The  greater  heat  thus  produced  causes  the  iron  to  combine  with  a  larger 
percentage  of  foreign  substances. 

Cast  Iron  for  purposes  requiring  great  strength  should  be  smelted  with 
a  cold  blast.  Pig-iron,  according  to  proportion  of  carbon  which  it  contains, 
is  divided  into  Foundry  Iron  and  Forge  Iron,  latter  adapted  only  to  conver- 
sion into  malleable  iron ;  while  former,  containing  largest  proportion  of  car- 
bon, can  be  used  either  for  castings  or  bars. 

High  temperature  in  melting  injures  gun-metal.    • 

There  are  many  varieties  of  Cast  Iron,  differing  by  almost  insensible 
shades ;  the  two  principal  divisions  are  gray  and  white,  so  termed  from 
color  of  their  fracture.  Their  properties  are  very  different. 

Gray  Iron  is  softer  and  less  brittle  than  white ;  it  is  in  a  slight  degree 
malleable  and  flexible,  and  is  insonorous ;  it  can  easily  be  drilled  or  turned, 
and  does  not  resist  the  file.  It  has  a  brilliant  fracture,  of  a  gray,  or  some- 
times a  bluish-gray,  color ;  color  is  lighter  as  grain  becomes  closer,  and  its 
hardness  increases.  It  melts  at  a  lower  heat  than  white,  and  preserves  its 
fluidity  longer.  Color  of  the  fluid  metal  is  red,  and  deeper  in  proportion  as 
the  heat  is  lower ;  it  does  not  adhere  to  the  ladle ;  it  fills  molds  well,  con- 
tracts less,  and  contains  fewer  cavities  than  white;  edges  of  its  castings 
are  sharp,  and  surfaces  smooth  and  convex.  It  is  used  for  machinery  and 
ordnance  where  the  pieces  are  to  be  bored  or  fitted.  Its  tenacity  and  specific 
gravity  are  diminished  by  annealing. 

White  Iron  is  very  brittle  and  sonorous ;  it  resists  file  and  chisel,  and  is 
susceptible  of  high  polish ;  surface  of  its  castings  is  concave ;  fracture  pre- 
sents a  silvery  appearance,  generally  fine  grained  and  compact,  sometimes 
radiating  or  lamellar.  When  melted  it  is  white,  throws  off  a  great  number 
of  sparks,  and  its  qualities  are  the  reverse  of  those  of  gray  iron  ;  it  is  there- 
fore unsuitable  for  machinery  purposes.  Its  tenacity  is  increased,  and  its 
specific  gravity  diminished,  by  annealing. 

3  H 


638 


METALS. — IKON. 


Mottled  Iron  is  a  mixture  of  white  and  gray ;  it  has  a  spotted  appear- 
ance ;  flows  well,  and  with  few  sparks ;  its  castings  have  a  plane  surface, 
with  edges  slightly  rounded.  It  is  suitable  for  shot,  shells,  etc.  A  fine  mot- 
tled is  only  kind  suitable  for  castings  which  require  great  strength.  The 
kind  of  mottle  will  depend  much  upon  volume  of  the  casting.  A  medium- 
sized  grain,  bright  gray  color,  fracture  sharp  to  touch,  and  a  close,  compact 
texture,  indicate  a  good  quality  of  iron.  A  grain  either  very  large  or  very 
small,  a  dull,  earthy  aspect,  loose  texture,  dissimilar  crystals  mixed  together, 
indicate  an  inferior  quality. 

Besides  these  general  divisions,  the  different  varieties  of  pig-iron  are  more 
particularly  distinguished  by  numbers,  according  to  their  relative  hardness. 

No.  i. — Fracture  dark  gray,  crystals  large  and  highly  lustrous,  alike  to 
new  surface  of  lead.  It  is  the  softest  iron,  possessing  in  highest  degree  the 
qualities  belonging  to  gray  iron ;  it  has  not  much  strength,  but  on  account 
of  its  fluidity  when  melted,  and  of  its  mixing  advantageously  with  scrap 
iron  and  with  the  harder  kinds  of  cast  iron,  it  is  of  great  use  to  a  foundry. 

No.  2  is  harder,  closer  grained,  and  stronger  than  No.  i ;  it  has  a  gray 
color  and  considerable  lustre.  It  is  most  suitable  for  shot  and  shells. 

No.  3  is  harder  than  No.  2.  Fracture  white,  crystals  larger  and  brighter 
at  centre  than  at  the  sides ;  color  gray,  but  inclining  to  white ;  has  consid- 
erable strength,  but  is  principally  used  for  mixing  with  other  kinds  of  iron 
and  for  large  castings. 

No.  4  or  Bright. — Fracture  light  gray,  with  small  crystals  and  little  lustre, 
and  not  being  sufficiently  fusible  for  castings  it  is  used  for  conversion  to 
wrought  iron. 

No.  5.  Mottled.  —  Fracture  dull  white,  with  gray  specks,  and  a  line  of 
white  around  edge  or  sides  of  fracture. 

No.  6.  White. — Fracture  white,  with  little  lustre,  granulated  with  radiat- 
ing crystalline  surface.  It  is  hardest  and  most  brittle  of  all  descriptions, 
and  is  unfit  for  use  unless  mixed  with  other  grades,  or  for  being  converted 
to  an  inferior  wrought  iron. 

Qualities  of  these  descriptions  depend  upon  proportion  of  carbon,  and  upon 
state  in  which  it  exists  in  the  metal ;  in  darker  kinds  of  iron,  where  propor- 
tion is  sometimes  7  per  cent.,  it  exists  partly  in  state  of  graphite  or  plumbago, 
which  makes  the  iron  soft.  In  white  iron  the  carbon  is  thoroughly  com- 
bined with  the  metal,  as  in  steel. 

Cast  iron  frequently  retains  a  portion  of  foreign  ingredients  from  the  ore, 
such  as  earths  or  oxides  of  other  metals,  and  sometimes  sulphur  and  phos- 
phorus, which  are  all  injurious  to  its  quality. 

Foreign  substances,  and  also  a  portion  of  the  carbon,  are  separated  by 
melting  iron  in  contact  with  air,  and  soft  iron  is  thus  rendered  harder  and 
stronger.  Effect  of  remelting  varies  with  nature  of  the  iron  and  character 
of  ore  from  which  it  has  been  extracted ;  that  from  hard  ores,  such  as  mag- 
netic oxides,  undergoes  less  alteration  than  that  from  hematites,  the  latter 
being  sometimes  changed  from  No.  i  to  white  by  a  single  remelting  in  an 
air  furnace. 

Color  and  texture  of  cast  iron  depend  greatly  upon  volume  of  casting  and 
rapidity  of  its  cooling ;  a  small  casting,  which  cools  quickly,  is  almost  always 
white,  and  surface  of  large  castings  partakes  more  of  the  qualities  of  white 
metal  than  the  interior. 

All  cast  iron  expands  at  moment  of  becoming  liquid,  and  contracts  in  cool- 
ing ;  gray  iron  expands  more  and  contracts  less  than  other  iron. 

Remelting  iron  improves  its  tenacity ;  thus,  a  mean  of  14  cases  for  two 
fusions  gave,  for  ist  fusion,  a  tenacity  of  29  284  Ibs. ;  for  2d  fusion,  33  790 
Ibs.  For  two  cases — for  first  fusion,  15  129  Ibs. ;  for  2d  fusion,  35  786  Ibs. 


METALS. — IRON.  639 

Malleable   Castings. 

Malleable  cast  iron  is  made  by  subjecting  a  casting  to  a  process  of  anneal- 
ing, by  enclosing  it  in  a  box  with  hematite  iron  ore  or  black  oxide  of  iron, 
and  maintaining  it  in  an  equable  heat  for  a  period  depending  upon  form  and 
volume  of  casting. 

"Wrought  Iron- 
Wrought  iron  is  made  from  pig-iron  in  a  Bloomeiy  Fire  or  in  a  Puddling 
Furnace— generally  in  latter.  Process  consists  in  melting  and  keeping  it 
exposed  to  a  great  heat,  constantly  stirring  the  mass,  bringing  every  part  of 
it  under  action  of  the  flame  until  it  loses  its  remaining  carbon,  when  it  be- 
comes malleable  iron.  When,  however,  it  is  desired  to  obtain  iron  of  best 
quality,  pig-iron  should  be  refined. 

Refining.— This  operation  deprives  iron  of  a  considerable  portion  of  its 
carbon ;  it  is  effected  in  a  Blast  Furnace,  where  iron  is  melted  by  means  of 
charcoal  or  coke,  and  exposed  for  some  time  to  action  of  a  great  heat ;  the 
metal  is  then  run  into  a  cast-iron  mold,  by  which  it  is  formed  into  a  large 
broad  plate.  As  soon  as  surface  of  plate  is  chilled,  cold  water  is  poured  on 
to  render  it  brittle. 

A  Bloomery  resembles  a  large  forge  fire,  where  charcoal  and  a  strong  blast 
are  used ;  and  the  refined  metal  or  pig-iron,  after  being  broken  into  pieces  of 
proper  size,  is  placed  before  the  blast,  directly  in  contact  with  charcoal ;  as 
the  metal  fuses,  it  falls  into  a  cavity  left  for  that  purpose  below  the  blast, 
where  the  "  bloomer  "  works  it  into  the  shape  of  a  ball,  which  he  places  again 
before  the  blast,  with  fresh  charcoal ;  this  operation  is  generally  again  re- 
peated, when  ball  is  ready  for  the  "  shingler." 

Shingling  is  performed  in  a  strong  squeezer  or  under  a  trip-hammer.  Its 
object  is  to  press  out  as  perfectly  as  practicable  the  liquid  cinder  which  a 
ball  contains ;  it  also  forms  a  ball  into  shape  for  the  puddle  rolls.  A  heavy 
hammer,  weighing  from  6  to  7  tons,  effects  this  object  most  thoroughly,  but 
not  so  cheaply  as  the  squeezer.  A  ball  receives  from  15  to  20  blows  of  a 
hammer,  being  turned  from  time  to  time  as  required :  it  is  now  termed  a 
Bloom,  and  is  ready  to  be  rolled  or  hammered ;  or  a  ball  is  passed  once 
through  the  squeezer,  and  is  still  hot  enough  to  be  passed  through  the  puddle 
rolls. 

A  Puddling  Furnace  is  a  reverberatory  furnace,  where  flame  of  bituminous 
coal  is  brought  to  act  directly  upon  the  melted  metal.  The  "  puddler  "  then 
stirs  it,  exposing  each  portion  in  turn  to  action  of  flame,  and  continues  this 
as  long  as  he  is  able  to  work  it.  When  it  has  lost  its  fluidity,  he  forms  it  into 
balls,  weighing  from  80  to  100  Ibs.,  which  are  then  passed  to  the  "shingler." 

Puddle  Rolls.  —  By  passing  through  different  grooves  in  these  rolls,  a 
bloom  is  reduced  to  a  rough  bar  from  3  to  4  feet  in  length,  its  term  convey- 
ing an  idea  of  its  condition,  which  is  rough  and  imperfect. 

Piling. — To  prepare  rough  bars  for  this  operation,  they  are  cut,  by  a  pair 
of  shears,  into  such  lengths  as  are  best  adapted  to  the  volume  of  finished  bar 
required ;  the  sheared  bars  are  then  piled  one  over  the  other,  according  to 
volume  required,  when  pile  is  ready  for  balling. 

Balling. — This  operation  is  performed  in  balling  furnace,  which  is  similar 
to  puddling  furnace,  except  that  its  bottom  or  hearth  is  made  up,  from  time 
to  time,  with  sand ;  it  is  used  to  give  a  welding  heat  to  piles  to  prepare 
them  for  rolling. 

Finishing  Rolls. — The  balls  are  passed  successively  between  rollers  of  va- 
rious forms  and  dimensions,  according  to  shape  of  finished  bar  required. 

Quality  of  iron  depends  upon  description  of  pig-iron  used,  skill  of  the 
kt  puddler,"  and  absence  of  deleterious  substances  in  the  furnace. 


640  METALS. — IRON. — LEAD. — STEEL. 

Strongest  cast  irons  do  not  produce  strongest  malleable  iron. 

For  many  purposes,  such  as  sheets  for  tinning,  best  boiler-plates,  and  bars 
for  converting  into  steel,  charcoal  iron  is  used  exclusively ;  and,  generally, 
this  kind  of  iron  is  to  be  relied  upon,  for  strength  and  toughness,  with  greater 
confidence  than  any  other,  though  iron  of  a  superior  quality  is  made  from 
pigs  made  with  other  fuel,  and  with  a  hot  blast.  Iron  for  gun-barrels  has 
been  lately  made  from  anthracite  hot-blast  pigs. 

Iron  is  improved  in  quality  by  judicious  working,  reheating,  hammering, 
or  rolling :  other  things  being  equal,  best  iron  is  that  which  has  been  wrought 
the  most. 

Best  quality  of  iron  has  greatest  elasticity. 

Tests. — It  will  not  blacken  if  exposed  to  nitric  acid.  Long  silky  fibres  in 
a  fracture  denote  a  soft  and  strong  metal ;  short  black  fibres  denote  a  badly 
refined  metal,  and  a  fine  grain  denotes  hardness  and  condition  known  as 
u  cold  short."  Coarse  grain  with  bright  and  crystallized  fracture,  with  dis- 
colored spots,  also  denotes  "  cold  short "  and  brittle  metal,  working  easily  and 
welding  well.  Cracks  upon  edges  of  a  bar,  etc.,  indicate  "  hot  short."  Good 
iron  heats  readily,  is  worked  easily,  and  throws  off  but  few  sparks. 

A  high  breaking  strain  may  not  be  conclusive  as  to  quality,  as  it  may  be 
due  to  a  hard,  elastic  metal,  or  a  low  one  may  be  due  to  great  softness. 

When  iron  is  fractured  suddenly,  a  crystalline  surface  is  produced,  and 
when  gradually,  a  fibrous  one.  Breaking  strain  of  iron  is  increased  by  heat- 
ing it  and  suddenly  cooling  it  in  water.  Iron  exposed  to  a  welding  or  white 
heat  and  not  reduced  by  hammering  or  rolling  is  weakened. 

Specific  gravity  of  iron  is  a  good  indication  of  its  quality,  as  it  indicates 
very  correctly  its  relative  degree  of  strength. 

LEAD. 

Sheet  Lead  is  either  Cast  or  Milled,  the  former  in  sheets  16  to  18  feet  in 
length  and  6  feet  in  width ;  the  latter  is  rolled,  is  thinner  than  the  former, 
is  more  uniform  in  its  thickness,  and  is  made  into  sheets  25  to  35  feet  in 
length,  and  from  6  to  7.5  feet  in  width. 

Soft  or  Rain  Water,  when  aerated,  Silt  of  rivers,  Vegetable  matter,  Acids, 
Mortar,  and  Vitiated  Air  will  oxidize  lead.  The  waters  which  act  with 
greatest  effect  on  it  are  the  purest  and  most  highly  oxygenated,  also  nitrites, 
nitrates,  and  chlorides,  and  those  which  act  with  least  effect  are  such  as  con- 
tain carbonate  and  phosphate  of  lime. 

Coating  of  Pipes,  except  with  substances  insoluble  in  water,  as  Bitumen 
and  Sulphide  of  lead,  is  objectionable. 

Lead-encased  Pipes. — An  inner  pipe  of  tin  is  encased  in  one  of  lead. 

STEEL. 

Steel  is  a  compound  of  Iron  and  Carbon,  in  which  proportion  of  latter 
is  from  I  to  5  per  cent.,  and  even  less  in  some  descriptions.  It  is  dis- 
tinguished from  iron  by  its  fine  grain,  and  by  action  of  diluted  nitric 
acid,  which  leaves  a  black  spot  upon  it. 

There  are  many  varieties  of  steel,  principal  of  which  are : 

Natural  Steel,  obtained  by  reducing  rich  and  pure  descriptions  of  iron 
ore  with  charcoal,  and  refining  cast  iron,  so  as  to  deprive  it  of  a  sufficient 
portion  of  carbon  to  bring  it  to  a  malleable  state.  It  is  used  for  files  and 
other  tools. 

Indian  Steel,  termed  Wootz,  is  said  to  be  a  natural  steel,  containing  a  small 
portion  of  other  metals. 


METALS.  — STEEL.  64 1 

Blistered  Steel,  or  Steel  of  Cementation,  is  prepared  by  direct  combination  of 
iron  and  carbon.  For  this  purpose,  iron  in  bars  is  put  in  layers,  alternating 
with  powdered  charcoal,  in  a  close  furnace,  and  exposed  for  7  or  8  days  to 
a  high  temperature,  and  then  put  to  cool  for  a  like  period.  The  bars,  on 
being  taken  out,  are  covered  with  blisters,  have  acquired  a  brittle  quality, 
and  exhibit  in  fracture  a  uniform  crystalline  appearance.  The  degree  of 
carbonization  is  varied  according  to  purposes  for  which  the  steel  is  intended, 
and  the  very  best  qualities  of  iron  are  used  for  the  finest  kinds  of  steel. 

Tilted  Steel  is  made  from  blistered  steel  moderately  heated,  and  subjected 
to  action  of  a  tilt  hammer,  by  which  means  its  tenacity  and  density  are  in- 
creased. 

Shear  Steel  is  made  from  blistered  or  natural  steel,  refined  by  piling  thin 
bars  into  fagots,  which  are  brought  to  a  welding  heat  in  a  reverberatory 
furnace,  and  hammered  or  rolled  again  into  bars ;  this  operation  is  repeated 
several  times  to  produce  finest  kinds  of  shear  steel,  which  are  distinguished 
by  the  terms  of  Half  shear,  Single  shear,  and  Double  shear,  or  steel  of  i,  2,  or 
3  mark*,  etc.,  according  to  number  of  times  it  has  been  piled. 

Spring  Steel  is  blister  steel  heated  to  an  orange  red  color  and  rolled  or 
hammered. 

Cast  or  Crucible  Steel  is  made  by  breaking  blistered  steel  into  small  pieces 
and  melting  it  in  close  crucibles,  from  which  it  is  poured  into  iron  molds ; 
ingot  is  then  reduced  to  a  bar  by  hammering  or  rolling.  Cast  steel  is  best 
kind  of  steel,  and  best  adapted  for  most  purposes ;  it  is  known  by  a  very 
fine,  even,  and  close  grain,  and  a  silvery,  homogeneous  fracture ;  it  is  very 
brittle,  and  acquires  extreme  hardness,  out  is  difficult  to  weld  without  use 
of  a  flux.  Other  kinds  of  steel  have  a  similar  appearance  to  cast  steel,  but 
grain  is  coarser  and  less  homogeneous ;  they  are  softer  and  less  brittle,  and 
weld  more  readily.  A  fibrous  or  lamellar  appearance  in  fracture  indicates 
an  imperfect  steel.  A  material  of  great  toughness  and  elasticity,  as  well  as 
hardness,  is  made  by  forging  together  steel  and  iron,  forming  the  celebrated 
Damasked  Steel,  which  is  used  for  sword-blades,  springs,  etc. ;  damask  ap- 
pearance of  which  is  produced  by  a  diluted  acid,  which  gives  a  black  tint  to 
the  steel,  while  the  iron  remains  white. 

With  cast  steel,  breaking  strength  is  greater  across  fibres  of  rolling  than 
with  them. 

Heath's  Process  is  an  improvement  on  this  method,  and  consists  in  adding  to 
molten  metal  a  small  quantity  of  carburet  of  manganese. 

Heaton's  Process  consists  in  adding  nitrate  of  soda  to  molten  pig-iron,  in  order  to 
remove  carbon  and  silica. 

Musket's  Process. — Malleable  iron  is  melted  in  crucibles  with  oxide  of  manganese 
and  charcoal. 

Puddled  Steel  is  produced  by  arresting  the  puddling  in  the  manufacture 
of  the  wrought  iron  before  all  the  carbon  has  been  removed,  the  small 
amount  of  carbon  remaining,  .3  to  i  per  cent.,  being  sufficient  to  make  an 
inferior  steel. 

Mild  Steel  contains  from  .2  to  .5  per  cent,  of  carbon ;  when  more  is  pres- 
ent it  is  termed  Hard  Steel. 

Bessemer  Steel  is  made  direct  from  pig-iron.  The  carbon  is  first  removed, 
in  order  to  obtain  pure  wrought  iron,  and  to  this  is  added  the  exact  quantity 
of  carbon  required  for  the  steel.  The  pig  should  be  free  from  sulphur  and 
phosphorus.  It  is  melted  in  a  blast  or  cupola,  and  run  into  a  converter  (a 
pear-shaped  iron  vessel  suspended  on  hollow  trunnions  and  lined  with  fire- 
brick or  clay),  where  it  is  subjected  to  an  air  blast  for  a  period  of  20  min- 
utes, in  order  to  dispel  the  carbon,  after  which  from  5  to  10  per  cent,  of  apie- 
geleisen  is  added. 

3  IP 


642  METALS. — STEEL. 

The  blast  is  then  resumed  for  a  short  period,  to  incorporate  the  two  metals, 
when  the  steel  is  run  off  into  molds.  The  moment  at  which  all  the  carbon 
has  been  removed  is  indicated  by  color  of  the  flame  at  mouth  of  converter. 
The  ingots,  when  thus  produced,  contain  air  holes,  and  it  becomes  necessary 
to  heat  them  and  render  them  solid  under  a  hammer. 

Siemens  Process.— Pig-iron  is  fused  upon  open  hearth  of  a  regenerative 
furnace,  and  when  raised  to  a  steel-melting  temperature,  rich  and  pure  ore 
and  limestone  are  added  gradually,  whereby  a  reaction  is  established  between 
the  oxygen  of  the  ferrous  oxide  and  the  carbon  and  silicon  in  the  metal.  The 
silicon  is  thus  converted  into  silicic  acid,  which  with  the  lime  forms  a  fusible 
slag,  and  the  carbon,  combining  with  oxygen,  escapes  as  carbonic  acid,  and 
induces  a  powerful  ebullition. 

Modification  of  this  process.— The  ore  is  treated  in  a  separate  rotatory  furnace 
with  carbonaceous  material,  and  converted  into  balls  of  malleable  iron,  which  are 
transferred  from  the  rotatory  to  the  bath  of  the  steel-melting  furnace. 

This  process  is  adapted  to  the  production  of  steel  of  a  very  high  quality,  because 
the  sulphur  and  phosphorus  of  the  ore  are  separated  from  the  metal  in  the  rotatory 
furnace. 

Siemens -Martin  Process. — Scrap-iron  or  steel  is  gradually  added  in  a 
highly  heated  condition  to  a  bath  of  about  .25  its  weight,  of  highly  heated 
pig,  and  melted.  Samples  are  occasionally  taken  from  the  bath,  in  order  to 
ascertain  the  percentage  of  carbon  remaining  in  the  metal,  and  ore  is  added 
in  small  quantities,  in  order  to  reduce  the  carbon  to  about  .1  per  cent. 

At  this  stage  of  the  process,  siliceous  iron,  spiegeleisen,  or  ferro-manganese 
is  added  in  such  proportions  as  are  necessary  to  produce  steel  of  the  required 
degree  of  hardness.  The  metal  is  then  tapped  into  a  ladle. 

Landore-Siemen's  Steel  is  a  variety  of  steel  made  by  the  Modification  of 
Siemens  Process.  Its  great  value  is  due  to  its  extreme  ductility,  and  its 
having  nearly  like  strength  in  both  directions  of  its  plates. 

Whitworth's  Compressed  Steel  is  molten  steel  subjected  to  a  pressure  of 
about  6  tons  per  square  inch,  by  which  all  its  cavities  are  dispelled,  and  it  is 
compressed  to  about  .875  of  its  original  volume,  its  density  and  strength  be- 
ing proportionately  increased. 

Chrome  and  Tungsten  Steel  are  made  by  adding  a  small  percentage  of 
Chromium  or  Tungsten  to  crucible  steel,  the  result  producing  a  steel  of 
great  hardness  and  tenacity,  suitable  for  tools,  such  as  drills,  etc. 

Homogeneous  Steel  is  a  variety  of  cast  steel  containing  .25  per  cent,  of 
carbon. 

Remarks  on  Manufacture  of  Steel,  and  Mode  of  Working  it. 
(D.  Chernoff,  1868). 

Steel,  when  cast  and  allowed  to  cool  quietly,  assumes  a  crystalline  structure. 
Higher  temperature  to  which  it  is  heated,  softer  it  becomes,  and  greater  is  liberty 
its  particles  possess  to  group  themselves  into  crystals. 

Steel,  however  hard  it  may  be,  will  not  harden  if  heated  to  a  temperature  lower 
than  what  may  be  distinguished  as  dark  cherry-red,  a,  however  quickly  it  is  cooled , 
on  contrary,  it  will  become  sensibly  softer,  and  more  easily  worked  with  a  file. 

Steel,  heated  to  a  temperature  lower  than  red,  but  not  sparkling,  &,  does  not 
change  its  structure  whether  cooled  quickly  or  slowly.  When  temperature  has 
reached  6,  substance  of  steel  quickly  passes  from  granular  or  crystalline  condition 
to  amorphous,  or  wax-like  structure,  which  it  retains  up  to  its  melting-point,  c. 

Points  a,  6,  and  c  have  no  permanent  place  in  scale  of  temperature,  but  their  posi- 
tions vary  with  quality  of  steel;  in  pure  steel,  they  depend  directly  on  quantity  of 
constituent  carbon.  Harder  the  steel,  lower  the  temperatures.  Tints  above  speci- 
fied have  reference  only  to  hard  and  medium  qualities  of  steel;  in  very  soft  kinds 
of  steel,  nearly  approaching  to  wrought  iron,  points  a  and  b  range  very  high,  and  in 
wrought  iron  point  6  rises  to  a  white  heat. 


METALS. — STEEL.  643 

Assumption  of  the  crystalline  structure  takes  place  entirely  in  cooling,  between 
temperatures  c  and  6;  when  temperature  sinks  below  6  there  is  no  change  of  struc- 
ture. For  successful  forging,  therefore,  heated  ingot,  after  it  is  taken  out  of  furnace, 
must  be  forged  as  quickly  as  practicable,  so  as  not  to  leave  any  spot  untouched  by 
hammer,  where  the  steel  might  crystallize  quietly,  as  formation  of  crystals  should 
be  hindered,  and  the  steel  should  be  kept  in  an  amorphous  condition  until  tem- 
perature sinks  below  point  6. 

Below  this  temperature,  if  piece  is  cooled  in  quiet,  mass  will  no  longer  be  disposed 
to  crystallize,  but  will  possess  great  tenacity  and  homogeneousness  of  structure. 

When  steel  is  forged  at  temperatures  lower  than  6,  its  crystals  or  grains,  being 
driven  against  each  other,  change  their  shapes,  becoming  elongated  in  one  direction, 
and  contracted  in  another;  while  density  and  tensile  strength  are  considerably  in- 
creased. But  available  hammer-power  is  only  sufficient  for  treatment  of  small  steel 
forgings;  and  object  of  preventing  coarse  crystalline  structure  in  large  forgings 
is  more  easily  and  more  certainly  effected,  if,  after  having  given  forging  desired 
shape,  its  structure  be  altered  to  an  homogeneous  amorphous  condition  by  heating 
it  to  a  temperature  somewhat  higher  than  6,  and  the  condition  be  fixed  by  rapid 
cooling  to  a  temperature  lower  than  &,  the  piece  should  then  be  allowed  to  finish 
cooling  gradually,  so  as  to  prevent,  as  far  as  practicable,  internal  strains  due  to 
sudden  and  unequal  contraction. 

Alloys  of  steel  with  Silver,  Platinum,  Rhodium,  and  Aluminum  have  been 
made  with  a  view  to  imitating  Damascus  steel,  Wootz,  etc.,  and  improving 
fabrication  of  some  finer  kinds  of  surgical  and  other  instruments. 

Properties  of  Steel. — After  being  tempered  it  is  not  easily  broken ;  it  welds 
readily ;  does  not  crack  or  split ;  bears  a  very  high  heat,  and  preserves  the 
capability  of  hardening  after  repeated  working. 

Hardening  and  Tempering. — Upon  these  operations  the  quality  of  manu- 
factured steel  in  a  great  measure  depends. 

Hardening  is  effected  by  heating  steel  to  a  cherry-red,  or  until  scales  of 
oxide  are  loosened  on  surface,  and  plunging  it  into  a  cooling  liquid ;  degree 
of  hardness  depends  upon  heat  and  rapidity  of  cooling.  Steel  is  thus  ren- 
dered so  hard  as  to  resist  files,  and  it  becomes  at  same  time  extremely 
brittle.  Degree  of  heat,  and  temperature  and  nature  of  cooling  medium, 
must  be  chosen  with  reference  to  quality  of  steel  and  purpose  for  which  it 
is  intended.  Cold  water  gives  a  greater  hardness  than  oils  or  like  sub- 
stances, sand,  wet-iron  scales,  or  cinders,  but  an  inferior  degree  of  hardness 
to  that  given  by  acids.  Oil,  tallow,  etc.,  prevent  cracks  caused  by  too  rapid 
cooling.  Lower  the  heat  at  which  steel  becomes  hard,  the  better. 

Tempering. — Steel  in  its  hardest  state  being  too  brittle  for  most  purposes, 
the  requisite  strength  and  elasticity  are  obtained  by  tempering — or  "  letting 
down  the  temper  " — which  is  performed  by  heating  hardened  steel  to  a  certain 
degree  and  cooling  it  quickly.  Requisite  heat  is  usually  ascertained  by  color 
which  surface  of  the  steel  assumes  from  film  of  oxide  thus  formed.  Degrees 
of  heat  to  which  these  several  colors  correspond  are  as  follows : 
At  430°,  very  faint  yellow..  (Suitable  for  hard  instruments;  as  hammer  -  faces, 
At  450°,  pale  straw  color. ...  \  drills,  lancets,  razors,  etc. 

At  470°,  full  yellow ( For  instruments  requiring  hard  edges  without  elastici- 

At  490°,  brown  color (     ty ;  as  shears,  scissors,  turning  tools,  penknives,  etc. 

knots'  br°WD'  WUh  PUrPl6  (For  tools  for  cuttin&  wood  and  soft  metals;  such  as 
At  538°,  purple  .'!!.'!!.'!!!!!(     Plane-irons>  saws»  knives,  etc. 

At  550°' dark  blue (For  tools  requiring  strong  edges  without  extreme 

At  560°,  full  blue I     hardness;  as  cold-chisels,  axes,  cutlery,  etc. 

At  600°,  grayish  blue,  verg-  j  For  spring- temper,  which  will  bend  before  breaking; 
ing  on  black \     as  saws,  sword-blades,  etc. 

If  steel  is  heated  to  a  higher  temperature  than  this,  effect  of  the  hardening 
process  is  destroyed. 

A  high  breaking  strain  may  not  be  conclusive  as  to  quality,  as  it  may  be 
due  to  a  hard,  elastic  metal,  or  a  low  one  may  be  due  to  great  softness. 


644  METALS. — TIN. — ZINC. — MODELS. 

Case-h.arden.ing. 

This  operation  consists  in  converting  surface  of  wrought  iron  into  steelT 
by  cementation,  for  purpose  of  adapting  it  to  receive  a  polish  or  to  bear  fric- 
tion, etc. ;  it  is  effected  by  heating  iron  to  a  cherry-red,  in  a  close  vessel,  in 
contact  with  carbonaceous  materials,  and  then  plunging  it  into  cold  water. 
Bones,  leather,  hoofs,  and  horns  of  animals  are  generally  used  for  this  pur- 
pose, after  having  been  burned  or  roasted  so  that  they  can  be  pulverized. 
Soot  is  also  frequently  used. 

The  operation  reduces  strength  of  the  iron. 

TIN. 

TIN  is  more  readily  fused  than  any  other  metal,  and  oxidizes  very  slowly. 
Its  purity  is  tested  by  its  extreme  brittleness  at  high  temperature. 
Tin  plate  is  iron  plate  coated  with  tin. 
Block  Tin  is  tin  plate  with  an  additional  coating  of  tin. 

ZINC. 

ZINC,  if  pure,  is  malleable  at  220°  ;  at  higher  temperatures,  such  as  400°, 
it  becomes  brittle.  It  is  readily  acted  upon  by  moist  air,  and  when  a  film 
of  oxide  is  formed,  it  protects  the  surface  from  further  action.  When,  how- 
ever, the  air  is  acid,  as  from  the  sea  or  large  towns,  it  is  readily  oxidized  to 
destruction. 

Iron,  Copper,  Lead,  and  Soot  are  very  destructive  of  it,  in  consequence  of 
the  voltaic  action  generated,  and  it  should  not  be  in  contact  with  calcareous 
water  or  acid  woods. 

The  best  quality,  as  that  known  as  "  Vielle  Montagne,"  is  composed  of  zinc 
.995,  iron  .004,  and  lead  .001.  Its  expansion  and  contraction  by  differences 
of  temperature  is  in  excess  of  that  of  any  other  metal. 


STRENGTH  OF  MODELS. 

The  forces  to  which  Models  are  subjected  are, 

i.  To-draw  them  asunder  by  tensile  stress.  2.  To  break  them  by  trans- 
verse stress.  3.  To  crush  them  by  compression. 

The  stress  upon  side  of  a  model  is  to  corresponding  side  of  a  structure  as 
cube  of  its  corresponding  magnitude.  Thus,  if  a  structure  is  six  times  greater 
than  its  model,  the  stress  upon  it  is  as  63  to  i  =  216  to  i :  but  resistance  of 
rupture  increases  only  as  squares  of  the  corresponding  magnitudes,  or  as 
62  to  i  =36  to  i.  A  structure,  therefore,  will  bear  as  much  less  resistance 
than  its  model  as  its  side  is  greater. 

To    Com.pu.te    Dimensions    of    a    Beam,  etc.,  -which    a 
Strvi.ctu.re    can   "bear. 

RULE. — Divide  greatest  weight  which  the  beam,  etc.  (including  its  weight), 
in  the  model  can  bear,  by  the  greatest  weight  which  the  structure  is  required 
to  bear  (including  its  weight),  and  quotient,  multiplied  by  length  of  beam, 
etc.,  in  model,  will  give  length  of  beam,  etc.,  in  structure. 

EXAMPLE. — A  beam  in  a  model  7  inches  in  length  is  capable  of  bearing  a  weight 
of  26  Ibs.,  but  it  is  required  to  sustain  only  a  weight  or  stress  of  4  Ibs. ;  what  is  the 
greatest  length  that  a  corresponding  beam  can  be  made  in  the  structure? 
26 -r- 4  =  6.5,  and  6. 5  X  7  =  45-5  »»*• 


In  structure. 


MODELS. — MOTION    OF   BODIES   IN    FLUIDS,          645 

Resistance  in  a  model  to  crushing  increases  directly  as  its  dimensions; 
but  as  stress  increases  as  cubes  of  dimensions,  a  model  is  stronger  than  the 
structure,  inversely  as  the  squares  of  their  comparative  magnitudes. 

Hemce,  greatest  magnitude  of  a  structure  is  ascertained  by  taking  square 
root  of  quotient,  as  obtained  by  preceding  rule,  instead  of  quotient  itself. 

EXAMPLE. — If  greatest  weight  which  a  column  in  a  model  can  sustain  is  26  Ibs., 
and  it  is  required  to  bear  only  4  Ibs. ;  height  of  column  being  18  ins.,  what  should 
be  height  of  it  in  structure  ? 

/( —  \  =  ^6.5  =  2.55,  and  2.55  X  18  =  45.9  ins.,  height  of  column  w 

If,  when  length  or  height  and  breadth  are  retained,  and  it  is  required  to 
give  to  the  beam,  etc.,  such  a  thickness  or  depth  that  it  will  not  break  in  con- 
sequence of  its  increased  dimensions, 

Then  /(  — )  =  V6-5  =  2-55>  which,  x  square  of  relative  size  of  model  =  thick- 
ness required. 

To    Compute   Resistance   of  a   Bridge   from,  a  Alodel. 
n2  W  —   —  (n  —  i)  w\  =  load  bridge  will  bear  in  its  centre. 

EXAMPLE.— If  length  of  the  platform  of  a  model  between  centres  of  its  repose 
upon  the  piers  is  12  feet,  its  weight  30  Ibs.,  and  the  weight  it  will  just  sustain  at  its 
centre  350  Ibs. ,  the  comparative  magnitudes  of  model  and  bridge  as  20,  and  actual 
length  of  bridge  240  feet ;  what  weight  will  bridge  sustain  ? 

2o2  X  350—  [^  X  (20—1)  X  30!  =  140000  —  3800  X  30  =  26000  Ibs. 


MOTION  OF  BODIES  IN  FLUIDS. 

If  a  body  move  "through  a  fluid  at  rest,  or  fluid  move  against  body  at 
rest,  resistance  of  fluid  against  body  is  as  square  of  velocity  and  density 
of  fluid  ;  that  is,  R  =  d  v*.  For  resistance  is  as  quantity  of  matter  or 
particles  struck,  and  velocity  with  wL\ch  they  are  struck.  But  quan- 
tity or  number  of  particles  struck  in  any  time  are  as  velocity  and  density 
of  fluid ;  therefore,  resistance  of  a  fluid  is  as  density  and  square  of 
velocity. 

— =  h,  and  =  R.    h  representing  height  due  to  velocity,  d  density  of  fluid, 

2  g  zg 

and  R  resistance  or  motive  force. 

Resistance  to  a  plane  is  as  plane  is  greater  or  less,  and  therefore  resistance 
to  a  plane  is  as  its  area,  density  of  medium,  and  square  of  velocity;  that  is, 
R  =  adv*. 

Motion  is  not  perpendicular,  but  oblique,  to  plane  or  to  face  of  body  in  any 
angle,  sine  of  which  is  *  to  radius  i ;  then  resistance  to  plane,  or  force  of 
fluid  against  plane,  in  direction  of  motion,  will  be  diminished  in  triplicate 
ratio  of  radius  to  sine  of  angle  of  inclination,  or  in  ratio  of  i  to  s3. 

Hence.  - — — —  —  R,  and  - — ^— ^-  =  F.    w  representing  weight  of  body,  and  F 

2  g  2  g  w 

retarding  force. 

Progression  of  a  solid  floating  body,  as  a  boat  in  a  channel  of  still  water, 
gives  rise  to  a  displacement  of  water  surface,  which  advances  with  an  un- 
dulation in  direction  of  body,  and  this  undulation  is  termed  Wave  of  Dis- 
placement. 


646 


MOTION   OF   BODIES   IN   FLUIDS. 


Resistance  of  a  fluid  to  progression  of  a  floating  body  increases  as  velocity 
of  body  attains  velocity  of  wave  of  displacement,  and  it  is  greatest  when  the 
two  velocities  are  equal. 

In  the  motion  of  elastic  fluids,  it  appears  from  experiments  that  oblique 
action  produces  nearly  same  effect  as  in  motion  of  water,  in  the  passage  of 
curvatures,  apertures,  etc. 

Resistance  to  an  Area  of  One  Sq..  Foot  moving  through. 
Water,  or    Contrariwise. 


Angle  of 

Angle  of 

Surface 
with 
Plane  of 

Pressure  per  Sq.  Foot  for  following  Ve- 
locities per  Foot  per  Minute. 

Surface 
with 
Plane  of 

Pressure  per  Sq.  Foot  for  following  Ve- 
locities per  Foot  per  Minute. 

Current. 

120 

240 

480 

900 

Current. 

1  20 

240 

480 

900 

0 

Lbs 

Lbs. 

Lb». 

Lbs. 

0 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

6 

.09 

•359 

'•435 

5.046 

45 

2.66 

10.639 

42.557 

149.614 

8 

•133 

•53 

2.122 

7-459 

50 

2-995 

11.981 

47-923 

168.48 

9 

.156 

.624 

2.496 

8-775 

55 

3-249 

12.995 

182.739 

10 

.179 

.718 

2.87 

10.091 

60 

3-455 

13.822 

55-286 

194.366 

15 

•355 

142 

5.678 

19.963 

65 

3.607 

14-43 

57-72 

202.922 

20 

.608 

2-434 

9-734 

34-222 

70 

3.728 

14.914 

59-654 

209.  722 

25 

94 

376 

15038 

52.869 

75 

3-8i 

15.241 

60.965 

214.329 

30 

1-353 

54J3 

21.653 

76.123 

80 

3-857 

15-428 

61.714 

216.926 

35 

1.798 

7.192 

28.766 

IOI.I32 

85 

3.892 

15-569 

62.275 

218.936 

40 

2.258 

9032 

36-13 

I27.0l8 

90 

3-9 

15-6 

62.4 

219.375 

Resistance  to  a  plane,  from  a  fluid  acting  in  a  direction  perpendicular  to 
its  face,  is  equal  to  weight  of  a  column  of  fluid,  base  of  which  is  plane  and 
altitude  equal  to  that  which  is  due  to  velocity  of  the  motion,  or  through 
which  a  heavy  body  must  fall  to  acquire  that  velocity. 

Resistance  to  a  plane  running  through  a  fluid  is  same  as  force  of  fluid  in 
motion  with  same  velocity  on  plane  at  rest.  But  force  of  fluid  in  motion  is 
equal  to  weight  or  pressure  which  generates  that  motion,  and  this  is  equal  to 
weight  or  pressure  of  a  column  of  fluid,  base  of  which  is  area  of  the  plane, 
and  its  altitude  that  which  is  due  to  velocity. 

ILLUSTRATION. — If  a  plane  \  foot  square  be  moved  through  water  at  rate  of  32.166 

o2  ^52 
feet  per  second,  then  — —  16.083,  space  a  body  would  require  to  fall  to  acquire 

a  velocity  of  32.166  feet  per  second;  therefore  i  x  62.5  (weight  of  a  cube  foot  of 
water)  x  ^^ —  =  1005  J&s.  =  resistance  of  plane. 

Resistance  of*  different  Figures  at  different  "Velocities  in 
Air. 


Veloci- 
ty per 
Second. 

Co 
Vertex. 

ne. 
Base. 

Sphere. 

Cylin- 
der. 

Hemi- 
sphere. 
Round. 

Veloci- 
ty per 
Second. 

Co 
Vertex. 

ne. 
Base. 

Sphere. 

Cylin- 
der. 

Hemi- 
sphere. 
Round. 

Feet. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Feet. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

3 

.028 

.064 

.027 

•05 

.02 

12 

.376 

-85 

•37 

.826 

•347 

4 

.048 

.109 

.047 

.09 

•039 

14 

.512 

1.166 

•505 

I-I45 

.478 

5 

.071 

.162 

.068 

•143 

.063 

15 

-589 

1.346 

.581 

1-327 

•552 

8 

.168 

-382 

.162 

-36 

.16 

16 

•673 

1.546 

.663 

1.526 

-634 

9 

10 

.211 

26 

.478 
-587 

•  205 
•255 

.456 
•565 

.199 
.242 

18 
20 

.858 
1.069 

2.002 
2-54 

.848 
1.057 

1.986 
2.528 

.818 
1.033 

Diameter  of  all  the  figures  was  6.375  ins.,  and  altitude  of  the  cone  6.625  ins. 
Angle  of  side  of  cone  and  its  axis  is,  consequently,  25°  42'  nearly. 

From  the  above,  several  practical  inferences  may  be  drawn. 
T.  That  resistance  is  nearly  as  surface,  increasing  but  a  very  little  above 
that  proportion  in  greater  surfaces. 


MOTION    OF   BODIES   IN   FLUIDS.  647 

9.  Resistance  to  same  surface  is  nearly  as  square  of  velocity,  but  gradu- 
ally increasing  more  and  more  above  that  proportion  as  velocity  increases. 

3.  When  after  parts  of  bodies  are  of  different  forms,  resistances  are  differ- 
ent, though  fore  parts  be  alike. 

4.  The  resistance  on  base  *  of  a  cone  is  to  that  on  vertex  nearly  as  2.3  to 
i.    And  in  same  ratio  is  radius  to  sine  of  angle  of  inclination  of  side  of  cone 
to  its  path  or  axis.     So  that,  in  this  instance,  resistance  is  directly  as  sine 
of  angle  of  incidence,  transverse  section  being  same,  instead  of  square  of  sine. 

Resistance  on  base  of  a  hemisphere  is  to  that  on  convex  side  nearly  as 
2.4  to  i,  instead  of  2  to  i,  as  theory  assigns  the  proportion. 

Sphere. — Resistance  to  a  sphere  moving  through  a  fluid  is  but  half  re- 
sistance to  its  great  circle,  or  to  end  of  a  cylinder  of  same  diameter,  moving 
with  an  equal  velocity,  being  half  of  that  of  a  cylinder  of  same  diameter. 

*  /  2  gx-  dx =  V.    d  representing  diameter  of  sphere,  and  N  and  n  spe~ 

V  3  n 

ciflc  gravities  of  sphere  and  relisting  fluid. 

—  x  -  d  —  S.     S  representing  space  through  which  a  sphere  passes  while  acquir- 
n       3 

ing  its  maximum  velocity,  in  falling  through  a  resisting  fluid. 

ILLUSTRATION.— If  a  ball  of  lead  i  inch  in  diameter,  specific  gravity  11.33,  be  set 
free  in  water,  specific  gravity  i,  what  is  greatest  velocity  it  will  attain  in  descend- 
ing, and  what  space  will  it  describe  in  attaining  this  velocity  ? 

g  =  32.166,       d  =  — foot,        N  =  n.33,  and  n  =  i. 


Then^2  X  32.166  x  ^-of  ^  x  *-±^— -  =v7.i48X  10.33  = 

Hence,  ^  x.  -  of  —  =  i.2$gfeet.         £2^L=f=  retardive  force  =  — . 
'     i          3       12  8#Nd     '  ags 

Cylinder.     =  R,  and  =/    a  representing  area  or p  r2,  and 

w  weight  of  body. 
ILLUSTRATION. — Assume  a  =  32  sq.  feet,  v  =  10  feet  per  second,  and  n  =  .0012. 

Then  —         — — —  =  .o6ofa  cube  foot  of  water  =  .  06  of  62. 5  =  3. 75  Ibs. 

nav2s3  npd2v2s*  .   npd*v2s2 

Conical  Surface.     =  R,  also  -±— =  R,  and  —?- — 

2  g  8  g  8  g  to 

=/.    *  representing  sine  of  inclination,  and  a  convex  surface  of  cone. 

Curved  End  as  a  Spliere  or  Hemispherical  End.     — — ^ 

io  g 
=  R,  and  Circle  .5  of  spherical  end. 

In  general,  when  n  is  to  water  as  a  standard,  result  is  in  cube  feet  of  water,  if 
a  is  in  sq.  feet;  and  in  cube  ins.  of  water,  if  a  is  in  sq.  ins.,  v  in  ins.,  and  g  in  ins. 

If  n  is  given  in  Ibs.  in  a  cube  foot,  a  is  in  sq.  feet,  v  and  g  are  in  feet,  result  is  in  Ibs. 

To  Compute  Altitude  of*  a  Column  of  Air,  ^Pressure  of 
-which,  shall  toe  ec^ual  to  Resistance  of  a  Body  moving- 
through  it,  -with  any  Velocity. 

^  X  —  =  »  =  altitude  in  feet.    ax  =  volume  of  column  in  feet,  and  —  ax  —  weight 

in  ounces,  a  representing  area  of  section  of  body,  similar  to  any  in  table,  perpen- 
dicular to  direction  of  motion,  r  resistance  to  velocity  in  table,  and  x  altitude  sought 
of  a  column  of  air,  base  of  which  is  a,  and  pressure  r. 

*  This  is  a  refutation  of  the  popular  assertion  that  a  taper  spar  can  be  towed  in  water  easiest  when 
the  base  is  foremost. 


648  MOTION   OF   BODIES   IK   FLUIDS. 

When  a  —  —  of  a  foot,  as  in  all  figures  in  table,  x  becomes  —  r  wfcen  r  =  re- 

9  4 

Distance  in  table  to  similar  body. 

ILLUSTRATION.—  Assume  convex  face  of  hemisphere  resistance  =  .634  oz.  at  a  Te- 
locity of  16  feet  per  second. 

Then  r  =  .634,  and  x  =  —  r  =  2.3775  feet  =  altitude  of  column  of  air,  pressure  of 
which  =  resistance  to  a  spherical  surface  at  a  velocity  of  16  feet. 

To  Compute  'when.  Pressure  of  Air  in  rear  of*  a  ^Projectile 
is  Inferior  to  Pressure  d.ue  to  its  "Velocity. 

Assume  height  of  barometer  =  2.  5  feet,  and  weight  of  atmosphere  =14.  7  Ibs. 

Weight  of  cube  inch  of  mercury  =  -^  =  .49  Ibs.,  and  weight  of  cube  inch  of  air 
=  .00004357  Ibs.;  hence,  .49-7-.  000043  57  =  11  246,  which  X2-5  feet  =  28  115  feet. 


Then  Vi6.o8  :  V^^S  '•'•  32-16  :  a?,  and  x  =  32'  l6  X,^2B  "5  =  1341.6  feet. 

VID 

To  Compute  "Velocity  -with,  which  a  Plane  Surface  must 
"be  projected,  to  generate  a  Resistance  just  eq.ua!  to 
^Pressure  of  Atmosphere  upon  it. 

By  table,  resistance  on  a  circle  with  an  area  of  .222  sq.  foot  (2  -1-9)  =  .051  oz.,  at  a 
velocity  of  3  feet  per  second.  Hence  32  :  i2  :  :  .051  :  .0056  oz.  at  a  velocity  of  i  foot, 
and  i  X  144  X  14-  7  X  16  X  2  ^-9=  7526.  4  oz.  Hence,  V-  0056  :  V/526.4  ::  i  :  n6ofeet. 

To    Compute   "Velocity   lost   toy   a   ^Projectile. 

If  a  body  is  projected  with  any  velocity  in  a  medium  of  same  density  with  itself, 
and  it  describes  a  space  =  3  of  its  diameters, 

Then  *  =  3d,  and  b  =  £j^  =  ±. 

Hence,  b  x  =  -|-  ,  and  —  ^--  =  ^-  =  velocity  lost  nearly  .66  of  projectile  velocity. 

c  =  base  of  Nap.  system  of  log.  ;  hence  c*  »  =  number  corresponding  to  Nap.  log. 
b  x.  Hence,  if  &  a;  x  -4343,  result  =.  com.  log.  of  c&  *. 

b  x  =|  =  1.  125,  which  x  .4343  =  .488  587  5,  and  number  to  this  com.  log.  =  3.0803. 

Hence,  velocity  lost  =  3'o8°3~I  =  ^. 
3.0803         3.08 

ILLUSTRATION.  —  If  an  iron  ball  2  ins.  diam.  were  projected  with  a  velocity  of  1200 
feet  per  second,  what  would  be  velocity  lost  after  moving  through  500  feet  of  space  ? 

d  =  —  =  -,    05  =  500,    N  =  7i.,    and  n  =  .  0012. 

3n»      3X12X500X3X6       81  1200 

Hence,  6»  =  ^=      8X  21  x  .000^~  =  —  .  **  •  =  ^  =  998  JU  per 

second,  having  lost  202  feet,  or  nearly  -J-  of  its  initial  velocity. 

o 

12    =  .0012,  —  and  —  =  ~  and  ~  inverted,  because  N  and  n  are  in  denominator. 

10  000  22  •  3  6 

To   Compute   Time   and   "Velocity. 


ILLUSTRATION.  —  If  an  iron  ball  2  ins.  in  diameter  were  projected  in  air  with  a  ve- 
locity of  1200  feet  per  second,  in  what  time  would  it  pass  over  1500  feet,  and  what 
its  velocity  at  end  of  that  time  ? 

6  3X«X3X6=  ,  6<B=i5oo  hflnce  '  27«6  JL  =  _L.  ,  and 

o  X  22  X  loooo  2710  2710  o  i  a  1200 

-i  =  —  =  i^Zl  =  J-nearlv.     /.  v  =  6QO  and  ^=2716  x  (  ^  ---  —}  —  1.67  see. 
o        a         1  200       ocv>  V>9o      i2oo/ 


NAVAL    ARCHITECTURE, 


649 


NAVAL  ARCHITECTURE. 

Results   of  Experiments   upon.   Form   of  "Vessels. 

(Wm.  Bland.) 

Cnbical  Models.  Head  Resistance. — Increases  directly  with  area 
of  its  surface.  Weight  Resistance. — Increases  directly  as  weight. 

Vessels'  Models.  Lateral  Resistance.  —  About  one  twelfth  of 
length  of  body  immersed,  varying  with  speed. 

Order  of  Superiority  of  Amidship  Section. — Rectangle,  Semicircular, 
Ellipse,  and  Triangle. 

Centre  of  lateral  resistance  moves  forward  as  model  progresses. 

Centre  of  gravity  has  no  influence  upon  centre  of  lateral  resistance. 

Relative    Speeds. 

Z^n^A.— Increased  length  gives  increased  speed  or  less  resistance. 

Depth  of  Flotation. — Less  depth  of  immersion  of  a  vessel,  less  the  resistance. 

A  midship  Section. — Curved  sections  give  higher  speed  than  angled. 

Sides. — Slight  horizontal  curves  present  less  resistance  than  right  lines. 
Curved  sides  with  one  fourth  more  beam  give  equal  speeds  with  straight 
sides  of  less  beam.  Keel. — Length  of  keel  has  greater  effect  than  depth. 
Stern. — Parallel-sided  after  bodies  give  greater  speed  than  taper-sided. 

FORM  OF  Bow.  Order  of  Speed. 


Isosceles  triangle,  sides  slightly  convex 

"     right  lines 

"     slightly  concave  at  entrance  and  running) 
out  convex J 

Spherical  equilateral  triangle  compared  to  Equilateral  triangle,  speed  is 
as  ii  to  12.  Equilateral  triangle ,  with  its  isosceles  sides  bevelled  off  at  an 
angle  of  45°,  compared  to  bow  with  vertical  sides,  is  as  5  to  4. 

When  bow  has  an  angle  of  14°  with  plane  of  keel,  compared  with  one  of 
7°,  its  speed  is  greater. 

Bodies  Inclined  Upwards  from  Amidship  Section. 

i.  Model  with  bow  inclined  from  &,  has  less  resistance  than  model  with- 
out any  inclination. 

a.  Model  with  stern  inclined  from  58,  has  less  resistance  than  model  with- 
out any  inclination. 

Model  i  had  less  resistance  than  model  2.  Model  with  both  bow  and 
stern  inclined  from  &,  has  less  resistance  than  either  i  or  2. 

Stability. 

Results  of  Experiments  npon  Stability  of*  Rectangular 
Blocks  of  Wood  of  Uniform  Length,  and  Depth,  toxit 
of  Different  Breadths.  (Wm.  Bland.) 

Length  15,  Depth  2,  and  Depression  i  inch. 


Width. 

Weight. 

AB  Observed. 

Ratio  o 
With  like 
Weights. 

f  Stability. 
By  Squares  of 
Breadth. 

By  Cubes  of 
Breadth. 

Int. 
3 
4-5 
6 

7 

Oz. 
24 

35 
45 
55 

i 
2-5 
7 
ii 

I 
2.4 

3-7 
4.8 

i 
2.25 

6.25 

i 

3-375 
8 
15-625 

650 


NAVAL   ARCHITECTURE. 


Hence  it  appears  that  rectangular  and  homogeneous  bodies  of  a  uniform 
length,  depth,  weight,  and  immersion  in  a  fluid,  but  of  different  breadths,  have 
stability  for  uniform  depressions  at  their  sides  (heeling)  nearly  as  squares 
of  their  breadth ;  and  that,  when  weights  are  directly  as  their  breadths, 
their  stability  under  like  circumstances  is  nearly  as  cubes  of  their  breadth. 

With  equal  lengths,  ratio  of  stability  is  at  its  limit  of  rapid  increase  when 
width  is  one  third  of  length,  being  nearly  in  cube  ratio ;  afterwards  it  ap- 
proaches to  arithmetic  ratio. 

Results    of*  Experiments    vipon    Stability   and.    Speed,    of 
Models  having  Amidship  Sections  of  different  Forms, 
t>\at  Uniform  Length,  Breadth,  and  "Weights.    (W.  Bland.) 
Immersion  different,  depending  upon  Form  of  Section. 


FORM  OF  IMMERSED  SECTION. 


Half-depth  triangle,  other  half  rectangle. 

Rectangle 

Right-angled  triangle  * 

Semicircle 


Stability. 


14 
7 
9 


Speed. 

4 
3 
3 


*•  Draught  of  water  or  immersion  double  that  of  rectangle. 


Fig.  i. 


Statical  Stability  is  moment  of  force  which  a  body  in  flotation  exerts  to 
attain  its  normal  position  or  that  of  equilibrium,  it  having  been  deflected 
from  it,  and  it  is  equal  to  product  of  weight  of  fluid  displaced  and  horizontal 
distances  between  the  two  centres  of  gravity  of  body  and  of  displacement,  or 
it  is  product  of  weight  of  displacement,  height  of  Meta-centre,  and  Sine  of 
angle  of  inclination. 

Dynamical  Stability  is  amount  of  mechanical  work  necessary  to  deflect  a 
body  in  flotation  from  its  normal  position  or  that  of  equilibrium,  and  it  is 
equal  to  product  of  sum  of  vertical  distances  through  which  centre  of  grav- 
ity of  body  ascends  and  centre  of  buoyancy  descends,  in  moving  from  ver- 
tical to  inclined  position  by  weight  of  body  or  displacement. 

To  Determine  Measure  of  Stability  of  Hull  of  a  Vessel 
or    Floating    Body.— Fig.  1. 

Measure  of  stability  of  a  floating  body  depends  essentially  upon  horizontal  dis- 
tance, G  s,  of  meta-centre  of  body  from  centre 
of  gravity  of  body;  and  it  is  product  offeree 
of  the  water,  or  resistance  to  displacement  of 
it,  acting  upward,  and  distance  of  G  5,  or  P  x 
G  s.  If  distance,  G  M,  represented  by  r,  and 
angle  of  rolling,  c  M  r,  by  M°,  measure  of  sta- 
bility, or  S  is  determined  by  P  r,  sin.  M°  =  8 ; 
and  this  is  therefore  greater,  the  greater  the 
weight  of  body,  the  greater  distance  of  meta- 
centre  from  centre  of  gravity  of  body,  and  the 
greater  the  angle  of  inclination  of  this  or  of 
cMr. 

Assume  figure  to  represent  transverse  section  of  hull  of  a  vessel,  G  centre  of 
gravity  of  hull,  w  I  water-line,  and  c  centre  of  buoyancy  or  of  displacement  of  im- 
mersed hull  in  position  of  equilibrium.  Conceive  vessel  to  be  heeled  or  inclined 
over,  so  that  ef  becomes  water-line,  and  s  centre  of  buoyancy;  produce  s  M,  and 
point  M  is  meta-centre  of  hull  of  vessel. 

Transverse  meta-centre  depends  upon  position  of  centre  of  buoyancy,  for  it  is  that  point  where  a 
yertical  line  drawn  from  centre  intersects  a  line  passing  through  centre  of  gravity  of  hull  of  vessel 
perpendicular  to  plane  of  keel. 

Point  of  mtta-centre  may  be  the  same,  or  it  may  differ  slightly  for  different  angles  of  heeling.  Angla 
of  direction  adopted  to  ascertain  position  of  meta- centre  should  be  greatest  which,  under  ordinary  cir- 
cumstances, is  of  probable  occurrence  ;  in  different  vessels  this  angle  ranges  from  20°  to  6oe. 

If  meta-centre  is  above  centre  of  gravity,  equilibrium  is  Stable;  if  it  coincides  with  it,  equilibrium  is 
Indifferent ;  and  if  it  is  below  it,  equilibrium  is  Unstable. 


W 


NAVAL   ARCHITECTURE.  65  I 

Comparative  Stability  of  different  hulls  of  vessels  is  proportionate  to  the  distance 
of  G  M  for  same  angles  of  heeling,  or  of  distance  G  s.  Oscillations  of  hull  of  a  ves- 
sel may  be  resolved  into  a  rolling  about  its  longitudinal  axis,  pitching  about  its 
transverse  axis,  and  vertical  pitching,  consisting  in  rising  and  sinking  below  and 
above  position  of  equilibrium. 

Tf  transverse  section  of  hull  of  a  vessel  is  such  that,  when  vessel  heels,  level  of 
centre  of  eravity  is  not  altered,  then  its  rolling  will  be  about  a  permanent  longi- 
tudinal axis  traversing  its  centre  of  gravity,  and  it  will  not  be  accompanied  by  any 
vertical  oscillations  or  pitchings,  and  moment  of  its  inertia  will  be  constant  while 
it  rolls  But  if  when  hull  heels,  level  of  its  centre  of  gravity  is  altered,  then  axis 
about  which  it  rolls  becomes  an  instantaneous  one,  and  moment  of  its  inertia  will 
vary  as  it  rolls;  and  rolling  must  then  necessarily  be  accompanied  by  vertical  os- 
cillations. 

Such  oscillations  tend  to  strain  a  vessel  and  her  spars,  and  it  is  desirable,  therefore, 
that  transverse  section  of  hull  should  be  such  that  centre  of  its  gravity  should  not 
alter  as  it  rolls,  a  condition  which  is  always  secured  if  all  water-lines,  as  w  I  and  ef, 
are  tangents  to  a  common  sphere  described  about  G;  or,  in  other  words,  if  point  of 
their  intersections,  o,  with  vertical  plane  of  keel,  is  always  equidistant  from  centre 
of  gravity  of  hull. 

To   Compute    Statical    Stability. 

D  c  M  sin.  M  =  S.  D  representing  displacement,  M  angle  of  inclination,  and  S 
stability. 

ILLUSTRATION  i.— Assume  a  ship  weighing  6000  tons  is  heeled  to  an  angle  of  9°, 
distance  c  M  =  3  feet, 

Sin.  9°  =  .1564.    Then  6000  X  3  X  .1564  =  2815.2  foot-tons. 

2._Weight  of  a  floating  body  is  5515  Ibs.,  distance  between  its  centre  of  gravity 
and  meta-centre  is  11.32  feet,  and  angle  M  =  20° 

Sin.  M  =  . 34202.     Hence  5515  X  11-32  X  34202  =21  352. 24 /oo<-Z5*. 
Statical    Surface    Stability. 

Moment  of  Statical  surface  stability  at  any  angle  is  c  z  D.  Assuming 
centre  of  gravity  of  vessel  coincided  with  c ;  coefficient  of  a  vessel's  stability 
at  any  angle  of  heel  is  expressed  when  the  displacement  is  multiplied  by 
vertical  height  of  the  meta-centre  for  given  angle  of  heel  above  centre  of 
gravity,  or  D  c  M. 

Approximately.  RULE.— Divide  moment  of  inertia  of  plane  of  flotation 
for  upright  position,  relatively  to  middle  line  by  volume  of  displacement ; 
and  quotient  multiplied  by  sine  of  angle  of  heel  will  give  result. 

Per  Foot  of  Length  of  Vessel,  -  (B3  sin.  M).    B  representing  half  breadth. 

Dynamical    Surface    Stability. 

Moment  of  Dynamical  surface  stability  is  expressed  by  product  of  weight 
of  vessel  or  displacement  and  depression  of  centre  of  buoyancy  during  the 
inclination,  that  is,  for  angle  M. 

To    Compute   Dynamical    Stability   of  a   Vessel. 
Approximately.    RULE.— Multiply  displacement  by  height  of  meta-centre 
above  centre  of  gravity,  and  product  by  versed  sine  of  angle  of  heel. 

!Or  multiply  statical  stability  for  given  angle  by  tangent  of  .5  angle  of  heel. 
To  Compute  Elements  of*  Stability  of*  a  Floating  Body. 

T  a  ~  **  sin  M  =  Tj  ~ — M  =  ^  ftnd  sin'  M  r  —  c-  A  representing  area  of 
immersed  section;  A'  section  immersed  by  careening  of  body,  as  fo  I;  s  horizontal 
distance,  c  r,  between  centres  of  buoyancy  ;  a  horizontal  distance  between  centres  of 
gravity,  i  i,  of  areas  immersed  and  emerged  by  careening;  g  distance,  c  M,  between 
centre  of  buoyancy  or  of  water  displaced  and  meta-centre  ;  r  distance,  G  M,  between 
centre  of  gravity  and  meta-centre;  c  horizontal  distance,  G  s,  between  centre  of  grav- 
ity and  of  line  of  displacement  of  it  when  careened ;  e  vertical  distance  between  centres 
of  gravity  and  buoyancy,  all  in  feet ;  and  M  angle  of  careening. 


652  NAVAL    ARCHITECTURE. 

NOTE.—  When  centre  of  gravity,  G,  is  below  that  of  displacement,  c,  then  e  is  +; 
when  it  is  above  c  it  is  —  ;  and  when  it  coincides  with  c  it  is  o;  or  e  is  —  when 

•p  <s;  and  a  body  will  roll  over  when  e  sin.  M  =  or  >s. 

Assumed  elements  of  figure  illustrated  are  A  =  86,  A'  =  21.  5,  6  =  21.  5,  and  e  =  .  5. 
The  deduced  arc  5  =  3.  7,  0  =  3.87,  #  =  10.82,  a  =14.9,  and  r  =  11.32.  b  repre- 
senting breadth  at  water-line  or  beam  in  feet,  and  P  weight  or  displacement  in  ibs. 


or  tons. 


=  io.82/eeJ,    c  =  .34202X  11.32  =  3.87  feet. 
Of  Hull  of  a  Vessel.      (       ^  ^  ±  e\  P,  sin.  M  =  S  ;  d  cos.  .5  M  =  d', 

b  -JL-(f_^=±c;          p(6ta  +  £iiiOl)=S;    and 

^ 


_  -  -  -_       ,     --_  t 

10.7  to  13  (11.93)  A        '     sin.  M\P       y  VA  ^ 

P  (s  db  e  sin-  M)  =S.    d  representing  depth  of  centre  of  gravity  of  displacement  un- 
der water  in  equilibrium,  and  d'  depth  when  out  of  equilibrium,  both  in  feet. 

ILLUSTRATION  i  —  Displacement  of  a  vessel  is  10000000  Ibs.  ;  breadth  of  beam,  50 
feet;  area  of  immersed  section,  800  sq.  feet;  vertical  distance  from  centre  of  grav- 
ity of  hull  up  to  centre  of  buoyancy  or  displacement,  1.9  feet,  and  horizontal  dis- 
tance a  between  centres  of  gravity  of  areas  immersed  and  emerged,  when  careened 
to  an  angle  of  9°  10'  —  33.4  feet,  immersed  area  being  50  sq.  feet. 

Sin.  9°  io'  =  .i593.    Then  s  =  -~  X  33.4  =  2.0875  feet,  800X2.0875  =  50X33.4, 


t= 


=       .  B     ..  =  g          . 

.  1593  ii  -93  X  800  \ii  .93  X  8oo/  T 

10  ooo  ooo  X.i  593  =  23  905  396  Ibs.,  and  e  =  —  —  (  23  9°5  39  --  2.0875)  =  1.9  feet. 

.1593  \iooooooo  / 

2.  —  Assume  a  ship  having  a  displacement  of  5000  tons,  and  a  height  of  meta-centre 
of  3.25  feet,  to  be  careened  to  6°  12'.    What  is  her  statical  stability  ? 

Sin.  6°  12'=.  1079.     Then  5000  X  3.25  X  -1079  =  1753.37  foot-tons. 

3.  —  Assume  a  weight,  W,  of  50  tons  to  be  placed  upon  her  spar  deck,  having  a 
common  centre  of  gravity  of  15  feet  above  her  load-line, 


Then  5000  X  3-  25  —  50  -j-  1 5  X  .  1079  —  J  747-  3^  foot-  tons. 

4.  — Assume  100  tons  of  water  ballast  to  be  admitted  to  her  tanks  at  a  common 
centre  of  gravity  of  15  feet  below  her  load-line. 


Then  5000  X  3-25  -J- 100  X  15  X  .1079  =  I91 5- 22  foot-tons. 
5.— Assume  her  masts,  weighing  6  tons,  to  be  cut  down  20  feet, 
Then  —    —  =  —  foot  =fall  of  centre  of  gravity,  and  5000  X  (3. 25  -f — j  X  .  1079 
=  1774.95  tons. 

To    Compixte    Elewnents    of  Power,    etc.,   required    to 
Careen    a   Body   or   "Vessel. 

Sin.  M  (h  -  n  sin.  M)  -f  n  sec.  M  -  s  =  I.     **    .  A    3/- —r=m. 

io-7toi3*A  V  64.125  L  A 

Wlr  =  Pc,  and  W 1  =  S.  W  representing  weight  or  power  exerted  and  I  distance 
at  which  weight  or  power  acts  to  careen  body,  taken  from  centre  of  gravity  of  displace- 
ment perpendicular  to  careening  force,  h  vertical  height  from  centre  of  gravity  of  dis- 
placement to  centre  of  weight  or  power  to  careen  body  tvhen  it  is  in  equilibrium, 
n  horizontal  distance  from  centre  of  vessel  to  centre  of  weight  or  power,  L  length  of 
vessel,  m  meta-centre,  and  S  as  in  preceding  case,  all  in  feet. 

*  Unit  for  section  of  a  parallelogram  ia  10.7  ;  of  a  semicircle  12,  and  of  a  triangle  12.8. 


NAVAL   ARCHITECTURE.  653 

ILLUSTRATION.— A  weight  is  placed  upon  deck  of  a  vessel  at  a  mean  height  of  3.87 
feet  from  centre  line  of  hull;  height  at  which  it  is  placed  is  11.32.  and  other  ele- 
ments as  in  tirst  case  given. 

Sec.  20°  =  .342.      Thenft=ii.32,  w  — 3.87,  and  ^  =  .342  (11.3  —  3.87  x.342)-f- 
3.87  X  1.0642  —  3.7  =  .342X16  +  4.12  —  3.7  =  3.84/<*rt. 
Assume  W  =  5515.    Then  5515  x  3.84  =  21  i8-,.6foot-lbs. 

Or  P  (w  cos.  M  -j-  h  sin.  M)  =  8.  w  representing  distance  of  weight  from  centre  of 
vessel,  and  h  height  ofw  above  water-line,  both,  in  feet. 

ILLUSTRATION.— If  a  weight  of  30  tons  placed  at  20  feet  from  centre  of  hull  or 
deck,  10  feet  above  water-line,  careens  it  to  an  angle  of  2°  9',  what  is  its  stability? 

cos.  2°  9'  =  .9993 ;  sin.  2°  9'  =  -0375. 
30  (20  X  -9993  + 10  X  .0375)  =  30  X  20.361  =  610.83  foot-tons. 

Bottom,  and.  Immersed  Surface  of  Hull  of  Vessels. 
To  Compute  Bottom  and  Side  Surface  of  Hull. 

Bottom  and  Side.  RULE.— Multiply  length  of  curve  of  amidship  section, 
taken  from  top  of  tonnage  or  main  deck  beams  upon  one  side  to  same  point 
upon  other  (omitting  width  of  keel),  by  mean  of  lengths  of  keel  and  be- 
tween perpendiculars  in  feet,  multiply  product  by  .85  or  .9  (according  to  the 
capacity  of  vessel),  and  product  will  give  surface  required  in  sq.  feet. 

EXAMPLE.— Lengths  of  a  steamer  are  as  follows:  keel  201  feet,  and  between  per- 
pendiculars 210  feet,  curved  surface  of  amidship  section  76  feet;  what  is  surface? 

Coefficient  .87.        2104-201  -=- 2  =  205.5,  and  76  X  205.5  X  .87  =  13  587  sq.  feet. 

NOTE.— Exact  surface  as  measured  was  13650  sq.  feet. 

Bottom  Surface.  RULE. — Multiply  length  of  hull  at  load-line  by  its 
breadth,  and  this  product  by  depth  of  immersion  (omitting  the  depth  of 
keel)  in  feet ;  and  this  product  multiplied  by  from  .07  to  .08  (according  to 
capacity  of  vessel)  will  give  surface  required  in  sq.  feet. 

EXAMPLE.— Length  upon  load-line  of  a  vessel  is  310  feet,  beam  40  feet,  depth  of 
keel  i  foot,  and  draught  of  water  20  feet;  what  is  bottom  or  wet  surface? 

Coefficient  assumed  .073.     310  X  4°  X  20  —  1  x  .073  =  17  199  sq.  feet 

To   Compute    Resistance   to    Wet    Surface   of  Hull. 

C  a  v2  =  R.  C  representing  a  coefficient  of  resistance,  a  area  of  wet  surface  in  sq. 
feet,  and  v  velocity  of  hull  in  feel  per  second. 

Values  of  C  f<007>  clean  c°PPer-        I         .014,  iron  plate. 

'(.or,    smooth  paint.       |         .019,  iron  plate,  moderately  foul. 
Power  required  to  propel  one  sq.  foot  of  immersed  amidship  section  at  gj  is  .073 
that  of  smooth  wet  surface. 

To  Compute  Elements  of  a  Vessel. 
Displacement   and    its    Centre    of  Grravity. 

Displacement  of  a  vessel  is  volume  of  her  body  below  water-line. 

Centre  of  Gravity,  or  Centre  of  Buoyancy  of  Displacement,  is  centre  of 
gravity  of  water  displaced  by  hull  of  vessel. 

For  Displacement.  RULE. — Divide  vessel,  on  half  breadth  plan,  into  a 
number  of  equidistant  sections,  as  one,  two,  or  more  frames,  commencing 
at  i&  and  running  each  side  of  it.  Add  together  lengths  of  these  lines  in 
both  fore  and  aft  bodies,  except  first  and  last,  by  Simpson's  rule  for  areas 
(see  page  344) ;  multiply  sum  of  products  by  one  third  distance  between 
sections,  and  product  will  give  area  of  water-line  between  fore  and  aft-sections. 

Then  compute  areas  contained  in  sections  forward  and  aft  of  sections  taken,  in- 
cluding stern  and  rudder-post,  rudder  and  stem,  and  add  sum  to  area  of  body-sec* 
tions  already  ascertained.  * 

*  To  Compute  Area  of  a  Water-line,  see  Mauau.atlon  of  Surfaces, 


654  NAVAL   ABCHITECTUEB. 

Compute  area  of  remaining  water-lines  in  like  manner.  Tabulate  results,  and 
multiply  them  by  Simpson's  rule  in  like  manner  as  for  a  water-line,  and  again  by 
consecutive  number  of  water-lines,  and  sum  of  products  between  water-line  and 
product  will  give  volume  between  load  and  lower  water-line. 

Add  area  of  lower  water-line  to  area  of  upper  surface  of  keel ;  multiply  half  sum 
by  distance  between  them,  and  product  will  give  volume;  then  compute  areas  con- 
tained in  sections  forward  and  aft  of  sections  taken  as  before  directed. 
If  keel  is  not  parallel  to  lower  water-line,  take  average  of  distance  between  them. 
Compute  volume  of  keel,  rudder-post  and  rudder  below  water-line ;  add  to  volume 
already  ascertained;  multiply  product  by  two,  for  full  breadth,  and  product  will 
give  volume  required  in  cube  feet,  all  dimensions  being  taken  in  feet. 
„.  EXAMPLE. -Assume 

a  vessel  100  feet  in 

length  by  20  feet  in 
extreme  breadth,  on 
load-line  of  8  feet  9 
inches  immersion. 
Figs.  2  and  3. 

Distance  between 
sections,  for  purpose 
of  simplifying  this 
example,  is  taken 
at  10  feet;  usually 
frames  are  18  to  30 
ins.  apart,  and  two  or  more  included  in  a  section.  Water-lines  2  feet  apart. 


Fig.  3. 


ist  Water-line. 

45                 =5 
3        7-7X4    =    30-8 

2             9.5    X       2      =       19 

i         9.9  X     4    =    39-6 

O          10         X       2      =      20 

A        9.6  X     4     =     38-4 
B         7.8  X     2    =     15.6 
C         6.8  X     4    =    27.2 
D         4                  =      4 
199.6 
10-^-3     =      3^ 

zd  Water-line. 

4        2-7               =      2-7 
3        6.9  X     4    =    27.6 
2        8.7  X     2    =    17.4 

i         9.5  X     4    =    38 

0            9.6    X       2      =      19.2 

A         9      X     4     =     36 
B        7      X     2    =     14 
C         5      x     4    =    20 

D            2                           =2 

176.9 

10  -i-  3     =      3^ 

3d  Water-line. 

4         i-5                =      »-5 
3         5      X     4     =    20 
2         6.6  X     2    =    13.2 
i         8.7  X     4     =     34-8 
o         8.9  X     2     =     17.8 
A         7.6  X     4     =     3<>-4 
B         7      X     2     =     14 
C         3      X     4     =     12 

D            J.  2                       =         1.2 

144.9 
xo-r-3     =       3£ 

665-3 
Abaft  section  4,  rud- 
der and  post  25 
Forward  section   D 

589.7 
Abaft  section  4,  rud- 
der and  post  13.2 
Forward  section   D 
and  stem  9.  i 

483 
Abaft  section  4,  rud- 
der and  post  7 
Forward   section    D 

7" 

4th  Water-line. 

4-7               =        -7 
3        2X4=8 
2        4.3  X     2    =      8.6 
i         6.5  X     4    =    26 
o        6.8  X     2    =    13.6 
A        5      X     4    =    90 
B        3.6  X     2    =      7.2 
C          -9  X    4    -=      3-6 
D          -3                =        -3 
88 

I0~*"3    =_M- 
293.3 
Abaft  section  4,  rud- 
der and  post  3.2 
Forward  section   D 
and  stem  8 

"6^2" 

Ke 

Half  breadth  =  .25  X  le 
Rudder-post  and  rudde 

Res' 

ist  water-line  711 
2d                  612     x  4  = 
3d                   495.4  X  2  — 
4th                    297.3  X  4  = 
Keel                 24.8 

495-4 

el. 
ngth  of  98  feet  =    24.  5 
r  3 

24.8 

ults. 
711 

2448     X  i  =  2448 

990.8   X    2  =  1981.6 

1189.2  X  3  =  3567-6 
24.8  X  4=      99-2 
5363"  8               8096.4 

2 

3)10727.6 
Displacement,  3575.9  X  2  =  jisi.Sculeft 

297-3 

NAVAL    ARCHITECTURE.  655 

To    Compute    Centre   of*  Q-ravity  of  Displacement. 

RULE. — Divide  sum  of  products  obtained  as  above,  by  consecutive  water- 
lines,  by  sum  of  products  obtained  in  column  of  products  by  Simpson's  mul- 
tipliers, and  quotient,  multiplied  by  distance  between  water-lines,  will  give, 
depth  of  centre  below  load  water-line. 

ILLUSTRATION  i.    8096.4,  from  above,  -4-  5363.8  =  1.5,  which  X  2  =  3  feet 

Or, n          =  d.    n  representing  draught  of  water  exclusive  of  any  drag  of 

2('-3Ti) 

keel,  a  area  of  immersed  surface  of  hull  in  sq.feet,  and  D  displacement  in  cube  feet. 

2.  —Assume  draught  of  water  8  feet,  displacement  7152  cube  feet,  and  area  of  Im- 
mersed surface  of  hull  noo  sq.  feet. 


(. 25! 

\  IIOO) 


X8/ 
To    Compute   Displacement  Approximately. 

Coefficient  of  Displacement  of  a  vessel  is  ratio  that  volume  of  displacement 
bears  to  parallelopipedon  circumscribing  immersed  body. 

y 

=  C.    V  representing  volume  of  displacement  in  cube  feet,  L  length  at  im- 

L  B  D 

mersed  water-line,  B  extreme  breadth,  and  D  draught  in  depth  of  immersion,  both 
in  feet. 

Coefficient  of  Area  of  A  midship  Section  in  Plane  of  a  Water-line  is  ratio 
which  their  areas  bear  to  that  of  circumscribing  rectangle. 
L  representing  length  of  water-line,  and  D  distance  between  water-lines,  both  in  feet. 

Coefficients.    (By  S.  M.  Pook,  Constructor  U.  S.  Navy. ) 
RULE. — Multiply  length  of  vessel  at  load-line  by  breadth,  and  product  by 
depth  (from  load-line  to  under  side  of  garboard-strake)  in  feet,  and  this 
product  by  coefficient  for  vessel  as  follows :  divide  by  35  for  salt  water,  36 
for  fresh  water,  and  quotient  will  give  displacement  in  tons. 

Amidship  sections  range  from  .7  to  .9  of  their  circumscribing  square,  and  mean 
of  horizontal  lines  from  .  55  to  .  75  of  their  respective  parallelograms.  Hence,  ranges 
for  vessels  of  least  capacity  to  greatest  are  -7  X  .55  =  .385,  and  .9  X  .75  =  .675. 


Merchant  ship,  very  full 6    to  .  7 

"          "     medium 5810.62 

River  steamer,  stern- wheel. . .  ,6   to  .65 

Ship  of  the  line 5-  to  6 

Naval  steamer,  first  class 5    to  .6 

"        52  to. 58 

Merchant  steamer,  sharp 54  to  .58 

Half  clipper 52  to  .56 

Brigs,  barks,  etc 52  to  .56 

River  steamer,  tug-boat,  med'm  .  52  to .  56 


Merchant  steamer,  medium. . .  .  52  to  .54 

Clipper 5    10.54 

Schooner,  medium 48  to  .52 

River  steamer,  tug-boat,  sharp  .45  to  .5 
"        medium 45  to. 5 

"         "       sharp 42  to  .45 

Schooner,  sharp 46  to  .5 

Yachts,  sharp 4   to  .45 

"       very  sharp 3    to  .4 

River  steamers,  very  sharp. . .  .36  to  .42 


In  steam  launch  Miranda,  when  making  16.2  knots  per  hour,  with  a  displace- 
ment of  58  tons,  her  coefficient  was  3. 

To   Compute   Change   of*  Trim. 
W  (I        T 
-=r-  X  —  =  d'.    D  representing  displacement  at  line  of  draught  in  tons,  L  length 

at  same  line  in  feet,  and  m  longitudinal  meta-centre. 

ILLUSTRATION.— "  Warrior,"  at  draught  of  25. 5  feet,  has  L  =  380  feet,  m  =  475  feet, 
and  D  =  8625  tons.    If,  then,  a  weight  of  20  tons  was  shifted  fore  and  aft  100  feet, 

^?X  *  =  .  , 

8625          475 


NAVAL    ARCHITECTURE. 


To    Compnte    Common   Centre  of  Oravity  of*  Hull,  Ar- 
mament,  Engine,   Boiler,  etc.,  of  a  Vessel. 

RULE. — Compute  moments  of  the  several  weights,  relatively  to  assigned 
horizontal  and  vertical  planes,  by  multiplying  weight  of  each  part  by  its 
horizontal  and  vertical  distance  from  these  planes. 

Add  together  these  moments,  according  to  their  position  forward  or  aft,  or 
above  or  below  these  planes,  and  difference  between  these  sums  will  give  po- 
sition forward  or  aft,  above  or  below,  according  to  which  are  greatest. 

Divide  results  thus  ascertained  by  total  weight  of  vessel,  and  product  will 
give  horizontal  and  vertical  distances  of  centre  of  gravity  from  these  planes. 

NOTE. — To  simplify  computation  in  table,  common  centre  of  gravity  of  hull,  ma- 
chinery, etc.,  is  taken,  instead  of  centres  of  individual  parts,  as  engine,  boiler,  pro- 
peller, etc. 

Illustration —  Vertical  Plane  at  &  and  Horizontal  at  Load-line. 


ELEMENTS  OF  A  STKAM 

FKIOATK. 

Weight. 

HORIJ 
Distances. 
Forward.  Abaft. 

ONTAL. 

Mom< 
Forward. 

nts. 
Abaft. 

Dista 
Above 

VEK 

nces. 
Below 

FICAL. 

Morr 
Above 

ents. 
Below 

Hull,  bunkers,  and  ce- 
ment in  bottom  

Tons. 
1075 

470 
252 

I3I-5 

24 
25 

3-25 

22 

30 

3° 
7-25 

Feet. 
1.6 

16 
62 

27 
40 

40 

1.2 
17 

Feet. 
29 

48 
40 

1720 

4032 

8l53 

648 
I  OOO 

880 
360 

510 

13630 

156 
290 

Feet. 

2 

31 

16 

5 
7 

Feet. 
I 

6-3 
4 

6 
3 

8 

263 
744 
52 
15° 

210 

1075 

3011 
1008 

ISO 
66 

"58 

Engines,  boilers,  water, 
and  stores     

Coal 

Battery  and  ammuni 
tion  

Masts,  spars,  sails,  and 
rigging    .  ... 

Anchors  and  cables  
Boats 

Water  and  ship's  stores 
Provisions  and  galley.  . 
Crew  and  effects  
Officers'  and  mess  stores 

Total..., 

2070 

1730-3 

14076 

1410  1  "5^68 

Moments  forward  g$,  17303  —  moments  abaft,  14076  —  3227-7-2070  tons  (weight) 
=  1.56  feet  =  distance  of  centre  forward  o/jgj. 

Moments  above  load-line,  5368  —  moments  below,  1419  —  3949  -4-  2070  tons  (weight) 
=  i.  91  feet  =  distance  of  centre  below  load-line. 

NOTE. — Rule,  in  Strength  of  Materials,  to  compute  common  centre  of  gravity, 
page  819,  would  apply  in  this  case. 


To     Compute     Depth    of    Centre    of    Grravity    or    Buoy- 
ancy   Belo-vv    ^teta-Centre. 

g 
-— — —  =  d.    S  representing  statical  stability,  D  displacement  in  tons,  and  sin.  M 

sine  of  angle  of  heel. 

ILLUSTRATION. — Elements  of  Fig.  2,  page  654,  are,  statical  stability  at  angle  of 
5.44°,  90  tons,  and  displacement  204.33  tons. 

Sin.  5. 44°  =  .0999.     Then  9° =4.41  feet. 

204. 33  X. 0999 


NAVAL   ARCHITECTURE. 


65; 


To  Compute  Centre  of  Q-ravity  or  Buoyancy  Approxi- 
mately. 

-  to  i-  of  mean  draught  of  hull,  using  larger  coefficient  for  full  bodied  vessels. 
5       20 

To   Delineate    Curve   of  Displacement. 
This  curve  is  for  purpose  of  ascertaining  volume  of  water  or  tons  weight, 
displaced  by  immersed  hull  of  a  vessel  at  any  given  or  required  draught ;  or 
weight  required  to  depress  a  hull  to  any  given  or  required  draught.     From 
results  of  computation  for  displacement  of  vessel,  proceed  as  follows,  Fig.  4 : 
Fig.  4.  On  a  vertical  scale  of  feet  and  ins., 

"  B  as  A  B,  set  off  depths  of  keel  and  water- 

lines,  draw  ordinates  thereto  represent- 
ing displacement  of  keel,  and  at  each 
water-line,  in  tons. 

Through  points  i,  2,  3,  4,  and  5  df 
lineate  curve  A  5,  which  will  represent 
displacement  at  any  given  or  required 
draught. 

Draw  a  horizontal  scale  correspond- 

1  3*5      mg  ^0  w  eight  due  to  displacement  at 

load-line,  as  A  C.  and  subdivide  it  into  tons  and  decimals  thereof,  and  a  ver- 
tical line  let  fall  from  any  point,  as  r»,  at  a  given  draught,  will  indicate 
weight  of  displacement  at  depth,  on  scale  A  C,  and,  contrariwise,  a  line  raised 
from  any  point,  as  z,  on  A  C  will  give  draught  at  that  weight. 

ILLUSTRATION. — Displacement  of  hull  (page  654)  at  load  1106  =  7151.8  cube  feet, 
which  -r-  35  for  salt  water  =  204. 3  tons,  hence  A  C  represents  tons,  and  is  to  be  sub- 
divided accordingly. 

Assume  launching  draught  to  have  been  4  feet,  then  a  vertical  let  fall  from  4  will 
indicate  weight  of  hull  in  tons  on  A  C. 

Coefficients.    (By  C.  Mackrow,  M.  I.  N.  A. ) 


DISCBIPTION  or  VESSEL. 

Length. 

Breadth. 

Mean 
Draught. 

Displace- 

Coefficient. 
Amidahip 
Section. 

Water- 
lines. 

225 

45 

15 

.715 

.932 

•755 

Mail  Steamers.  < 

325 

350 
38< 

59 
35 
42 

24-75 
21 
22 

.64 
.687 
•  6^0 

.81 

-85 
.88 

$ 

.8 

368.27 
220 

42.5 
27 

18.71 

8 

.516 
.702 

.812 
.912 

•635 
•742 

Gunboats  | 

90 
125 

15 
23 

8 

.637 
•536 

.914 
.87 

.704 
.616 

Troop  Ships  < 

160 

350 

3i-3 
49.12 

12 
23-5 

.466 
•47 

-745 
-674 

.603 
•7Q 

Swift  Naval  Steamers.  .  .  .  j 
Fast  Steamers.  R.  N.... 

340-5 
337-3 
270 

300 

46.13 
50.28 
42 

40.27 

15-75 
22-75 

'9 

14 

:;83 

•497 
.414 

.68 
.787 
.792 

.711 

.582 
.6I4 
.628 

•  711 

Cnrve   of  "Weight. 

To  Compute  Number  of*  Tons  required  to  Depress  a 
Vessel  One  Inch,  at  any  Draught  of  Water  Parallel 
to  a  "Water-line. 

RULE.— Divide  area  of  plane  by  12,  and  again  by  35  or  36,  as  may  be 
required  for  salt  or  fresh  water. 

EXAMPLE.— Area  of  load  water-line  of  a  vessel  is  1422  sq.  feet;  what  is  its  ca« 
pacity  per  inch  in  salt  water? 

1422-7-12  =  118.5,  which  -=-35  =  3.38  tom. 


658 


NAVAL   ARCHITECTURE. 


To    Compute    Centre    of   Gravity    of    Bottom    Plating 
of  a    Vessel. 

Longitudinal. 

RULE. — Measure  half  girths  of  plating  at  equidistant  sections,  as  at  two 
or  more  frames.  Multiply  these  in  accordance  with  Simpson's  rule  for 
areas  and  add  products  together. 

Multiply  each  of  these  products  in  their  order,  by  number  representing 
number  of  intervals  of  section  forward  and  abaft  of  &.  Divide  difference 
of  these  moments  by  sum  of  products  of  half  girths,  previously  obtained. 

Multiply  product  by  common  distance  between  sections,  and  result  will 
give  distance  of  centre  of  gravity  from  JS  in  a  horizontal  plane. 

ILLUSTRATION.— Assume  half-girths  as  in  following  table,  and  distance  between 
sections  10  feet. 


Sec- 
tion. 

No. 

f. 

B. 
C. 
D. 
E. 

Half- 
Girths. 

FOR1? 

Multi- 
pliers. 

fARD. 
Prod- 
uct. 

Multi- 
pliers. 

Mo- 
ments. 

Sec- 
tion. 

Half- 
Girths. 

ABJ 
Multi- 
pliers. 

LFT. 
Prod- 
uct. 

Multi- 
pliers. 

Mo- 
ments. 

92 
80 
216 

128 
70 

Feet. 
25 
23 
21 

19 
17 
15 

A 

4 

2 

4 

2 

I 

25 
92 
42 
76 
34 
15 

I 

2 

3 
4 
5 

i* 

84 

228 

75 

No 
I  . 
2  . 

3  • 
4  • 
5  ~ 

Feet. 
23 
20 
18 
16 
14 

4 

2 

4 

2 

X 

92 
40 
72 
32 
14 

S 
2 

3 
4 
5 

534 

586 

615 

Moments  forward,  615  —  moments  abaft,  586  =  29  -4-  sum  of  product  534  =  .054, 
which  x  10  feet  =  .54  feet  forward  of  gj. 

Centre   of  Lateral    Resistance. 

Centre  of  Lateral  Resistance  is  centre  of  resistance  of  water,  and  as  its  po- 
sition is  changed  with  velocity  of  vessel,  it  is  variable.  It  is  generally  taken 
at  centre  of  immersed  vertical  and  longitudinal  plane  of  vessel  when  upon 
an  even  keel. 

If  vessel  is  constructed  with  a  drag  to  her  keel,  the  centre  will  be  moved 
proportionately  abaft  of  longitudinal  centre. 

Yacht  A  merica  had  a  drag  to  her  keel  of  2  feet,  and  centre  of  lateral  re- 
sistance of  her  hull  was  8.08  feet  abaft  of  centre  of  her  length  on  load-line. 

Centre   of  Effort. 

Centre  of  Effort  is  centre  of  pressure  of  wind  upon  sails  of  a  vessel  in  a 
vertical  and  longitudinal  plane.  Its  position  varies  with  area  and  location 
of  sails  that  may  be  spread,  and  it  is  usually  taken  and  determined  by  the 
ordinary  standing  sails,  such  as  can  be  carried  with  propriety  iii  a  moderately 
fresh  breeze. 

In  computing  this  position,  the  yards  are  assumed  to  be  braced  directly  fore 
and  aft  and  the  sails  flat. 

NOTE. — Centre  of  effort  of  sails,  to  produce  greatest  propelling  effect,  must  accord 
with  capacity  of  vessel  at  her  load  line,  compared  with  fullness  of  her  immersed 
body  at  its  extremities.  Thus,  a  vessel  with  a  full  load-line  and  sharp  extremities 
below,  will  sustain  a  higher  centre  of  effort  than  one  of  dissimilar  capacity  and  con- 
struction. 


NAVAL    ARCHITECTURE. 


659 


To,  Compute    Location    of*  Centre   of  Effort. 

RULE. — Multiply  area  of  each  sail  in  square  feet  by  height  of  its  centre  of 
gravity  above  centre  of  lateral  resistance  in  feet,  divide  sum  of  these  prod- 
ucts (moments)  by  total  area  of  sails  in  square  feet,  and  quotient  will  give 
height  of  centre  in  feet. 

2.  Multiply  area  of  each  sail  in  square  feet,  centre  of  which  is  forward  of 
a  vertical  plane  passing  through  centre  of  lateral  resistance,  by  direct  dis- 
tance of  its  centre  from  that  plane  in  feet,  and  add  products  together. 

3.  Proceed  in  like  manner  for  sails  that  are  abaft  of  this  plane,  add  their 
products  together,  and  centre  of  effort  will  be  on  that  side  which  has  greatest 
moment  of  sail. 

EXAMPLE. — Assume  elements  of  yacht  America  as  rigged  when  in  U.  S.  Service. 


SAIL. 

Area. 

Height  of 
Cent,  of  Grav- 
ity of  Sails. 

Vertical 
Moments. 

Distance  o 
of  Gravity 
Foreward. 

f  Centre 
of  Sails. 
Abaft. 

Mome 
Foreward. 

nta. 
Abaft. 

Flying  Jib  

Sq.  Feet. 
656 
1087 
1455 
2185 

Feet. 
28 
26 
34 
35 

18368 
28262 
49470 
76475 

52 
32 

3 
40 

34112 
34784 

4365 
87400 

Jib  

Foresail  

Mainsail  

5383 

172575 

68896 

9i765 

Vertical  moments  17 


5  I72  575  ==  32.06  =  height  of  centre  above  centre  of  lateral  re- 
Area  of  sails 5  383      • 

sistanee, 

Moments  {^^ 
sistanee. 


53°3 


-  ^-  =  4.25  =  distance  of  centre  abaft  centre  of  lateral  re- 


Relative   Positions   of  Centre   of*  Effort   and.   of  Lateral 
Resistance. 


Square  Riff. 
4A. 


Fare  and  Aft  Rig. 


and  —•  —  E'.     L  representing  length  of  load-line,  d  distance  of  centre  of  buoyancy 

of  vessel  below  it,  d'  distance  of  centre  of  lateral  resistance  abaft  centre  of  it,  d"  dis- 
tance of  centre  of  buoyancy  before  centre  of  it,  E  distance  of  centre  of  effort  before 
centre  of  lateral  resistance,  and  E'  distance  of  centre  of  effort  above  centre  of  lateral 
resistance. 

Meta-Centre. 

Meta-centre  of  a  vessel's  hull  is  determined  by  location  of  centre  of  grav- 
ity or  buoyancy  of  immersed  bottom  of  hull,  for  it  is  that  point  in  transverse 
section  of  hull,  where  a  vertical  line  raised  from  its  centre  of  gravity  or 
buoyancy  intersects  a  line  passing  through  centre  of  gravity  of  hull,  as 
Fig.  i,  page  650. 

To   Compute   Height   of  Meta-Centre. 

By  Moment  of  Inertia.  =r  =  M.  I  representing  moment  of  inertia  of  area 
of  water-line  or  plane  of  flotation,  and  D  volume  of  displacement  in  cube  feet. 

NOTE.—  Moment  of  Inertia  of  an  area  is  sum  of  products  of  each  element  of  that 
area,  by  square  of  its  distance  from  axis,  about  which  moment  of  area  is  to  be 
computed, 

To   Ascertain    Moment   of  Inertia  approximately. 

Rectangle  =  CLB3;    C  =  —  when  L  =  4B;    C  =  —  when  L  =  sB;    and  C  » 
12  50 

—  when  L  =  6  B.  With  very  fine  lines  and  great  proportionate  length  C  =  — . 
200  25 

L  and  B  measured  at  load-line. 


66o 


NAVAL    ARCHITECTURE. 


ILLUSTRATION.— Assume  length  of  vessel  233  feet,  breadth  43,  draught  16,  and 
displacement  2700  tons.  Length  =  5.65  beams;  hence  C  is  taken  at  — .  Volume 
of  displacement  =  2700  X  35  =  92  500  cube  feet. 


Then 


21  X  233  X  433 
400  X  92  500 


=  10.51.    Exact  height  of  moment  was  10.44  feet. 


By  Ordinates.  RULE. — Divide  a  half  longitudinal  section  of  load  water- 
line  by  ordinates  perpendicular  to  its  length,  of  such  a  number  that  area 
between  any  two  may  be  taken  as  a  parallelogram.  Multiply  sum  of  cubes 
of  ordinates  by  respective  distances  between  them,  and  divide  two  thirds 
of  product  by  volume  of  immersion,  in  cube  feet. 

ILLUSTRATION.— Take  dimensions  from  Figs.  2  and  3,  page  654. 

Cube. 
51460 

2 

3)102920 
7I5I.8)   34306.6  =  4.77  ft. 

If  there  are  more  ordinates,  their  coefficients  must  be  taken  in  like  manner,  as 
i— 4  — 2  — 4  — 2  — 4  — i. 

For  operation  of  this  method,  see  Simpson's  rule  for  areas,  page  342. 

Or,  —  i  y  X  =  M.  y  representing  ordinates  of  half -breadth  sections  at  load- 
line,  d  x  increment  of  length  of  load-line  section  or  differential  ofx,  and  D  displace- 
ment of  immersed  section  in  cube  feet. 

I 


Length. 

Cube. 

A  ... 

Length. 

Cube. 
.  885 

3  ••• 

2  ... 

7.7... 
....   9.5-. 

456 
857 

B  ... 
C  ... 
D... 

...7-8  
...6.8  

•  475 

•3;t 

ffi... 

....  9.9.. 

....10      .. 

97° 

•5146  X  10 

By   Areas. 


-  =  M.     a,  6,  c,  d, 


and  e  representing  ordinates  of  ist  or  load  water-line,  F  area  of  irregular  section 
between  ist  frame  and  stem,  and  A  area  of  like  section  between  last  frame  and 
stern-post,  both  in  sq.feet,  D  displacement,  in  cube  feet,  and  I  distance  between  frames 
or  sections  of  water-line,  as  may  be  taken,  in  feet. 

To  Ascertain  Areas  of  F  and  A. 


—  abXbc3-±-A=:F,  and  -de 

3  3 

Elements    of  Capacity  and   Speed   of  Several   Types  of 
Steamers   of  R.  N.    (W.  H.  White.) 


CLASSES. 

Length. 

Length 
to 
Breadth. 

Displacement. 

Speed. 

HP 

Displace- 
ment. 

to 

Displace- 
ment §. 

IRON-CLADS. 
Recent  types, 
do.    twin  sc. 
UNARMORED. 
Swift  cruisers 
Corvettes  
Ships  

Feet. 

300  to  330 
280  to  320 

270  to  340 

2OO  tO  22O 
1  60 
I25tOl70 

80  to  90 

400  to  500 
300  to  400 
250  to  350 

200  tO  300 

5.25105.75 
4-5  tO  5 

6.  5  to  6.  75 
6 

5.5106.25 
3    to  3.  25 

9    to  ii 
8    to  10 
7.  5  to  10 
7    to  9 

Tons. 
7500  to   9000 
6000  to   9000 

3000  to   5  500 
1800  to    2  ooo 
850  to      950 
420  to     800 

2OO  tO        250 
7OOO  tO  IOOOO 

5000  to   7  ooo 
3000  to  6000 
1500  to  4000 

Knots. 

14      to  15 
14      to  15 

15      toi6 
12.751013.25 
ii 
9.  5   to  ii 
8      to  9 

14      to  15 
13      to  14 

II        tO  15 

9      to  ii 

.9  to  i 
.7  to  .9 

1.3*01.5 
I       tO  I.  2 
I       tO  I.  2 

.8toi-4 
.8  to  1.  1 

.5  to  .6 
.4  to  .5 
3  to  .5 

,3  tQ    .4 

1  6  to  20 
15  tO  19 

20  tO  24 
13  to  14 
10  tO  1  1 

7  to  ii 
5  to   7 

10  to  ii 
7  to  10 
5  to  o 
3to  6 

Gun-vessels.. 
Gun-boats  .  .  . 
MERCHANT. 
Mail,  Inrge... 
44     smaller. 
Cargo,  large.. 
"    smaller. 

NAVAL    ARCHITECTURE. 


66 1 


To  Compute  3?o\ver  Required,  in  a  Steam  Vessel,  capac- 
ity  of  anotlier   "Vessel   "being   given. 

,      >  .    >,  j  ,        v  A      __      S3V  rV  .     C 

In  vessels  of  similar  models.     —  =  V;    — —  =  V  ;   — —  =  C;  and  — >  =  R; 

u  and  V  representing  product  of  volumes  of  given  and  required  cylinders  and  revo- 
lutions in  cube  feet,  a  and  A  areas  of  immersed  section  of  given  and  required 
vessel  in  sq.  feet  at  like  revolutions  and  speed  of  given  vessel,  s  and  S  speeds  of  given 
and  required  vessel  at  revolutions  of  given  vessel,  both  in  feet  per  minute,  r  and  r' 
revolutions  of  given  and  required  vessel  per  minute,  and  C  product  of  volume  of  com,' 
bined  cylinder  and  revolutions  for  required  vessel. 

ILLUSTRATION. — A  steam  vessel  having  an  area  of  amidshlp  section  of  675  sq.  feet, 
has  two  cylinders  of  a  combined  capacity  of  533.33  cube  feet,  and  a  speed  of  10.5 
knots  per  hour,  with  15  revolutions  of  her  engines.  Required  volume  of  steam 
cylinders,  with  a  stroke  of  10  feet,  for  a  section  of  700  feet  and  a  speed  of  13  knots 
with  14.5  revolutions. 


v  =  533. 33  X  15  =  8000  cube  feet, 
15  X  15  745.2  _ 


8000X700 


15745.2  cube  feet, 


675 
=  16  288.  i  cube  feet, 


=  8296. 3  cube  feet, 


and 


16288.; 


133  x  8296. 3  _ 
10.53 

=  561.66  cube 


J4-5  2X14-5 

feet,  which  -r-  10  stroke  of  piston,  12  for  ins.,  and  X  1728  ins.  in  a  cube  foot=. 

^-1  -  ^Zl_=  8087.9  52-  tm-  area  of  each  cylinder  —  diameter  0/101.5  ins. 
Approximate  Rules  to  Compute  Speed  and  IBP  of  Steam 


C  representing  coefficient  of  vessel,  A  area  of  immersed  amidship  section  in  sq.feet, 
V  velocity  of  vessel  in  knots  per  hour,  and  D  displacement  of  vessel  in  tons. 

NOTE.—  When  there  exists  rig,  an  unusual  surface  in  free  board,deck-houses,  etc., 
or  any  element  that  effects  coefficient  for  class  of  vessel  given,  a  corresponding  ad- 
dition to,  or  decrease  of,  following  units  is  to  be  made: 

Range  of  Coefficients  as  deduced  from  observation  is  as  follows  : 


SIDE-WHEEL. 


PPOPELLER. 


C 

C 

VESSEL. 

A 

D 

V 

Vs  A 

V'D! 

VESSEL. 

A 

D 

V 

V3  A 

V3D 

IH» 

IIP 

"IH?" 

"Tip" 

Steamboat. 

Sq.F. 

T's. 

K'ts. 

Steamboat. 

Medium  lines  .... 

43 

73 

10 

470 

212 

Medium  lines.. 

45 

— 

12 

— 

500 

Fine  lines  

150 
136 

465 
300 

13 
19 

570 
540 

219 

200 

T^ine  lines  

150 

— 

15 

— 

53» 

Steamer. 

Steamer. 

Medium  full  lines* 

675 

3600 

IO 

650 

214 

Medium  full... 

55° 

2532 

9 

194 

570 

390 

1475 

10 

180 

470 

Fine  linesf  

880 

5233 

15 

650 

211 

Torpedo  boat.. 

3600 

13 

20 

210 

170 

*  Full  rigged.  t  Bark  rigged. 

Coefficients  as  Determined  by  Several  Steamers  of  H.  B.  M. 
(C.  Mackrow,  M.  I.  N.  A.) 


Service. 


Length. 

Length 
Beam. 

Area  of 
Section  at  gj. 

Displace- 
ment. 

IIP 

Speed. 

T15  =  c- 

Feet 

Sq.  Feet. 

Tons. 

Knots. 

185 

6-53 

775 

782 

10.34 

333 

212 

589 

377 

1554 

1070 

10.89 

456 

270 

7-33 
6.43 

814 
632 

5898 
3057 

2084 
2046 

"•5 
12.3 

598 
574 

380 

6.  52 

1308 

9487 

3205 

12.05 

7M 

400 

6-73 

1198 

9152 

5971 

13.88 

536 

^62 

400 

7-33 
6-73 

778 
1185 

5600 
9071 

3945 
6867 

14.06 
15-43 

548 
634 

3K 

662  NAVAL   ARCHITECTUKE. 

Approximate    R-vile    for    Speed,    of   Screw    Propellers. 

(Molesworth.) 

IO'V_N.      PN  ioiV  88  «  PN  and88"-P 

~P~-N>      To7-V'       ~N~--P>       -p--N'      "88"- Wj  l~N~- 

V  and  v  representing  velocities  in  knots  and  miles  per  hour,  P  pitch  of  propeller  in 

feet,  and  N  number  of  revolutions  per  minute. 
This  does  not  include  slip,  which  ranges  from  10  to  30  per  cent. 

Pitch    of  Screw    Propeller. 

Pitch  ranges  with  area  of  circle  described  by  diameter  of  screw  to  that  of 
amidship  section. 


Area  of  screw  circle  to  amidship  H    *    \          I 
section  =  i  to 1,1  5       4<5 


3-53       2-5 


Two  Blades. 

Pitch  to  diameter  of  screw  =  i  to   |    ,8    |  1.02  |  i.n  |  1.2   |  1.27  1 1.31  |  1.4  |  1.47 
Four  Blades.  |  1.08  |  1.38  |  1.5   |  1.62  |  1.71  |  1.77  1  1.89  |  1.98 

Length  = .  166  diameter. 

Slip   of  Side-wlieels. 

Radial  Blades.     2  (A^-C)-S>       Feathering.     *'5  (^~c)  =  S.    A  representing 

length  of  arc  of  immersed  circumference  of  blades,  c  length  of  chord  of  immersed  arc. 
and  S  slip,  all  in  feet. 

Area  of  Blades. 

JTT>  TTP 

River  Service,  '-—r —  =  A.  Sea  Service,  ~y—  =  A,  D  representing  diameter 
of  wheel  in  feet,  and  A  area  of  each  blade  in  square  feet. 

Length  of  Blades.    .7  in  River  service  and  6  in  Sea  service. 

Distances  between  Radial  Blades.  2.25  in  River  service  and  3  feet  in  Sea  service; 
between  Feathering  blades,  4  to  6  feet. 

Proportion   of  Power   Utilized   in   a    Steam   Vessel. 

p g 

Side  Wheel.    ^r— -  =  C.      P  representing  gross  HP,  z  loss  of 

.00000259  <P  r2 

effect  by  slip  and  oblique  action  of  wheels,  d  diameter  of  wheels  at  centre  of  effect, 
r  revolutions  per  minute,  and  C  coefficient  for  vessel. 

ILLUSTRATION.— IIP  of  engines  of  a  side-wheel  steamer  is  1120;  slip  of  wheels 
and  loss  by  oblique  action,  33.37  per  cent. ;  diameter  of  centre  of  effect  of  wheels  is 
29.5  feet,  and  number  of  revolutions  13.5  per  minute;  what  is  coefficient,  and  what 
power  applied  to  propel  vessel  ? 

NOTE. — Slip  of  wheels  from  their  centre  of  effect  in  this  case  is  15.37  Per  cent., 
and  loss  by  oblique  action  18  per  cent.  Hence,  representing  total  power  by  100, 
100  —  (i8-f- 15.37)  =66.63  per  cent,  of  power  applied  to  wheels. 

As  assumed  power  that  operates  upon  wheels  in  this  case  is  taken  at  86  12  per 
cent,  of  power  exerted  by  engines,  86.12  X  33.37  =  28.74  per  cent,  for  sum  of  loss 
by  wheels. 

II2O (lI2O  X  28.74-:-  ICO)          7O8.II 

— ^ — — ^ — £- !-  =  '-?—-r  =  65.63  coefficient. 

.00000259  X  29.53  x  13. 52       12.16 

Speed  of  vessel  being  10  knots  per  hour  =  17.05  feet  per  second,  power  applied 
to  propel  vessel  at  this  speed  =  65. 63  x  17-05 2=  19076.13,  and  IP  exerted  = 
,9076.13X17.05X60  _ 

33°°°  H».  of'poTer 

Friction  of  engines  1.5  Ibs.  upon  3848  sq.  ins.  X  13.5  revolu-i  -j 

tions  XioX2-=-33oooX2 }      94-451 

Friction  of  load  6  per  cent,  upon  pressure  of  steam,  less  2  Ibs. )      6        f 

for  friction  of  engine,  as  above )         '*->  J 

Oblique  action  of  wheels 201.6  18 

Slip  of  wheels 172.14  15.37 

Absorbed  by  propulsion  of  vessel 591. 36          52. 8 

II2O  IOO 


NAVAL    ARCHITECTURE. 


663 


•rp  Per  cent 

"•  of  Power 

Screw  Propeller.    Friction  of  engines 06.06)  QQ 

Friction  of  load 81.48  } 

"       of  screw  surface  and  resistance  of  edges  of  blades 53.44  6.83 

Slip  of  propeller. 205.55  26.27 

Absorbed  by  propulsion  of  vessel 375-92  48.04 

782.45  100 

NOTE.— From  experiments  of  Mr  Froude,  he  deduced  that,  as  a  rule,  only  37  to 
40  per  cent,  of  whole  power  exerted  was  usefully  employed. 

With  an  auxiliary  propeller,  essential  differences  are  in  friction  of  surfaces  and 
edges  of  blades  of  propeller  and  slip  of  propeller,  being  as  12  to  6-83  in  excess  in  first 
case,  and  as  13.7  to  26.27  m  second  case,  or  50  per  cent  less. 

Resistance  of  Bottoms  of  Hulls  at  a  Speed  of  one  Knot  per  Hour. 


Smooth  wood  or  painted. 01    Ib. 

Smooth  plank 016  " 

Iron  bottom,  painted 014  " 


Copper. 007  Ib. 

Moderately  foul 019  " 

Grass  and  small  barnacles 06    " 


Sailing. 

Ratio   of  Effective  Area  of  Sails  and  of  Vessel's  Speed 
under    Sail   to   Velocity  of  Wind. 


Ratio  of 

Ratio  of 

Ratio  of 

Ratio  of 

Effective 

Speed  of 

Effective 

Speed  of 

COURSE. 

.   Area 

Vessel 

COURSE. 

Area 

Vessel 

of  Sails. 

to  Wind. 

of  Sails. 

to  Wind. 

5  points  of  wind  
2      "     abaft  beam 

•59 

-33 

Wind  abeam 

.82 

.6 

e 

'  '     astern       ...  . 

6      "     of  wind,,. 

:§5 

•5 
.«; 

**     on  quarter... 

X.o6 

:& 

Propulsion    and    Area   of  Sails. 

Plain  sails  of  a  vessel  are  standing  sails,  excluding  royals  and  gaff  topsails. 
Resistance  of  vessels  of  similar  models  but  of  different  dimensions  for  equal 
speeds  —  D§ 

Hence  -7  =  (=y)    •    a  and  a'  representing  areas  of  sails  of  known  and  given  ves- 
tels,  and  D  and  D'  their  displacements  in  tons. 
ILLUSTRATION.—  Assume  D  and  D'  =  24oo  and  1600. 


Then 


i6oo/ 


—  ^/I52_I<3Ij  hence  area  of  sails  a'  =  ——  =  .763  per  cent 

I-I 


Tn  Vessels  of  Dissimilar  Models.—  Plain  sail  area  should  be  a  multiple 
of  Df. 

Multiples  for  Different  Classes  of  Vessels,  R.  N. 


Sailing. 

Ships  of  Line 100  to  120 

Frigates ) 

Sloops >    120  to  160 

Brigs ) 


Steamers. 

Ships,  iron-clad 60  to  80 

Frigates , ) 

Sloops J    80  to  120 

Brigs ) 


English  Yachts,  designed  for  high  speed,  have  multiples  from  180  to  200, 
and  when  designed  for  ordinary  speed  from  130  to  180. 

When  Area  of  Sail  to  Wet  Surface  of  Hull  is  taken.—  American  yacht  Sappho  had  a 
ratio  of  2.7  to  i,  and  several  English  yachts  nearly  the  same,  while  in  some  others 
it  was  but  2  to  i. 


664 


NAVAL   AKCHITECTURE. 


Location    of  IVlusts,  etc.     Load-line  =  100. 


VESSEL, 

D 

Fore. 

stance  from  Ste 
Main. 

n. 
Miuen. 

Foot  of  Sail.* 

Height  of  Centre 
of  Effect  above 
Water-line  = 
Breadth.* 

Ship  

10  to  20 

12  tO  20 

17  to  20 

l6  tO  22 

53  to  58 
54  to  60 
64  to  65 
55  to  61 

^6  tO  A.2 

80  to  90 
81  to  91 

125  to  160 
130  to  160 
160  to  165 
160  to  170 

170  tO  IQO 

.5     102 

•5    to  1.95 
•5    to  1.75 
•  5    to  i.  75 
.2q  to  i.  75 

Bark  

Brig  

Schooner  .... 
Sloop... 

*  Measured  from  Tack  of  Jib  to  Clew  of  Spanker  or  Mainsail. 

Rake  of  Masts. 

Ships. — Foremast  o  to  .28  of  length  from  heel,  Main  and  Mizzen  o  to  .25. 
Schooners.— Foremast  .1  to  .25,  Mainmast  .63  to  .77.     Sloops.— .08  to  .u. 


SAILS. 

3  Yards  upon 
each  Mast. 

Area,   c 

4  Yards  upon 
each  Mast. 

>f  Sails. 

SAILS. 

3  Yards  upon 
'  each  Mast. 

4  Yards  upon 
each  Mast. 

jib  

.08 

.08 

Mizzenmast.  .  .  . 

.  127 

Foremast  
Mainmast  

•295 
.417 

•295 
.417 

Spanker  or  } 
Driver.  .  .  )  " 

.081 

,068 

Proportional  A 
SAIL. 

rea  of  Sails  u^ 

Fore. 

oon  each  Mast 
Main. 

under  above  D 
Mizzen. 

ivisions. 

Proportion  to  x. 

Course    

•"5 
.105 
•075 

.08 

.097 
.00 

.063 
•045 

.08 

.162 

;$ 

.138 

.127 

.089 
.063 

•075 
.052 

.081 

.063 
•045 
.032 
.068 

.389 
.358 
•253 

•33 
•303 
.215 
.152 

Topsail  

Topgallant  sail        .   . 

Roval  

Spanker  or  Driver  
Jib  

375 

•375 

.417 

.417 

.208 

.208 

I 

i 

Balance  of  Sails. — Effect  of  jib  is  equal  to  that  of  all  sails  upon  main- 
mast, and  sails  upon  mizzenmast  balance  those  of  foremast. 

Areas  of  sails  upon  masts  of  a  ship  should  be  in  following  proportion : 
Fore 1.414  |  Main 2  |  Mizzen i 

When,  therefore,  main  yard  has  a  breadth  of  sail  of  100  feet,  fore  yard 
should  have  70.71  feet,  and  mizzen  50  feet  j  topgallant  and  royal  yards  and 
sails  being  in  same  proportion. 

Angles   of  Heel   for   Different  Vessels. 

Approximately.     — — — =  S.      D  representing  displacement  of  vessel  in  Ibs., 

M  height  of  meta- centre  above  centre  of  gravity  in  feet,  a  angle  of  heel  of  vessel  in  cir- 
cular measure,*  and  H  height  of  centre  of  effect  above  centre  of  lateral  resistance, 
infeet. 

Moment  of  sail  should  be  equal  to  moment  of  stability  at  a  defined  angle 
of  heel. 

Angle, 

Frigates,  etc 4° 

Corvettes 5° 

ILLUSTRATION.  — Assume  displacement  170  tons,  height  of  meta-centre  6.75  feet> 
H  =  36  feet,  and  angle  of  heel  9° ;  what  should  be  area  of  sails  ? 
170  X  2240  —  380  800  Ibs.     9°  =  .  107. 


._.        Circular 
An&le'    Measure. 
.07 
.087 


Circular 
Measur*. 

Schooners,  etc 6°         .  105 

Yachts 6°  to  9°    .  105  to  .  107 


*  S«e  rule,  page  113. 


NAVAL   AKCHITECTURE. 


665 


Trimming    of  Sails. 

That  a  vessel's  sail  may  have  greatest  effect  to  propel  her  forward,  it  should 
be  so  set  between  plane  of  wind  and  that  of  her  course,  that  tangent  of  angle 
it  makes  with  wind  may  be  twice  tangent  of  angle  it  makes  with  her  course. 

Or,  tan.  a  =  2  tan.  b.  a  representing  angle  of  sail  with  wind,  and  b  angle  of  sail 
and  course  of  vessel. 

Angles   of  Course   and    Sails   \vith.    Wind. 


Wind 
Ahead. 

Angle 
Course. 

Tan- 
gent. 

Half 
gei". 

.281 

.365 
.461 
.707 

Angles 
with 
Wind. 

of  Sail 
with 
Course. 

Wind 

Abaft. 

Angle 
of 
Course. 

Tan- 
gent. 

Half 
Tan- 
gent. 

1.082 
1.368 
1.781 
3-754 

Angle, 
with 
Wind. 

of  Sail 
with 
Course. 

Points. 
4 
5 
6 
Abeam 

45° 
56°  15' 
67°  30' 
900 

.562 
•732 
•923 
I-4I5 

29°  1  8' 
36°  12' 
42°  43' 
54°  45' 

15°  42' 

20°      3' 

24°  45' 
35°  16' 

Points. 

2 

3 

I 

112°  30' 
123°  45' 
135° 
157°  30' 

2.166 
2-737 
3o62 
7-5" 

65°  13' 
69056; 
74°  17 
82°  25' 

47°  17' 
53°  49' 
60043' 

75°    5' 

Fig.  6. 


Effective   Impulse   of  "Wind. 

Let  P  o,  Fig.  6,  represent  direction  by  com- 
pass and  force  of  wind  on  sail,  AB;  from  P 
draw  P  C  parallel  to  A  B,  from  o  draw  o  C  per- 
pendicular to  AB;  o  C  is  eft'ective  pressure 
of  wind  on  sail  A  B,  and  r  C,  perpendicular  to 
plane  of  vessel,  is  component  of  o  C,  which  pro- 
duces lateral  motion,  as  heel  and  leeway,  and 
r  o  is  component  of  o  C,  which  propels  vessel. 
I  sin.  a  •=.  P  ;  P  cos.  x  —  L  ;  and  P  sin.  x  —  E. 
I  representing  direct  impact  and  P  effective 
pressure  of  wind  on  sail,  L  effective  impact 
producing  leeway,  and  E  effective  impact  which 
propels  vessel. 

NOTE.—  The  law  as  usually  given  is  sin.2.    This  is  manifestly  incorrect,  as  it  gives 
results  less  than  normal  pressure  for  angles  of  small  incidence.     At  an  angle  of  in- 
cidence of  wind  of  25°,  the  law  of  sin.  is  exact.     Hence,  although  it  may  not  be 
exact  at  all  angles,  it  is  sufficiently  so  for  practical  purposes. 
ILLUSTRATION  i.  —  Assume  wind  5  points  ahead,  and  I  =  100  IDS. 
By  preceding  table  angle  of  course  with  wind  56°  15';  hence  angle  of  sail  a,  with 
wind  36°  12',  as  tan.  36°  12.'  =  2  tan.  20°  3',  and  angle  x  56°  15'  —  36°  12'  =  20°  3'. 
Then,  100  X  sin.  36°  12'  =  100  X  .5906  =  59.06;      59.06  X  cos.  20°  3'  =  59.06  X 
.9394  =  55.48,  and   59.06  x  sin.  20°  3'  =  59.06  x  .3426  =  20.23  Ibs. 
2.  —Assume  wind  4  points  abaft,  and  I  =  100  Ibs. 

Then,  iooxsin.274°  17'=  100  X  •  g6262  =r  92.  66  ;    92.66  X  cos.  180°  —  74°  17'+  45° 
=  60°  43'  =  92.  66  X-  49  =  45.41,  and  92.  66  X  sin.  60°  43'  =  92  66  x  .  8722  =  80.  82  Ibs. 

To    Compute    Sailing    Power   of  a   Vessel. 

F/sin.  w,  sin.  s  =  P. 
To   Compute    Careening    IPower  of  a    Sailing  Vessel. 

F/sin.  w,  cos.  s  =  P.     F  representing  area  of  sails  in  sq.  feet,  f  force  of  wind  in 
Ibs.  per  sq.foot,  w  angle  of  wind  to  sails,  and  s  angle  of  sails  to  course  of  vessel. 

To   Compute   Angle   of  Steady   Heel. 

Within  a  Range  0/8°. 


a  P  E 
„  ., 

JJ  .M 


=  sin.  H.    a  representing  area  of  plain  sail  in  sq.feet,  P  pressure  of  wind 

in  Ibs.  per  sq.foot,  E  height  of  centre  of  effect  above  mid-draught,  in  feet,  D  displace- 
ment of  hull,  in  Ibs.,  and  M  height  of  meta-centre  in  feet. 

P  assumed  at  i  Ib.  per  sq.  foot,  or  that  due  to  a  brisk  wind. 

ILLUSTRATION.—  Assume  a  =15  600,  draught  =  20,  and  £  =  62;  hence  62-}-—  = 
7*,  D  =  6  800000,  and  M  =  3. 


Then 


15  6oo  X  i  X  72 1 123  200 

6  800  ooo  X  3    ~~  20  400  ooo  ~ 


;  =  3°  10'. 


3K* 


666 


NAVAL    ARCHITECTURE. 


Course   and   Apparent   Course   of  \Vind. 

Apparent  course  of  a  wind  against  sails  of  a  vessel  is  resultant  of  normal 
course  of  wind  and  a  course  equal  and  directly  opposite  to  that  of  vessel. 


Fig.  7- 


ILLUSTRATION.  —  If  P,  Fig.  7,  repre- 
sent direction  by  compass  and  force  of 
wind,  and  a  b  direction  and  velocity  of 
vessel,  from  P  draw  P  c  parallel  and 
equal  to  a  6,  join  c  a  and  it  will  repre- 
sent direction  and  force  of  apparent 
wind. 

Or,  —  =  ratio  of  velocity  of  apparent 


aP 


wind  to  that  of  vessel,  —=  ==.  ratio  of  velocity  oj  wind  to  that  of  vessel. 

Resistance   of  Air.     (Mr.  Froude.) 

Resistance  of  wind  to  a  vessel  is  estimated  as  equivalent  to  square  of  its 
velocity. 

In  a  calm,  resistance  of  air  to  a  steamer  =  one  thirty-fourth  part  of  resist- 
ance of  water,  and  when  a  steamer's  course  is  head-to,  and  combined  veloc- 
ity of  vessel  and  wind=  15  knots,  resistance  is  one  ninth  of  that  of  the  water. 

Resistance  of  air  to  a  sq  foot  of  surface  at  right  angles  to  course  of  a  ves- 
sel is  about  .33  lb.,  and  when  surface  is  inclined  to  direction  of  wind,  press- 
ure varies  as  sine  of  angle  of  incidence, 

Mean  of  angles  of  surface  of  a  steamer  exposed  to  wind  may  be  taken  at 
45° ;  hence  their  resistance  is  about  .25  lb.  per  sq.  foot  when  wind  has  a  ve- 
locity of  10  knots  per  hour. 

If  sectional  area  of  a  steamer's  hull  above  water  is  750  sq.  feet,  resistance 
to  air  at  a  speed  of  10  knots  in  a  calm  would  be  750  X  .25  =  187.5  Iks.,  and 
resistance  to  smoke-pipe,  spars,  and  rigging  (brig  rigged)  would  be  201  Ibs. 

Leeway. 

Angle  of  Leeway  in  good  sailing  vessels,  close  hauled,  varies  from  8°  to 
12°,  and  in  inferior  vessels  it  is  much  greater. 

Ardency  is  tendency  of  vessel  to  fly  to  the  wind,  a  consequence  of  the 
centre  of  effort  being  abaft  centre  of  lateral  resistance. 

Slackness  is  tendency  of  vessel  to  fall  off  from  the  wind,  a  consequence  of 
the  centre  of  effort  being  forward  centre  of  lateral  resistance. 

Results  of  Experiments  upon  Resistance  of  Screw-propellers,  at  High  Velocities 
and  Immersed  at  Varying  Depths  of  Water. 


Immersion  of 
Screw. 

Resistance. 

Immersion  of 
Screw. 

Resistance. 

Immersion  of 

Screw. 

Resistance. 

Surface, 
i  foot. 

I 
5 

2  feet. 

3    u 

7 
7-5 

4  feet, 

5     " 

Is 

Slip  of Propeller,  15  per  cent. ;  of  Side-wheel  (feathering  blades),  and  tak- 
ing axes  of  blades  as  the  centre  of  pressure,  23  per  cent. 

ITree"board. 

Measured  from  Spar-deck  stringer  to  surface  of  water.    Depth  of  Hold  from  under- 
side of  spar  deck  to  top  of  ceiling. 


Hold. 

Per  Ft. 

Hold. 
Feet. 

12 

M 

Per  Ft. 

Hold. 

Per  Ft. 

Hold. 

Per  Ft. 

Hold. 

Per  Ft. 

Hold. 

Per  Ft. 

Feet. 
8 

10 

Ins. 
*  5 

2 

Ins. 
2.25 

2-5 

Feet. 
16 
18 

Ins. 
2-75 
3 

Feet. 
20 
22 

Ins. 
3-125 
3-25 

Feet. 
24 
26 

Ins. 

3-375 
3-5 

Feet. 
28 
30 

Ins. 
3-625 
3-75 

NAVAL    ARCHITECTURE. 


667 


PL 

Soobd 


Plating    Iron    Hulls. 

=  T.     D  representing  displacement  in  torn,  L  length  of  hull,  b  breadth,  and 

d 


d  depth.     Or,  .o$f^d  =  T.    /representing  distance  between  centres  of  frames,  and 
d  depth  of  plate  below  load-line,  all  in  feet,  and  T  thickness  of  plate  in  ins. 


Masts    and    Spars 

Lower  masts at  spar  deck. 

Bowsprit "  stem. 

Topmasts "  lower  cap. 

Topgallant  masts "  topmast  cap. 


Diameter  for  Dimensions. 

Jib-boom at  bowi 

Yards in  middle. 

Gaffs at  inner  end. 

Main  and  Spanker  booms  at  taffrail. 


Fore  and  main  masts,  when  of  pieces,  i  inch  for  each  3  to  3.25  feet  of  whole 
length.  Mizzenmast  .66  diameter  of  mainmast.  Masts  of  one  piece  i  inch  for  each 
3-5  to  3-75  feet  of  whole  length. 

Bowsprit,  depth,  equal  diameter  of  mainmast;  width,  diameter  equal  to  foremast. 


Main  and  fore  topmasts 

Mizzen  topmast 

Topgallant  masts 

Royal  masts 

Topgallant  poles 

Jib  boom 

Fore  and  main  yards 

Topsail  yards 

Cross -jack,  Topgallant,   and) 

Royal  yards ) 

Main  and  Spanker  booms 

Gaffs 

Studding-sail  yards  and  booms. 


inch  for  each  3      to  3.25  ) 
"     "      "    3-25  " 


875 


feet  of  whole  length. 


2  ft.  of  length  beyond  bowsprit  cap. 

4 

4 

5 


3-5 

3-5  to  4 
4-5  to  4.75 


feet  of  whole  length. 


Pd_ 


2400 


Rudder   Head.     (Mackrow.) 
l  =  T:    .  196  C  D3  —  M ;     3  / — — -  —  D ;    and  — —  =  P.   P  representing  press- 

V  •  196  c  2400 

ure  on  rudder  when  hard  over,  in  tons,  d  distance  of  geometrical  centre  of  rudder  from 
axis  of  motion,  in  ins.,  T  stress  on  head,  and  M  moment  of  resistance  of  head,  both  in 
inch-ions,  A  immersed  area  of  rudder  in  sq.  feet,  v  velocity  of  water  passing  rudder 
in  knots  per  hour,  and  G  coefficient  —  3. 5  per  sq.  inch  for  Iron,  and  .  125  for  Oak. 

ILLUSTRATION. — Assume  area  of  wooden  rudder  24  sq.  feet,  distance  of  its  geomet- 
rical centre  from  centre  of  pintles  2  feet,  and  velocity  of  water  10  knots. 

=  i  ton.     i  X  2  X  12  =  24  inch-tons.     3  I 1  —  =  9-93  tw*- 

Memoranda. 

Weights.  —A  man  requires  in  a  vessel  a  displacement  or  488  Ibs.  per  month,  for 
baggage,  stores,  water,  fuel,  etc.,  in  addition  to  his  own  weight,  which  is  estimated 
at  175  Ibs.  A  man  and  his  baggage  alone  averages  225  Ibs. 

A  ship,  150  feet  in  length,  32  beam,  and  22.83  in  depth,  or  664  tons,  C.  H.  (O.  M.), 
has  stowed  2540  square  and  484  round  bales  of  cotton.  Total  weight  of  cargo 
i  254448  Ibs.,  equal  to  4.57  bales,  weighing  1889  Ibs.,  per  ton  of  vessel. 

A  full  built  ship  of  1625  tons,  N.  M.,  can  carry  1800  tons'  weight  of  cargo,  or  stow 
4500  bales  of  pressed  cotton. 

Hull  of  iron  steamboat  John  Stevens  —  length  245  feet,  beam  31  feet,  and  hold 
ii  feet;  weight  of  iron  239440  Ibs.  And  of  one  other — length  175  feet,  beam  24 
feet,  and  8  feet  deep;  weight  of  iron  159 190  Ibs. 

Weight  of  hull  of  a  vessel  with  an  iron  frame  and  oak  planking  (composite),  com- 
pared with  a  hull  entirely  of  wood,  is  as  8  to  15. 

An  iron  hull  weighs  about  45  per  cent,  less  than  a  wooden  hull. 

Iron  ship,  254  feet  in  length,  42  beam,  and  23.5  hold,  1800  tons  register,  has  a  stow- 
age of  3200  tons  cargo  at  a  draught  of  22  feet.  Weight  of  hull  in  service  1450  tons. 

Loss  by  Weight  per  Sq.  Foot  per  Month  of  Metalling  of  a  VesseVs  Bottom  in  Service. 
Copper  .0061  Ib. ;  Muntz  metal  .0045  Ib. ;  Zinc  .007  Ib. ;  and  Iron  .0204  Ib. 
Comparison  between  Iron  and  Steel  plated  Steamers. — In  a  vessel  of  5000  tons 
displacement,  hull  of  steel-plated  will  weigh  320  tons  less  =  6.66  per  centum  less. 


668 


OPTICS. 


Fig.  2. 


OPTICS. 

IVTirrors,  in  Optics,  are  either  Plane  or  Spherical.  A  plane  mirror  is  a 
plane  reflecting  surface,  and  a  spherical  mirror  is  one  the  reflecting  surface 
of  which  is  a  portion  of  surface  of  a  sphere.  It  is  concave  or  convex,  ac- 
cording as  inside  or  outside  of  surface  is  reflected  from.  Centre  of  the 
sphere  is  termed  Centre  of  curvature. 

Focus — Point  in  which  %  number  of  rays  meet,  or  would  meet  if  produced. 
Fig.  i.  Principal  Focal  Distance  is  half  radius 

of  curvature,  and  is  generally  termed  the 
focal  distance.     Line  a  c  is  termed  the 
principal  axis,  and  any  other  right  line 
through  c  which  meets  the  mirror  is  termed 
a  Secondary  axis.     When  the  incident 
rays  are  parallel  to  the  principal  axis,  the 
reflected  rays  converge  to  a  point,  F. 
Conjugate  Foci  are  the  foci  of  the  rays  proceeding  from  any  given  point 
in  a  spherical  concave  mirror,  and  which  are  reflected  so  as  to  meet  in  an- 
other point,  on  a  line  passing  through  centre 
of  sphere.    Hence,  their  relation  being  mu- 
tual, they  are  termed  conjugate. 

Let  P  be  a  luminous  point  on  principal  axis, 
Fig.  2,  and  P  i  a  ray ;  draw  the  normal  line  c  i, 
whichfis  a  radius  of  the  sphere;  then  c  i  P  is  an 
gle  of  incidence,  and  c  i  O  the  angle  of  reflection, 
equal  to  it ;  hence  c  i  bisects  an  angle  of  triangle 

t  P      c  P 

P  i  0,  and  therefore,  —  =  — 
'  i  0      cO 

When  conjugate  focus  is  behind  a  mirror,  and  reflected  rays  diverge,  as 
if  emanating  from  that  point,  such  focus  is  termed  Virtual,  and  a  focus  in 
which  they  actually  meet  is  termed  Real. 

Fig.  3.  ^  As  a  luminous  point,  as  P,  Fig.  3,  is 

^"^  moved  to  the  mirror,  the  conjugate  focus 

moves  up  from  an  indefinite  distance  at 
back,  and  meets  it  at  surface  of  mirror. 
If  an  incident  ray  converges  to  a  point 
s,  at  back  of  mirror,  it  will  be  reflected 
to  a  point  P  in  front.  The  conjugate 
foci  P  s  having  changed  places. 

Pencil— Kays  which  meet  in  a  focus  and  are  taken  collectively. 

Objects. — As  regards  comparative  dimensions  or  volumes,  it  follows,  from 
similar  triangles,  that  their  linear  dimensions  are  directly  as  their  distances 
from  centre  of  curvature. 

To    Compute   Dimension,   or  "Vol-urne   of  an    Image. 

When  Dimensions  and  Position  of  Object  are  Given,  and  for  either  Convex 
or  Concave  Mirrors. 

—  =z  — - ,  or  —  =  ^r .    L  and  I  representing  lengths  of  image  and  object,  F  fooal 

IV  Li        r 

length,  and  D  and  d  respectively,  distances  of  image  and  object  from  principal  focux. 

Refraction . 

Deviation.— Angle  at  which  a  ray  is  diverted  from  its  original  or  normaJ 
course  when  subjected  to  refraction  is  thus  termed. 

Indices^of Refraction. — Katio  of  sine  of  angle  of  incidence  to  sine  of  angle 
of  refraction,  when  a  ray  is  diverted  from  one  medium  into  another,  is  termed 
relative  index  of  refraction  from  former  to  latter. 


OPTICS.  669 

When  a  ray  is  diverted  from  vacuum  into  any  medium,  the  ratio  is  greater 
than  unity,  and  is  termed  absolute  index  or  index  of  refraction. 

Mean  Indices  of  Refraction. 


Eye,  vitreous  humor i. 339 

u    crystalline  lens,  under 1.379 

"  "  "    central......  1.4 

Diamond 2.6 

Glass,  flint. . '. i-57 


Glass,  lead,  3  flint 2.03 

"     lead  2,  sand  i 1.99 

"       "    i,  flint  i 1.78 

Ice 1.31 

Quartz 1.54 


For  indices  of  other  substances,  see  page  584. 
Heat  increases  refractive  power  of  fluids  and  glass. 

Critical  A  ngle.  —  Its  sine  is  reciprocal  of  index  of  refraction,  the  incident 
ray  being  in  the  less  refractive  medium. 


Visual  Angle  is  measure  of  length  of  image  of  a  straight  line  on  the  retina. 

Total  Reflection  is  when  rays  are  incident  in  the  more  refractive  medium, 
at  an  angle  greater  than  the  critical  angle. 

Mirage.—  An  appearance  as  of  water,  over  a  sandy  soil  when  highly  heated 
by  the  sun. 

Caustic  Curves  or  Lines  are  the  luminous  intersections  from  curve  lines,  as 
shown  on  any  reflective  surface  in  a  circular  vessel. 

To   Compute    Index   of  Refraction. 

-|5:  —  —  Index.    I  representing  angle  of  incidence,  and  R  that  of  refraction. 
Sin.  R 

Xo    Compute    Refraction. 

Concave-Convex  and  Meniscus.  —  Effect  of  a  concave-convex  in  refracting 
light  is  same  as  that  of  a  convex  lens  of  same  focal  distance,  and  that  of  a 
meniscus  is  same  as  a  concave  lens  of  same  focal  distance. 

Meniscus,  with  parallel  rays  ^—  -  =  P. 

Magnifying  Power.  —  In  Telescopes  the  comparison  is  the  ratio  in  which  it 
apparently  increases  length.  In  Microscopes  the  comparison  is  between  the 
object  as  seen  in  the  instrument  and  by  the  eye,  at  the  least  distance  of 
vision,  which  is  assumed  at  10  ins.,  and  the  magnifying  power  of  a  micro- 
scope is  equal  to  the  distance  at  which  an  object  can  be  most  distinctly  ex- 
amined, divided  by  the  focal  length  of  the  lens  or  sphere. 

Linear  power  is  number  of  times  it  is  magnified  in  length,  and  Super- 
ficial, number  of  times  it  is  magnified  in  surface. 

Magnifying  power  of  microscopes  varies,  according  to  object  and  eye- 
glass, from  40  to  350  times  the  linear  dimensions  of  object,  or  from  1600  to 
122500  times  its  superficial  dimensions. 

Apparent  Area.  —  As  areas  of  like  figures  are  as  the  squares  of  their  linear 
dimensions,  the  apparent  area  of  an  object  varies  as  square  of  visual  angle 
subtended  by  its  diameter. 

The  number  expressing  Magnification  of  Apparent  Area  is  therefore 
square  of  magnifying  power  as  above  described. 

ILLUSTRATION.  —If  diameter  of  a  sphere  subtends  i°  as  seen  by  the  eye,  and  100 
as  seen  through  a  telescope,  the  telescope  is  said  to  have  a  power  of  10  diameters. 


6/O  OPTICS. 

To    Compute    Elements   of  Mirrors   and.   Lenses. 

Mirrors.    Spherical  Concave.*    —  =  D;  —  =  L. 

r  —  2  I  r  —  2  I 

Or  Lr  d2 

Spherical  Convex.^     —=— — =D;     — =— ; —  =  I.        Parabolic  Concave.    -—r  —  F 
2L-J-r  2L-j-r  i6h 

Unequally  Convex,  $     /*   .     —  F.       Piano-  Convex.  §    2  B  — .  66  t  =  F. 

Hyperbolic  Concave.]]      Elliptic  Concave.  1[  Sphere.     — = =  F. 

O  representing  object—  i,  r  radius  of  convexity,  I  and  L  length  or  distance  of  object 
from  vertex  of  curve,  and  from  external  vertex,  D  dimension  of  object,  d  diameter  of 
base,  F  focal  distance,  and  h  depth  of  mirror  in  like  dimensions,  1  index  of  refraction, 
and  t  thickness  of  lens. 

ILLUSTRATION  i.— Before  a  concave  mirror  of  5  feet  radius  is  set  an  object  at  1.5 
feet  from  vertex  of  curve ;  what  is  ratio  of  apparent  dimension  of  image,  and  what 
is  length  of  and  distance  of  object  from  external  vertex  ?  Object  =  i. 

1X5      =  2. 5  feet,  and  -ii-5-*-5-  =  3. 75  feet. 
5—2X15  5—2X15 

a.— If  object  is  set  at  4.5  feet  from  vertex  of  a  like  mirror,  what  is  length  of  and 
distance  of  inverted  object  from  internal  vertex  ? 

=  1.25  feet,  and  —  5.625  feet. 

8X4-5  —  5  2X4-5  —  5 

3.—  Before  a  convex  mirror  of  3.5  feet  radius  is  set  an  object  at  3  feet  from  ver- 
tex of  curve;  what  is  length  of  and  distance  of  object  from  external  curve? 

IX3'5     =  .368  foot,  and      3  X  f  5     =  1. 105  feet. 
2X3  +  3-5  2X3  +  3-5 

4.— A  parabolic  reflector  has  a  depth  of  1.25  feet  and  a  diameter  of  2  feet;  what 
is  its  focal  distance  from  vertex  of  internal  curve  ? 

_2 

=  .2  feet  or  2.4  ins. 


16X125 

TJ  y 

Lenses.    Double  Convex. —  —  F.     When  R  —  r  =  F; 

m-ixR  +  r  2m  — i 

^_L  =  D-     JJL-L-      ?-dbZ_p.       OF  _      and     sj?  _0 

Double  Concave.   r          —  F ; 


Optical  centres  are  in  centres  of  lens.  Piano  -  Convex  and  Piano  -  Concave. 
m__  —  F-  Optical  centres  are  respectively  centres  of  convex  and  concave  sur- 
faces. Convex  Concave  (Meniscus)  and  Concavo-  Convex.  Rr  =F. 

m  — i  XR  —  r 

Optical  Centres.  Convex  Concave.  Delineate  lens  in  half  section,  draw  R  from 
its  centre  to  circumference  of  lens  (intersection  of  radii),  draw  r  parallel  thereto 
and  extending  to  its  circumference,  connect  R  and  r  at  these  external  points  of 
contact  with  circumference  and  external  curve,  extend  line  to  axis  of  lens,  and  point 
of  contact  is  centre  required.  Concavo- Convex.  Proceed  in  like  manner,  but  in 
this  case  r  extends  to,  or  delineates,  the  inner  surface  of  the  lens,  and  point  of  con- 
tact with  axis  is  centre  required. 

*  D  or  image  disappears  when  I  —  .5  r. 

Z  ±J T 

*nd  2  p  i  ~  =  I-       t  When  equally  convei  F  =  R.       §  When  convex  side  is  exposed  to  parallel  rays 

and  when  parallel  rays  fall  upon  plane  side,  F  =  2  R.  ||  Rays  of  light,  heat,  or  sound,  reflected  from 
focus  of  a  hyperbola,  will  diverge  from  its  concave  surface,  f  and  when  from  the  focus  of  an  ellipw, 
will  be  refracted  by  surface  of  the  other. 


OPTICS. — PILE-DRIVING.  67 1 


Wken  object  ii  beyond  focal  distance  (F),  its  image  (D)  will  be  inverted,  as  °F  =  D,  and    LF    =  /. 

P  representing  magnifying  power  of  lens  ,  S  limit  of  normal  sight,  to  to  12  ins.  for 
far-sighted  eyes  and  6  to  S  for  near-sighted,  ordinarily  10  ins.,  V  limit  of  distinct 
vision,  0  extreme  distance  of  object  from  optical  centre  at  distinct  vision,  and  m  index 
of  refraction. 

ILLUSTRATION  i.—  If  a  double  convex  lens  of  flint  glass  has  radii  of  6  and  6.25  ins. 
what  is  its  focal  distance?  Index  of  refraction  =  1.57,  see  page  584.   ' 

6x6.25 

-  =.  =  5-37  ins- 
1.57  —  1X6-1-6.25 

2.—  If  a  double  concave  lens  has  a  focal  distance  of  2  ins.,  and  object  is  6  ins.  from 
vertex  of  curve,  what  is  its  dimension  and  what  is  its  distance  from  vertex  of  inner 
curve  ? 

6X2  .  4X  2 

—  r—  =  2  ins.,  and  —  1_  —  I>33  in8, 

2  +  4  4  +  2 

3.—  If  focal  distance  of  a  single  microscope  is  4  ins.,  what  is  its  limit  of  distinct 
vision,  and  what  its  magnifying  power?  O  =  2.857  tn*- 


Telescopes,  Opera-glasses,  etc. 

D:o  =  F:/;    o/-r-F  =  D,  and  ^-  =  Z;     rA±-^  =  F-f/    f  represent- 

j  ij  —  /      s-\-x 

ing  length  of  focal  distance  from  object  lens. 

ILLUSTRATION.—  Principal  focal  distance  of  ocular  lens  of  a  telescope  is  .9  in.,  of 
objective  lens  90  ins.  ;  what  is  its  magnifying  power? 

90  -r-  .9  =  ioo  times  the  object. 


PILE-DRIVING. 

Effect  of  the  impact  of  the  ram  of  a  pile-driver  is  as  the  square  root  of 
its  velocity  or  height  of  its  fall.  Thus  the  theoretical  velocity  of  fall  is  aa 

Vz  ff  h  or  8  Vh. 

The  impact  or  dynamic  effect  of  the  blow  of  a  ram  on  a  pile  cannot  be 
determined  with  exactness,  so  long  as  it  yields  under  the  blow,  as  the  yield- 
ing cushions  it  and  reduces  its  effect. 

By  my  experiments  in  1852  to  determine  the  dynamic  effect  of  a  falling  body,  I 
found  it  to  be  far  greater  than  that  given  by  the  formula  ^/2  g  h,  and  upon  a  late 
repetition  of  them,  under  improved  conditions  of  the  instrument  of  registry,  I  find 
it  to  be  for  one  pound  falling  two  feet,  52  pounds.  One  pound  falling  2  feet  has  a 
velocity  of  11.31  feet  per  second,  but  its  dynamical  effect  or  vis  viva  was  52  pounds, 
or  4.6  times  the  velocity. 

Observation  and  tests  of  the  sustaining  power  of  piles,  at  different  locations  and 
under  different  conditions,  gave  it  as  2,  3,  and  3.7  to  i  times  that  deduced  by  the 
formula  8  ^/h,  which  was  but  the  net  effect,  or  capacity,  of  ram,  less  the  friction  of 
its  operation. 

Wm.  J.  McAlpine  in  his  operation  on  the  foundations  of  the  dry-dock  in  the  Navy 
Yard,  Brooklyn,  estimated  the  effect  of  a  ram  weighing  2240  Ibs.,  falling  30  feet  to  a 
refusal,  at  224000  Ibs.,  or  2.28  times  that  given  by  the  formula  w  8  ^/h. 

Essayists  present  a  variety  of  formula,  which  differ  in  form.  Some  are  com- 
paratively simple,  while  others  embrace  diameter,  length,  weight,  and  sectional 
area,  depth  driven  by  last  blow  in  feet  or  in  inches,  and  Modulus  of  Elasticity  of  the 
material  of  the  pile,  together  with  various  factors  for  results. 

When  the  losses  of  effect  in  the  operation  of  a  pile-driver  are  duly  considered — 
viz.,  friction  of  ram  in  the  guides  of  the  leader,  and  of  the  hoisting  line  of  ram  in 
the  sheave  and  over  drum  {ascertained  by  experiment  with  a  very  heavy  ram  to 

*  +  for  telescopes  and  —  for  op«ra-glasses,  etc. 


672 


PILE-DRIVING. 


be  equal  to  .2  foot  of  penetration:  with  a  light  ram  it  would  be  materially  more), 
the  cushioning  of  it  on  head  of  a  pile,  however  square  it  may  be  dressed  off,  the 
want  of  verticality  both  of  ram  in  falling  and  of  plane  of  the  pile  to  the  blow  and 
consequent  lateral  vibration  of  it,  the  buckling  of  it  in  driving,  the  frequent  split- 
ting of  it  on  a  boulder,  and  the  condition  of  soil,  whether  dry,  moist,  or  wet;  if  it 
is  imbedded  or  partially  exposed  to  the  air,  or  wholly  immersed  in  wet  soil  and 
water,  and  the  integrity  of  the  driving— they  furnish  the  elements  in  determina, 
tion  of  a  coefficient  of  safety. 

Opposed  to  these  effects  is  that  of  the  subsidence  of  the  soil  around  a  pile  thai 
has  been  disturbed  in  driving,  the  effect  of  which,  under  favorable  conditions  of 
soil,  has  approached  to  that  of  the  resistance  of  the  pile  at  its  final  blow. 

The  following  formula  is  constructed  on  the  basis  of  a  pile  being  driven 
to  a  depression  of  one  inch  or  less,  as  all  estimates  based  upon  a  greater  de- 
pression are  not  only  comparatively  valueless,  in  consequence  of  the  cush- 
ioning of  the  ram,  but  if  piles  are  not  driven  to  such  depression  their  utility 
is  decreased,  and  a  greater  number  are  rendered  necessary  to  support  the 
weight  to  be  imposed  upon  them,  and  in  it  I  have  omitted  an  element  which 
is  universally  given  in  others,  that  of  the  last  depression  of  a  pile  as  a  divi- 
sor, as  I  not  only  fail  to  recognize  its  connection,  but  hold  its  introduction 
erroneous. 

rJ?o  Compute   Safe    Load,  of  a   Pile   Driven  to  a   Depres- 
sion,   of  1    Inch,    or    Less. 

4  W  8  v^  =  L-  W  representing  weight  of  ram,  and  L  load,  both  in  Ibs.,  and  h 
height  of  fall  in  feet. 

From  which  result  is  to  be  deducted  a  factor  of  safety  representing  the  friction 
and  losses  of  effect. 

Hence,  the  formula:  - — ——  ==  L,  or  safe  load  in  pounds. 
\j 

For  C,  or  coefficient  of  safety,  in  consideration  of  the  several  losses  of  effect  re- 
cited, and  especially  that  of  brooming  of  the  heads  of  a  pile,  it  is  assumed  at  from 
3  to  6,  according  to  the  soil  and  the  integrity  of  the  driving. 
Eliminating  the  numerator  4  and  correspondingly  reducing  the  3  and  6,  the  for- 
W  8  -Jh 

mula  is, * —  =  L. 

'    -75  to  1.5 

ILLUSTRATION.— Assume  an  ordinary  pile  driven  in  firm  soil  by  a  ram  of  2000  Ibs. 
weight,  falling  25  feet,  with  a  final  depression  of  .5  inch,  and  coefficient  of  1.25; 
what  would  be  its  safe  load? 

2000  X  8  ^25      2000  X  8X5      ,•          7r 

— -  = =  64  ooo  Ibs. 

1.25  1.25 

In  practice,  in  the  determining  the  capacity  of  a  range  of  piles,  it  is  proper 
to  reduce  the  result  obtained  by  the  formula,  to  meet  incidental  effects,  as 
negligence  in  driving,  in  the  superintendence  of  it,  and  the  frequent  and  un- 
observed splitting  or  crushing  of  a  pile  on  a  stone  or  boulder. 

A  heavy  ram  and  a  low  fall  is  most  effective  condition  of  operation  of  a 
pile-driver,  provided  height  is  such  that  force  of  blow  will  not  be  expended 
in  merely  overcoming  friction  of  leader  and  inertia  of  pile,  and  at  same  time 
not  from  such  a  height  as  to  generate  a  velocity  which  will  be  essentially 
expended  in  crushing  fibres  of  head  of  pile. 

When  the  soil  is  very  soft  or  wet,  concrete  should  be  laid  between  the 
heads  of  the  piles  to  a  depth  of  from  1.5  to  3  feet. 

When  the  soil  is  of  fine  sand  or  light  gravel,  piles  may  be  set  two  feet 
from  their  centres,  but  if  it  is  saturated  with  moisture,  a  greater  distance  is 
necessary,  otherwise  small  piles  are  liable  to  be  disturbed  by  large, 

(Continued  on  page  972.) 


PILE-DRIVING. — PNEUMATICS. — AEROMETRT.       6/3 

Pile-sinking. 

MitchelV*  Screw  Piles  are  constructed  of  a  wrought-iron  shaft  of  suitable 
diameter,  usually  from  3  to  8  ins.,  with  1.5  turns  of  a  cast-iron  thread  of 
from  1.5  to  3  feet  diameter. 

Hydraulic  Process  is  effected  by  the  direction  of  a  stream  of  water  under 
pressure,  within  a  tube  or  around  the  base  of  a  pile,  by  which  the  sand  or 
earth  is  removed. 

Pneumatic  and  Plenum  Process. — For  illustration  and  details,  see  Traut- 
wine's  Engineer's  Pocket-book,  647-8.  New  Edition. 

Dr.  Whewell  deduced  the  following  results : 

1.  A  slight  increase  in  hardness  of  a  pile  or  in  weight  of  a  ram  will  con- 
siderably increase  distance  a  pile  may  be  driven. 

2.  Resistance  being  great,  the  lighter  a  pile  the  faster  it  may  be  driven. 

3.  Distance  driven  varies  as  cube  of  the  weight  of  ram. 

Relative  Resistance  of  Formations  to  Driving  a  Pile. 

Coral 100  I  Hard  clay 60  I  Light  clay  and  sand. . .  35 

Clay  and  gravel 83  |  Clay  and  sand. 45  |  River  silt 25 


PNEUMATICS.— AEROMETRY. 

Motion  of  gases  by  operation  of  gravity  is  same  as  that  for  liquids. 
Force  or  effect  of  wind  increases  as  square  of  its  velocity. 

If  a  volume  of  ah*  represented  by  i,  and  of  32°,  is  heated  t  degrees  without 
assuming  a  different  tension,  the  volume  becomes  (i  -f-  .002088  t)  =V;  and 
if  it  requires  a  temperature  in  excess  of  t1  32°,  it  will  then  assume  volume 
(i  +  .002088  t'  —  32°).  All  aeriform  fluids  follow  this  law  of  dilatation  as 
well  as  that  of  compression  proportional  to  weight. 

When  air  passes  into  a  medium  of  less  density,  its  velocity  is  determined 
by  difference  of  its  densities.  Under  like  conditions,  a  conduit  will  discharge 
30.55  times  more  air  than  water. 

To  Compnte  the  Degree  of  Rarefaction  that  may  "be  ef- 
fected,  in.  a  "Vessel. 

Let  quantity  of  air  in  vessel,  tube,  and  pump  be  represented  by  i,  and 
proportion  of  capacity  of  pump  to  vessel  and  tube  by  .33 ;  consequently,  it 
contains  .25  of  the  air  in  united  apparatus. 

Upon  the  first  stroke  of  piston  this  .25  will  be  expelled,  and  .75  of  original 
quantity  will  remain ;  .25  of  this  will  be  expelled  upon  second  stroke,  which 
is  equal  to  .1875  of  original  quantity;  and  consequently  there  remains  in 
apparatus  .5625  of  original  quantity.  Proceeding  in  this  manner,  following 
Table  is  deduced : 


No.  of  Strokes. 

Air  Expelled  at  each  Stroke. 

Air  Remaining  in  Vessel. 

.      i 

2 

3 

•25  =  -25 
3  _      3 
16      4X4 
9           3X3 

•75  =  -75 
9  _3X3 
16      4X4 
27      3X3X3 

64      4X4X4 

64      4X4X4 

And  so  on,  multiplying  air  expelled  at  preceding  stroke  by  3,  and  dividing 
it  by  4 ;  and  air  remaining  after  each  stroke  is  ascertained  by  multiplying 
air  remaining  after  preceding  stroke  by  3,  and  dividing  it  by  4. 

3  L 


6/4 


PNEUMATICS. AEROMETEY. 


Distances    at    which    Different    Sounds    are    Audible. 


Feet. 

A  full  human  voice  speaking  in  open  air,  calm 460 

In  an  observable  breeze,  a  powerful  human  voice  with  the)  a 

wind  can  be  heard J  I5  *40 

Report  of  a  musket 16  ooo 

Drum 10  560 

Music,  strong  brass  band 15  840 

Cannonading,  very  heavy 575  ooo 


Miles. 
.087 

3 
3.02 

2 

3 
90 

In  Arctic  Ocean,  conversation  has  been  maintained  over  water  a  distance 
of  6696  feet. 

In  a  conduit  in  Paris,  the  human  voice  has  been  heard  3300  feet. 

For  an  echo  to  be  distinctly  produced,  there  must  be  a  distance  of  55  feet. 

Coefficients  of  Efflux  of  Discharge  of  Air.     (D' Aubuisson.) 

Orifice  in  a  thin  plate 65  .751 

Cylindrical  ajutage 93  .958 

Slight  conical  ajutage 94  1.09 

To  Compute  "Volume  of  Air  Discharged,  th.rou.gh  an  Open- 
ing into  a  "Vacuum,  per  Second. 

a  C  -\/2  g  h  =  V  in  cube  feet,    a  representing  area  of  opening  in  square  feet,  C  co- 
efficient of  efflux,  and  Vz  g  h=  1347.4,  as  shown  at  page  428. 
ILLUSTRATION. — Area  of  opening  i  foot  square,  and  C  =  .707. 
Then  i  X  .707  X  1347.4  =  952.61  cube  feet. 
Inversely,  V  -f-  a  =.  velocity  in  feet  per  second. 

Velocity   and   Pressure    of  \Vind. 

Pressure  varies  as  square  of  velocity,  or  P  oc  V2. 

V2x.oo5  =  P;        V2ooP  =  V;        w2X-oo23  =  P;    and    .0023  v2  sin.  a  —  P. 
V  representing  velocity  in  miles  per  hour,  v  in  feet  per  second,  P  pressure  in  Ibs. 
per  sq.foot,  and  x  angle  of  incidence  of  wind  with  plane  of  surface. 

Table   deduced   from   a"bove   Formulas. 


Vel 
Hou'r. 

>city 
per 
Minute. 

Pressure 
on  a 
Sq.  Foot. 

Character  of  the  Wind. 

Vel 
Hour. 

ocity 
Minute. 

Pressure 
on  a 
Sq.  Foot. 

Character  of  th« 
Wind. 

Miles. 

Feet. 

Lbs. 

Miles. 

Feet. 

Lbs. 

88 

.005 

Barely  observable. 

25 

2  2OO 

3-125 

Very  brisk. 

2 

3 

*i 

264 

.02     } 
•045) 

Just  perceptible. 

30 
35 

2640 
3080 

4-5      \ 
6.125} 

High  wind. 

4 

352 

.08 

Light  breeze. 

4° 

3520 

8 

Very  high  wind. 

5 
6 

440 
528 

•125) 
.18 

Gentle,      pleasant 
wind 

45 
5° 

3960 
4400 

10.125 
12.5 

Gale. 
Storm. 

8 

704 

•32    ) 

60 

5280 

18 

Great  storm. 

10 

880 

•  5 

Fresh  breeze. 

80 

7040 

32 

Hurricane. 

15 

20 

1320 
1760 

1.125 

2 

Brisk  blow. 
Stiff  breeze. 

9° 

100 

792O 
8800 

40-5} 
So    J  " 

Tornado. 

ILLUSTRATION.— What  is  pressure  per  sq.  foot,  when  wind  has  a  velocity  of  18 
miles  per  hour?  l82  x  ^  =  1<6a  lbs 

To    Compute    Force   of  "Wind   upon   a   Surface, 

—  ( —  ^— - — J  =  P.  v  representing  velocity  of  wind  in  feet  per  second,  a  area, 
oj  surface  in  sq.feet,  and  x  angle  of  incidence  of  wind. 

At  Mount  Washington  wind  has  been  observed  to  have  had  a  velocity  of  150  milei 
per  hour—  112.5  lbs.  per  sq.  foot. 

Extreme  pressure  of  wind  at  Greenwich  Observatory  for  a  period  of  20  years  was 
41  lbs.  per  eq.  foot. 


PNEUMATICS. — AEROMETK  Y.  6/  5 

Force  of  wind  upon  a  surface,  perpendicular  to  its  direction,  has  been  ob- 
served as  high  as  57.75  Ibs.  per  sq.  foot ;  velocity  =  159  feet  per  second. 

Dr.  Hutton  deduced  that  resistance  of  air  varied  as  square  of  velocity 
nearly,  and  to  an  inclined  surface  as  1.84  power  of  sine  x  cosine. 

Figure  of  a  plane  makes  no  appreciable  difference  in  resistance,  but  con- 
vex surface  of  a  hemisphere,  with  a  surface  double  the  base,  has  only  half 
the  resistance. 

At  high  velocities,  experiments  upon  railways  show  that  the  resistance 
becomes  nearly  a  constant  quantity. 

Ccmrse   of  "Wind.. 

Direction  in  Cvolon^s  Direction  in 

Northern  Hemisphere.  ^y^  Southern  Hemisphere 

Wind  has  its  direction  nearly  at 

right  angles  to  line  between  points  of 

highest  and  lowest  pressure  of  air,  or 

barometer  readings,  and  its  course  is 

with  the  point  of  lowest  pressure  at 

its  left,  and  its  velocity  is  directly  as 

difference  of  the  pressures. 
In  Northern  Temperate  zone,  winds  course  around  an  area  of  low  pressure 
in  reverse  direction  to  course  of  hands  of  a  watch,  and  they  flow  away  frorii 
a  location  of  high  pressure,  and  cause  an  apparent  course  of  the  winds  in  di* 
rection  of  course  of  the  hands. 

To   Compute   Resistance   of  a   IPlane    Surface   to   Air. 

.0023  a  v2  =  P  in  Ibs.  a  representing  area  of  plane  in  sq.feet,  v  velocity  in  direc 
tion  of  wind  in  feet  per  second,  -{-when  it  moves  opposite,  and  —  when  with  the  wind 

When  Barometer  Pressure =  30  Lbs. 

(C.  F.  Martin,  U.S.  S.  S.) 

.004  a  V»  =  P.  V  representing  velocity  of  wind  in  miles  per  hour,  and  a  area  of 
pressure  in  sq.  feet. 

To  Compute  Height  of  a  Column  of  Mercury  to  induce 
an    ICfnux    of  Air   through    a   given    Nozzle. 

Barometer  assumed  at  2. 46  feet  =  29.52  ins.,  and  Temperature  52°. 

pa 

-g-^ — f-^  —  H,  and  48.073  d2  ^/R  =  P.     d  representing  diameter  of  nozzle  and  H 

height  of  column  of  mercury,  loth  in  feet,  and  P  volume  of  air  in  Ibs.  per  one  second. 
ILLUSTRATION.— Assume  d  =  .19,  and  P  ==  .7  Ibs. 

<-<  f^      48.073  X.i92V.x6a6  = -7- 


To  Compute  Pressure  or  \Veight  of  Air  under  a  given 
Height  of  Barometer  and  Temperature,  Discharged  in 
One  Second 


30. 787  d* ^E  -^——pressure  in  Ibs.    Or,  48.073  d2  ^B  =  Ibs.    I  representing 

height  of  barometer  in  external  air,  B  manometer  or  pressure  of  air  in  reservoir  in 
mercury,  both  in  feet,  and  t  temperature  of  air  or  gas  in  degrees. 

ILLUSTRATION.  —Assume  b  =  2. 5  feet ;  d  = .  25  foot ;  B  = .  i  foot ;  and  t  =  1.0550. 
Then  30.787  X  .0625  ./.i  X  2'5"t"'1  =  1.924  X  V-2465  =  -9543  «*• 


1 


6/6 


PNEUMATICS.  —  AEKOMETBY. 


To  Compute  Teinperature  for  a,  given  Latitude  and.  Ele- 
vation. 


82. 8  cos.  I  — .  ooi  98 1  E  —  .4  —  t.     E  representing  elevation  in  feet. 
ILLUSTRATION.—  Assume  1  =  45°;  cos.  =.707;  and  E  =  656  feet. 

Then  82.8  X  .707  —  .001 981  X  656  —  .4  =  58.54  —  1.299  —  .4  =  58.54— .899  = 
57.641. 

To  Compute  Volume  of  Air  or  Q-as  Discharged  through 
an  Opening  and  vinder  a  Pressure  above  that  of  Ex- 
ternal Air. 


A  ir.    1347.  4  C  —  VB  (6'  -f  B)  T  =  V  in  cube  feet  per  second. 

T  =  i  -j-  .002  22  (t  —  32°),  and  6'  =  2.5  —  .00009  elevation. 

Or,  621.28  c?2  VB  =  V- 

ILLUSTRATION.—  What  would  be  volume  of  air  that  would  flow  through  a  nozzle 
.246  foot  in  diam.  from  a  reservoir  under  a  pressure  of  .098  foot  of  mercury,  into 
air  under  a  barometric  pressure  of  2.477  ^ee^  temperature  of  air  55.4°,  location  45° 
of  latitude,  and  at  an  elevation  of  650  feet  above  level  of  sea? 

C  =  -75;    6'  =  2.5  —  .00009  X  650  =  2.4415  (2.44);  and   T  =  1.0502. 


Then  1347.4  X  .75  :^—  ^.098  (2.44  -f-  .098)  X  1.0502  =  24.689  X  V-26l7  =  12.63 
cube  feet. 

When  Densities  of  External  Air  and  that  in  Reservoir  are  Equal. 


1 347. 4  C  —,  \/  B  (6  -f- B)  T '  =  V.    &'  representing  height  of  mercury  in  reservoir. 

Gas.    ^j^ .  /=-] — -,  —  V.p  representing  specific  gravity  of  gas  compared 

vP  V  lj  H-  42  X  d 
with  atr,  and  L  length  of  pipe  or  conduit  in  feet. 

ILLUSTRATION. — If  a  pipe  .05  feet  in  diameter  and  420  feet  in  length,  communi- 
cates with  a  gasometer  charged  with  carburetted  hydrogen  (illuminating  gas),  under 
a  water  pressure  as  indicated  by  a  manometer  of .  1088  foot,  what  would  be  the  dis- 
charge per  second  ? 

d  =  .05  foot ;  L  =  420  feet ;  and  B  =  'I0     =  .008  foot.    Specific  gravity  of  gas 

13.6* 
.5625. 

4231          /   .008  X -05*         4231       /. 0000000025000 

;  •;         / 3     =3-^-     / - — ^ —  =  .01371  cwfte /oof. 

V-5625   V  420+  42  X- 05        -75    V        420-1-2.1 

Resistance  of  Curves  and  Angles. — Curves  and  angles  increase  resistance 
to  discharge  of  air  or  gas  very  materially.  By  experiment  of  D'Aubuisson 
7  angles  of  45°  reduced  discharge  of  gas  one  fourth. 

To  Compute   Diameter  of  Discharge-pipe   or   Nozzle. 

When  Length  and  Diameter  of  Pipe,  Volume,  and  Pressure  are  given. 

4/        f-^d',_.  =  yfcA* 

V  42302  B  d5  —  L  V2 

ILLUSTRATION.— If  a  pipe  1000  feet  in  length,  and  .4  foot  in  diameter,  leads  to  a 
reservoir  of  air,  under  a  mercurial  manometric  pressure  of  .18  foot,  what  diameter 
must  be  given  to  a  nozzle  to  discharge  4  cube  feet  per  second  ? 

./  42X42X.4S  ./         6.88128 

Then   */ •  =  4  / _ _ —  =  .#.0004052  = 

V423o*Xi8X. 45-ioooX42      V  32 980. 19 -16000 


.i4i8jfoo<=  1.703  ins. 

Volumes  of  two  gases  flowing  through  equal  orifices,  and  under  equal  pressures, 
are  in  inverse  ratio  of  square  roots  of  their  respective  densities. 

*  Specific  gravity  of  mercury  compared  with  waUr. 


BAILWAYS. 


6/7 


RAILWAYS. 
To  Define  a   Curve.— Fig.  1.    (Moksworth.) 

I7'9°  or  *  tan.  *  =  R ;       R  (cotan.  «)  =  * 


'7i9c 
R 


=  a;       R  (cosec.  x  —  i)  =  d 


R  (cosin.  x)  =  s\        R  (coversin.  x)  =  V; 
^i°^ =  n,  and   (5400  —  x)  .000582  R  =  i 

c  representing  any  chord,  t  length  of  tangent,  d  distance  of  centre  of  curve  from  in- 
tersection of  tangents,  s  half  chord  of  curve,  and  I  length  of  curve,  all  in  like  dimensions, 
a  tangential  angle  ofc  in  minutes,  n  number  of  chords  in  curve,  and  x  half  angle  oj 
intersection,  but  in  formulas  for  number  of  chords  and  length  of  curve  to  be  expressed 
in  minutes. 

ILLUSTRATION.— Assume  radius  900  and  chord  400  feet;  angle  of  intersection  = 
12°  44'  =  764  minutes,  and  x  =  56°  15'  5". 

Tangent  of  56°  15'  5"  =  1.496  73.    Cotangent  =  .668 14. 

-  =  R  =  goo  feet ;       ^^ — —  =  764  minutes  ;        900  X  .  668 14  =  t  = 
900  X  1.20269  —  1  =  d  =  182.42  feet ;       900  X  . 555  55  =  s  —  500  feet; 
5400  —  3379  _  2  645  times^  and  .000  582  X  900  X 


764 
601.33  feet; 


5400  —  3379  =  1058.6/66*. 


Tangential  Angles  for  Chords  of  One  Chain. 


Radius  of 
Curve. 

Tangential 
Angle. 

Radius  of 
Curve. 

Tangential 
Angle. 

Radius  of 
Curve. 

Tangential 
Angle. 

Radius  of 
Curve. 

Tangential 
Angle. 

Chains. 

Chains. 

Chains. 

5°  43-8', 
3°  34-87 

IS 

20 

1°  54-6' 
i°  25-95' 

40 
45 

42.  97,' 
38.2' 

i  mile 
i.  25  mil's 

21.48' 
17.19' 

9 

3°  "' 

25 

1°     8.76' 

50 

3t'l8 

1.5  miles 

14-33' 

10 

2°  51.9', 

30 

57-3' 

60 

28.65' 

i-75    " 

12.28 

12 

2°   23.25 

35 

49.11 

70 

24-55' 

2 

10.74' 

NOTE.— Angle  for  2  chain  chords  is  double  angle  for  i  chain  chords.  Angle  for  .5 
chain  chords  is  .5  the  angle  for  i  chain  chords. 

Curves  of  less  than  20  chains  radius  should  be  set  out  in  .  5  chain  chords.  Curves 
of  more  than  i  mile  radius  may  be  set  out  in  2  chain  chords. 

Angles  in  above  Table  are  in  degrees,  minutes,  and  decimals  of  minutes. 

Fig.  2. 


I 


Sidings. 

2  VdR.  —  (.5  d)2  =  1.  R  representing  radius  of 
curve,  I  length  of  curve  over  points,  and  d  distance 
between  tracks, 
aU  in  feet. 


Turn-out  of  TJneq.vi.al   Radii 


R  and  r  representing  radii  of  the  curves  re- 
spectively as  to  length,  x  distance  between  outer 
rails  of  tracks  and  other  symbols  as  shown,  att 
to  feet. 


678  RAILWAYS. 

IPoints    and    Crossings. 

fin      . 

' 


senting  radius  of  curves,  G  gauge  of  road,  a  angle  of  crossing, 
~  and  x  =.  R  —  G,  all  in  feet. 

In  horizontal  curves,  width  required  for  clearance  of 
flange  of  wheel,  and  for  width  of  rail  at  heel  of  switch, 
render  it  necessary  to  make  an  allowance  in  length  of  /, 
as  ascertained  by  formula. 

For  other  diagrams  and  formulas,  see  Molesworth's  Pocket- 
book,  pp.  208-18,  2ist  edition. 

1719  c 
To   Compute    Tangential   A~ngle  .^br   Curves.     — — -  =  a.   c 

representing  chord  in  feet,  and  a  angle  in  minutes. 

ILLUSTRATION.— What  is  angle  for  a  curve  with  a  radius  of  900  feet,  and  a  chord 
of  400  feet? 

,7,9  X  400  =  inutes 

900 

Curving   of  Rails. 

— 5- —  =  v.     I  representing  length  of  rail  in  feet,  v  versed  sine  at  centre,  when 

R 

€urved,  in  ins. 
ILLUSTRATION.— What  is  curve  for  a  rail  20  leet  in  length,  with  a  radius  of  900  feet? 

I.5X20*_ 

900 
Curves   t>y   Offsets   in    Equal    Chorda, 

Fig.  5.  .#  Chord2  Chord2 

^*  2  R     =  o  offset.         — - —  =  2,  o  offset 

ILLUSTRATION.— Assume  chords  150,  andra 
dius  900  feet. 
22500 


To   Compute  'Versed    Sines   and  Ordinates   of  Curves. 

.;         LL^  +  ,,  =  D;      and 

\  R2  —  »2  —  (R  —  v)  =  o.      D  representing  diameter  of 

I  'xs                            \  circle,  and  v  versed  sine  of  curve. 

»      ^  "Xv                      \  ILLUSTRATION.  —  Assume  radius  900,  and  chord  400  feet. 

D  900  —  VSioooo  —  40000  =  900  —  877.5  =22.5  ./«e£. 

Relation  of  Base  of  Driving  or  Rigid  Wheels  to  Curve. 
—  ^  =:  B.    R  representing  minimum  radius  of  curve,  G  gauge  of  road,  and  B  base, 
all  in  feet. 

To   Compute   Elevation   of  Outer   Rail. 
For  any  Radius  or  Combination  of  Curve  with  Straight  Line. 


=  c.     V  representing  velocity  of  train  in  feet  per  second,  G  gauge  of  road, 
and  c  length  of  a  chord,  both  in  feet,  the  versed  sine  of  which  =  elevation  in  ins. 

On    Curves. 
V2 

-  G  =  E.    E  representing  elevation  of  outer  rail  in  ins, 


1.25  R 


RAILWAYS.  679 

Radii   of  Curves   set   out   in   Tangential    Angles. 


Angle  for 
Chord  of 
100  Feet. 

Radius 
Curve. 

Angle  for 
Chord  of 
loo  Feet. 

Radius 
of 
Curve. 

Angle  for 
Chord  of 
100  Feet. 

Radius 
Curve. 

Angle  for 
Chord  of 
ipo  Feet. 

Radius 
of 
Curve. 

o    ' 

30 

I 
i   30 
2 

Feet. 
5729.6 
2864.8 
1909.9 
1432.4 

o    ' 
2    30 

3 

3  30 
4 

Feet. 
"45-9 
954-9 
818.5 
716.2 

o    ' 

4  30 
5 
5  3° 
6 

Feet. 
636.6 

573 
520.9 

447-5 

0     ' 

6  30 
7 
7  3<> 

8 

Feet. 
440.7 

409-3 
382 
358.1 

NOTE.— If  chords  of  less  length  are  used,  radius  will  be  proportional  thereto. 
To  Ascertain  Radius  of  Curve  in  Inches  for  Scale,  in  Feet  per  Inch. 
Divide  radius  of  curve  in  feet  by  scale  of  feet  per  inch. 

To    Compute    Required    \Veight   of   Rail. 

RULE. — Multiply  extreme  load  upon  one  driving-wheel  in  Ibs.  by  .005, 
and  product  will  give  weight  of  rail  in  Ibs.  per  yard. 

To    Compute    Radius   of  Curve   and   "Wheel    Base. 
P 

9  B  G  =  R.     — —  =  B.    B  representing  maximum  rigid  wheel  base  of  cars,  and  G 

gauge  of  way,  both  in  feet, 

To  Determine  Elevation  of  Outer  Rail. 

For  any  Radius  or  Construction  of  Curve  with  Straight. — Fig.  7. 

Fig.  7.  V  .5  v'G  =  c.    V  representing  speed  of  train  in  feet  per  sec- 

ond,  G  gauge  of  rails  in  feet,  and  c  length  of  chord,  versed  sine 
v  of  which  will  give  at  its  centre  the  elevation  required. 

Thus,  determine  chord  c,  align  it  on  inner 
side  of  rail,  and  distance  of  rail  from  it  at 
centre  of  its  length  will  give  elevation  re- 
quired,  whatever  tke  radius  of  rail. 

ForCums.    I.7«.V(NDW)]-4PB  or,W-I!-  =  E.    » representing 

r»  1)  K  1.25  K 

diameter  of  wheels,  W  width  of  gauge,  P  lateral  play  between  flange  and  rail,  and 
R  radius  of  curve,  all  in  feet,  i  ^-  N  ratio  of  inclination  of  tire,  V  velocity  of  train  in 
miles  per  hour,  and  E  elevation  of  outer  rail  in  ins.  (Molesworth.) 

'      =  resistance  due  to  curve,  and  W  representing  weight  of  body,  both  in 

2  R 

Ibs.,  C  coefficient  of  friction  of  wheels  upon  rails  =  .\  to  .27,  according  to  condition  of 
weather,  d  distance  of  rails  apart,  I  length  of  rigid  wheel  base,  and  R  radius  of  curve, 
all  in  feet.  (Morrison.) 

ILLUSTRATION. — Assume  weight  of  locomotive  30  tons,  radius  of  curve  1000  feet, 
distance  of  rails  apart  4  feet  8.75  ins.,  length  of  base  10  feet,  and  rails,  dry,  C  =  i. 

)  =  ^ 


To    Compute    Resistance    due    to    GJ-ravity   upon    an    In- 
clination. 

2240     —  Ibs.  per  ton  of  train. 
gradient 

Rise   per   Mile,  and    Resistance  to  GJ-ravity,  in   Lbs.  per 
Ton. 


Gradient  of  i  inch.. 


Rise  in  feet 

Resistance 


264 


176 


64 


45 


60 


80  |  90 

i 

66    59 
28    24.8 


68o 


RAILWAYS. 


To  Compute  Load,  -which,   a  Locomotive   -will   Draw   up 
an    Inclination. 


T  -i-  r  -j-  r'  —  W  =  L.  T  representing  tractive  power  of  locomotive  in  Ibs. ,  r  re- 
sistance due  to  gravity,  and  r'  resistance  due  to  assumed  velocity  of  train  in  Ibs.  per 
ton,  W  weight  of  locomotive  and  tender,  and  L  load  locomotive  can  draw,  in  tons,  ex- 
clusive of  its  own  weight  and  tender. 

Coefficients  of  Traction  of  Locomotives.—  Railroads  in  good  order,  etc.,  4  to  6  Ibs. ; 
in  ordinary  condition,  8  Ibs. 

In  coupled  engines  adhesion  is  due  to  load  upon  wheels  coupled  to  drivers. 

To  Compute  Traction,  Retraction,  and.  Adhesive  IPower* 
of  a    Locomotive    or    Train. 

When  upon  a  Level.  asP-r-D  =  T.  a  representing  area  of  one  cylinder  in 
sq.  ins.,  s  stroke  of  piston  and  D  diameter  of  driving-wheels,  both  in  feet,  P  mean 
pressure  of  steam  in  Ibs.  per  sq.  inch,  and  T  traction,  in  Ibs. 

When  upon  an  Inclination.  asP-r-p  —  rwh  =  T.  r  representing  resistance 
per  ton,  w  weight  of  locomotive  upon  driving-wheels,  in  tons,  h  height  of  rise  in  feet 
per  zoo  of  road,  and  R  =  r  w  h  =  retraction,  in  Ibs. 

C  w  b  -f- 100  =  A.     6  representing  base  of  inclination  in  feet  per  100  of  road. 

C  w  =  A.    C  =  coefficient  in  Ibs.  per  ton,  and  A  adhesion,  in  Ibs. 

When  Velocity  of  a  Train  is  considered. 

When  upon  a  Level,  W  (C  +  VV)  =  R.  When  upon  an  Inclination, 
W  (r  h  +  C  +  •y/V)  =  R.  V  representing  velocity  of  train  in  miles  per  hour. 

ILLUSTRATION. — A  train  weighing  200  tons  is  to  be  driven  up  a  grade  of  52.8  feet 
per  mile,  with  a  velocity  of  16  miles  per  hour;  required  the  retractive  power? 
52.8  per  mile  =  i  in  100  feet  =  r  =  22.4  Ibs.        C  =  5. 
=  6280/65. 


200  (22-4  X  I-f  5  +  Vl6)  =  20°X 

Velocity  of  Trains. 


Miles  per  hour  

10 

15 

20 

30 

40 

50 

60 

70 

Lbs. 

Lba. 

Lba. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Resistance  upon  straight 
line  per  ton  . 

8-5 

9-25 

10.25 

I3-25 

I7-25 

22.5 

29 

36.5 

Do.,  with  sharp  curves 
and  strong  wind*.  .  .  . 

13 

14 

15-5 

20 

26 

34 

43-5 

55 

*  Equal  to  50  per  cent,  added  to  resistance  upon  a  straight  line. 

Friction  of  locomotive  engines  is  about  9  per  cent.,  or  2  Ibs.  per  ton  of  weight. 
Case-hardening  of  wheel-tires  reduces  their  friction  from   14  to  .08  part  of  load. 

To  Compute  Maximum  Load  that  can  Ibe  drawn  "by-  an 
Engine,  "up  the  ^Eaximnm  Q-rade  that  it  can  .Attain, 
\Veight  and  G-rade  being  given.  (Maj.  McClellan,  U.S.A.) 


.2  A 


-  =  L,  and 


*—  =  G.    A  representing  adhesive  weight  of  engine, 


.424204-8  .4242  L 

in  Ibs. ,  G  grade  in  feet  per  mile,  and  L  load,  in  tons. 

NOTE  i.— When  rails  are  out  of  order,  and  slippery,  etc.,  for  .2  A,  put  .143  A. 

2.  —With  an  engine  of  4  drivers,  put  .6  as  weight  resting  upon  drivers;  with  6 
drivers  the  entire  weight  rests  upon  them. 

ILLUSTRATION. — An  engine  weighing  30  tons  has  6  drivers;  what  are  the  maximum 
loads  it  can  draw  upon  a  level,  and  upon  a  grade  of  250  feet,  and  what  is  its  maxi- 
mum grade  for  that  load  ? 

..  X  «MO  X  3»       .34JQ  fon,  g  fewt        .*X2*4°X3C •        «34g  = 

.4242-f-8  8.4242  .4242X250-}-8  114.05 

,       .  .2X2240X30 8XlI7.8  12497  ,.       . 

117. 8  tons  up  a  grade  of  250  feet.     — — '—  =.  — -^-  =  250.  i  feet. 

Adhesion  of  a  4- wheeled  locomotive,  compared  with  one  of  6  wheels,  i?  as  5  to  & 


RAILWAYS. 


68 1 


OPERATION    OF    LOCOMOTIVES.      (O.  Chanute,  Am.  Soc.  C.  E.) 
.Axlliesion.. 

Adhesion  of  a  locomotive  is  friction  of  its  driving-wheels  upon  the  rails, 
rarying  with  condition  of  the  surface,  and  must  exceed  traction  of  the  engine 
upon  them,  otherwise  the  wheels  will  slip. 

Improvements  heretofore  made  in  the  construction  of  locomotives  and 
tracks  have  gradually  increased  the  proportion  which  the  adhesion  bears  to 
the  insistent  weight  upon  the  driving-wheels. 

The  first  accurate  experiments  were  those  of  Mr.  Wood  upon  the  early  English 
coal  railways.  He  deduced  the  adhesion  to  be  as  follows: 

Upon  perfectly  dry  rails 14  of  weight  on  drivers. 

"    damp  or  muddy  rails 08  " 

"     very  greasy  rails 04"       "       " 

In  1838,  B.  H.  Latrobe  indicated  .13  as  a  safe  working  adhesion,  while  modern 
European  practice  assumes  about  .2  of  weight  as  maximum,  and  .  n  as  a  minimum, 
except  perhaps  in  some  mountainous  regions,  subject  to  mists.  Thus,  on  the  Scem- 
mering  line,  adhesion  is  generally  .16,  and  between  Pontedecimo  and  Busalla,  in 
Italy,  it  never  exceeds  .12  in  open  cuttings,  or  .1  in  tunnels. 

Extensive  experiments  made  upon  French  railways,  1862-67,  bv  Messrs.  Vuille- 
min,  Guebhard,  and  Dieudonne  gave  following  coefficients  in  actual  working;  dry 
weather,  extreme,  .105  to  .2;  damp,  .132  to  .139;  wet,  .078  to  .164;  light  rain,  .09; 
extreme  rain,  .109  to  .2,  mean, .  13;  rain  and  fog,  .115  to  .14;  heavy  rain,  16. 

Materially  better  results  are  obtained  in  United  States,  partly,  perhaps,  in  con- 
sequence  of  greater  dryness  of  the  weather,  and  certainly  because  of  the  American 
method  of  construction  and  equalizing  the  weight  between  the  drivers,  and  of  mak- 
ing the  locomotive  so  flexible  as  to  adapt  itself  to  inequalities  in  the  track. 

Modern  engines  in  America  can  safely  be  relied  upon  to  operate  up  to  an  adhesion 
equal  to  .222  in  summer  and  .2  in  winter,  of  weight  upon  the  driving  wheels. 

From  these  data  the  following  tables  have  been  computed: 

Coefficients  of  Adhesion  xipoii  Driving  "Wheels  per  Ton. 


Condition  of  Rails. 

European 
Practice. 

American 
Practice. 

Condition  of  Rails. 

European 
Practice. 

American 
Practice. 

C. 

Lbs. 

C. 

Lbs. 

C. 

Lbs. 

C. 

Lbs. 

Rails  very  dry  

•  3 

670 

•33 

667 

In  misty  weather  . 

.015 

350 

.2 

400 

Rails  very  wet  

.27 

600 

•25 

500 

In  frost  and  snow. 

.09 

200 

.16 

333 

Ordinary  working.  . 

.2 

450 

.222 

444 

Adhesion  of  Locomotives,  in  Lbs.  (.222  in  Summer  and  .2  in  Winter). 

Type  of  Locomotive. 

No.  of  Drivers. 

Wei 
Locomotive. 

At. 

On  Drivers 

Adh 

esion. 
Winter. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

American  

4  wheels  coupled.  .  .  . 

64000 

42000 

9350 

8400 

Ten  wheeled  

6 

i 

jonne 

cted.. 

ii  6 

Mogul  

6 

88000 

72  ooo 

16 

r>nr» 

14  ooo 

Consolidation  

8 

ii 

\'m 

100  000 

88000 

I955° 

17600 

Tank  switching.... 

6 

u 

68000 

68000 

15100 

13600 

"         .... 

4 

w 

48000 

48000 

10650 

9600 

Tractive    Power. 

Traction  of  a  locomotive  is  the  horizontal  resultant  on  the  track  of  the 
pressure  of  the  steam,  as  applied  in  the  cylinders. 

D2  P  L-:-  W  =:T.  D  representing  diameter  of  cylinder,  L  length  of  stroke,  and  W 
diameter  of  driving  wheels,  all  in  ins. ,  P  mean  pressure  in  cylinder,  in  Ibs.  per  »q. 
inch,  and  T  tractive  force  on  rails,  in  Ibs. 

ILLUSTRATION. — Assume  a  locomotive,  cylinders  18  ins.  in  diam.,  22  ins.  stroke, 
wheels  68  ins.  in  diam.,  and  average  steam  pressure  in  cylinders  50  Ibs.  per  sq.  inck 
Then  18  X  18  X  50  X  22  -=-  68  =  5241  /6s. 


682 


RAILWAYS. 


Train   Resistances. 

Usual  formula  for  train  resistances,  on  a  level  and  straight  line,  is 

yz  V2 

1-  8  =  R  per  ton  of  train,  and \-  6  =  R  per  ton  of  train  alone.    V  renre- 

171  240 

senting  velocity  in  miles  per  hour,  and  8  constant  axle  friction.     (D.  K.  Clark.) 

NOTE.— To  meet  the  unfavorable  conditions  of  quick  curves,  strong  winds,  and 
imperfection  of  road,  Mr.  Clark  estimates  results  as  obtained  by  above  formula 
should  be  increased  50  per  cent. 

ILLUSTRATION.— At  20  miles  per  hour,  the  resistance  would  be: 
20 2  ~r  171  -p-  8  —  10.3  Ibs.  per  ton  of  train. 

This  formula,  however,  is  empirical.  It  gives  results  which  are  too  large  for 
freight  trains  at  moderate  speeds,  and  too  small  for  passenger  trains  at  high  speeds. 

Engineers  are  not  agreed  as  to  exact  measure  and  value  of  each  of  the  elements 
of  train  resistances,  but  following  approximations  are  sufficient  for  practical  use: 

Analysis  of  Train  Resistances. 

Resistance  of  trains  to  traction  may  be  divided  into  four  principal  ele- 
ments :  i st.  Grades ;  2d.  Curves ;  3d.  Wheel  friction ;  4th.  Atmosphere. 

ist.  Grades.  —  Gradients  generally  oppose  largest  element  of  resistance 
to  trains.  Their  influence  is  entirely  independent  of  speed.  The  meas- 
ure of  this  resistance  is  equal  to  weight  of  train  multiplied  by  rate  of  in- 
clination or  per  cent,  of  grade.  Thus,  a  gradient  of  .5  per  100  feet  (26.4 
feet  per  mile)  offers  a  resistance  of  5  X  2240 ._.  I12  lbs>  t  or  JQ  j^s 

10  X  ioo 
per  2000  Ibs.,  which  is  to  be  multiplied  by  weight  in  tons  of  entire  train. 

Following  table  shows  resistance,  due  to  gravity  alone,  for  the  most  usual  grades, 
in  Ibs.  per  ton  of  train  : 

i  st.  Resistance  due  to  Grades. 


Lbs.  per  ton  of  2240  Ibs.  .  . 
Rate  per  mile 

si 

4-48 
" 

<% 

16 

& 

o 

II.  2 

26 

13-44 

,5^8 

17.92 

Lbs.  per  ton  of  2000  Ibs.  .  . 
Rate  per  ioo  feet  

; 

.0 

6 

8 

10 

32 

12 

H 

?6 
1.6 

Lbs.  per  ton  of  2240  Ibs.  .  . 
Rate  per  mile 

*i 

20.10 

22.4 

24.64 
s8 

26.88 
61. 

i-O 

29.12 
68 

3-36 

*•  D 

33-6 

35-84 

ge 

Lbs.  per  ton  of  2000  Ibs.  .  . 

JJ 
2O 

0° 
22 

UJ 
24 

26 

74 

28 

yy 
3° 

°j 
32 

2d.  Curves.— Recent  European  formula  is  that  given  by  Baron  von  Weber. 
.  6504  -T-  R  —  55  =  W.  R  representing  radius  of  curve  in  metres. 

This  formula  assumes  that  resistance  due  to  curve  increases  faster  than  radius 
diminishes.  It  gives  results  varying  from  a  resistance  of  .8  Ib.  per  2000  Ibs.  per 
degree  for  a  curve  of  1000  metres  radius  (3310  feet,  or  i°  44')  to  a  resistance  of  1.67 
Ibs.  per  2000  Ibs.  per  degree  for  curves  of  ioo  metres  radius  (331  feet,  or  17°  20'). 

Messrs.  Vuillemin,  Guebhard,  and  DieudonnS  found  curve-resistance  to  European 
rolling-stock  to  be  from  8  to  i  Ib.  per  2000  Ibs.  per  degree,  on  a  gauge  of  4  feet  8. 5 
ins.,  while  Mr.  B.  H.  Latrobe,  in  1844,  found  that  with  American  cars  resistance  on 
a  curve  of  400  feet  radius  did  not  exceed  .56  Ib.  per  2000  Ibs.  per  degree. 

Resistance  of  same  curve  varies  with  coning  given  tires  of  wheels,  elevation  of 
outer  rail,  and  speed  of  train  running  over  it,  but  both  reasoning  and  experiment 
indicate  that  the  general  resistance  of  curves  increases  very  nearly  in  direct  pro- 
portion to  degree  of  curvature,  or  inversely  to  the  radius. 

Recent  American  experiments  show  that  a  safe  allowance  for  curve  resistance 
may  be  estimated  at  .125  of  a  Ib.  per  2000  Ibs.  for  each  foot  in  width  of  gauge. 
Thus,  for  3  feet  gauge  resistance  would  be  .375  Ib.  per  degree  of  curve;  for  standard 
gauge  of  4  feet  8.5  ins.  .589,  say  .60,  and  for  6  feet  gauge  .75  Ib.  per  degree. 

For  standard  gauge,  when  radius  is  given  in  feet,  resistance  due  to  this  element  is: 
.60  X  5730  -i-  R  =  C  in  Ibs.  per  ton  of  train. 


RAILWAYS.  683 

This  is  somewhat  reduced  when  curve  coincides  with  that  for  which  wheels  are 
coned  (generally  about  3°),  and  when  train  runs  over  it,  at  precise  speed  for  which 
outer  rail  is  elevated,  an  allowance  of  .5  Ib.  per  ton  per  degree  is  found  to  give  good 
results  in  practice. 

2d.  Resistance  on  Curves. 

It  follows  from  above  estimate  of  curve  resistance  that,  in  order  to  have  the  same 
resistance  on  a  curve  as  on  a  straight  line,  the  gradient  should  be  diminished  by 
.03  per  ioo  feet  of  each  degree  of  curve.  Thus  a  3°  curve  requires  an  easing  of  the 
grade  by  .09  per  ioo  feet,  a  10°  curve  an  easing  of  .3  per  ioo,  etc. 

This,  however,  need  only  be  done  upon  the  limiting  gradients,  and  when  sum  of 
grade  and  curve  resistances  exceeds  resistance  which  has  been  assumed  as  limiting 
the  trains. 

3d.  Resistance  due  to  Wheel  Friction. 

Experimenters  are  not  agreed  whether  friction  of  wheels  increases  simply  with 
weight  which  they  carry,  but  also  in  some  ratio  with  the  speed.  Originally  as- 
sumed as  a  constant  at  8  Ibs.  per  ton,  improvements  in  condition  of  track  (steel 
rails,  etc.)  and  in  construction  and  lubrication  of  rolling  stock  have  reduced  it  to 
3.5  and  4  Ibs.  per  ton  for  well-oiled  trains.  Under  ordinary  circumstances,  in  sum- 
mer, it  will  be  safe  to  estimate  it  at  5  Ibs.  per  ton  on  first-class  tracks,  and  6  Ibs. 
per  ton  on  fair  tracks.  It  may  run  up  to  7  or  8  Ibs.  per  ton  on  bad  tracks  (iron 
rails)  in  summer,  and  all  these  amounts  should  be  increased  from  25  to 50  percent, 
in  cold  climates  in  winter,  to  allow  for  inferior  lubrication. 

4th.  Resistance  due  to  A  tmosphere. 

Atmospheric  resistance  to  trains,  complicated  as  it  is  by  the  wind  which  may  be 
prevailing,  has  not  been  accurately  ascertained  by  experiment.  It  consists  of— 
ist.  Head  resistance  of  first  car  of  train,  which  is  presumably  equal  to  its  exposed 
area,  in  sq.  feet,  multiplied  by  air  pressure  due  to  speed. 

2d.  Head  resistance  of  each  subsequent  car.  This  varies  with  distance  they  are 
coupled  apart,  and  so  shield  each  other  from  end  air  pressure  due  to  speed. 

3d.  Friction  of  air  against  sides  of  each  car  depending  upon  the  speed.  This  is 
generally  so  small  that  it  may  be  neglected  altogether. 

4th.  Effect  due  to  prevailing  wind,  which  modifies  above  three  items  of  resistance. 
A  head  wind  retards  the  train,  a  rear  wind  aids  it,  while  a  side  wind  increases  re- 
sistance by  pressing  flanges  of  wheels  against  one  rail,  and,  in  consequence  of  curves, 
a  train  may  assume  all  of  these  positions  to  same  wind. 

Recent  experiments  on  Erie  Railway  seem  to  indicate  that  in  a  dead  calm  re- 
sistance of  first  car  of  a  freight  train  may  be  assumed  at  an  exposed  surface  of  63 
sq.  feet,*  multiplied  by  a'ir  pressure  due  to  speed,  and  that  each  subsequent  car  may 
be  assumed  to  offer  a  resistance  of  20  per  cent,  of  that  of  first  car,  while  in  a  pas- 
senger train  first  car  may  be  assumed  at  an  area  of  90  sq.  feet,t  multiplied  by  air 
pressure  due  to  speed,  and  that  each  subsequent  car  adds  an  increment  equal  to  40 
per  cent,  that  of  first  car,  in  consequence  of  greater  distance  they  are  coupled  apart. 

This  resistance  is,  of  course,  entirely  independent  of  cars  being  loaded  or  empty. 
In  practice  it  has  been  found  that  an  allowance  of  1.5  to  2  Ibs.  per  ton  of  weight  of 
&  freight  train  covers  atmospheric  resistance,  except  in  very  high  winds. 

In  consequence  of  complexity  of  elements  above  enumerated,  exact  formulas  can- 
not probably  be  now  given  for  train  resistances,  but  following,  if  applied  with  judg- 
ment (and  modified  to  fit  circumstances),  will  be  found  to  give  fairly  accurate  results 
in  practice.  They  are  for  standard  gauge,  and  in  making  them,  curve  resistance  has 
been  assumed  at  .5  Ib.  per  degree,  wheel  friction  at  5  Ibs..  exposed  end  area  of  first 
car  at  90  sq.  feet  for  passenger  cars  and  63  feet  for  freight  cars,  and  increment  for 
succeeding  cars  at  .4  for  passenger  trains  and  .2  for  freight  trains. 

Passenger   Train.     W  (&  +  —  +  s)  +  (r  +  ~~)  9°  P=::B- 
Freight   Train.     W  (G  -f^  +  5)  +  ('  +^Zl1)  63  P  =  R 

*  This  is  less  than  area  of  car,  which  generally  measures  about  71  sq.  feet ;  but  part  is  shielded  by 
tender,  and  parts  being  convex,  as  wheels,  bolts,  etc.,  offer  lees  resistance  than  a  flat  plane. 

t  Not  only  is  end  area  of  passenger  cars  greater  than  that  of  freight  cars,  but  in  consequence  of  the 
projecting  roof  the  end  forms  a  hood  in  nature  of  a  concave  surface,  and  so  opposes  greater  resistance 
than  a  flat  plan*. 


684 


EAILWAYS. 


W  representing  weight  of  train,  without  engine,  in  tons  (2000  Ibs.  ),  G  resistance  of 
gradient  per  ton  (2000  Ibs.;  see  table,  page  683),  C°  curve  in  degrees,  n  number  of  cars 
in  train,  P  pressure  per  sq.foot  due  to  speed,  to  which  an  allowance  must  be  made  for 
wind,  if  existing,  R  resistance  of  train,  and  5,  wheel  friction,  both  in  Ibs. 

ILLUSTRATION  i.  —  Assume  a  passenger  train  of  5  cars,  weighing  136  tons  (2000  Ibs.), 
ascending  a  grade  .5  per  100  (26.4  feet  per  mile),  with  curves  of  4°,  at  a  speed  of  60 
miles  per  hour  (for  which  the  pressure  is  18  Ibs.  per  sq.  foot),  resistance  will  be: 

136  (10+  2  -f-  5)  +  f  i  +  —  J  (90  X  18)  =  6524  Ibs.,  of  which  2312  Ibs.  are  due  t& 

grade,  curve,  and  wheels,  and  4212  Ibs.  to  atmospheric  resistance. 

2.—  Assume  a  freight  train  of  31  cars,  weighing  620  tons  (2000  Ibs.),  turning  a  curvt 
of  3°,  up  a  grade  of  52.8  feet  per  mile  (i  foot  per  100),  at  a  speed  of  21  miles  per  hour 
(pressure  2  Ibs.  per  sq.  foot),  resistance  will  be: 


620(20+1. 


~    (63X  2)  =  17  312  Ibs.,  requiring  a  "Consolidation 


engine  to  haul  it,  allowance  being  made  for  possible  winds,  etc. 

Assume  conversely,  it  is  desired  to  know  how  many  tons  an  American  engine, 
with  an  adhesion  of  10650  Ibs.,  will  draw  up  a  grade  of  .9  per  100  (47  feet  per  mile), 
with  curves  of  4°.  assuming  atmospheric  resistance  between  1.5  to  2  Ibs.  per  ton  of 
train. 

Resistance  from  grade  .9  x  2000-4-  100  .............  =  18  Ibs.  ) 

u    curve  4-^-2  .......................  =   2"    }  27  Ibs. 

"    wheel  friction  5,  atmosphere  2  .....  —    7  "    ) 

Hence,  10  650  -r-  27  :=  395  tons,  or  about  20  cars,  and  in  winter  same  engine  will 
haul  9600-7-27  —  355  tons  (2000  Ibs.),  or  about  18  cars. 

Following  table  approximates  to  best  modern  practice.  For  freight  trains  it  gives 
aggregate  resistance,  in  Ibs.  per  ton  (2000  Ibs  ),  for  various  grades  and  curves.  In 
usiiig  it,  it  is  sufficient  to  divide  the  adhesion  in  Ibs.  of  locomotive  used  by  number 
found  in  table,  in  order  to  obtain  number  of  tons  of  train  that  it  will  haul  at  or- 
dinary speeds  on  gradient  and  curve  selected.  Of  course,  if  grade  has  been  equated 
for  curves,  only  number  found  in  first  column  (for  straight  lines)  is  to  be  used  in 
computing  tons  of  train  on  limiting  gradient. 

Approximate   !F"reiglit-train.   Resistances. 

Gauge  4  feet  8.  5  ins. 

In  Lbs.  per  2000  Ibs.  at  Ordinary  Speeds. 

Curve  Resistance  assumed  at  .5  Ibs.  per  °,  Wheel  Friction  at  5  Ibs.,  Atmospheric  Re- 
sistance at  2  Ibs.  per  Ton. 


GRADB. 

i 

CURVE. 

Per 

Per 

* 

Cent. 

Mile. 

qo 

i° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

i5° 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs". 

Ibs. 

IbsT 

Ibs. 

IbsT 

Ibs. 

IbsT 

Ibs. 

Ibs. 

Ibs. 

Ibs.    Ibs. 

Level. 

Feet. 

7 

7-5 

8 

8-5 

9 

9-5 

10 

10.5 

ii 

"•5 

12 

12.5 

13 

13-5 

14    14-5 

.1 

5 

9 

9-5 

10 

10.5 

ii 

"•5 

12 

12.5 

13 

13-5 

14 

14-5 

15 

15-5 

16    16.5 

.2 

ii 

ii 

"•5 

12 

12.5 

13 

13-5 

H 

14-5 

15 

15-5 

r6 

I6.5 

i? 

17-5 

18   18.5 

•3 

16 

13 

i3-5 

14 

14-5 

15 

15-5 

16 

16.5 

i? 

17-5 

18 

l8.5 

19 

19.5 

20 

20.5 

•4 

21 

15 

i5-5 

16 

16.5 

i? 

i7-5 

18 

18.5 

'9 

19-5 

20 

20.5 

21 

21.5 

22 

22.5 

•5 

26 

17 

i7-5 

18 

18.5 

i9 

19-5120 

20.5 

21 

21.5 

22 

22-5 

23 

23-5 

24 

24-5 

.6 

32 

»9 

i9-5 

20 

20.5 

21 

2I.5|  22 

22.5 

23 

23-5 

24 

24-5 

25 

25-5 

26 

26.5 

•7 

37 

21 

21.5 

22 

22.5 

23 

23.5,24 

24-5 

25 

25-5 

26 

26.5 

2? 

27-5 

28 

28.5 

.8 

42 

23 

23-5 

24 

24-5 

25 

25.5(26 

26.5 

27 

27-5 

28 

28.5 

29 

29-5 

30 

30-5 

•9 

47 

25 

25-5 

26 

26.5 

27 

27-5 

28 

28.5 

29 

29-5  1  30 

30-5 

31 

31-5 

32 

32-5 

i 

53 

27 

27-5 

28 

28.5129 

29-5 

30 

30-5 

31 

3i-5    32 

32-5 

33 

33-5 

34 

34-5 

i.i 

58 

29 

29-5 

30 

30-  5  i  3i 

31-5 

32 

32.5 

33 

33-5134 

34-5 

35 

35-5 

36 

36.5 

1.2 

63 

31 

3i-5 

32 

32-5    33 

33-5 

34 

34-5 

35I35.5    36 

36.5 

37    37-5 

38 

38.5 

i-3 

68 

33 

33-5 

34 

34-3    35 

35-5 

36 

36.5 

37  '37-  5 

38 

38.5 

39 

39-5 

40 

40-5 

1.4 

74 

35 

35-5 

36 

36-5    37 

37-5 

38 

38.5 

39    39-5 

40 

40-5 

4i 

4i-5 

42 

42-5 

i-J 

79 

37 

37-5 

38 

38-  5    39 

39-5 

40 

40-5 

41    41.5   42 

42-5 

43 

43-5 

44 

44-5 

1.6 

85 

39  1  39-  5 

40   40.5:41 

4i-5 

42 

42-5!  43    43-5    44 

44-5 

45 

45-5 

46 

46.5 

ILLUSTRATION.—  Assume  a  "Mogul"  engine  to  have  an  adhesion  of  16000  Ibs.  ; 
what  weight  will  it  haul  up  a  grade  of  74  feet  per  mile,  combined  with  a  curve  of  9°  ? 

16000-4-39.5  =405  tons  (2000  Ibs.). 

EA1LWATS. 


685 


Hence,  To  Compute  Adhesion  on  a  Given  Grade  and  Curve,  having  Weight 
of  Train. 

RULE.— Multiply  tabular  number  by  weight  of  train  in  tons  (2000  Ibs.), 
and  product  will  give  adhesion,  in  Ibs. 

EXAMPLE.— Assume  preceding  elements.    Then  39. 5  X  405  =  16  ooo  Ibs. 

NOTE.— A  "Consolidation"  engine,  by  its  superior  adhesion  (19550  Ibs.)  would 
haul  up  a  like  grade  and  curve  495  tons. 

Memoranda  .011   English.   Railways. 

Regulations  (Board  of  Trade). 

Cast-iron  girders  to  have  a  breaking  weight  =  3  times  permanent  load,  added  to 
6  times  moving  load. 

Wrought-iron  bridges  not  to  be  strained  to  more  than  5  tons  per  sq.  inch. 

Minimum  distance  of  standing  work  from  outer  edge  of  rail  at  level  of  carriage 
steps,  3.5  feet  in  England  and  4  feet  in  Ireland. 

Minimum  distance  between  lines  of  railway,  6  feet. 

Stations. — Minimum  width  of  platform,  6  feet,  and  12  at  important  stations. 
Minimum  distance  of  columns  from  edge  of  platform,  6  feet.  Steepest  gradient  for 
stations,  i  in  260.  Ends  of  platforms  to  be  ramped  (not  stepped).  Signals  and  dis- 
tant signals  in  both  directions. 

Carriages.— Minimum  space  per  passenger  20  cube  feet.  Minimum  area  of  glass 
per  passenger,  60  sq.  ins.  Minimum  width  of  seats,  15  ins.  Minimum  breadth  of 
seat  per  passenger,  18  ins.  Minimum  number  of  lamps  per  carriage,  2. 

Requirements.—  Joints  of  rails  to  be  fished.  Chairs  to  be  secured  by  iron  spikes. 
Fang  bolts  to  be  used  at  the  joints  of  flat-bottomed  rails. 


Construction.  v^T 

Width,  single  line  ...................................     18 

'*      double  line  ...................................    30 

ii  My.    top  of  ballast,  single  line  ......................    13 


Broad. 
Feet.  Ins. 

P 

6 


15 

29 


double  line  .....................    24 

Slope  of  cuttings  from  centre,  i  in  30.  Width  of  land  beyond  bottom  of  slope, 
o  to  12  feet.  Ditch  with  slopes,  i  foot  at  bottom,  i  to  i.  Quick  mound,  18  ins.  in 
height.  Post  and  rail-fence  posts,  7  feet  6  ins.  x  6  ins.  x  3.5  ins.,  9  feet  apart,  3  feet 
in  ground.  Intermediate  posts,  5  feet  6  ins.  X  4  ins.  X  1.5  ins.,  3  feet  apart.  Rails 
4  of  4X  1.5  ins. 

Parliamentary   Regulations   for   Crossing   Roads. 


Turnpike 
Road. 

Public 
Road. 

Occupation 
Road. 

Clear  width  of  under  bridge,  or  approach  .... 
Clear  height  of  under  bridge  for  a  width  of  12  ft. 

((                         U                         U                      U                   U           IQ  U 

a                tc                u              «            u        g  a 
"             "             "        at  springing  

Feet.  Ins. 
3I     ~~ 

16    — 

12      — 

4    — 
i  in  30 

3    — 

ach  side 

Feet.  Ins. 
25     — 

15      — 
12      — 

4    — 
i  in  20 

3    — 

of  centre 

Feet.  Ina. 
12      — 

14      — 

i  in  16 
3    — 

line.    In 

Limits  of  Deviation.  —  In  towns,  10  yards  € 

country,  100  yards,  or  5  chains  nearly. 

Level.— In  towns,  2  feet.    In  country,  5  feet. 

Gradient.  —  Gradients  flatter  than  i  in  100,  deviation  10  feet  per  mile 
steeper.    Do.,  steeper,  3  feet  per  mile. 

Curve. — Curves  upwards  of  .5  a  mile  radius,  may  be  sharpened  to  .5  mile 
radius.    Curves  of  less  than  .5  mile  radius  may  not  be  sharpened. 

3M 


686 


EOADS,  STEEETS,  AND   PAVEMENTS. 


ROADS,  STREETS,  AND    PAVEMENTS. 

Classification.   o±'  Roads. 

i.  Earth.  2.  Corduroy.  3.  Plank.  4.  Gravel.  5.  Broken  stone  (Marv 
ftdam).  6.  Stone  sub-pavement  with  surface  of  broken  stone  (Telford). 
7.  Stone  sub-pavement  with  surface  of  broken  stone  and  gravel,  or  gravel 
alone.  8.  Rubble  stone  bottom  with  surface  of  broken  stone  or  gravel,  or 
both.  9.  Concrete  bottom  with  surface  of  broken  stone  or  gravel,  or  both. 

Oracle    of  Roads. 

Limit  of  practicable  grade  varies  with  character  of  road  and  friction  of  ye- 
hide.  For  best  carriages  on  best  roads,  limit  is  i  in  35,  or  150  feet  in  a  mile. 

Maximum  grade  of  a  turnpike  road  is  i  in  30  feet.  An  ascent  is  easier 
for  draught  if  taken  in  alternate  ascents  and  levels,  than  in  one  continuous 
rise,  although  the  ascents  may  be  steeper  than  in  a  uniform  grade. 

Ordinary  angle  of  repose  is  i  in  40  if  roads  are  bad,  and  i  in  30,  to  i  in  20. 

When  roads  have  a  greater  grade  than  i  in  35,  time  is  lost  in  descending, 
in  order  to  avoid  unsafe  speed.  Grade  of  a  road  should  be  less  than  its  angle 
of  repose.  Minimum  grade  of  a  road  to  secure  effective  drainage  should  be 
i  in  80.  In  France  it  is  i  in  125. 

In  construction  of  roads  the  advantage  of  a  level  road  over  that  of  an  in- 
clined one,  in  reduction  of  labor,  is  superior  to  cost  of  an  increased  length 
of  road  in  the  avoiding  of  a  hill. 

Alpine  roads  over  the  Simplon  Pass  average  i  in  17  on  Swiss  side,  i  in  22 
on  Italian  side,  and  in  one  instance  i  in  13. 

In  deciding  upon  a  grade,  the  motive  power  available  of  ascent  and  avoid- 
able of  waste  of  power  in  descending  are  to  be  first  considered. 

When  traffic  is  heavier  in  one  direction  than  the  other,  the  grade  in  as- 
cent of  lighter  traffic  may  be  greatest. 

When  axis  of  a  road  is  upon  side  of  a  hill,  and  road  is  made  in  parts  by 
excavation  and  by  embankment,  the  side  surface  should  be  cut  into  steps, 
in  order  to  afford  a  secure  footing  to  embankment,  and  in  extreme  cases, 
sustaining  walls  should  be  erected. 

Construction, 

Estimate  of  Labor  in  Construction  of  Roads.    (M.  Ancelin.) 
A  day's  work  of  10  hours  of  an  average  laborer  is  estimated  as  follows: 
In  Cube  Yards. 

Loose 
Earth. 


WOBK. 

Ordinary 
Earth. 

Picking  and  digging  
Excavation  and  pitching  ) 
6  to  12  feet  ) 

18  to  23 
8  to  12 

Loading  in  barrows  
Wheeling  in  barrows  per) 
100  feet           J 

22 

20  to  33 

Loading  in  carts 

16  to  48 

Spreading  and  levelling.  .  . 

44  to  88 

Mud. 

Clay  and 
Earth. 

Gravel. 

Blasting 
Rock. 

— 

9 

7  to  ii 

2.4 

7  to  16 

4 

— 

2.2 

8 

— 

J9 

— 

— 

— 

24  to  28 

— 

_ 



17  to  27 



25 

— 

30  to  80 

— 

Time  of  pitching  from  a  shovel  is  one  third  of  that  of  digging. 

Ditches. — All  ditches  should  lead  to  a  natural  water-course,  and  their  min- 
imum inclination  should  be  i  in  125. 

Depressions  and  elevations  in  surface  of  a  roadway  involve  a  material  loss  of 
power.  Thus,  if  elevation  is  i  inch,  under  a  wheel  4  feet  in  diameter,  an  inclined 
plane  of  i  in  7  has  to  be  surmounted,  and,  as  a  consequence,  one  seventh  of  weight 
has  to  be  raised  x  inch. 


ROADS,  STREETS,   A^D    PAVEMENTS.  68/ 


An  unyielding  foundation  and  surface  are  indispensable  for  a  perfect  roadway. 
Earth  in  embankment  occupies  an  average  of  one  tenth  less  space  than  in  natural 
bank,  and  rock  about  one  third  more. 

Ruts.  —  Surface  of  a  roadway  should  be  maintained  as  intact  as  prac- 
ticable, as  the  rutting  of  it  not  only  tends  to  a  rapid  destruction  of  it,  but 
involves  increased  traction. 

The  general  practice  of  rutting  a  road  displays  a  degree  of  ignorance  of 
physical  laws  and  mechanical  effects  that  is  as  inexplicable  as  it  is  injurious 
and  expensive. 

On  compressible  roadways,  as  earth,  sand,  etc.,  resistance  of  a  wheel  decreases  as 
breadth  of  tire  increases. 

Depressing  of  axles  at  their  ends  increases  friction.  Long  and  pliant  springs  de- 
crease effect  of  shock  in  passing  over  obstacles  in  a  very  great  degree. 

Transverse  Section.—  Best  profile  of  section  of  roadway  is  held  to  be  one 
formed  by  two  inclined  planes  meeting  in  centre  of  road  and  slightly 
rounded  off  at  point  of  junction. 

Roads  having  a  rough  surface  or  of  broken  stone  should  have  a  rise  of 
i  in  24,  equal  to  a  rise  on  crown  of  6  ins.,  and  on  a  smooth  surface,  as  a 
block-stone  or  wood  pavement,  the  rise  may  be  reduced  to  i  in  48. 

On  roads,  when  longitudinal  inclination  is  great,  the  rise  of  transverse 
section  should  be  increased,  in  order  that  surface  water  may  more  readily 
run  off  to  sides  of  roadway,  instead  of  down  its  length,  and  consequently 
gullying  it. 

Stone  Breaking.  A  steam  stone-breaking  machine  will  break  a  cube  yard 
of  stone  into  cubes  of  1.5  ins.  side,  at  rate  of  i  to  1.5  IP  per  hour. 

Macadainized.    Roads. 

In  construction  of  a  Macadamized  road,  the  stones  (  road  metal  )  used 
should  be  hard  and  rough,  and  cubical  in  form,  the  longest  diameter  of  which 
exceed  2.5  ins.,  but  when  they  are  very  hard  this  may  be  reduced  to  1.25 
and  1.5  ins. 

The  best  stones  are  such  as  are  difficult  of  fracture,  as  basaltic  and  trap, 
and  especially  when  they  are  combined  with  hornblende.  Flint  and  sili- 
ceous stone  are  rendered  unfit  for  use  by  being  too  brittle.  Light  granites 
are  objectionable,  in  consequence  of  their  being  brittle  and  liable  to  disinte- 
gration ;  dark  granites,  possessing  hornblende,  are  less  objectionable.  Lime- 
stones, sandstones,  and  slate  are  too  weak  and  friable. 

Dimensions  of  a  hammer  for  breaking  the  stone  should  be,  head  6  ins.  in 
length,  weighing  i  lb.,  handle  18  ins.  in  length;  and  an  average  laborer  can 
break  from  1.5  to  2  cube  yards  per  day. 

Stones  broken  up  in  this  manner  have  a  volume  twice  as  great  as  in  their 
original  form.  100  cube  feet  of  rock  will  make  190  of  1.5  ins.  dimension, 
182  of  2  ins.,  and  170  of  2.5  ins. 

A  ton  of  hard  metal  has  a  volume  of  1.185  cut>e  yards. 

Construction  of  a  Roadway.  —  Excavate  and  level  to  a  depth  of  i  foot, 
then  lay  a  "  bottom  "  12  ins.  deep  of  brick  or  stone  spalls  or  chips,  clinker 
or  old  concrete,  etc.,  roll  down  to  9  ins,  then  add  a  layer  of  coarse  gravel  or 
small  ballast  5  ins.  deep,  roll  down  to  3  ins.,  and  then  metal  in  2  equal  lay- 
ers of  3  ins.,  laid  at  an  interval,  enabling  first  layer  to  be  fully  consolidated 
before  second  is  laid  on  and  rolled  to  a  depth  of  4  ins.  ;  a  surface  or  "  blind'' 
of  .75  inch  of  sharp  sand  should  be  laid  over  last  layer  of  metal  and  rolled 
in  with  a  free  supply  of  water. 


688 


ROADS,  STREETS,-  AND   PAVEMENTS. 


Proportion  of  Getters,  Fillers,  and  Wheelers  in  different  Soils, 
at  a  Run  of  50  Yards.     (Molesworth.) 


Wheelers  computed 


Getters. 

Fillers. 

Wheelers. 

Getters. 

Fillers. 

Wheelers. 

Loose  earth  ) 

1.25 

1.25 

Sand,  etc.  }  •  •  •  • 
Compact  earth  .  .  . 
Marl... 

I 

X 
I 

I 

2 

2 

X 

2 
2 

Compact    ) 
gravel    J  "•' 
Rock  .  . 

X 

7 

2 
I 

X 

I 

Telford    Roads. 

In  construction  of  a  Telford  road,  metalling  is  set  upon  a  bottom  course  of 
stones,  set  by  hand,  in  the  manner  of  an  ordinary  block  stone  pavement, 
which  course  is  composed  of  stones  running  progressively  from  3  inches  in 
depth  at  sides  of  road  to  4,  5,  and  7  inches  to  centre,  and  set  upon  their 
broadest  edge,  free  from  irregularities  in  their  upper  surface,  and  their  in- 
terstices filled  with  stone  spalls  or  chips,  firmly  wedged  in. 

Centre  portion  of  road  to  be  metalled  first  to  a  depth  of  4  ins.,  to  which, 
after  being  used  for  a  brief  period,  2  ins.  more  are  to  be  added,  and  entire 
surface  to  be  covered,  "  blinded,"  with  clean  gravel  1.5  ins.  in  depth. 

Telford  assigned  a  load  not  to  exceed  i  ton  upon  each  wheel  of  a  vehicle, 
with  a  tire  4  ins.  in  breadth. 

Gravel   or   Earth.   Roads. 

In  construction  of  a  gravel  or  earth  road,  selection  should  be  made  between 
clean  round  gravel  that  will  not  pack,  and  sharp  gravel  intermixed  with 
earth  or  clay,  that  will  bind  or  compact  when  submitted  to  the  pressure  of 
traffic  or  a  roll. 

Surface  of  an  ordinary  gravel  roadway  should  be  excavated  to  a  depth  of 
from  8  to  12  ins.  for  full  width  of  road,  the  surface  of  excavation  conforming 
to  that  of  road  to  be  constructed. 

The  gravel  should  then  be  spread  in  layers,  and  each  layer  compacted  by 
the  gradual  pressure  due  to  travel  over  it,  or  by  a  roller,  the  weight  of  it  in- 
creasing with  each  layer.  One  of  6  tons  will  suffice  for  limit  of  weight. 

If  gravel  is  dry  and  will  not  readily  pack,  it  should  be  wet,  and  mixed 
with  a  binding  material,  or  covered  with  a  thin  layer  of  it,  as  clay  or  loam. 

In  rolling,  the  sides  of  road  should  be  first  rolled,  in  order  to  arrest  the 
gravel,  when  the  centre  is  being  rolled,  from  spreading  at  the  side. 

To  re-form  a  mile  of  gravel  or  earth  road,  30  feet  in  width  between  gutters, 
material  cast  up  from  sides,  there  wili  be  required  1640  hours'  labor  of  men, 
and  20  of  a  double  team. 

Cordiiroy    Roads. 

A  Corduroy  road  is  one  in  which  timber  logs  are  laid  transversely  to  its  plane. 
TMarilz  Roads. 

A  single  plank  road  should  not  exceed  8  feet  in  width,  as  any  greater  width 
involves  an  expenditure  of  material,  without  any  equivalent  advantage. 

If  a  double  track  is  required  it  should  consist  of  two  single  and  independ- 
ent tracks,  as  with  one  wide  track  the  wear  would  be  mostly  in  the  centre, 
and  consequently,  wear  would  be  restricted  to  one  portion  of  its  surface. 

Materials. — Sleepers  should  be  as  long  as  practicable  of  attainment,  in  depth  3  or 
4  ins.,  according  to  requirements  of  the  soil,  and  they  should  have  a  width  of  3  ins. 
for  each  foot  of  width  of  road. 

Pine,  oak,  maple,  or  beech  are  best  adapted  for  economy  and  wear. 

Planks  should  be  from  3  to  3.5  ins.  thick,  and  not  less  than  9  ins.  in  width,  or 
more  than  12  if  of  hard  wood,  or  15  if  of  soft. 

A  plank  road  will  wear  from  7  to  12  years,  according  to  service,  material, 
and  location,  and  its  traction,  compared  with  an  ordinary  Macadamized  road, 
is  2.5  to  3  times  less,  and  with  a  common  country  road  in  bad  order  7  times. 

For  other  elements,  see  Earth- work,  page  466. 


ROADS,  STREETS,  AND    PAVEMENTS.  689 

-A.sph.alt. 

Asphalt  pavements  are  made  in  two  ways,  either  from  a  mixture  of  asphaltum 
with  sand  and  a  little  powdered  limestone,  or  from  natural  asphaltic  limestone, 
called  sometimes  "rock  asphalt,"  which  contains  from  6  to  12  per  cent,  of  asphal- 
tum. 

The  asphalt  pavements  of  America  are  principally  made  by  the  former,  and  those 
of  Europe  by  the  latter  method.  The  composition  of  one  is  of  about  12.5  per  cent, 
refined  asphaltum,  2. 5  residuum  oil  of  petroleum  or  soft  bitumen  termed  u  maltha," 
5  powdered  limestone,  and  80  sharp  sand,  by  weight,  mixed  at  about  300°. 

The  rock  asphalt  pavement  is  made  by  powdering  the  natural  asphaltic  lime- 
stone, heating  the  powder,  and  compressing  it  in  place. 

Asphaltic  mastic,  for  floors,  roofs,  and  sidewalks,  is  made  from  rock  asphalt,  by 
adding  asphaltum  to  it  as  a  flux  and  incorporating  60  per  cent,  more  or  less  of  sand 
and  gravel,  according  to  the  density  needed  and  the  temperature  of  the  place,  cel- 
lar or  walk,  and  whether  exposed  to  the  sun  or  not.  The  roadway  needs  a  convex- 
ity of  at  least .  15  of  its  breadth. 

Artificial  Asphalt. — Heated  sand,  gravel,  and  powdered  limestone,  with  gas  tar  or 
coal  tar,  when  mixed,  possess  some  of  the  properties  of  asphalt  mastic,  but  are 
much  inferior. 

Bituminous  Road  may  be  made  by  breaking  up  asphaltic  limestone,  laying  it  2 
ins.  thick,  covering  with  coal  tar  and  ramming.  Useful  in  country  districts  near 
such  deposits. 

'Wood.    3?avexnent. 

Close-grained  and  hard  woods  only  are  suitable,  such  as  oak,  elm,  ash, 
beech,  and  yellow  pine,  and  they  should  be  laid  on  a  foundation  of  concrete. 
Block    Stone   I?avexnent. 

Paving-blocks,  as  the  Belgian,  etc.,  where  crest  of  street  or  area  of  pave- 
ment does  not  exceed  i  inch  in  7.5  feet,  should  taper  slightly  toward  the 
top,  and  the  joints  be  well  filled,  "  blinded,"  with  gravel.  The  common 
practice  of  tapering  them  downward  is  erroneous. 

The  foundation  or  bottoming  of  a  stone  pavement  for  street  travel  should 
consist  either  of  hydraulic  concrete  or  rubble  masonry  in  hydraulic  mortar. 
The  practice  in  this  country  of  setting  the  stones  in  sand  alone  is  at  variance 
with  endurance  and  ultimate  economy,  but  when  resorted  to,  there  should  be 
a  bed  of  12  ins.  of  gravel,  rammed  in  three  layers,  covered  with  an  inch  of 
sand.  Granite  or  Trap  blocks  should  be  4  x  9  X  12  ins. 
Ru."b~ble  Stone  I?avement. 

Bowlders  or  Beach  stone  of  irregular  volumes  and  forms,  set  in  a  bed  of 
sand,  involves  great  resistance  to  vehicles  and  frequent  repairs ;  it  is  wholly 
at  variance  with  requirements  of  heavy  traffic  or  city  use. 
Concrete    Ttoads. 

Concrete  roads  are  constructed  of  broken  stones  (road  metal)  4  volumes, 
clean  sharp  sand  1.25  to  .33  volumes,  and  hydraulic  cement  i  volume.  The 
mass  is  laid  down  in  a  layer  of  3  or  4  ins.  in  depth,  and  left  to  harden  during 
a  period  of  3  days,  when  a  second  and  like  layer  is  laid  on  and  well  rolled, 
and  then  left  to  harden  for  a  period  of  from  10  to  20  days,  according  to 
temperature  and  moisture  of  the  weather. 

Roads.     (Molesworth.) 

Ordinary  turnpike  roads. —  30  feet  wide,  centre  6  ins.  higher  than  sides ; 
4  feet  from  centre,  .5  inch  below  centre ;  9  feet  from  centre,  2  ins.  below 
centre;  15  feet  from  centre,  6  ins.  below  centre. 

Foot-paths — 6  feet  wide,  inclined  i  inch  towards  road,  of  fine  gravel,  or 
sifted  quarry  chippings,  3  ins.  thick. 

Cross-roads— 20  feet  wide.        Foot-paths— 5  feet. 

Side  drains — 3  feet  below  surface  of  road. 

Road  material — bottom  layer  gravel,  burned  clay  or  chalk,  8  ins.  deep. 
Top  layer,  broken  granite  not  larger  than  1.5  cube  ins.,  6  ins.  deep. 

3  M* 


690  ROADS,  STREETS,  AND    PAVEMENTS. 

Miscellaneous   !N"otes. 

Metalling  should  be  from  6  ins.  to  i  foot  in  depth,  and  in  cubes  of  1.5  to  1.75  ins. 

One  layer  of  material  of  a  road  should  be  spread  and  submitted  to  traffic  or  roll- 
ing before  next  is  laid  down,  and  this  process  should  be  repeated  in  2  or  3  layers 
of  3  ins.  each. 

When  new  metal  is  laid  on  old,  the  surface  of  the  old  should  be  loosened  with  a 
pick.  Patching  is  termed  darning. 

Sand  and  Gravel,  Blinding,  should  not  be  spread  over  a  new  surface,  as  they  tend 
to  arrest  binding  of  metal.  Mud  should  be  scraped  oft"  of  surface. 

Hoggin  is  application  of  a  binding  of  surface  of  a  metal  road,  composed  of  loam, 
fine  gravel,  and  coarse  sand. 

Metalled  Roads  should  be  swept  wet. 

Rolling. — Steam  rolls  are  most  effective  and  economical.  1000  sq.  yards  of  metal- 
ling will  require  24  hours'  rolling  at  i.  5  miles  per  hour.  A  roller  of  15  tons'  weight 
will  roll  looo  sq.  yards  of  Telford  or  Macadam  pavement  in  from  30  to  40  hours,  at 
a  speed  of  1.5  miles  per  hour,  equal  .675  and  .9  ton  mile  per  sq.  yard. 

Sprinkling. — 60  cube  feet  of  water  with  one  cart  will  cover  850  sq.  yards.  100 
cube  feet  per  day  will  cover  1000  sq.  yards;  ordinarily  two  sprinklings  are  necessary. 

Granite  Pavement.— The  wear  of  granite  pavement  of  London  Bridge  was  .22  inch 
per  year,  and  from  an  average  of  several  streets  in  London,  the  wear  per  100  vehicles 
per  foot  of  width  per  day  is  equal  to  one  sixteenth  of  an  inch  per  year. 

Sweeping  and  Watering  of  granite  pavement  and  Macadam  road,  for  equal  areas 
and  under  alike  conditions  in  every  respect,  costs  as  i  for  former  toj  of  latter. 

By  men,  with  cart,  horse,  and  driver,  costs  3.25  times  more  than  by  a  machine, 
one  of  which  will  sweep  16000  sq.  yards  of  street  per  period  of  6  hours. 

Asphalt  Pavement.  — Average  cost  per  sq.  yard  in  London:  foundation,  50  cents; 
surface,  $3.25;  cost  of  maintenance  per  sq.  yard  per  year,  40  cents.  Wear  varies 
from  .2  to  42  near  curb,  and  .17  to  .34  inch  on  general  surface  per  year. 

Washing. — Surface  cleaning  of  stone  or  asphalt  pavement  by  a  jet  can  be  effected 
at  from  i  to  2  gallons  per  sq.  yard. 

Wood  Pavement.—  Wear  of  wood  pavement  in  London,  per  100  vehicles  per  day 
per  foot  of  width,  .083  inch  per  year. 

Macadamized  Roads.  —  Annual  cost  of  maintenance  of  several  such  roads  in 
London  was  62  cents  per  sq.  yard. 

Block  Stone  Pavement.—  Stones  should  be  set  with  their  tapered  or  least  ends  up- 
wards, with  surface  joints  of  i  inch. 

Fascines,  when  used,  should  be  in  two  layers,  laid  crosswise  to  each  other  and 
picketed  down. 

Bituminous  road  may  be  made  by  breaking  up  asphalt,  laying  it  2  ins.  thick, 
covering  with  coal  tar,  and  ramming  it  with  a  heavy  beetle.  To  repair  a  bitumi- 
nous surface,  dissolve  one  part  of  bitumen  (mineral  tar)  in  three  of  pitch  oil  or  resin 
oil,  spread  .625  of  a  Ib.  of  solution  over  each  sq.  yard  of  road,  sprinkle  2  Ibs.  pow- 
dered asphalt  (bituminous  limestone)  and  then  sand,  and  sweep  off  the  surplus. 

Slipping.—  Granite  safest  when  wet,  and  asphalt  and  wood  when  dry. 

Gravel,  alike  to  that  of  Roa  Hook,  from  its  uniformity,  will  bear  an  admixture 
of  from  .2  to  .25  of  ordinary  gravel  or  coarse  sand. 

Annual  cost  of  a  Telford  pavement  4.2  cents  per  sq.  yard,  including  sprinkling, 
repairs,  and  supervision. 

Voids  in  a  Cube  Yard  of  Stone. 

Broken  to  a  gauge  of  2. 5  ins 10      cube  feet.  I  Shingle 9    cube  feet. 

2      u 10.66     "     "     I  Thames  ballast. ...  4.5  "     " 

i-S    " "-33     '       ' 

For  further  and  full  information,  see  Law  and  Clarke  on  Roads  and  Streets,  New 
York,  1867;  Weale's  Series,  London,  1861  and  1877;  Roads,  Streets,  and  Pavements, 
by  Brev.  Maj.-Gen.  Q.  A.  Gilmore,  U.  S.  A.,  New  York,  1876;  Engineering  Notes,  by 
F.  Robertson,  London  and  New  York,  1873;  and  Construction  and  Maintenance  of 
Roads,  by  Ed.  P.  North,  C.  E.,  see  Transactions  Am.  Soc.  of  C.  E.,  vol.  viii.,  May,  1879. 


SEWERS. 


691 


',  and  D  =  r.     x  representing 


SEWERS. 

Sewers  are  the  courses  from  a  series  of  locations,  and  are  classed  as 
Drains,  Sewers,  and  Culverts. 

Brains  are  small  courses,  from  one  or  more  points  leading  to  a  sewer. 
Ctdverts  are  courses  that  receive  the  discharge  of  sewers. 
Greatest  fall  of  rain  is  2  ins.  per  hour  =  54  308.6  galls,  per  acre. 
Inclination  of  sewers  should  not  be  less  than  i  foot  in  240,  and  for 
house  or  short  lateral  service  it  should  be  i  inch  in  5  feet. 
Fig.  i.  Circular.    55  Vx  zf=  v,  and  u  a  =  V. 

_r      Egg.     -  =  w,  —  = 

area  of  sewer  -4-  wetted  perimeter,  f  inclination  of  sewer 
per  mile,  and  v  velocity  of  flow  of  contents  in  feet  per 
minute  ;  a  area  of  flow,  in  sq.feet,  V  volume  of  discharge, 
in  cube  feet  per  minute  ;  D  height  of  sewer,  w  and  w' 
width  at  bottom  and  lop,  and  r  radius  of  sides,  in  feet. 

For  diameter  of  sewer  exceeding  6  feet.    (T.  Hawksley.) 

D  --  —  w'.    D  diameter  of  a  circular  sewer  of  area  required. 
9 

Elliptic.  —  Top  and  bottom  internal  should  be  of  equal  diam- 
eters. Diameter  .66  depth  of  culvert  ;  intersections  of  top 
and  bottom  circles  form  centres  for  striking  courses  connect- 
ing top  and  bottom  circles. 

Pipes  or  Small  Sewers.  —  Height  of  section  =  i  ;  diameter 
of  arch  =r  .66  ;  of  invert  =  .33,  and  radius  of  sides  =  i. 

In  culverts  less  than  6  feet  internal  depth,  brickwork  should  be  9  ins.  thick  ; 
when  they  are  above  6  feet  and  less  than  9  feet,  it  should  be  14  ins.  thick. 

If  diameter  of  top  arch  =  i,  diameter  of  inverted  arch  =  .5,  and  total 
depth  =  sum  of  the  two  diameters,  or  1.5  ;  then  radius  of  the  arcs  which  are 
tangential  to  the  top,  and  inverted,  will  be  1.5. 

From  this  any  two  of  the  elements  can  be  deduced,  one  being  known. 

Drainage   of  I^ands   toy   3?ipes. 


SOILS. 

Depth 
of  Pipes. 

Distance 
apart. 

SOILS. 

Depth 
of  Pipes. 

Distance 
apart. 

Coarse  gravel  sand  
Light  sand  with  gravel 

Ft.   Ins. 
4     6 

Feet. 
60 

Loam  with  gravel  .  .  . 
Sandy  loam  

Ft.   Ins. 
3     3 

3Q 

Feet. 
27 

Light  loam  

3    6 

•J-3 

Soft  clav  

2      O 

21 

Loam  with  clav  .  .  . 

3       2 

21 

Stiff  clav... 

2      6 

1C 

"Velocity   and    <3-rade    of   Servers    and.    Drains 
in   Cities.     iWickstted.) 


Diam. 

Vel. 
per 
Minute. 

Grade, 
tin 

Grade 

MTle. 

Diam. 

Vel. 
per 

Minute. 

Grade, 
i  in 

Grade 
jfXe. 

Diam. 

Vel. 
per 
Minute. 

Grade, 
i  in 

Grade 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

240 

36 

146.7 

15 

180 

244 

21.6 

42 

180 

686 

77 

6 

220 

65 

81.2 

18 

180 

294 

18 

48 

1  80 

784 

6.8 

8 

220 

87 

60.7 

24 

1  80 

392 

'3-5 

54 

1  80 

882 

6 

10 

210 

119 

44-4 

30 

1  80 

490 

10.8 

60 

180 

980 

5-4 

12 

190 

175 

30.2 

36 

1  80 

588 

9 

Area  of  Sewers  or  Pipes. — An  area  of  20  acres,  miles,  etc.,  will  not  re- 
quire 20  times  capacity  of  pipes  for  one  acre,  mile,  etc.,  as  the  discharge  from 
the  19  acres,  etc.,  will  not  flow  into  the  main  simultaneously  with  that  from 
one  acre,  etc.  Ordinarily  in  this  country  an  area  of  sewer  or  pipe  that  will 
discharge  a  rainfall  of  i  inch  per  hour  (3630  cube  feet  per  acre)  is  sufficient. 


092 


SEWERS. 


Sewage. — The  excreta  per  annum  of  100  individuals  of  both  sexes  and 
all  ages  is  estimated  at  7250  Ibs.  solid  matter  and  94  700  fluid,  equal  to  1020 
Ibs.  per  capita,  and  in  volume  16  cube  feet,  to  which  is  to  be  added  the 
volume  of  water  used  for  domestic  purposes.  A  velocity  of  flow  of  from  2.5 
to  3  feet  per  second  will  discharge  a  sewer  of  its  sewage  matter  and  prevent 
deposits.  The  minimum  velocity  should  not  be  less  than  1.3  feet  per  second. 

Surface  from  -which  Circular  Sewers  -with  proper  Curves 
•will  discharge  that  Proportion  of*  \Vater  from  a  ITall 
of  One  Inch  in  Depth  per  Hour  which  would,  reach 
them.,  including  City  Drainage.  (John  Roe.) 


INCLINATION  IN  FEET. 

2 

D 

2-5 

IAMETEK 

3 

OF  SEW 
4 

ERS  IN   ] 

5 

^EKT. 

6 

7 

8 

None    

Acres. 

08  7  K 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

in  480  .   . 

4.8 

108 

57° 

5 

i  n  240  •  

87 

' 

1318 

5° 
63 

460 

2871; 

78 

g 

rgoe 

in  80  

jfie 

1188 

2486 

6625 

in  60.  .  . 

I2S 

l82 

318 

73O 

moo 

267"; 

4^0 

7125 

Surface  Of  a  Town  from  which  small  Circxilar  Drains 
will  discharge  \Vater  equal  in  "Volume  to  Two  Inches 
in  Depth  per  Hour.  (John  Roe.) 


INCLINATION. 
Fall  of  one  in. 

Di 
3 

iMETE 

4 

B  OF 

5 

DRAIN 
6 

IN  INS. 

7     i     8 

INCLINATION. 
Fall  of  one  in. 

DlAMI 

9 

TER   OF 
12 

DRAIN  i 

s  INS. 
18 

Acres. 

Feet. 

F^el 

Feet. 

Feet. 

Feet. 

Feet. 

Acres. 

Feet. 

Feet. 

Feet. 

Fe«t. 

•  125 

120 

— 

— 

— 

— 

— 

2.1 

120 

— 

— 

— 

•25 

20 

120 

— 

— 

— 

— 

2-5 

80 

— 

— 

— 

•4375 

— 

40 

— 

— 

— 

— 

2-75 

60 

— 

— 

— 

•  5 

— 

30 

80 

— 

— 

— 

4-5 

— 

1  20 

— 

— 

.6 

— 

20 

60 

— 

— 

— 

5-3 

— 

80 

— 

— 

i 



— 

2O 

60 

— 

— 

5-8 

— 

60 

240 

— 

1.3 

— 

— 



40 

20 

— 

7.8 

— 

— 

120 

— 

1.5 



— 

— 

20 

60 

120 

9 

— 

— 

80 

— 

1.8 

•»    T 

— 

— 

— 

— 

— 

80 
60 

10 

17 

— 

— 

60 

240 
1  20 

Dimensions,  Areas,  and  "Volume  of  Material  per   Lineal 
IToot  of  Egg-shaped  Sewers  of  different  Dimensions. 


Depth. 


Feet. 

2.25 

3 

3-75 

45 

I'5 
6.75 

L55 

9 


INTERNAL  D 
Diam.  of 
Top  Arch. 

IMKNSIONS. 

Diam.  of 
Invert. 

Area. 

VOLU 

4.5  Ins. 
thick. 

ME  OF  BRICK-TI 
9  Ins. 
thick. 

FORK. 

13.5  Ins. 
thick. 

Feet. 

Feet. 

Sq.  Feet. 

Cube  Feet. 

Cube  Feet. 

Cube  Feet. 

1.5 

•75 

2-53 

2.81 

— 

— 

2 

i 

4-5 

3-56 

— 

— 

2-5 

1.25 

7-03 

4-31 

9-56 

— 

3 

IO.  12 

5-o6 

10.87 

— 

3-5 
4 

2 

I3-78 

18 

5-8i 
6.56 

12.75 
14.25 



4-5 

2.25 

22.78 

7-31 

15-75 

24-75 

5 

2-5 

28.12 

— 

17.06 

27 

5-5 

2-75 

34-03 

— 

18 

28.41 

6 

3 

40-5 

— 

19.69 

30-94 

Area  =  product  of  mean  diameter  x  height. 

Sewer  Pipes  should  have  a  uniform  thickness  and  be  uniformly  glazed, 
both  internally  and  externally. 
Fire-clay  pipes  should  be  thicker  than  those  of  stone-clay. 


STABILITY.  693 

STABILITY. 

STABILITY,  Strength,  and  Stiffness  are  necessary  to  permanence  of  a 
structure,  under  all  variations  or  distributions  of  load  or  stress  to  which 
it  may  be  subjected. 

Stability  of  a  Fixed  Body — Is  power  of  remaining  in  equilibrio  without 
sensible  deviation  of  position,  notwithstanding  load  or  stress  to  which  it 
may  be  submitted  may  have  certain  directions. 

Stability  of  a  Floating  Body. — A  body  in  a  fluid  floats,  or  is  balanced, 
when  it  displaces  a  volume  of  the  fluid,  weight  of  which  is  equal  to  weight 
of  body,  and  when  centre  of  gravity  of  body  and  that  of  volume  of  fluid  dis- 
placed are  in  same  vertical  plane. 

When  a  body  in  equilibrio  is  free  to  move,  and  is  caused  to  deviate  in  a 
small  degree  from  its  position  of  equilibrium,  if  it  tends  to  return  to  its 
original  position,  its  equilibrium  is  termed  Stable ;  if  it  does  not  tend  to  de- 
viate further,  or  to  recover  its  original  position,  its  equilibrium  is  termed 
Indifferent  •  and  when  it  tends  to  deviate  further  from  its  original  position, 
its  equilibrium  is  Unstable. 

A  body  in  equilibrio  may  be  stable  for  one  direction  of  stress,  and  unstable 
for  another. 

Moment  of  Stability  of  a  body  or  structure  resting  upon  a  plane  is  mo- 
ment or  couple  of  forces,  which  must  be  applied  in  a  plane  vertically  inclined 
to  the  body  in  addition  to  its  weight,  in  order  to  remove  centre  of  resistance 
of  body  upon  plane,  or  of  the  joint,  to  its  extreme  position  consistent  with 
stability.  The  couple  generally  consists  of  the  thrust  of  an  adjoining  struct- 
ure, or  an  arch  and  pressure  of  water,  or  of  a  mass  of  earth  against  the 
structure,  together  with  the  equal  and  parallel,  but  not  directly  opposed,  re- 
sistance of  plane  of  foundation  or  joint  of  structure  to  that  lateral  thrust. 
It  may  differ  according  to  position  of  axis  of  applied  couple. 

Couple. — Two  forces  of  equal  magnitude  applied  to  same  body  or  struct- 
ure in  parallel  and  opposite  directions,  but  not  in  same  line  of  action,  consti- 
tute a  couple. 

NOTE.— For  Statical  and  Dynamical  Stability,  see  Naval  Architecture,  page  649. 

To  Ascertain  Stability  of"  a  Body  on  a  Horizontal  Plane. 
—  Fig.  1. 

Fig.  i.  ILLUSTRATION.— Stability  of  a  body,  A,  Fig.  i,  when  a 

7)  L  _       ^  thrust  is  applied  as  at  o,  to  turn  it  on  a,  is  ascertained  by 

multiplying  its  weight  by  distance  a  s,  from  fulcrum  a  to 
line  of  centre  of  gravity,  c  s. 

Hence,  if  cubical  block  weighed  10  tons  and  its  base  is 
6  feet,  its  moment  would  be  10  x  —  =  30  tons. 

If  upper  part,  a  b  d  c,  was  removed,  remainder,  a  e  d, 
would  weigh  but  5  tons,  but  its  centre  of  gravity  •  would  be  —  a  e  •=.  4  feet  Hence 
its  moment  would  be  5  x  4  =  20  tons,  although  it  is  but  half  the  weight 

To   Compute   Weight   of  a   GHven   Body   to    Sustain  a 
Q-iven    Tlirnst. 

F  h 

—  =  W.    F  representing  thrust  in  Ibs. ,  h  height  of  centre  of  gravity  of  body — c  *, 

and  I  distance  of  fulcrum  from  centre  of  gravity  =  as. 

ILLUSTRATION. — Assume  figure  to  be  extended  to  a  height  of  20  feet,  and  required 
to  be  capable  of  resisting  the  extreme  pressure  of  wind. 


694 


STABILITY. — KEVETMENT   WALLS. 


Pressure  estimated  at  50  Ibs.    F  =  6  X  20  X  50  =  6000  Ibs.  at  centre  of  gravity  of 
*~turface  of  body. 

6000  X  10 

Then =  20000  Ibs. 

3 

NOTK  i.— This  result  is  to  be  increased  proportionately  with  the  factor  of  safety 
due  to  character  of  its  material  and  structure. 

2.— If  form  of  body  has  a  cylindrical  section,  as  a  round  tower,  the  thrust  of  wind 
would  be  but  one  half  of  that  of  a  plane  surface. 

When  the  Body  is  Tapered,  as  Frustum  of  Pyramid  or  Cone.  —  Ascertain 
centres  of  gravity  of  surface  for  pressure  or  thrust,  and  of  body  for  its  sta- 
bility, and  proceed  as  before. 


Fig.  2. 


To  Ascertain  Stability  of  a  Body  on. 
an.    Inclination.— Fig.  2. 

ILLUSTRATION.— Stability  of  body,  Fig.  2,  when  thrust 
is  applied  at  c,  is  ascertained  by  multiplying  its  weight 
by  distance  a  b  from  fulcrum,  6,  to  line  of  centre  of 
gravity,  a  g. 

If  thrust  was  applied  at  o,  stability  would  be  ascer- 
tained by  distance  s  r  from  fulcrum  r. 

Angles  of  EcLnili"bri\xm  at  -which,  varions  Sn~bstances  Avill 
Repose,  as  determined,  "by  a  Clinometer. 

Angle  measured  from  a  Horizontal  Plane,  and  fatting  from  a  spout. 

Degrees 

Common  mold. . .  37 
Common  gravel. .  35  to  36 
Stones  or  Coal...  43 


Degrees. 

Lime-dust 45 

Dry  sand 40 

Moist  sand 41 


Degrees. 

Sand,  less  dry 39.6 

Wheat 37 

Corn 37 


"Weight  of  a  Cnbe  Foot  of  Materials  of  Embankments, 
"Walls,  and    Dams. 


Concrete  in  cement.  .  . 
Stone  masonry  

^ 

Gravel.  .  . 
Loam  .  . 

125 
I26 

Clay  
Marl 

120 

Brick  »  

1x2 

Sand  

120 

Revetment  "Walls. 

When  a  wall  sustains  a  pressure  of  earth,  sand,  or  any  loose  material,  it 
is  termed  a  Revetment  wall,  and  when  erected  to  arrest  the  fall  or  subsidence 
of  a  natural  bank  of  earth,  it  is  termed  a  Face  wall. 

When  earth  or  banking  is  level  with  top  of  wall,  it  is  termed  a  Scarp  re- 
vetment, and  when  it  is  above  it,  or  surcharged,  a  Counterscarp  revetment. 

When  face  of  wall  is  battered,  it  is  termed  Sloping,  and  when  back  is  bat- 
tered, Countersloping. 

Thrust  of  earth,  etc.,  upon  a  wall  is  caused  by  a  certain  portion,  in  shape 
of  a  wedge,  tending  to  break  away  from  the  general  mass.  The  pressure 
thus  caused  is  similar  to  that  of  water,  but  weight  of  the  material  must  be 
r«duced  by  a  particular  ratio  dependent  upon  angle  of  natural  slope,  which 
varies  from  45°  to  60°  (measured  from  vertical)  in  earth  of  mean  density. 

Or,  natural  slope  of  earth  or  like  material  lessens  the  thrust,  as  the  cosine 
of  the  slope. 

Angle  which  line  of  rupture  makes  with  vertical  is  .5  of  angle  which  line 
of  natural  slope,  or  angle  of  repose,  makes  with  same  vertical  line.  When 
earth  is  level  at  top,  its  pressure  may  be  ascertained  by  considering  it  as  a 
fluid,  weight  of  a  cube  foot  of  which  is  equal  to  weight  of  a  cube  foot  of  the 
earth,  multiplied  by  square  of  tangent  of  .5  angle  included  between  natural 
slope  and  vertical. 


STABILITY. — REVETMENT   WALLS.  695 

Therefore  squares  of  the  tangents  of  .5  of  45°  and  .5  of  60°  =  .  1716  and 
•3333?  which  are  the  multipliers  to  be  used  in  ordinary  cases  to  reduce  a 
cube  foot  of  material  to  a  cube  foot  of  equivalent  fluid,  which  will  have 
same  effect  as  earth  by  its  pressure  upon  a  wall. 

3?ressu.re    of  Earth   against    Revetment  "Walls. 
Fig.  3.  Let  A  B  C  D,  Fig.  3,  be  vertical  section  of  a  revetment 

•HA  0        wall,  behind  which  is  a  bank  of  earth,  A  D/e  ;  let  D  o 

-? — 6  represent  angle  of  repose,  line  of  rupture,  or  natural  slope 


c          /'          which  earth  would  assume  but  for  resistance  of  wall. 
j\  /  In  sandy  or  loose  earth  angle  o  D  A  is  generally  30° ; 

in  firmer  earth  it  is  36°;  and  in  some  instances  it  is  45°. 

If  upper  surface  of  earth  and  wall  which  supports  it  are 
both  in  one  horizontal  plane,  then  the  resultant,  I  n,  of 
, —  f      pressure  of  the  bank,  behind  a  vertical  wall,  is  at  a  dis- 
u      -i)  tance,  D  n,  of  one  third  A  D. 

Line  of  Rupture  behind  a  wall  supporting  a  bank  of  vegetable  earth  is  at 
a  distance  A  o  from  interior  face,  A  D  =  .618  height  of  it. 

When  bank  is  of  sand,  A  o  =  .677  h ;  when  of  earth  and  small  gravel  == 
.646  h ;  and  when  of  earth  and  large  gravel  =  .618  h. 

The  prism,  vertical  section  of  which  is  A  D  o,  has  a  tendency  to  descend 
along  inclined  plane,  o  D,  by  its  gravity;  but  it  is  retained  in  its  place  by 
resistance  of  wall,  and  by  its  cohesion  to  and  friction  upon  face  o  D.  Each 
of  these  forces  may  be  resolved  into  one  which  will  be  perpendicular  to  o  D, 
and  into  another  which  will  be  parallel  to  o  D.  The  lines  c  i,  i  I  represent 
components  of  the  force  of  gravity,  which  is  represented  by  vertical  line  c  J, 
drawn  from  centre  of  gravity,  c,  "of  prism.  Lines  nr,lr  represent  compo- 
nents of  forces  of  cohesion  and  friction,  which  is  represented  by  horizontal 
line  n  /.  Force  that  gives  the  prism  a  tendency  to  descend  is  i  /,  and  that 
opposed  to  this  is  r  /,  together  with  effects  of  cohesion  and  friction. 

Thus,  i  I  =  r  I  -f  cohesion  -f-  friction.  Consequently,  exact  solution  of  prob- 
lems of  this  nature  must  be  in  a  great  measure  experimental. 

It  has  been  found,  however,  and  confirmed  experimentally,  that  angle 
formed  with  vertical,  by  prism  of  earth  that  exerts  greatest  horizontal  stress 
against  a  wall,  is  half  the  angle  which  angle  of  repose  or  natural  slope  of 
earth  makes  with  vertical. 

Memoranda. 

Natural  slope  of  dry  sand  =  39°,  moist  soil  =  43°,  very  fine  sand  =  21°,  wet  clay 
=  14°,  and  gravel  =  35°. 

In  setting  or  founding  of  retaining  walls,  if  earth  upon  which  wall  is  to  rest  is 
clayey  or  wet,  coefficient  of  friction  between  wall  and  earth  falls  to  .3;  hence  it  is 
necessary,  in  order  to  meet  this,  that  the  wall  should  be  set  to  such  a  depth  in  the 
earth  that  the  passive  resistance  of  it  on  outer  face  of  wall,  combined  with  its  fric- 
tion on  its  bottom,  may  withstand  the  pressure  or  thrust  on  its  inner  face. 

Moment  of  a  Retaining  Wall  is  its  weight  multiplied  by  distance  of  its  centre  of 
gravity  to  vertical  plane  passing  through  outer  edge  of  its  base. 

Moment  of  Pressure  of  Earth  against  a  retaining  wall  is  pressure  multiplied  by 
distance  of  its  centre  of  pressure  to  horizontal  plane  passing  through  base  of  wall. 

Equilibrium  of  Retaining  Wall  is  when  respective  moments  of  wall  and  earth  are 
equal. 

Stability  of  a  Retaining  Wall  should  be  in  excess  of  its  equilibrium,  according  to 
character  of  thrust  upon  it,  and  the  line  of  its  resistance  should  be  within  wall  and 
at  a  distance  from  vertical  passing  through  centre  of  gravity  of  wall,  at  most  .44  of 
distance  of  exterior  axis  of  wall  from  this  line. 

Coefficient  of  Stability  varies  with  character  of  earth,  location,  exposure  to  vibra- 
tions, floods,  etc. ;  hence  thickness  of  base  of  wall  will  vary  from  1.4  to  2  b. 

Backs  of  retaining  walls  should  be  laid  rough,  in  order  to  arrest  lateral  subsidence 
of  the  filling. 


696 


STABILITY. KEVETMENT   WALLS. 


When  filling  is  composed  of  bowlders  and  gravel,  the  thickness  of  wall  must  be 
increased,  and  contrariwise;  when  of  earth  in  layers  and  well  rammed,  it  may  b& 


Courses  of  dry  wall  should  be  inclined  inwards,  in  order  to  arrest  the  flow  of 
water  of  subsidence  in  filling  from  running  out  upon  face  of  wall. 

Less  the  natural  slope,  greater  the  pressure  on  wall. 

Sea  walls  should  have  an  increased  proportion  of  breadth,  as  the  earth  backing 
is  not  only  subjected  to  being  flooded,  but  the  walls  have  at  times  to  sustain  the 
weight  of  heavy  merchandise. 

Buttress.— An  increased  and  projecting  width  of  wall  on  its  front,  at  intervals  in 
its  length. 

Counterfort.— An  increased  and  projecting  width  of  wall  at  its  back  and  at  in- 
tervals. 

Coefficient  of  Friction  of  masonry  on  masonry  .67,  of  masonry  on  dry  clay  .51, 
and  on  wet  clay  .  3. 

Face  of  wall  should  not  be  battered  to  exceed  i  to  1.25  ins.  in  a  foot  of  height,  in 
consequence  of  the  facility  afforded  by  a  greater  inclination  to  the  permeation  of 
rain  between  the  joints  of  the  courses. 

Footing  of  a  wall,  projecting  beyond  its  faces,  is  not  included  in  its  width. 

Pressure.—  Limit  of  pressure  on  masonry  12  500  to  16  500  Ibs.  per  sq.  foot  wall. 

t  i 

Thickness  of  Walls,  in  Mortar,  Faces  vertical.    For  Railways  or  Like  Stress. 

Cut  stone  or  Ranged  rubble ,35    j    Brick  or  Dressed  rubble 4 

When  laid  dry,  add  one  fourth. 

Friction  in  vegetable  earths  is  .5;  pressure  in  sand  .4. 
When  vegetable  earths  are  well  laid  in  courses,  the  thrust  is  reduced  .5. 
When  bank  is  liable  to  be  saturated  with  water,  thickness  of  wall  should  be 
doubled. 

Centre  of  Pressure  of  earthwork,  etc.,  coincides  with  centre  of  pressure  of  water, 
and  hence,  when  surface  is  a  rectangle,  it  is  at  .33  of  height  from  base. 

The  theory  of  required  thickness  of  a  retaining  wall,  as  before  stated,  is,  that  the 
lateral  thrust  of  a  bank  of  earth  with  a  horizontal  surface  is  that  due  to  the  prism 
or  wedge-shaped  volume,  included  between  the  vertical  inner  face  of  the  wall  and 
a  line  bisecting  the  angle  between  the  wall  and  the  angle  of  repose  of  the  material. 

To   Compute   Elements   of  Revetment  Walls.— Fig.  4:. 

n*  *    A  Let  A  Do  represent  angle  of  repose  of  material,  against 

-^ -j£ ?°       a  wall,  ABC  D.      ADn  =  .sADo.      Tan.  A  D  w  =  -492. 

/        /  h         h2 

!      /  Tan.  A  D  n.  h  —  ,  or  —  tan.  ADn  =  V; 

f        /  2*2 

/       / 


, 
=  m:    ——  tan.aADn  =  E;     -  tan.3ADn  =  S;    h* 


h  tan.  A  D  n/^  —  x,  and  h  tan.  A  D  n  */-^  =  *'•      *  representing  height  of 

watt  in  feet,  V  volume  of  section  of  prism  of  material  A  D  n  one  foot  in  length  in  cube 
fttt,  W  and  w  weights  of  a  cube  foot  of  wall  and  of  material,  P,  p,  and  p'  lateral 
and  moments  of  pressure  of  prisms  of  earth  A  D  o  and  A  D  n  upon  wall,  M  and  m 
moments  of  pressure  and  weight  on  and  of  wall,  E  and  S  equilibrium  and  stability  of 
wall,  all  in  Ibs.,  and  x  and  x',  C  D/or  weights  of  wall  for  equilibrium  and  stability. 
ILLUSTRATION.—  A  revetment  wall,  Fig.  4,  of  125  Ibs.  per  cube  foot  and  40  feet  in 
height,  sustains  a  bank  of  earth  having  a  natural  slope  of  52°  24',  and  a  weight  of 
89.  25  Ibs.  per  cube  foot  ;  what  is  pressure  or  thrust  against  it,  etc.  ? 


STABILITY. — REVETMENT   WALLS. 


697 


Tan.2 


Then  .492  X  40  X  —  =  393.6  cube  feet. 


89.25X40* 


X  .4922  =  17278.8  Ibs. 


125  X  40  X  — —  =  230400  Ibs. 

''\XJ*?  =  **s*feet- 


,  and   4oX.49V-T^ 

V  j    XN    **3 

For  Rubble  Walls  in  Mortar  or  Dry  Hubble,  add  respectively  to  base  as  above 
obtained,  .14  and  .42  part. 

NOTE  i. — When  coefficient  of  friction  is  known,  use  it  for  tan.2  A  D  n. 
S  X  C  D  fig.  5  =  moment  of  stability.     (Molesworth.) 

2. —When  either  relative  weights  of  equal  volumes  of  wall  and  bank  of  earth  or 
their  specific  gravities  are  given,  S  and  s  may  be  taken  for  W  and  w. 

These  equations  involve  simply  the  operation  of  a  lever,  the  fulcrum  being  at 
the  outer  edge  of  wall  C.  The  moment  of  pressure  of  bank  is  product  of  lateral 
pressure  and  perpendicular  distance  from  fulcrum  to  line  of  direction  of  pressure. 

The  moment  of  weight  of  wall  is  product  of  weight  of  wall  and  perpendicular 
distance  from  fulcrum  to  vertical  line  drawn  through  centre  of  gravity  of  wall. 

When  Weights  of  Embankment  and  Wall  are  equal  per  Cube  Foot. 
C  for  clay  =  .336,  and  for  sand  .267. 

Wlien  Weights  are  as  4  to  5.    C  for  clay  = .  3,  and  for  sand  .239. 
When  Watt  has  an  Exterior  Slope  or  Batter.— Fig.  5. 

. ( c  D  -f-  E  C ^j  =  M.       M  representing 


r 


moment  of  weight  of  wall  in  Ibs. 

ILLUSTRATION.— Assume  weight  of  wall  120  Ibs.  per 
cube  foot,  and  C  D  and  E  C  respectively  10  and  2. 5  feet, 
and  all  other  elements  as  in  preceding  case. 

Hence,  ?20X4°- 


JX  (10+2.5—  — M  =  3700 


/ 1 —  tan.  2ADn  —  nh  =  x.    x  representing  A  B  or  C  D.    n  ratio  of 

3       3  *» 

difference  of  widths  of  base  and  top  to  height.     In  absence  of  tan.2  A  D  n  put  C,  co- 
efficient of  material. 

C  =  .0424  for  vegetable  or  clayey  earth,  mixed  with  large  gravel;  .0464  if  mixed 
with  small  gravel;  .1528  for  sand,  and  .166  for  semi-fluid  earths. 

ILLUSTRATION.— Assume  elements  of  preceding  case,    n  =  one  fortieth,  and  tan. 


4o\/; 


V3^+T^7x-49**-'  =  '2-6/^ 


Hence,  thickness  of  wall  at  base  =  12.6-)-  i  (one  fortieth  of  height)  =  13.6  feet. 


NOTE.— If  n  =  one  twentieth,  40 

V  3  X  202        3X  125 

Hence,  wall  at  base  =  11.63-1-2  (one  twentieth  of  height)  =  13.63  feet.     IfC  was 
used,  ii.  32  feet. 

3N 


STABILITY. — REVETMENT   WALLS. 


When  Wall  has  an  Interior  Slope  or  Batter,  B  E.— 
Fig.  6. 

w  h^  o  E  r 

J^_Xtan.2  l±JL  =  Mo/ 


/  /       earth  for  equilibrium;  — 

/,''  2        ,  j      , 

M  of  wall ;  and  —  —  X  tan.2  o  E  n  =  M  of  earth  for  sta- 
bility. 
Coefficients  for  Batter  of  following  Proportions. 

Base  =  Height  X  Tab.  number. 

Weight  of  Earth  to  Wall.  Weight  of  Earth  to. Wall. 

BATTKR  OF  As  4  to  5.  As  i  to  i.  BATTER  OF  As  4  to  5. 

Clay.      Sand.      Clay.      Sand. 


WALL. 


Clay. 


As  i  to  i. 
Sand.      Clay.      Sand. 


.083 

.122 

.149 


.029 
.065 
.092 


•"5 

•155 
.183 


•054 
.092 
.118 


i  in  8  ...... 

I    "    12  ..... 

Vertical  ____ 


.184 
.221 

•3 


.125 
.16 
•239 


.218 
.256 
.336 


•'53 
.189 
.267 


To  Compute  Pressure  Perpendicular   to  Baclt  of  "Wall. 
—Fig.  7. 

\ n o      P  #  =  —  or  — ,  and/*  at  right  angle  to  back  of  wall, 

/     whether  vertical  or  inclined. 


LxAn 
—2- 

n 


,or  L  x  tan.  A  D  w,  or 


2x  tan.2ADn 


,  or 


=/  *.     L  representing  weight  of  triangle  of  em- 

'  bankment,  as  A  D  n. 
This  is  pressure  independent  of  friction  between  surfaces  of  wall  and  earth. 

To  Ascertain,  and  Compute  Amount  and  Effect  of  Fric- 

tion   of  \Vall    and    Earth.—  Fig.  8. 

Fig.  8.  Draw/  *  by  scale  to  computed  pressure  at  right  angle 

i.  ----  i]L  -----  ,0    to  back  of  wall,  draw  angle/*  r  =  mDo  of  natural  slope 
-   -'  -      -'        of  earth  with  horizon,  draw/r  at  right  angle  to/  #,  make 
r  c  =/  *,  then  c  r  will  represent  by  scale  effect  of  friction 
against  back  of  wall. 

Assume  friction  to  act  at  point  #,  then  r  *  will  give  by 
scale  resultant  of  the  two  forces  of  pressure  and  friction, 
equal  to  pressure  in  force  and  direction,  which  bears 
=m  against  wall. 

This  resultant  is  also  equal  to/  #  x  sec.  m  D  o. 


L  x  A  n  X  sec.  m  D  o 


-  =  r  #,  or 


-  X  sec.  m  D  o,  or  L  x  tan.  A  D  n 


X  sec.  w  D  o. 

To   Ascertain  Point   of  Moment  of  Pressure  of  a 
Fif.  9._ 

(P 


By  its  resisting  lever  la,added  to  its  weight 
Weight  of  wall  as  computed  assumed  as  concentrated  at  its 
centre  of  gravity  • 

Draw  a  vertical  line  .  o  through  its  centre  of  gravity,  and  con- 
tinue line  of  pressure  P  *  to  I,  take  any  distance  r  o  by  scale  rep- 
resenting weight  of  wall,  and  r  w,  by  same  scale,  for  amount  of 
pressure  or  thrust  against  wall,  complete  parallelogram  r  o  w  M, 
then  diagonal  ru  will  give  resultant  of  pressure  in  amount  and 
direction  to  overturn  wall. 

For  stability  this  diagonal  should  fall  inside  of  base  at  a  point 
not  less  than  one  third  of  its  breadth. 


STABILITY. — EEVETMENT   WALLS. 


699 


Surcharged.  Revetments. 

Fig.  lo.  / r o      When  the  earth  stands  above  a  wall,  as  A  B  e, 

~~~7    Fig.  10,  with  its  natural  slope,  A/,  A  B  C  is  termed 
a  Surcharged  Revetment. 

If  C  r  is  line  of  rupture,  A/r  C  is  the  part  of  earth 
that  presses  upon  wall,  which  part  must  be  taken  into 
the  computation,  with  exception  of  portion  A  Be, 
which  rests  upon  wall;  that  is,  the  computation  must 
be  for  part  C  efr,  which  must  be  reduced  by  multiply- 
ing weight  of  a  cube  foot  of  it  by  square  of  tangent  of 
angle  e  C  r  =  angle  of  line  of  rupture,  or  half  angle 
eC  o,  which  natural  slope  makes  with  vertical,  and 
then  proceed  as  in  previous  cases  for  revetments. 

— — — =  breadth  or  C  D.     W  and  w  representing  weights  of  watt  and 

3  ft  W 

embankment  in  Ibs.  per  cube  foot,  and  h'  height  of  embankment,  as  C  e. 

ILLUSTRATION.— Height  of  a  surcharged  revetment,  BC,  Fig.  10,  is  12  feet,  weight 
130  Ibs.  per  cube  foot;  what  is  its  width  or  base  to  resist  pressure  of  earth  of  a  weight 
of  ioo  Ibs.  per  cube  foot,  and  a  height,  C  e,  of  15  feet,  angle  of  repose  45°  ? 


Tan.2  (45°  -^  =  . 


Then  15 


=  I5  ^.055  =  3 


To   Ascertain   IPoint  of  MLoment  of  DPressnre   of  a  Sur- 
charged.  'Wall.— ITig.  11. 

F*g.  ii.  „/      Draw  a  line,  P  *,  parallel  to  slope,  C  r,  through  centre 

•"'       of  gravity  of  sustained  backing,  B  C  r. 

When,  as  in  this  case,  this  section  is  that  of  a  triangle, 
point  *  will  be  at  .33  height  of  wall. 

~~°         When  natural  slope  is  1.5  in  length  to  i  in  height,  as 
with  gravel  or  sand,  w  x  .  64  =  pressure  P  #. 

In  a  surcharged  revetment,  as/B  o,  at  its  natural  slope, 
the  maximum  pressure  is  attained  when  the  backing 
reaches  to  r.  When  slope  of  maximum  pressure,  Cnr, 
intersects  face  of  natural  slope,  B/,  so  that  if  backing  is 
raised  to  /,  or  above  it,  there  is  theoretically  no  addi- 
tional stress  exerted  at  back  of  or  against  wall,  but  prac- 
tically there  is,  from  effect  of  impact  of  vibration  of  a 
passing  train,  proximity  to  percussive  action,  alike  to  that  of  a  trip  hammer,  etc. 

When  backing  rests  on  top  of  wall,  as  A  B  e.  Fig.  10,  small  triangle  of  it  is  omitted 
in  computations.  Direction  of  pressure  against  wall  is  same  as  when  wall  is  not 
surcharged. 

When  Wall  is  set  below  Surface  of  Earth.— Tig.  12. 
h 


Fig.  12. 


i.  4  tan.  45°  —  — 


W 


-=d 


B 


a  representing  angle  of  repose  of  earth,  w  and  W  weightt 
of  earth  and  wall  per  cube  foot,  f  friction  of  wall  on  base 
A  B,  and  V  weight  of  wall. 

ILLUSTRATION.  —  If  a  wall  of  masonry,  Fig.  12,  8  feet  in  thickness 
and  13  in  height,  is  to  sustain  earth  level  with  its  upper  surface, 
earth  weighing  ioo  Ibs.  per  cube  foot,  weight  of  wall  150  Ibs.  per 
cube  foot  =15600  Ibs.,  and  angle  of  repose  of  earth  30°;  what 
should  be  the  depth  of  wall  below  surface  of  earth  ? 


Tan.  45  —  30  -=-  2  = .  5774,  and  /= .  3. 

ST^.x.ax.sto,.,^ 


/936o^ 
V  i5 


Then  1.4  X- 5774X/— 
=  4.o2jfeet. 

NOTE.— Coefficient  of  stability  is  assumed  by  French  engineers  for  walls  of  forti- 
fications 1.4  h,  and  if  ground  is  clayey  or  wet/=.3- 


7OO      STABILITY. — EMBANKMENT  WALLS  AND  DAMS. 


Fig.  13. 


In  Computing  Stability  of  a  Surcharged  Wall,  Fig.  13,  nub- 
stitute  dfor  ft,  as  in  following  illustration.     (Molesworth.) 
d,  representing  depth  at  distance  I,  =  h. 

In  slopes  of  i  to  i,  d  =  1.71  h;  of  1.5  to  j,=  1.55;  of  2  to  i,= 
1.45;  of  3  to  i,=  1.31,  and  4  to  1,=:  1.24. 

To  Determine  Form  of*  a  3?ier  to  Snstain 
equal  Pressure  per  Unit  of  Surface  at  all 
its  Horizontal  Sections,  or  any  Height. 
=  a,  or  AN  =  a.  A  and  a  representing  areas  of  sections  at  summit  of  pier 
and  at  any  depth,  d,  measured  from  summit,  n  a  number  the  hyp.  log.  of  which  =  i  -4- 
height,  H,  of  a  column  of  the  material  of  which  pier  is  constructed,  due  to  required 

pressure,  and  N  the  number,  com.  log.  of  which  =  43^3     • 

ILLUSTRATION.— Height  of  a  pier  is  20  feet,  and  area  of  section  of  its  summit  = 
i  foot;  what  should  be  its  areas  at  10  feet  and  base? 

i  -7-  20  =  .05,  and  number  =1.0513;  i  X  i. 0513 10  =  i. 649  feet ;  and  i  x  i.o5i320  = 
a.  jig  feet. 

Counterforts  are  increased  thicknesses  of  a  wall  at  its  back,  at  intervals  of 
its  length. 

Em.T3an.ls:rn.en.t   ^Walls   and.   Dams. 

Thrust  of  water  upon  inner  face  of  an  Embankment  wall  or  Dam  is 
horizontal. 

When  Both  Faces  are  Vertical,  Fig.  14. 

Assume  perpendicular  embankment  or  wall,  A  B  C  D,  Fig.  14,  to  sustain 
pressure  of  water,  B  C  ef. 


Fig.  14. 


Let  lei  be  a  vertical  line  passing  through  o,  centre 
of  gravity  of  wall,  c  centre  of  pressure  of  water,  dis- 
tance C  c  being  =  .33  B  C.  Draw  c  I  perpendicular 
to  B  C ;  then,  since  section  A  C  of  wall  is  rectangular, 
centre  of  gravity,  o,  is  in  its  geometrical  centre,  and 
therefore  D  i  =  .5  DC.  Now  I  D  i  is  to  be  consid- 
ered as  a  bent  lever,  fulcrum  of  which  is  D,  weight  of 
wall  acting  in  direction  of  centre  of  gravity,  o,  on  arm 
D  *,  and  pressure  of  water  on  arm  D  /,  or  a  force  equal 
to  that  pressure  thrusting  in  direction  c  I. 

Then  P  x  D I  =  P  X =  W  x ,  or  P  =  - —    '      .    P  representing  pressure 

3  2  2  r>  O 

of  water. 

NOTE.  —When  this  equation  holds,  a  wall  or  embankment  will  just  be  on  the 
point  of  overturning;  but  in  order  that  they  may  have  complete  stability,  this 
equation  should  give  a  much  larger  value  to  P  than  its  actual  amount. 

The  following  formulas  are  for  walls  or  embankments  one  foot  in  length; 
for  if  they  have  stability  for  that  length  they  will  be  stable  for  any  other 
length. 

h2 

P  —  —  iv,  also  W  =  h  b  W,  each  value  being  for  i  foot  in  length,  which,  being  sub- 
stituted in  the  equations,  there  will  result 

h3         ibxhb W'  AW  /  w 

—  w  — ,  or  h2  w  =  i  o2  W;  b  A  /- —  =  h,  and  A     /— ^r  =  6.    h  rep- 

2  2  A  V™  V  3  w 

resenting  depth  of  water  and  wall  or  embankment,  which  are  here  assumed  to  be 
equal,  b  breadth  of  wall  or  embankment,  and  W  and  w  weights  of  wall  and  wattr 
per  cube  foot  in  Ibs. 

Which  gives  breadth  of  a  wall  or  embankment  that  will  just  sustain 
pressure  of  the  water. 


STABILITY. — EMBANKMENT  WALLS  AND  DAMS.       70 1 

To    Compute    Eq.uililt>riu.m.     h     /—==&. 

ILLUSTRATION  i.—  Height  of  a  wall,  B  C,  equal  to  depth  of  water,  is  12  feet,  and  re- 
spective weights  of  water  and  wall  are  62.5  Ibs.  and  120  Ibs.  per  cube  foot;  required 
breadth  of  wall,  so  that  it  may  have  complete  stability  to  sustain  the  pressure  of 
water. 

I2    / —    —  =  12  X  .4166  =  sfeet,  breadth  that  will  just  sustain  pressure  of  the 

water. 

Therefore  an  addition  should  be  made  to  this  to  give  the  wall  complete  stability, 
say  2  feet;  hence  5  -|-  2  :=  7,  required  width  of  wall. 

2.— Width  of  a  wall  is  3  feet,  and  weight  of  a  cube  foot  of  it  is  150  Ibs. ;  required 
height  of  wall  to  resist  pressure  of  fresh  water  to  the  top. 
H  X  i  ^o 


To   Compute    Stability.    ^A= 

V  3  w 
ILLUSTRATION.— Take  elements  of  preceding  case. 


Or,  Divide  i,  2,  or  3,  etc.,  according  as  the  nature  of  the  ground,  the  mate- 
rial, and  the  character  of  the  thrust  of  the  water  requires,  by  .05  weight  of 
material  of  wall,  per  cube  foot,  extract  the  square  root  of  quotient,  and  mul- 
tiply result  by  extreme  height  of  water. 

EXAMPLE.  —What  should  be  the  thickness  of  a  vertical  faced  wall  of  masonry, 
having  a  weight  of  125  Ibs.  per  cube  foot,  to  sustain  a  head  of  water  of  40  feet,  and 

to  have  stability  ?  

V(2  -r-  .05  X  125)  40  =  -V/-32  X  40  =  22.63  feet. 

Or,  *i/^y  =  4°  V-3472  =  23.56 feet. 

When  Dam  has  an  Exterior  Slope  or  Batter,  as  A  D.— Fig.  15. 
Fig.  15*     A        B^j        Assume  prismoidal  wall,  A  B  C  D,  to  sustain  press- 
ure of  water,  B  C  ef. 

Draw  A  E  perpendicular  to  D  C ;  h  =  B  C,  the  top 
breadth  A  B  =  E  C  =  &,  and  bottom  breadth,  D  E, 
of  sloping  part,  A  E  D  =.  S. 

Then  weights  of  portions  A  C  and  A  E  D  respec- 
tively for  one  foot  in  length  are  hbW  and  .5  W  S  h, 
these  weights  acting  at  points  n  and  i  respectively. 

To    Compute    IVIoinent. 

h  b  W  X  (s  -^ — )  =  moment  for  A  C,  and  —  —  x  —  =  moment  for  A  E  D. 

•yy  fa  / 2      S^ 

Hence, (S-f-6 )—  moment  of  dam.    S  representing  batter  or  base  E  D. 

ILLUSTRATION.— Height  of  a  dam,  B  C,  Fig.  15,  is  9  feet,  base  C  E  3,  and  E  D  4  feet; 
what  is  its  moment? 

A  C  =  9  X  3  X  120  X  U  +  |j  =  3240  X  5-5  =  17820  Ibs. 

2      '   3 

Hence,  17  820  +  5760  =  23  580  Ibs.  moment.  Or,  I2°  X  9  /4  -f  3—  i-\  =  54o  x  43$ 
=  23  580  Ibs.  moment. 


JQ2      STABILITY.  —  EMBANKMENT  WALLS  AND  DAMS. 


To    Cornpxite    Elements    of  "Walls    or    Dams    with.    an 
Exterior    Batter.—  Fig.  15. 

To    Compute    Width    of  Top. 


When  Width  of  Batter  is  Given. 


+      _  s  =  6. 


ILLUSTRATION.  —  Assume  height  of  wall  9  and  batter  3  feet,  and  W  and  w  120  and 
62.5  Ibs.  per  cube  foot. 


To    Compute    Widtli   of  Base. 


When  Width  of  Batter  is  Given. 


+  —  =  B. 


To   Compute   Width   of  Batter. 
When  Width  of  Top  is  Given. 


+        -      =  S. 


/gx^qT^_3)^  =  v 

V        120  4  2 

TFAcw  JFicM  </  Bottom  is  Given.     ^/3  B2  —  2-^?  =  S. 

To  IDeteriniiie  Stability  of  a  Retaining  Wall  or  Dam  by 

IPro  traction.  —ITig.  16. 

Assume  A  B  C  D,  section  of  a  wall.  On  horizontal 
line  of  centre  of  thrust  or  pressure,  with  a  suitable 
scale,  lay  off,  from  vertical  line  of  centre  of  gravity  • 
of  wall,  line  o  r  =  thrust  against  wall,  and  on  vertical 
line  at  centre  of  gravity  of  wall,  at  its  intersection,  o, 
with  centre  of  thrust,  let  fall  o  s  =  weight  of  wall. 

Complete  parallelogram,  and  if  diagonal  o  u  or  its 
prolongation  falls  within  C,  the  wall  is  stable,  and 
W  '  X  distance  from  line  05  =  moment  of  wall. 

W  representing  whole  weight  of  wall  in  Ibs. 

To    Determine    Centre    of  Oravity    of  a  Wall   or   Dam.— 
Fig.  16. 


To    Compute    Base   of  Darn. 

When  Height,  Rate  of  Batter,  and  Weight  of  Materials  are  given.  RULE. 
—  Multiply  square  of  width  of  batter  by  .0166  weight  of  material  per  cube 
foot,  add  i,  2,  or  3  times  square  of  depth  of  water,  according  as  resistance 
due  to  equilibrium  is  required,  divide  result  by  .05  weight  of  material  per 
cube  foot,  and  extract  square  root  of  quotient. 


Or,    / 

V 


-  —  -    - 


-°5 


=  6.    x  =  number  of  times  of  resistance  required. 


EXAMPLE.  —  Assume  a  dam  40  feet  in  height,  constructed  of  masonry  weighing 
120  Ibs.  per  cube  foot,  to  batter  3  ins.  per  foot,  and  to  have  twice  the  resistance  due 
to  its  equilibrium  ;  what  should  be  its  breadth  at  its  base,  DC? 

,  and      AO'  X  .  +  ^  X  .0.66  X  ..o  =     73^9  = 

V  .05  X  120  V      6  J      J 


STABILITY. EMBANKMENT  WALLS  AND  DAMS.       703 


When  Section  of  Dam  is  a  Triangle,  Fig.  17.  —  As- 
sume dam,  A  B  C,  to  sustain  a  head  of  water,  ef. 

RULE.— Proceed  as  by  Rule  for  Fig.  14 ;  multiply  by 
if      .033  instead  of  .05. 

EXAMPLE. — As  before. 


V(2  -7-  .033  X  125)  40  =  V-485  X  40  =  27-  84  fat- 
Or,  Formula  for  S  (C  B),  Fig.  15.       /  —  =^  =  28.28  feet. 

To  Determine  Section  of  a  Vertical  Wall  which  shall  have  Equal  Resist- 
ance of  one  having  Section  of  a  Triangle.    (See  J.  C.  Trauticine,Phila.,  1872.) 

To   Compute   Thickness    of   Base    of  a   Wall    or    Dam.— 
Fig.  18. 

Fig.  18.  RULE.  —  Divide  i,  2,  or  3  times  square  of  depth  of  water 

by  .05  weight  of  material,  add  quotient  to  .5  batter  on  one 
face,  and  square  root  of  this  sum,  added  to  half  batter  on 
other  side,  will  give  thickness. 


Or, 


~)  H  —  =  Base,      b  and  b'  representing 


.05  W 

exterior  and  interior  batters,  and  st,  as  before,  number  of  timet 
=    of  resistance  or  square  of  depth. 

~jfo        r     C  "^     EXAMULB.— Assume  a  dam  40  feet  in  height,  to  batter  5  feet 
on  each  side,  constructed  of  masonry  weighing  120  Ibs.  per  cube 
foot,  and  to  have  twice  the  resistance  due  to  its  equilibrium;  what  should  be 
breadth  of  base,  DC? 


High   Masonry   Dams. 

Rubble  Masonry,  well  laid  in  strong  cement,  will  bear  with  safety  a  load 
equivalent  to  weight  of  a  column  of  it  160  feet  in  height.    Assuming  such 
masonry  as  twice  weight  of  water,  it  is  equivalent 
to  a  pressure  of  20  ooo  Ibs.  per  sq.  foot. 

Log.  B  -f  .434  294  X  g-  =  b.    B  representing  width  of 

wall  at  top,  and  d  depth  at  any  desired  point  below  top, 
both  in  feet. 

Ordinarily,  B  may  be  taken  at  18  feet,  and  in  cases 
of  extreme  and  exposed  heights  of  dam  at  20  and  more, 
and  when  6  is  determined,  .9  of  it  is  to  be  <w  outer  face 
of  wall,  as  A  B,  and  .  i  on  inner  face. 

ILLUSTRATION.— Determine  section  of  a  dam,  Fig.  19, 
)T6eS  80  feet  in  height,  at  depths  of  10,  20,  40,  60,  and  80  feet. 
Log.  B  =  1.2553. 


^  =  log.  1.2553  + .0543  =  20.4,  which  X-9  =  18.36. 

DO 

"  I.2553  +  -4343  X^  =  log.  1.2553  +  . 1086  =  23.11,  which  X. 9  =  20.a 

"  i. 2553  +  -4343  X  |^  =  log.  1.2553  +  .2172  =  29.68,  which  X  .9  =  26.81. 

*'  i.«553  + -4343  X  ^=  log.  1.2553  +  . 3257  =  38.  ii,  which  X. 9  =  34-3- 

"  i-2553-f -4343X  ~  =  log.  i. 2553  +  . 4343  =  50.07,  which  X  .9  =  45-06. 


704 


STEAM. 


STEAM. 


STEAM  is  generated  by  heating  of  water  until  it  attains  temperature 
of  ebullition  or  vaporization,  and  elevation  of  its  temperature  is  sensible 
to  indications  of  a  thermometer  up  to  point  of  ebullition ;  it  is  then 
converted  into  steam  by  additional  temperature,  which  cannot  be  in- 
dicated by  a  thermometer,  and  is  termed  latent.  (See  Heat,  page  508.) 

Pressure  and  density  of  steam,  which  is  generated  in  free  contact  with  water, 
rises  with  the  temperature,  and  reciprocally  its  temperature  rises  with  the  press- 
ure and  density,  and  higher  the  temperature  more  rapid  the  pressure.  There  is 
but  one  and  a  corresponding  pressure  and  density  for  each  temperature,  and  steam 
generated  in  free  contact  with  water  is  both  at  its  maximum  density  and  pressure 
for  its  temperature,  and  in  this  condition  it  is  termed  saturated,  from  its  being  in- 
capable of  vaporizing  more  water  unless  its  temperature  is  raised. 

Saturated  Steam  is  the  normal  condition  of  steam  generated  in  free  contact  with 
water,  and  same  density  and  same  pressure  always  exist  in  conjunction  with  same 
temperature.  It  therefore  is  both  at  its  condensing  and  generating  points;  that 
is,  it  is  condensed  if  its  temperature  is  reduced,  and  more  water  is  evaporated  if 
its  temperature  is  raised. 

If,  however,  the  whole  of  the  water  is  evaporated,  or  a  volume  of  saturated  steam 
is  isolated  from  water,  in  a  confined  space,  and  an  additional  quantity  of  heat  is 
supplied  to  the  steam,  its  condition  of  saturation  is  changed,  the  steam  becomes 
superheated,  and  both  temperature  and  pressure  are  increased,  while  its  density  is 
not  increased.  Steam,  when  thus  surcharged,  approaches  to  condition  of  a  gas. 

With  saturated  steam,  pressure  does  not  rise  directly  with  the  temperature. 

Steam,  at  its  boiling-point,  is  equal  to  pressure  of  atmosphere,  which  is  14.723  307 
Ibs.  (page  427),  at  60°  upon  a  sq.  inch. 

In  all  computations  concerning  steam,  it  is  necessary  to  have  some  or  all  of  fol- 
lowing elements,  viz.  : 

Its  Pressure,  which  is  termed  its  tension  or  elastic  force,  and  is  expressed  in  Ibs. 
per  sq.  inch.  Its  Temperature,  which  is  number  of  its  degrees  of  heat  indicated  by 
a  thermometer.  Its  Density,  which  is  weight  of  a  unit  of  its  volume  compared 
with  that  of  water.  Its  Relative  volume,  which  is  space  occupied  by  a  given  weight 
or  volume  of  it,  compared  with  weight  or  volume  of  water  that  produced  it. 

Under  pressure  of  the  atmosphere  alone,  temperature  of  water  cannot  be  raised 
above  its  boiling-point. 

Expansive  force  of  steam  of  all  fluids  is  same  at  their  boiling-point. 

A  cube  inch  of  water,  evaporated  under  ordinary  atmospheric  pressure,  is  convert- 
ed into  1642*  cube  ins.  of  steam,  or,  in  a  unit  of  measure,  very  nearly  i  cube  foot, 
and  it  exerts  a  mechanical  force  equal  to  raising  of  14.723307  X  144  =  2120.156208 
Ibs.  i  foot  high. 

A  pressure  of  i  Ib.  upon  a  sq.  inch  will  support  a  column  of  mercury  at  a  tem- 
perature of  60°,  1-7-4907769  (page  427)^=2.037586  ins.  in  height;  hence  it  will 
raise  a  mercurial  siphon  gauge  one  half  of  this,  or  1.018793  ins. 

Velocity  of  steam,  when  flowing  into  a  vacuum,  is  about  1550  feet  per  second  when 
at  a  pressure  equal  to  the  atmosphere ;  when  at  10  atmospheres  velocity  is  increased 
to  but  1780  feet;  and  when  flowing  into  the  air  under  a  similar  pressure  it  is  about 
650  feet  per  second,  Increasing  to  1600  feet  for  a  pressure  of  20  atmospheres. 

Boiling-points  of  Water,  corresponding  to  different  heights  of  barometer,  see 
Heat,  page  517. 

Volume  of  a  cube  foot  of  water  evaporated  into  steam  at  212°  is  1642  cube  feet; 
hence  i  -r- 1642  =  .000609013,  which  represents  density  or  specific  gravity  of  steam 
at  pressure  of  atmosphere. 

Elasticity  of  vapor  of  alcohol,  at  all  temperatures,  is  about  2. 125  times  that  of  steam. 

Specific  Gravity,  compared  with  air,  is  as  weight  of  a  cube  foot  of  it  compared 
witn  equal  volume  of  air.  Thus,  weight  of  a  cube  foot  of  steam  at  212°  and  at 
pressure  of  atmosphere  is  266.124  grains;  weight  of  a  like  volume  of  air  at  32°  is 
565.096  grains,  and  at  62°  532.679  grains.  Hence  266. 124  -f-  532.679  =  .499  59,  specific 
gravity  of  steam  compared  with  air  at  32°,  and  with  water  it  is  .000609013. 

*  Pole's  Formuia  makes  it  1712. 


STEAM.  7O5 

Total   Heat   of  Saturated.    Steam.     (Regnault.) 

From  Water  at  32°. 

1081 .4  -f-  .305  T  =  total  heat.     T  representing  initial  temperature  of  water. 
LELUSTRATION. — What  is  total  heat  of  steam  at  212°? 

1081.4-)-  .305  X  212  =  1146.06. 

As  Specific  heat  of  water  is  .9  greater  at  212°  than  at  32°,  hence  the  212°  would 
be  212.9,  and  I1[46-33  the  result. 

Total  Heat  of  Gaseous  Steam  from  Water  at  32°  =  1074.6  -f-  .475  T. 

Absorption    of  Heat. 

In  Generation  of  i  Lb.  of  Steam  at  212°  from  Water  at  32°. 
Sensible  heat,  or  heat  to  raise  temperature  of  water          Thermal  Units.         Foot-lbs. 

from  32°  to  212° 181.8  X  772=  139655 

Latent  heat  to  produce  steam 892.9 

"        "    to  resist  atmospheric  pressure  14.7  Ibs. 

per  sq.  inch 71.4    JE^M  •  3  X  772  =  745 '34 

Total  or  constituent  heat 1146.1  884789 

In  Generation  of  j  Lb.  of  Steam  at  175  Ibs.  from  Water  at  32°. 

Thermal  Units.     Foot-lbs. 

Sensible  heat  as  in  preceding  case  from  32°  to  370.8° 342.4         275  333 

Latent  heat  to  produce  steam 768.2         593  050 

"        "    to  resist  external  pressure  =  175  Ibs 83.8          64694 

Total  heat  from  32° "94-4         933°77 

Mechanical  Equivalent  of  Heat  Contained  in  Steam. 
i  Ib.  water  heated  from  32°  to  212°  requires  as  much  heat  as  would  raise 

180  Ibs.  i°.     Hence 181 . 8° 

i  Ib.  water  at  212°,  converted  into  steam  at  212°  (=14.7  Ibs.  pressure), 

absorbs  as  much  heat  as  would  raise  966.6  Ibs.  water  i °.    Hence 964 . 3° 

1146.1° 

Mechanical  Equivalent,  or  maximum  theoretical  duty  of  quantity  of  heat  in  one 
thermal  unit  or  one  degree  in  i  Ib.  of  water,  is  772  foot-lbs.,  which  X  1146.1  units 
of  heat  =  884  789. 2  Ibs.  raised  i  foot  high. 

To    Compute    Pressxire    of  Steam 

Above  Perfect  Vacuum. 

When  Height  of  Column  of  Mercury  it  will  Support  is  given.  RULE.— Di- 
vide height  of  column  of  mercury  in  ins.  by  2.037  5^6? an(^  quotient  will  give 
pressure  per  sq.  inch  in  Ibs. 

EXAMPLE.  —Height  of  a  column  of  mercury  is  203.7586  ins. ;  what  pressure  per 
sq.  inch  will  it  contain  ? 

203. 7586  -r-  2.037  586  =  ioo  Ibs. 

To   Compute   Weight  of  a   Cu.be   Foot   of  Steam. 
RULE.— Multiply  its  density  by  62.425. 
Ex  AMPLE.— Density  of  a  volume  of  steam  is  .000609013;  what  is  its  weight? 

.000609013  X  62.425  =  .038  016825  tt>s- 
NOTB.— See  table,  page  708. 

i  atmosphere  or  14.723307  Ibs.  per  sq.  inch  =  30  ins.  of  mercury. 

To    Compute    Temperature   of  Steam. 
RULB. — Multiply  6th  root  of  its  force  in  ins.  of  mercury  by  177.2,  sub- 
tract ioo  from  product,  and  remainder  will  give  temperature  in  degrees. 

EXAMPLE.— When  elastic  force  of  steam  is  equal  to  a  pressure  of  64  ins.  of  mer- 
cury, what  is  its  temperature? 

NOTE.— To  extract  6th  root  of  a  number,  ascertain  cube  root  of  Its  square  root 
-^64  =  8,  and  -g/8  =  2.    Hence,  2  x  177-  2  —  ioo  =  254. 4°  t. 

Or, 2Q38'^ 371. 85  =  t.    p  rejn-esenting  pressure  in  Ibt.  per  tq.  inch. 


706  STEAM. 


To  Compute  "Volume  of  "Water  contained,  in  a  given  "Vol- 
ume   of  Steam. 

When  its  Density  is  given.  RULE. — Multiply  volume  of  steam  in  cube 
feet  by  its  density,  and  product  will  give  volume  of  water  in  cube  feet. 

EXAMPLE. — Density  of  a  volume  of  16420  cube  feet  of  steam  is  .000609;  what  is 
the  weight  of  it  in  Ibs.  ? 

16  420  X  .000609  =  10  =  volume  of  water,  which  X  62.425  =  624.25  Ibs. 

To  Compute   Pressure  of  Steam    in  Ins.  of  Mercury,  or 

L"bs.  per    Sq..  Inch. 

When  Temperature  is  given.    RULE  i.— Add  100  to  temperature,  divide 
sum  proportionally  by  177.2  for  temperature  of  212°,  and  by  160  for  tem- 
peratures up  to  445° ;  or,  177.6  for  sea-water,  and  185.6  for  sea-water  sat- 
urated with  salt,  and  6th  power  of  quotient  will  give  pressure. 
EXAMPLE. — Temperature  of  steam  is  254°;  what  is  its  pressure? 
loo -{-254-:- 177.2  =  1.998,  and  i.  9986  =  63. 62  ins. 

When  Ins.  of  Mercury  are  given.  2. — Divide  ins.  of  mercury  by  2.037  5^6, 
and  quotient  will  give  pressure. 

When  Pressure  in  Lbs.  is  given.    3. — Multiply  pressure  by  2.037  586. 

To  Compute  Specific  Q-ravity  of  Steam   compared,  -with 

Air. 

RULE.  —  Divide  constant  number  829.05  (1642  x  .5049)  by  volume  of 
steam  at  temperature  of  pressure  at  which  gravity  is  required. 
EXAMPLE.— Pressure  of  steam  is  60  Ibs.,  and  volume  437 ;  what  its  specific  gravity  ? 
829.05-7-437  =  1.898. 

To  Compute  Volume  of  a  Cutoe  Foot  of  "Water  in  Steam. 

When  Elastic  Force  and  Temperature  of  Steam  are  given.  RULE. — To 
430.25  for  temperature  of  212°,  and  332  for  temperatures  up  to  445°,  add 
temperature  in  degrees ;  multiply  sum  by  76.5,  and  divide  product  by  elastic 
force  of  steam  in  ins.  of  mercury. 

NOTE.— When  force  in  ins.  of  mercury  is  not  given,  multiply  pressure  in  Ibs.  per 
sq.  inch  by  2.037  586- 

EXAMPLE. —Temperature  of  a  cube  foot  of  water  evaporated  into  steam  is  386°, 
and  elastic  force  is  427.5  ins. ;  what  is  its  volume? 

Assume  369  for  proportionate  factor.  369  +  386  x  76.5-7-427.5  =  135.1  cube  feet. 
Or,  for  i  Ib.  of  steam,  2. 519  —  .941  log.  p  —  log.  V  in  cube  feet. 

Assumep  =  i4.7  Ibs.  2.519  —  .941  log.  14.7  =  2.519  — 1.098  =  1.421  =  log.  26.34 
cube  feet,  which  X  62.425  =  164  feet. 

Or,  When  Density  is  given.—  Divide  i  by  density,  and  quotient  will  give  volume 
in  cube  feet. 

To   Compute    Density   or   Specific   Q-ravity   of  Steam. 
When  Volume  is  given.    RULE. — Divide  i  by  volume  in  cube  feet. 
EXAMPLE.— Volume  is  210;  what  is  density? 

i -r- 210=:. 004  761.    Or,  for  i  Ib.  of  steam,  .941  log.  p  —  2. 519  =  log.  D. 

When  Pressure  is  given.— Take  temperature  due  to  pressure,  and  proceed 
as  by  rule  to  compute  volume,  which,  when  obtained,  proceeds  as  above. 

To  Compute  Volume  of  Steam  required  to  raise  a  Qiven 
Volume   of  \Vater    to   any    Qiven    Temperature. 

RULE. — Multiply  water  to  be  heated  by  difference  of  temperatures  between 
it  and  that  to  which  it  is  to  be  raised,  for  a  dividend ;  then  to  temperature 
of  steam  add  965.2°,  from  that  sum  take  required  temperature  of  water  for 
a  divisor,  and  quotient  will  give  volume  of  water. 


STEAM. 


707 


EXAMPLE.—  What  volume  of  steam  at  212°  will  raise  100  cube  feet  of  water  at  80° 

tO  212° ? 


IPO  X  212 _      6g  cube  feet  water;  or,  (13.68  X  1642  —  212)  =22  250  of  steam. 

212-4-965.2 — 212 

To  Compute  "Volume  of  "Water,  at  any  Given  Temper- 
atui*e,  that  must  toe  Mixed  -with  Steam  to  liaise  or  Re- 
duce   the    Mixture    to    any    Required    Temperatiire. 
KULE.— From  required  temperature  subtract  temperature  of  water ;  then 
ascertain  how  often  remainder  is  contained  in  required  temperature  sub- 
tracted from  sum  of  sensible  and  latent  heat  of  the  steam,  and  quotient  will 
give  volume  required. 

Sum  of  Sensible  and  Latent  Heats  for  a  range  of  temperatures  will  be  found  under 
Heat,  pages  508  and  509. 

EXAMPLE.— Temperature  of  condensing  water  of  an  engine  is  80°,  and  required 
temperature  100° ;  what  is  proportion  of  condensing  water  to  that  evaporated  at  a 
pressure  of  34  Ibs.  per  sq.  inch  ? 
Sum  of  sensible  and  latent  heats  930.12°-!-  257.6°  =  1187.72°. 

100  —  80  —  20.     Then,  1187.72  —  100-7-20=  54.386  to  i. 

2-i-T  —  t 
When  Temperature  of  Steam  is  given.        t_w    —  v-    *  representing  latent  heat, 

T  and  t  temperatures  of  steam  and  required  temperature,  w  temperature  of  condensing 
water,  and  V  volume  of  condensing  water  in  cube  feet. 

ILLUSTRATION.— Temperature  of  steam  in  a  cylinder  is  257.6°,  and  other  elements 
same  as  in  preceding  example  ;  required  volume  of  injection  water?  Latent  heat 
of  steam  at  230°  =  930. 12°. 

930. 1 2°  +  257. 6  —  ioo 


100  —  80  20 

To    Compute    Temperature    of  "Water    in.    Condenser    or 
Reservoir    of  a    Steam-engine. 


H-TJ-_VXW> 
v-f-i 


=  t.     ILLUSTRATION.— Assume  elements  as  preceding. 


930. 1 2°  -f-  257. 6  -f  54. 39  X  80  _  5539  _  IOQO 

54-39  +  1  55-39 

To    Compute    Latent    Heat    of   Saturated    Steam. 
1112.5  —  '7°8  t  =  l.    ILLUSTRATION. — Assume  temperature  257.6°  as  preceding. 

1112.5  — .708  X  257.6  =  930.12°. 

To   Compute    Total    Heat   of  Saturated    Steam. 
•3°5  ^ H-  1081.4  =  H.     ILLUSTRATION. — Assume  temperature  as  preceding. 
.305  X  257.6  -f- 1081.4  =  1160. 

Elastic  Force  and  Temperature  of  "Vapors  of  -A-lcoliol, 
Either,  Sulphuret  of  Carbon,  Petroleum,  and  Tur- 
pentine. 

Force  in  Ins.  of  Mercury. 


o 

Ins. 

0 

Ins. 

0      |        Ins. 

o 

Ins. 

o 

In*. 

ALCOHOL. 

ALCOHOL. 

ETHER. 

SULPHURET  OF 

PETROLEUM. 

3* 

•4 

140 

13-9 

34 

6.2 

CARBON. 

316 

30 

50 

.86 

160 

22.6 

54 

15-3 

53-5 

•    7-4 

345 

44.1 

60 

1.23 

'73 

3° 

74 

16.2 

72-5 

12-55 

375 

64 

70 
80 

1.76 
2.45 

1  80 

200 

34-73 
53 

96) 

24.7 

no 

212 

3° 
126 

OIL  OF 

9° 

3-4 

212 

67.5 

104) 

30 

279.5 

300 

TURPENTINE. 

IOO 

4-5 

22O 

78.5 

120 

39-47 

347 

606 

315 

3° 

1  20 

8.1 

240 

111.24 

I5O 

67.6 

357 

4778 

130 

10.6 

264 

166.1 

212 

178 

37° 

62.4 

;o8 


STEAM. 


Saturated.    Steam. 

Pressure,  Temperature,  Volume,  and  Density. 


ISSURE 

| 

Ii. 

"o 

•*••£ 

PRESSURE 

£ 
2 

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112.5 

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141.6 

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1170.7 

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89.62 

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124.19  ;  293.8 

1171.1 

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.1447 

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162.3 

130.9 

72.66 

.0138 

62 

126.23 

294.8 

1171.4 

6.81 

.1469 

12.22 

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133-3 

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.0163 

63 

128.26 

295-9 

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6.7 

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176.9 

135-3 

52.94 

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64 

130.3 

296.9 

1172 

6.6 

.1516 

16.29 

182.9 

137-2 

46.69 

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65 

132.34 

298 

1172.3 

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.1538 

18.32 

188.3 

138-8 

41.79 

.0239 

66 

*34-37 

299 

1172.6 

6.41 

.156 

20.36 

193-3 

140.3 

37.84  .0264 

67 

136.4 

300 

1172.9 

6.32 

•1583 

22.39 

197.8 

141.7 

34.63  .0289 

68 

138.44 

300.9 

1173.2 

6.23 

.1605 

24-43 

202 

143 

31.88  .0314 

69 

140.48 

301.9 

"73-5 

6.15 

.1627 

26.46 

205.9 

144.2 

29-57 

•0338 

70 

142-52 

302.9 

1173.8 

6.07 

.1648 

28.51 

209.6 

145-3 

27.61 

.0362 

144-55 

303-9 

1174.1 

5-99 

.167 

29.92 

212 

146.1 

26.36 

.03802 

72 

146.59 

304.8 

"74-3 

5-91 

.  1692 

30.54 

2I3.I 

146.4 

25-85 

.0387 

73 

148.62 

305-7 

1174.6 

5-83 

.1714 

32-57 

216.3 

147.4 

24-32 

.041  1 

74 

150.66 

306.6    1174.9 

5-76 

.1736 

34.61 

219.6 

148.3 

22.96 

•0435 

75 

152.69 

307-5    "75-2 

5-68 

•1759 

36.65     222.4 

149.2 

21.78 

•°459 

76 

154-73 

308.4 

"75-4 

5-6i 

.1782 

38.68     225.3 

150.1 

20.7 

.0483 

77 

156-77 

309-3 

"75-7 

5-54 

.1804 

40.72     228 

150.9 

19.72 

.0507 

78 

153.8 

310.2 

1176 

5-48 

.1826 

42-75 

230.6 

151-7 

18.84 

•0531 

79 

160.84 

3"-i 

1176.3 

5-41 

.1848 

44-79 

233-1 

152-5 

18.03 

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80 

162.87 

312 

1176.5 

5-35 

.1869 

46-83 

235-5 

153-2 

17.26 

.058 

81 

164.91 

312.8 

1176.8 

5-29 

.1891 

48.86 

237.8 

J53-9 

16.64 

.0601 

82 

166.95 

313.6 

1177.1 

5-23 

.1913 

50.9 

240.1 

154.6 

15.99 

.0625 

83 

168.98 

3I4-5 

1177.4 

•1935 

52.93 

242.3 

155-3 

15.38  .065 

84 

171.02 

3I5-3 

1177.6 

5-" 

•1957 

54-97 

244.4 

155-8 

14.86  .0673 

85 

I73-05 

316.1 

1177.9 

5-05 

.198 

57-01 

246.4 

156.4 

14.37   -°696 

86 

175-09 

316.9 

1178.1 

5 

.2002 

59-04 

248.4 

i57-i 

.0719 

87 

I77-I3 

3I7-8 

1178.4 

4-94 

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61.08 

250.4        157.8 

13.46 

•°743 

88 

179.16 

318.6 

178.6 

4.89 

.2044 

63-11 

252.2 

158.4 

13-05 

.0766 

89 

181.2 

3I9-4 

178-9 

4.84 

.2067 

65-15 

254.1 

158.9 

12.67 

.0789 

90 

183.23 

320.2 

179.1 

4-79 

.2089 

67.19 

255-9 

J59-5 

12.31 

.0812 

91 

185.27 

321 

179-3 

4-74 

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69.22 

257.6 

160 

11.97 

•0835 

92 

187.31 

321-7 

179-5 

4-69 

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71.26 

259-3 

160.5 

11.65 

.0858 

93 

189.34 

322.5 

179.8 

4.64 

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73-29 

260.9 

161 

"•34 

.0881 

94 

191-38 

323-3 

180 

4.6 

.2176 

75-33 
77-37 

262.6 
264.2 

161.5 
162 

11.04 
10.76 

.0905 
.0929 

95 
96 

I93-4I 

324.1 
324.8 

180.3 
180.5 

4-55 
4-5i 

.2198 
.2219 

79-4 

265.8 

162.5 

10.51 

.0952 

97 

197.49 

325-6 

180.8 

4.46 

.2241 

81.43 

267-3 

162.9 

10.27 

.0974 

98 

199.  52 

326.3 

18 

4.42 

.2263 

83.47 

268.7 

163.4 

10.03 

.0996 

99 

201.56 

327-1 

18  .2 

4-37 

.2285 

85-5 

270.2 

163.8 

9.81 

.102 

100 

203-  59 

327-9 

18  .4 

4-33 

.2307 

87-54 

271.6 

164.2 

9-59 

.1042 

101 

205.63 

328.5 

18  .6 

4.29 

2329 

89.58 

273 

164.6 

9-39 

.1065 

102 

207.  66 

329.1 

18  .8 

4-25 

.2351 

91.61 

274.4 

165-1 

9.18 

.1089 

103 

209.7 

329-9 

182 

4.21 

2373 

93-65 

275.8 

165-5 

9 

.in  i 

I04 

211.74 

33°-6 

182.2 

4.18 

•2393 

95-69 

277.1 

165.9 

8.82 

•"33 

105 

213-77 

331-3 

182.4 

4.14 

2414 

97.72 

278.4 

166.3 

8.65 

.1156 

106 

215.81. 

33J-9 

182.6 

4.11 

2435 

99.76 

279-7 

166.7 

8.48  j.  1179 

107 

217.84 

332.6 

182.8 

4.07 

2456 

01.8 

28l 

167.1 

8.31  !  .1202 

108 

219.88 

333-3 

183 

4.04 

2477 

03-83 

282.3 

167.5 

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109 

221.92 

334 

183-3 

4 

2499 

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283.5 

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no 

223-95 

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284.7 

168.3 

7.88    .1269 

in 

225-99 

335-3 

183-7 

3-93 

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09-94 
11.98 

285.9 
287.1 

1  68.  6 
169 

7-74    -1291 
7.61  1.1314 

112 

"3 

228.02 
230.06 

336 
336.7 

1x83.9 
1184.1 

2564 
2586 

14.01 

288.2 

169.3 

7.48    .1336 

232.1    |  337.411184.3 

3-83 

2607 

16.05 

289.3 

169.7 

7-  36  1-136  4 

"5 

3-8 

2628 

STEAM. 


709 


per 

& 

ESSURK 

in 
Mer- 

i 

£ 

Ik 

if 

Volume  of 
i  Lb. 

Density, 
or  Weight  of 
one  Cube  Foot. 

PR 

. 

BSSURB 

in 
Mer- 
cury. 

Temperate  r«. 

Total  H*at 
from  Water 
atjja6. 

fe 

r 

Density, 
or  Weight  of 
one  Cube  Foot. 

Lbs. 

Ins. 

0 

0 

Cub.  ft 

Lb. 

Lbs. 

Ins. 

o 

0 

Cub.  ft 

Lbs. 

116  j  236.17 

338.6 

1184.7 

3-77 

.2649 

149 

303-35 

357-8 

1190.5 

2.98 

•3357 

117 

238.2 

339-3 

1184.9 

3-74 

.2652 

305-39 

358.3 

1190.7 

2.96 

•3377 

118 

240.24 

339-9 

1185.1 

3-71 

.2674 

155 

3I5-57 

361 

1191.5 

2.87 

.3484 

119  242.28 

340.5 

"85.3 

3-68 

.2696 

160 

325-75 

363.4 

1192.2 

2-79 

•359 

120  j  244.31 

341.1 

1185-4 

3-65 

•2738 

165 

335-93 

366 

1192.9 

2.71 

.369^ 

121   246.35 

341-8 

1185.6 

3-62 

•2759 

170 

346-11 

368.2 

II93-7 

2.63 

•3798 

122  I  248.38 

342-4 

1185.8 

3-59 

.278 

175 

356.29 

370.8 

1194.4 

2.56 

•3899 

I23 

250.42 

343 

1186 

3-56 

.2801 

180 

366,47 

372.9 

1195.1 

2-49 

.4009 

124 

252.45 

343-6 

1186.2 

3-54 

.2822 

185 

376-65 

375-3 

1195-8 

2-43 

.4117 

125 

254-  49  '344-  2 

u86.4 

3-5i 

.2845 

190 

386.83 

377-5 

1196.5 

2-37 

.4222 

126 

256.53 

344-8 

1186.6 

3-49 

.2867 

397-01 

379-7 

1197.2 

2.31 

•4327 

I27 

258.56 

345-4 

1186.8 

3-46 

.2889 

200 

407.  19 

381-7 

1197.8 

2.26 

•4431 

128  |26o.6 

346 

1186.9 

3-44 

.2911 

2IO 

427-54 

386 

1199.1 

2.16 

•4634 

129 

262.64 

346.6 

1187.1 

•2933 

220 

447-9 

389-9 

1200.3 

2.06 

.4842 

130 

264.67  347.2 
266.71  347-8 

187-3 
187-5 

3-38 
3-35 

•2955 
.2977 

230 
240 

468.26 
488.62 

393-8 
397-5 

1201.5 

1202.6 

1.98 
1.9 

•5052 
.5248 

132 

133 

268.74  348.3 
270.78  348.9 

187.6 
187.8 

3-33 

.2999 
.302 

250 
260 

508.98 
529-34 

401.1 
404-5 

1203.7 
1204.8 

1.83 
1.76 

.'5660 

134 
135 

272.81 
274.85 

349-5 
35o.i 

188 
1188.2 

3-29 
3-27 

g 

270 
280 

549-7 
570-06 

407.9 
411.2 

1205.8  '  1.7 

1206.  8  j  1.64 

.*6o8i 

136 

276.89 

350.6 

1188.3 

3-25 

290 

590-42 

414.4 

1207.8 

i-59 

•6273 

137 

278.92 

351-2 

1188.5 

3-22 

3101 

300 

610.78 

4I7.5 

1208.7 

i-54 

.6486 

138 

280.96 

351-8 

1188.7 

3-2 

3121 

350 

712.57 

430.1 

I2I2.6 

i-33 

.7498 

J39 

282.99 

352-4 

1188.9 

3H2 

400 

8i4-37 

444-9 

I2I7.I 

1.18 

.8502 

140 

285.03 

352-9 

1189 

3.16 

3162 

450 

916.17 

456.7 

1220.7 

1.05 

•9499 

141 

287.07 

353-5 

1189.2 

3.14 

3184 

500 

1018 

467-5 

1224 

•95 

.049 

142 

289.1 

354 

1189.4 

3-12 

3206 

550 

1119.8 

477-5 

1227 

•87 

.148 

143 

291.14 

354-5 

1189.6 

3228 

600 

I22I.6 

487 

1229.9 

.8 

•  245 

144 

293-I7 

355 

1189.7 

loB 

325 

650 

I323-4 

495-6 

1232.5 

•74 

•342 

145 

295-21 

355-6 

1189.9 

3-o6 

3273 

700 

H25.8 

504.1 

I235-I 

.69 

•4395 

146 

297-25 

356.I 

1190 

3-04 

3294 

800 

1628.7 

5I9-5 

1239.8 

.61 

.6322 

147 

299.  28 

356.7 

1190.2 

3-02 

3315 

900 

1832.3 

533-6 

1244.2 

•55 

•8235 

«48 

301.32 

357-2 

190.3 

3- 

3336 

1000 

2035.9 

546.5 

1248.1 

•  5 

2.014 

Saturated    Steam   from   32°  to   S1S°.    (Claudel) 


Tem- 
pera- 
ture. 

PRK 

Mercu- 
ry. 

5SURB. 

Per 

Sq.  Inch. 

Weight 
of  ICQ 
Cub.  Feet. 

Volume 
of 
iLb. 

Tem- 
pera- 
ture. 

PR* 
Mercu- 
ry. 

5SUHK. 

Per 
Sq.  Inch. 

Weight 
of  zoo 
Cub.  Feet. 

Volum* 
of 
iLb. 

o 

Ins. 

Lbs. 

Lb. 

Cub.  Feet. 

0 

Ins. 

LI-. 

Lbs. 

Cub.  Feet. 

32 
35 

.181 
.204 

.089 
.1 

.031 
.034 

3226 
2941 

125 
130 

3-933 
4-509 

1.932 
2.215 

•554 
•63 

180.5 
158-7 

40 

.248 

.122 

.041 

2439 

135 

5-174 

2.542 

.714 

140.1 

45 

.299 

.147 

.049 

2041 

140 

5-86 

2.879 

.806 

124.1 

50 

.362 

.I78 

•059 

1695 

145 

6.662 

3-273 

.909 

no 

55 

.426 

.214 

.07 

1429 

ISO 

7-548 

3.708 

1.022 

97.8 

60 

•Si? 

•254 

.082 

I22O 

8-535 

4-193 

I-I45 

87-3 

65 

.619 

•304 

.097 

1031 

1  60 

9-63 

4-731 

1-333 

75 

70 

•733 

•36 

.114 

877.2 

165 

10.843 

1-432 

69.8 

75 

•427 

•I34 

746.3 

170 

12.183 

5^985 

1.602 

62.4 

80 

1.024 

•503 

.156 

64I 

175 

13-654 

6.708 

1-774 

56.4 

85 

1.205 

•592 

.182 

549-5 

1  80 

15.291 

7-5" 

1.97 

50.8 

90 
95 

1.41 

1.647 

.693 
.809 

.212 
•245 

408.2 

190 

17.041 
19.001 

8-375 
9-335 

2.181 
2.411 

45-9 

100 

1.917 

.942 

.283 

353-4 

195 

21.139 

0-385 

2.662 

37-6 

105 

2.229 

1.095 

•325 

3°7-7 

200 

23.461 

1.526 

2-933 

no 

"5 

2-579 
2.976 

1.267 
1.462 

•373 
.426 

268.1 
234-7 

205 

2IO 

25.994 
28.753 

2-77 
4.127 

3.225 
3-543 

|2 

120 

3-43 

1.685 

.488 

204.9 

212 

29.922 

4-7 

3-683 

27.2 

7IO  STEAM. 

GASEOUS  STEAM. 

When  saturated  steam  is  surcharged  with  heat,  or  superheated,  it  is  termed 
gaseous  or  steam-gas.  The  distinguishing  feature  of  this  condition  of  steam 
is  its  uniformity  of  rate  of  expansion  above  230°,  with  the  rise  of  its  tem- 
perature, alike  to  the  expansion  of  permanent  gases. 

To    Compute    Total    Heat   of  G-aseous    Steam. 

1074.6 -j- .475  t  =  H.  t  representing  temperature,  and  H  total  heat  in  degrees. 
Hence,  total  heat  at  212°,  and  at  atmospheric  pressure  =  1175.3°. 

Specific  gravity  =  .622. 

To    Compute   Velocity    of  Steam. 

Into  a  Vacuum.  RULE. — To  temperature  of  steam  add  constant  459,  and 
multiply  square  root  of  sum  by  60.2 ;  product  will  give  velocity  in  feet  per 
second. 

Into  Atmosphere.  3.6  ^h  =  V.  V  representing  velocity  as  above,  and  h  height  in 
feet  of  a  column  of  steam  of  given  pressure  and  uniform  density,  weight  of  which  is 
equal  to  pressure  in  unit  of  base. 

ILLUSTRATION.— Pressure  of  steam  100  Ibs.  per  sq.  inch,  what  is  velocity *of  its 
flow  into  the  air? 

Cube  foot  of  water  =  62. 5  Ibs. ,  density  of  steam  at  100  Ibs.  =  270  cube  feet.  Henc«, 
62.5  :  ioo  ::  270  :  432  =  volume  at  100  Ibs.  pressure,  and  432  X  144  =  62208  feet=sz 
height  of  a  column  of  steam  at  a  pressure  q/ioo  Ibs.  per  sq.  inch. 

Then  3.6  -^62  208  —  898  feet. 

EXPANSION. 

To    Compute   Point   of  Cutting    off   to   Attain    Limit   of 
Expansion. 

b-\-f!j-±-P  =  point  of  cutting  off.  b  representing  mean  back  pressure  for  entire 
stroke,  in  lbs.&er  sq.  inch,  f  friction  of  engine,  P  initial  pressure  of  steam,  all  in  Ibs 
per  sq.  inch,  and  L  length  of  stroke,  in  feet. 

ILLUSTRATION.— Assume  stroke  of  piston  9  feet,  pressure  30  Ibs. ,  mean  back  press- 
ure 3  Ibs. ,  and  friction  2  Ibs. 

3  +  2X9-:-  30  =  i-  5  feet. 
To  Compute   Actual    Ratio   of  Expansion. 

,  *"  °  =  R.    c  representing  clearance  or  volume  of  space  between  valve  seat  and 

l-\-c 

mean  surface  of  piston,  at  one  or  each  end  in  feet  of  stroke,  I  length  of  stroke  at  point 
of  cutting  off,  excluding  clearance  in  feet,  and  R  actual  ratio  of  expansion. 

ILLUSTRATION.— Assume  length  of  stroke  2  feet,  clearance  at  each  end  1.2  ins., 
and  point  of  cutting  off  i  foot. 

1.2  ins.  =  .x.     Then  —  —  =  i.g  ratio. 

To    Compute    Pressure    at    any    Point    of   Period    of  Ex- 
pansion. 

When  Initial  Pressure  is  given.  Pl^-s—p.  p  representing  pressure  at  period 
of  given  portion  of  stroke,  both  in  Ibs.  persq.  inch,  and  s  any  greater  portion  of  stroke 
than  I. 

When  Final  Pressure  is  given.  P'  x  I/  -r-  s  =  p.  P'  representing  final  pressure, 
in  Ibs.  per  sq.  inch,  and  L'  length  of  stroke,  including  clearance,  in  feet. 

ILLUSTRATION  i.— Assume  length  of  stroke  6  feet,  clearance  at  each  end  1.2  ins., 
pressure  of  steam  60  Ibs.,  point  of  cutting  off  one  third;  what  is  pressure  at  4  feet? 

1.2  ins.  —.ifoot.     60  x  2  +  .i+- 4 +  -1  —  30.73  Ibs. 

t.— What  is  pressure  in  above  cylinder  at  2.8  feet,  when  final  pressure  is  21  Iba  ? 
21  X  6-4- .  i  -i-  2. 8  -j- .  i  =  44. 17  Ibt, 


STEAM.  711 

To   Compute    Mean    or   Total   Average   Pressure. 


P  (I'  i  +  hyp.  log.  R  —  c)        , 

-  -  -  =  p  or  mean  or  average  pressure.     I   length  of  stroke  at 
Li 

point  of  cutting  off,  including  clearance. 
ILLUSTRATION.  —  Assume  elements  of  preceding  cases:  i  -f-  hyp.  log.  R=  2.065. 

60  (2.  x  X  2.065  -  1)  =  254^  6    ^ 

6  6 

To    Compute    Final    Pressure. 


ILLUSTRATION.  —Assume  elements  of  preceding  cases,  steam  cut  off  at  2  feet. 

60  x  2  +  .  i  -=-6  +  .  i  =  20.65  MS- 
To    Compute    Mean.    Effective    Pressure. 


j-  —  V,    VI     V/7      L.;. 

ILLUSTRATION.— Assume  elements  of  preceding  cases,  b  —  2  Ibs.  per  sq.  inch. 


_2  = 


To   Compute   Initial   Pressure   to  Produce   a  Q-iven 
erage    Effective    or    Net    Pressure. 


_  _  _ 

r(i+hyp.log.R)—  c 

ILLUSTRATION.—  Assume  elements  of  case  i. 
6 


=P 


.  __ 

2-|--I  (2.1X2.065)—.!         4.2365 

To  Compute  Point  of  Cutting  off  for  a  GHven   Ratio  of 
Expansion. 

L'-=-R  —  c.     Or,  L-f-c-=-R  —  c  —  I 

ILLUSTRATION.—  Assume  elements  of  preceding  cases:  R  =  -^t--  ~  2.o,  and     '"'I 

2  +  .1  2.9 

—  .1  =  2  /ee«. 

To  Compute  Pressure  in  a  Cylinder,  at  any  Point  of  Ex- 
pansion, or    at    End    of  Stroke. 


ILLUSTRATION.  —  Assume  elements  of  preceding  cases: 

60  X  2.  i  .60 

—  ;  -  =  60  Ibs..  and  —  =  20.60  ibs. 

2  +  .I  2.9 

To  Compute  Initial  Pressure  for  a  Required  Net  Effec- 
tive   Pressure    for    a    GHven.    Ratio    of  Expansion. 

6  L  -       Or,  P   -  —     —  =  P.     W  representing  net- 


a  (I'  i  +  hyp.  log.  K  —  c)  I'  i  +  hyp.  log.  R  —  c 

work  infoot-lbs.  =  a  L  p'  —  6,  and  a  area  of  piston,  in  sq.  ins. 

ILLUSTRATION. — Assume  elements  of  preceding  cases:  area  of  piston  =  100  sq. 
ins.,  back  pressure  2  Ibs.,  and  net  effective  pressure  =  42.365  Ibs. 

loo  X  6  X  42. 365  —  2  =  24  219  foot-lbs. 


242194-100X2X6   _       25419       =6Q  ^         42.365X6      __254.i9__6o  ^ 
100X2.1  X  2.065  — . i       "00X4-2365  '    2.1X2.065—.!      4.2365" 


f\2  STEAM. 

IPoints   of  Expansion. 

Relative  points  of  expansion,  including  clearance  5  per  cent.,  assuming 
stroke  of  piston  to  be  divided  as  follows,  and  initial  pressure  =  i. 

Point i      .75     .6875     .625    .5625    .5      .4375    .375    .333    .25    .2    .125    .1 

Ratio i     1.31    1.43       1.55    1.71      1.912.15      2.43    2.74    3.5    4.46.        7. 

Hyp.  Log.  of  above  Ratios. 

.0          1.27          1.36          1.44          1.54          1.65          1.77          1.9          2          2.25          2-43          2.79          2-95 

Receiver   of  Compound. 


Volume  —  Into  which  the  HP  cylinder  exhausts,  should  be  from  i  to  1.5 
times  the  volume  of  it,  plus  that  of  the  clearance  in  it,  when  the  cranks  are 
set  at  angles  of  120°  and  90°.  When  the  cranks  are  opposite  (180°)  or  very 
nearly  so,  the  volume  may  be  proportionately  decreased. 

Pressure  —  In  a  Receiver  should  not  exceed  one  half  that  of  the  boiler  pres- 
sure, and  usually  it  is  operated  lower. 

Receiver  —  Of  a  Triple  compound  engine  need  not  have  as  great  a  vol- 
ume, as  the  cranks  are  set  at  angles  of  120°  to  each  other. 

If  a  receiver  is  insufficient  in  volume,  the  result  is  back  pressure  in  the 
B?  cylinder.  If  otherwise,  it  has  too  great  a  volume,  the  result  is  that  of  a 
material  reduction  of  the  pressure,  when  the  exhaust  port  of  the  cylinder  is 
opened,  a  consequent  loss  of  external  work  and  of  efficiency. 

rJ?o    Compute   Volume   of  a    Receiver. 

Single  Compound. 
A2  —  81.5 

Cranks  at  go0.  •  ^—-  —  =  volume  in  cube  feet.  A  representing  area  of  BP 
cylinder,  and  S  stroke  of  piston,  both  in  sq.  ins. 

ILLUSTRATION.  —  Assume  a  compound  engine  having  a  B?  cylinder  of  28  ins., 
cranks  at  90°,  and  stroke  of  piston  36  ins.  ;  what  should  be  volume  of  receiver? 


. 

1728  1728 

Triple  Compound. 

A2  —  S  i 

Cranks  at  120°      ---  ?—  —  —  volume  in  cube  feet. 

1728 

ILLUSTRATION.  —  Assume  a  triple  compound  engine  having  a  B?  cylinder  of  28  ins., 
cranks  at  120°,  and  stroke  of  piston  36  ins.  ;  what  should  be  volume  of  receiver? 


1 
-  '—1  -  -^  -  ±-  =i7.i  cube  feet. 


-  —  -  -     —  -  - 
1728  1728 

(The  Practical  Engineer.) 

The  practice  with  some  is  to  give  the  receiver  an  equal  volume  with  that  of  the 
cylinder  from  which  it  receives  the  steam. 

To  Compute  Mean.  ^Pressure  of  Steam  upon  a  IPiston, 
"by  Hyperbolic  J-jogari.th.ms. 

RULE.  —  Divide  length  of  stroke  of  a  piston,  added  to  clearance  in  cylinder 
at  one  end,  by  length  of  stroke  at  which  steam  is  cut  off,  added  to  clearance 
at  that  end,  and  quotient  will  express  ratio  or  relative  expansion  of  steam  or 
number. 

Find  in  table,  logarithm  of  number  nearest  to  that  of  quotient,  to  which 
add  i.  The  sum  is  ratio  of  the  gain. 

Multiply  ratio  thus  obtained  by  pressure  of  steam  (including  the  atmos- 


STEAM.  7 1  3 

phere)  as  it  enters  the  cylinder,  divide  product  by  relative  expansion,  and 
quotient  will  give  mean  pressure. 

NOTE.— Hyp.  log.  of  any  number  not  in  table  may  be  found  by  multiplying  a 
common  log.  by  2.302  585,  usually  by  2.3. 

Proceed  by  referring  to  table  pp.  331-334. 

EXAMPLE.— Assume  steam  to  enter  a  cylinder  at  a  pressure  of  50  Ibs.  per  sq.  inch, 
and  to  be  cut  off  at  .25  length  of  stroke,  stroke  of  piston  being  10  feet;  what  wUl  be 
mean  pressure? 

Clearance  assumed  at  2  per  cent.  =  .2  feet. 

10  +  2  =  10. 2  feet,  stroke  10  -f-  4-^.2  =  2. 38  feet.  Then  10. 2  -r-  2. 38  =  4. 29  rela- 
tive expansion. 

Hyp.  log.  4.29  (p.  332)  =  1.4563,  which  -f- 1  =  2.4563,  and  2'4s63  x  5<>  _  2g  6z  lbg 

4.29 

Relative  Effect  of  steam  during  expansion  is  obtained  from  preceding  rule. 

Mechanical  Effect  of  steam  in  a  cylinder  is  product  of  mean  pressure  in 
Ibs.,  and  distance  through  which  it  has  passed  in  feet. 

Effects    of  Expansion.    (Essentially  from  D.  K.  Clark.) 
Back  Pressure  is  force  of  the  uncondensed  steam  in  a  cylinder,  consequent 
upon  impracticability  of  obtaining  a  perfect  vacuum,  and  is  opposed  to  the 
course  of  a  piston.     It  varies  from  2  to  5  Ibs.  per  sq.  inch. 

It  must  be  deducted  from  average  pressure.  Thus:  assume  pressure  60  Ibs., 
stroke  of  piston  as  in  preceding  case,  and  back  pressure  2  Ibs. 

At  termination  of. ist,    2d,    3d,    4th,    sth,    and  6th  foot  of  stroke. 

Pressure 60     30      20      15       12  10  Ibs.  per  inch. 

Backpressure 22222  2    "     "      " 

Effective  pressure 58      28      18      13       10  8    "     "      " 

Total  work  done  by  expansion  at  termination  of  each  foot  or  assumed 
division  of  stroke  of  piston  is  represented  by  hyp.  log.  of  ratio  of  expansion, 
initial  work=i. 

Thus,  for  a  stroke  of  10  feet  and  a  pressure  of  10  Ibs. : 
At  end  of ist,    2d,    3d,    4th,    sth,    6th,    yth,    Sth,    gth,  and  loihfoot. 

Steam  is  expanded ) 

into   vols.,  hyp.  >=  .69     I.I      1.39      I.6l      1.79      1.95      2.08      2.2  2.3 

log.  of  which...  ) 


Initial  duty  

i 

Total  duty  

...    i    1.69 

2.1 

2-39 

2.61 

2,79 

2-95 

3.08 

3-2 

3-3 

Initial  duty  is  rep-  \ 
resented  by  10..  ] 

[      10  16.9 

21 

23-9 

26.1 

27.9 

29-5 

30.8 

32 

33 

Resistance  for  each  i 
foot  of  stroke...  j 

[=    .     4 

6 

8 

10 

12 

i4 

16 

18 

20 

Total  effective  ] 
duty  ] 

|=    8  12.9 

'5 

15-9 

16.1 

15-9 

15-5 

14.8 

14 

13 

Gain  by  expansion    061.2587.598.75101.2598.7593.7585        75  62.5 

The  same  results  would  be  produced  if  expansion  was  applied  to  a  non-condens- 
ing engine,  exhausting  into  the  atmosphere. 

Again,  assume  total  initial  pressure  in  a  non-condensing  cylinder  75  Ibs.  per  sq. 
inch,  expanded  5  times,  or  down  to  15  Ibs.,  and  then  exhausted  against  a  back  press- 
ure of  atmosphere  and  friction  of  15  Ibs. 

At  termination  of. ist,  2d,  3d,  4th,    and     $ih  foot  of  stroke. 

Total  duty. i  1.69  2.1  2.39  2.61 

"       "     performed...    75         126.75        157.5        179.25          195.75/00^0*. 
u     backpressure....   15  30  45  60  75         '•    " 

11     effective  duty 60  96.75        112.5        "9-25          120.75    "    " 

Gain  by  expansion o          61.25         87.5         98.75          101^25  per  cent. 

From  which  it  appears  that  the  total  duty  performed  by  expanding  steam  5  times 
its  initial  volume  is  full  2.5  times,  or  as  75  to  195.75. 

30* 


STEAM. 


Relative    Effect  of  Equal   "Volumes   of  Steam. 

Relative  total  effect  or  work  of  steam  is  directly  as  its  mean  or  average  pressure 
(A),  and  inversely  as  its  final  pressure  (B),  or  volume  of  steam  condensed. 

If  former  is  divided  by  latter,  quotient  will  give  relative  total  effect  or  work  (C) 
of  a  given  volume  of  steam  as  admitted  and  cut  off  at  different  points  of  stroke  of 
piston,  with  a  clearance  of  3.125  per  cent. 

In  following  computations  resistance  of  back  pressure  is  omitted.  If  this  press- 
ure is  uniform  with  all  the  ratios  of  expansion,  it  is  a  uniform  pressure,  to  be  de- 
ducted from  the  total  mean  pressure  in  column  (A). 


Cut  off  at 

Pres 

JA) 
Mean. 

ure. 

& 

Relative 
Effect. 

Cut  off  at 

Pres 

(A) 
Mean. 

ure. 
(B) 
Final. 

(C) 

Relative 
Effect. 

I 

^6875 
.625 
•5625 
•  5 

I 

.946 
.924 
.889 
.857 

V 

21 

.636 

.576 
.501 

.28 
•35 
•45 
•54 
•7i 

•375 
•33 
•25 

.2 
.125 
.1 

.761 
.702 
.628 
•559 
•435 
.418 

•394 
•335 
•273 
.224 

•15 

•!3 

1-93 
2.09 

2-3 

2.05 
2.9 
3.21 

To  Compvite  Total  Effective  Worlz  in  One  Strolre  of  I>is- 
ton,  or   as    Griven   Tt>y    an    Indicator    Diagram. 


a  P  (I'  i  -f-  hyp.  log.  R  —  c)  =  w,  and  a  6  L  =  w'.  w  representing  total  work,  and 
w'  back  pressure. 

NOTE. — Pressure  of  atmosphere  is  to  be  included  in  computations  of  expansion; 
it  is  therefore  to  be  deducted  from  result  obtained  in  non-condensing  engines.  In 
condensing  engines,  the  deduction  due  to  imperfect  vacuum  must  also  be  made, 
usually  2. 5  Ibs.  per  sq.  inch. 

ILLUSTRATION. — Assume  cylinder  of  a  condensing  engine  26.1  ins.  in  diameter,  a 
stroke  of  2  feet,  pressure  of  steam  95  Ibs.  (8o.3-}-i4-7)persq.  inch,  cut  off  at  .5  stroke, 
with  an  average  back  pressure  of  2  Ibs.  per  sq.  inch,  and  a  clearance  of  5  per  cent. 

Area  of  piston,  deducting  half  area  of  rod  ==  530  sq.  ins.  2X5-;-  too  = .  i  clear- 
ance, and  2  -{-.  i-7-  i  +  .1  =  1.9  =  ratio  of  expansion,  and  i  -j-hyp.  log.  1.9  =  1.642. 

Then  530  X  95  X  1. 1  X  i. 642  — . i  —  530  X  2  X  2  =  50  350 X  i. 706  —  2120  =  83  777  Ibs. 

ILLUSTRATION. —  Assume  cylinder  of  a  non-condensing  engine  having  an  area  of 
2000  sq.  ins.,  a  stroke  of  8  feet,  steam  at  a  pressure  of  50  Ibs.  (35.3 -f- 14-7)>  cut  °ff  a^ 
.25  of  stroke,  and  clearance  .25  foot. 

Ratio  of  expansion  3.66,  back  pressure  17  Ibs.,  and  i  -f  hyp.  log.  3.66  =  2. 297. 

2000  X  50(2.25  X  i-}- hyp.  log.  3.66  — .25)  =  loooooX  2.25  X  1  +  1.297 —  .25  = 
460573  foot-to*- 

2000  X  17  X  S  =  272  ooo  foot-lbs.  or  negative  effect,  and  460575  —  272000=188575 
foot-lbs. 

Total   Effect   of  One    Lt>.  of  Expanded    Steam. 

If  i  Ib.  of  water  is  converted  into  steam  of  atmospheric  pressure  =  14.7  Ibs.  per 
sq.  inch,  or  2116.8  Ibs.  per  sq.  foot,  it  occupies  a  volume  equal  to  26.36  cube  feet ; 
and  the  effect  of  this  volume  under  one  atmosphere  =  2116. 8  Ibs.  x  26.36  feet  = 
55799  foot-lbs.  Equivalent  quantity  of  heat  expended  is  i  unit  per  772  foot-lbs., 
—  55  799  -J-  772  =  72. 3  unit*.  This  is  effect  of  i  Ib.  of  steam  of  a  pressure  of  one  at- 
mosphere on  a  piston  without  expansion. 

Gross  effect  thus  attained  on  a  piston  by  i  Ib.  of  steam,  generated  at  pressures 
varying  from  15  to  100  Ibs.  per  sq.  inch,  varies  from  56000  to  62  ooo  foot-lbs. ,  equiv- 
alent to  from  72  to  80  units  of  heat. 

Effect  of  i  Ib.  of  steam,  without  expansion,  as  thus  exemplified,  is  reduced  by 
clearance  according  to  proportion  it  bears  to  volume  of  cylinder.  If  clearance  is  5 
per  cent,  of  stroke,  then  105  parts  of  steam  are  consumed  in  the  work  of  a  stroke, 
which  is  represented  by  too  parts,  and  effect  of  a  given  weight  of  steam  without  ex- 
pansion, admitted  for  full  stroke,  is  reduced  in  ratio  of  105  to  100.  Having  deter- 
mined, by  this  ratio,  effect  of  work  by  i  Ib.  of  steam  without  expansion,  as  reduced 
by  clearance,  effect  for  various  ratios  of  expansion  may  be  deduced  from  that,  in 
terms  of  relative  operation  of  equal  weights  of  steam. 


STEAM.  7 1  J 

Volume  of  i  Ib.  of  saturated  steam  of  100  Ibs.  per  sq.  inch  is  4.33  cube  feet,  and 
pressure  per  sq.  foot  is  144X100=  14  400  Ibs.;  then  total  initial  work  =  14400X4-33 
—  62  is-zfoot-lbs.  This  amount  is  to  be  reduced  for  clearance  assumed  at  7  per  cent. 

Then  62  352  X  ioo-r- 107  =  58273/00^65.,  which,  divided  by  772  (Joule's  equiva- 
lent), =  75.5  units  of  heat. 

Total  or  constituent  heat  of  steam  of  100  Ibs.  pressure  per  sq.  inch,  computed  from 
a  temperature  of  212°,  is  1001.4  units;  and  from  102°  (temperature  of  condenser 
under  a  pressure  of  i  Ib.)  the  constituent  heat  is  1111.4  units. 

Equivalent,  then,  of  net  simple  effect  75.5  units  is  7.5  per  cent,  of  total  heat  from 
212°,  or  6.7  per  cent,  from  102°. 

When  steam  is  cut  off  at 

i  .75          .5          .33          .25          .2          .125  and .  i  of  stroke, 

comparative  effects  are  as 

i         1.26         1.616      1.92         2.14         2.27       2.51  and  2.6. 

Total  effects  as  given  in  table,  page  718. 

Effect  of  i  Ib.  of  steam,  without  deduction  for  back  pressure  or  other  effects,  vanes 
from  about  6oooofoot-lbs.,  without  expansion,  to  about  double  that,  or  1 20000  foot- 
Ibs.,  when  expanded  3  times,  cutting  off  at  about  27  per  cent,  of  stroke;  and  to 
about  1 50  ooo  foot- Ibs.  when  expanded  about  6  times,  and  cut  off  at  about  10  per 
cent,  of  stroke. 

Effect   of  Clearance. 

Clearance  varies  with  length  of  stroke  compared  with  diameter  of  cylinder, 
with  form  of  valve,  as  poppet,  slide,  etc. 

With  a  diameter  of  cylinder  of  48  ins.,  and  a  stroke  of  10  feet,  and  poppet 
valves,  clearance  is  but  3  per  cent.,  and  with  a  diameter  of  34  ins.  and  a 
stroke  of  4.5  feet  and  slide  valves,  it  is  7  per  cent. 

ILLUSTRATION  OF  EFFECT.  —Assume  steam  admitted  to  a  cylinder  for  .25  of  its 
stroke,  with  a  clearance  of  7  per  cent 

Mean  pressure  for  i  Ib.  =  .637,  and  loss  by  clearance  =  7  -=- 100  =  .07,  which,  added 
to  .637,  =.707,  which  is  effect  of  a  given  volume  of  steam,  if  there  was  not  any  loss 
by  clearance,  or  a  gain  of  n  per  cent. 

When  steam  is  cut  off  at i      .75      .5      .33      .25      .125  and  .1  stroke. 

Loss  at  7  per  cent,  clearance..  =7    7.2      8.1    9.6    u       15.3         17     percent. 

To  Compute  !N"et  Volume  of  Cylinder  for  Q-iven  "Weight 
of  Steam,  Ratio  of  Expansion  and  One   Strolte. 

RULE. — Multiply  volume  of  i  Ib.  of  steam,  by  given  weight  in  Ibs.,  by 
ratio  of  expansion  and  by  100,  and  divide  product  by  100,  added  to  per  cent, 
of  clearance. 

EXAMPLE.— Pressure  of  steam  95  Ibs.,  cut  off  at  .5,  weight  .54  Ibs.,  volume  of  i  Ib. 
steam  4.55,  and  weight^  .2198  Ibs.,  stroke  of  piston  2  feet,  and  clearance  7  per  cent 

Ratio  of  expansion  2  -f- .  14  -f.  i-j-.I4  —  ^gg. 

4.55  X. 54  X.. 88  X. 00  =  46iS!  = 
100+7  107 

To  Compute  "Volxime  of  Cylinder  for  Given  Effect  witli 
a  Q-iven   Initial  Pressure  and   Ratio  of  Expansion. 

RULE.  —  Divide  given  effect  or  work  by  total  effect  of  i  Ib.  of  steam  of 
like  pressure  and  ratio  of  expansion,  and  quotient  will  give  weight  of  steam, 
from  which  compute  volume  of  cylinder  by  preceding  rule. 

EXAMPLE.— Assume  given  work  at  50766  foot-lbs.,  and  pressure  and  expansion  as 
preceding. 

Total  work  by  i  Ib.,  too  Ibs.  steam,  cut  off  at  .5,  =by  table  94200/00^6*.,  and  by 
table  of  multipliers  for  95  Ibs.  =  .998,  which  x  94200  =  94012/00^6*. 

Then  5iZ_  =  .54  ibs.  weight  of  steam. 
94012 


STEAM. 

Consumption    of  Expanded   Steam  per  UP    of  Effect  per 
Hour. 

IP  =  33  ooo,  which  X  60=  i  980  ooo  foot  -  Ibs.  per  hour,  which  -4-  i  Ib. 
steam,  the  quotient  =  weight  of  steam  or  water  required  per  IP  per  hour. 

ILLUSTRATION. — Effect  of  i  Ib.,  100  Ibs.  steam,  without  expansion,  with  7  per  cent, 
of  clearance  =  58273  foot-lbs.,  and  *  9  °  cgg  __  ^  ^5  steam  =  weight  of  steam  con- 
sumed for  the  effect  per  IP  per  hour. 

When  steam  is  expanded,  the  weight  of  it  per  IP  is  less,  as  effect  of  i  Ib.  of  steam 
is  greater,  and  it  may  be  ascertained  by  dividing  i  980000  by  the  respective  effect, 
or  by  dividing  34  Ibs.  by  quotient  of  total  mean  pressure  by  final  pressure,  as  given 
in  table,  page  718. 

When  steam  is  cut  off  at  i  .75  .5  .375  .33  .25  and  .2  of  stroke. 
our  |  —34  26-9  21  18.5  17.6  16  14.9/65. 


Hence,  assuming  10  Ibs.  steam  are  generated  by  combustion  of  i  Ib.  coal  per  IP 
of  total  effect  per  hour, 

The  coal  consumed  per  \  0  ,       , 

EPperhour  .....  ...j-3'4    2'69      2>I      x-85        *-76     x.6          1.49  to*. 

SATURATED    STEAM. 
To  Compute  Energy  and  Efficiency  of  Saturated  Steam. 


l»-p'XaRS  =  X;  _XD  =  H";  = 

j,      Y  —  *'•         h    —  P"-  —       ' 

•*'  - 


P"  T         '       —  E—       i9oooo-  =     ; 

•  —  =  e;       nlap—  p'  =  x,  and  -  =  IBP;       Rp—  jp'a=:a5;        FCx6o=/; 

^—  =  cube  feet.     '9  °°gg  _  cuiefeei  water  evaporated  per  hour  per  IP. 

pa  —  p  a  62.5  X 

V  and  v  representing  volumes  of  mass  of  steam  entering  cylinder  and  of  it  at 
termination  of  stroke  of  piston;  S  and  s  volumes  of  i  Ib.  steam  when  admitted  and 
when  at  termination  of  expansion  ;  C  volume  of  cylinder  per  minute  for  each  IIP  ; 
R  and  r  ratios  of  expansion  and  effective  cut-off;  F  feed  water  per  cube  foot  of  vol- 
ume of  cylinder  per  stroke  of  piston,  and  f  per  IIP  per  hour,  all  in  cube  feet.  D  den- 
sity or  weight  of  i  cube  foot  of  steam  at  temperature  of  operation,  in  Ibs.  ;  p  mean 
pressure  ;  p'  mean  back  pressure  ;  I  initial  pressure  ;  P  mean  effective  pressure,  or 
energy  per  cube  foot  of  volume  of  cylinder  ;  P'  pressure  per  sq.  inch  or  that  equivalent 


of  volume  of  cylinder,  or  pressure  equivalent  to  heat  expended  per  sq.foot;  H""heat 
rejected  per  cube  foot  of  steam  admitted;  H'"  heat  rejected  per  cube  foot  of  volume 
of  cylinder  ;  A  available  heat  per  IIP  per  hour ;  e  energy  per  cube  foot  of  volume  of 
cylinder  to  point  of  cutting  off,  or  of  steam  admitted;  h  and  h'  heat  expended  and 
rejected,  and  X  energy  exerted,  all  per  Ib.  of  steam  and  infoot-lbs.  E  efficiency ;  x  en- 
ergy exerted  per  minute  and  per  cube  foot  of  steam  admitted ;  a  area  of  piston  in 
sq.  ins.  ;  I  length  of  stroke  of  piston  in  feet,  and  f  feed  water  per  IIP  per  hour*  in 
cube  feet. 

ILLUSTRATIONS. — Assume  volume  of  cylinder  and  clearance  (5  percent.  =  .6  inch) 
i  cube  foot,  steam  (86.34- 14.7)  100  Ibs.  per  sq.  inch,  cut  off  at  .5,  mean  pressure  by 
rule  (page  711)  86  Ibs.,  and  back  pressure  3  Ibs. 

V=i.       v  =  2.       8  =  4.33.       5  =  8.31.       p  =  B6.       p'  —  3.       a  =  144  ins. 
0-f  46i.2°and  ioo°  +  46i.2°.       l  =  2feet.       n=i.      1^157748. 


STEAM. 


»-T-J  =  a  ratio. 

33000 


86-3X144 


4-  33  •*•& 3*  =  -52i  effective  cut-off. 
=  2.76  cube  feet. 


4-33 


772  X-  231  (789.1°  —  561.  2°) 
=  99  195  foot-lbs. 


2      "  2X4-33 
157  748  =  198  389/00^6*. 


86  —  3X144  =  11952  Ibs. 
.1154  cube  feet. 


86  —  3  X  144  X  2  X  4-  33  =  103  504  foot-lbs. 

198  389  -r-  .231  —  103  504  X  .231  =  174  479  foot-lbs.     174  479  -f-  2  =  87  239  foot-lbt. 
1 5. 5  X  ioo  X  144  X  4-  33  =  966  456  foot-lbs.  966  456  — 103  504  =  862  952  foot-lbs. 

966456  144X86  —  144X3      „   1980000 

— — ^—  =  in  6oolbs.     -— — — — -=.io7E. 

2X4-33  111600 

Or  i  980000  X  966456  =  18  504  673  foot-lbs. 

103504  33000 

i  X  2  X  144  X  86  —  3  =  23  904  foot-lbs.         ^  \  9^° '     —  = .  306  cube  feet. 


=  18  504  (>T$foot-lb\ 
23  904  _ 


2  X  86  —  3  X  144  =  23  904  foot-lbs. 


62. 5  X  103  504 
.1154X2.76x60  =  19.11  cube  feet. 


103  504 


=  ii  952  foot-lbs. 


33000 


-  =  2.761  cube  feet. 


2X4-33"  86X144-3X144" 

In  illustration  of  connection  of  expenditure  of  available  heat  (A)  and  consumption 
of  fuel,  assume  coal  to  have  a  total  heat  of  combustion  of  iooooooo*/oo£-/&s.,  cor- 
responding to  an  equivalent  evaporative  power  under  i  atmosphere  at  212°  of  13.4 
los  water  and  efficiency  of  furnace  .  5 ;  then  available  heat  of  combustion  of  i  Ib. 
coal  —  5  ooo  ooo  foot-lbs. 

Hence,  consumption  of  coal  per  IEP  in  an  engine  of  like  dimensions  and  opera- 
tion with  that  here  given  would  be  19223000---  5000000  =  3.8444  Ibs. 

Properties   of  Steam  of  Maximum   Density.    (Rankine.) 
Per  Cube  Foot. 


L 

0 



o 

O 

O 

O 

o 

32 

248 

95 

1999 

158 

9687 

221 

33180 

284 

88740 

347 

197700 

41 

348 

104 

2571 

167 

II  760 

230 

38700 

293 

100500 

356 

219000 

50 

481 

"3 

3277 

176 

14200 

239 

44930 

302 

113400 

365 

242000 

59 

655 

122 

4136 

185 

17  oio 

248 

51920 

3" 

127500 

374 

266600 

68 

88  1 

131 

5178 

I94 

20280 

257 

59720 

320 

143000 

383 

293  ioo 

77 

1171 

140 

6430 

203 

24020 

266 

68420 

329 

159800 

392 

321400 

86 

1538 

I49 

7921 

212 

28  3IO 

275 

78050 

338 

178000 

401 

351600 

L  representing  latent  heat  of  evaporation  per  cube  foot  of  vapor  in  foot-lbs.  of  en- 
ergy.    To  reduce  this  to  units  of  heat  divide  by  772,  or  Joulejs  equivalent. 

SUPERHEATED    STEAM. 

The  results  attained  by  imparting  to  steam  a  temperature  moderately  in 
excess  of  that  due  to  the  volume  or  density  of  saturated  steam  are : 

1.  An  increase  of  elasticity  without  a  corresponding  increase  of  water  evaporated. 

2.  Arresting  or  reducing  passage  of  water,  in  suspension,  to  cylinder  (foaming),  as 
the  heat  contained  in  that  water  is  wholly  lost  without  affording  any  elastic  effect. 

Both  of  these  results,  by  increasing  effect  of  the  steam,  economize  fuel. 
Superheated  steam  should  be  treated  as  a  gas. 

The  product  of  its  pressure,  p  in  Ibs.  per  sq.  foot,  and  volume  v  of  i  Ib.  of  it  in  cube 
feet,  in  the  perfectly  gaseous  condition,  is  obtained  by  following  formula: 
42  140  T-r-«=pt>  =  8s.44T.    T  temperature  of  steam  4-461.2°,  and  t  32° -j-  461. 2°. 
ILLUSTRATION.— Assume  temperature  of  steam,  327.9°,  superheated  to  341.1°. 

Then  42  140  X  461. 2° +  341.1°  -f-  324-461.2°  =  68  549  foot-lbs. 

Hence,  as  pressure  of  steam  at  327.9°=  ioo  Ibs.  per  sq.  inch,  and  at  341.1°  120. 

120  -T-  ioo  =  1.2  to  i  =  a  gain  of  one  fifth. 

*  Coal  of  average  composition,  14 133  x  772  =  10910676. 


STEAM. 


To    Compute     Energy     and.     Efficiency    of    Superheated 
Steam. 

In  following  illustrations  elements  are  same  as  those  in  preceding  cases  for  satu- 
rated steam,  with  addition  of  the  steam  being  superheated,  so  that 
1  =  115  lbs.t  <  =  338°-f  461.  2°=  799.  2°,  t'  —  290  +  461.  2°  =  751.  2°,  8  =  3.8,  5  =  7.4. 


Efficiency  of  saturated  steam  (p.  716) .  107,  and,  as  above, .  109 ;  hence  —  =  1.02  to  i. 

If,  then,  available  heat  of  combustion  of  efficiency  of  furnace  is  assumed  at  5  ooo  ooo 
,  as  above,  consumption  of  coal  per  IIP  18 183  486  -r-  5  oooooo  ^  3.637  Ibs. 

NOTE.— For  further  illustrations  Rankine's  "  Steam-engine,1'  London,  1861,  p.  436. 
"Wire-drawing. 

Wire  drawing  of  steam  is  difference  between  pressure  in  boiler  and  pressure  in 
cylinder,  and  is  occasioned  as  follows: 

Resistance  or  friction  in  steam-pipe  to  passage  of  steam  to  steam-chest  and  piston. 

Resistance  of  throttle- valve  to  passage  of  steam,  when  it  is  partly  closed  or  of  in- 
sufficient area  in  proportion  to  steam -pipe. 

Resistance  from  insufficient  area  of  valves  or  ports. 

Mr.  Clark,  from  his  experimental  investigation,  declared,  that  resistance  in  a 
steam-pipe  is  inappreciable,  when  its  sectional  area  is  not  less  than  .  i  area  of  piston, 
and  its  velocity  not  exceeding  600  feet  per  minute. 

When  velocity  of  a  piston  is  from  200  to  240  feet  per  minute,  area  of  steam  may 
be  .04th  of  piston. 

Effect  of*  Expansion,  xvitn.  Equal  "Volumes,  and  Effect  of 
One  Llo.  of  1OO  Llt)s.  Pressvire  per   Sci.  Inch. 

Clearance  at  each  End  of  Cylinder,  including  Volume  of  Steam  Openings,  7  per  cent, 
of  Stroke,  and  iooper  cent,  of  Admission  =  i. 


Ratio 
of  Ex- 
pansion . 

Initial 
Volume 
=  i. 

Point 
of 
Cut-off. 

Stroke 
=  i. 

TOTA 

Final. 

Initial 
Pressure 
=  i. 

L  PRESSU 
Mean. 

Initial 
Pressure 
=  i. 

RES. 

Initial. 

Mean 
Pressure 
=  i. 

Weight 
of  Steam 
of  100  Lbs. 
for  one 
Stroke  per 
Cube  Foot. 

ACTUAL 

By  i  Lb. 
of  100  Lbs. 
Steam. 

EFFECT. 
Per 
Sq.  Inch 
per  Foot 
of  Stroke 
by  too  Lbs. 
Steam. 

Volume 
of  Steam 
expended 
per  IP 
of  Work 
per  Hour. 

Heat 
con- 
verted. 

Lbs. 

Foot-lbs. 

Foot-lbs. 

Lba. 

Units. 

X 

j 

i 

I 

.247 

58273 

100 

34 

75-5 

i.i 

•9 

.909 

.996 

004 

.225 

63850 

99.6 

31 

82.7 

1.18 

.83 

.847 

.986 

014 

.209 

67836 

98.6 

29.2 

87.9 

1.23 

.8 

.813 

.98 

O2 

.201 

70246 

98 

28.2 

91 

i-3 

•75 

.769 

.969 

032 

•190 

73513 

96.9 

26.9 

95-2 

i-39 

•  7 

.719 

•953 

049 

.178 

77242 

95-3 

25-6 

100.  1 

1.45 

.66 

.69 

.942 

062 

•17 

79555 

94-2 

24.9 

102.9 

i-54 

.625 

.649 

•925 

081 

.l6l 

83055 

92-5 

23.8 

107.6 

1.6 

.6 

.625 

•9I3 

095 

•155 

85125 

9r-3 

23-3 

110.3 

1.88 

.5 

•532 

.86 

163 

•131 

94200 

86 

21 

122 

2.28 

•4 

•439 

.787 

271 

.108 

104  466 

78.7 

19 

132.5 

2-4 

•375 

.417 

.766 

3°5 

.103 

107  050 

76.6 

l8.5 

138.6 

2.65 

•33 

•377 

.726 

377 

•093 

1  12  22O 

72.6 

17.7 

145-4 

2.9 

.3 

•345 

.692 

445 

.085 

116855 

69.2 

16.9 

I5I-4 

3-35 

•25 

.298 

.637 

•57 

.074 

124066 

63.7 

16 

160.7 

4 

.2 

•25 

.567 

.764 

.062 

I32770 

56-7 

14.9 

I7I.9 

4-5 

.16 

.322 

.526 

.901 

•055 

138130 

52.6 

14-34 

178.8 

5 

.14 

.2 

.488 

.049 

.049 

142  1  80 

48.8 

13.92 

184.2 

5-5 

.125 

.182 

•457 

.188 

•045 

146325 

45-7 

13-53 

l89.5 

5-9 

.11 

.169 

•432 

•315 

.042 

148  940 

43-2 

13.29 

192.9 

6-3 

.1 

•159 

•4*3 

.421 

•039 

I5I370 

4i-3 

13.08 

196.  I 

6.6 

.09 

.152 

.398 

•5i3 

•037 

152955 

39-8 

12.98 

197.7 

7 

.083 

•143 

•  381 

.625 

•035 

155200 

38-1 

12.75 

2OI.I 

7-8 

.066 

.128 

.348 

.874 

.032 

158414 

34-8 

12.5 

205.2 

8 

.0625 

•  125 

•342 

.924 

.031 

159433 

34'  2 

11.83 

206.5 

STEAM. 


Multipliers    for    Actual    Weight    and     Effect    for    other 
Pressures    than    IOO    Lbs. 


Pressure 

Multipliers. 

Pressure 

Multipliers. 

Pressure 

Multipliers. 

per 
Sq.  Inch. 

Weight. 

Actual 
Effect. 

per 
Sq.  Inch. 

Weight. 

Actual 
Effect. 

per 
Sq.  Inch. 

Weight. 

Actual 
Effect. 

Lbs. 

Lba. 

Lbs. 

6S 

.666 

•975 

9° 

.901 

•995 

130 

1.28 

.015 

70 

.714 

.981 

95 

.952 

.998 

140 

i-37 

.022 

£ 

s 

.986 
.988 

100 

no 

I 
1.09 

I 
1.009 

150 
160 

1.46 
1-55 

.025 
.031 

85 

.855 

.991 

120 

HZ 

i.  on 

170 

1.64 

•033 

In  this  illustration,  in  connection  with  preceding  table,  no  deductions  are  made 
for  a  reduction  of  temperature  of  steam  while  expanding,  or  for  loss  by  back 
pressure. 

When  steam  is  cut  off  at  .0625,  or  one  sixteenth,  its  expansion  is  16  times,  but  as 
7  per  cent,  of  stroke  is  to  be  added  to  it  (.0625 -K°7)  —  -1325  —  132. 5  per  cent ,  or 
nearly  double  of  16,  or  only  a  little  over  7  times,  as  in  3d  column  of  table  on  pre- 
ceding page. 

Column  7  is  product  of  58  273  and  ratio  of  total  effect  of  equal  weights  of  steam 
when  expanded,  or  average  total  pressure  divided  by  average  final  pressure. 

Thus,  if  steam  is  cut  off  at  .5,  with  a  clearance  of  7  per  cent,  - — X  I0°~ = 

'•532X  100  =  53.2 
1.6165,  and  58273  X  1.6165  =  94200/00^65. 

Column  9  gives  volume  of  steam  consumed  per  HP  per  hour.  Thus,  assume  cyl- 
inder to  have  an  area  of  292  sq.  ins.,  a  stroke  of  2  feet,  and  pressure  of  steam  100  Ibs. 
without  expansion. 

292  X  ioo  x  2  =  584oo/oo<-Z&*.,  and  292  +  7  per  cent,  of  stroke  for  clearance  = 
.  14 ;  hence,  292  X  2. 14  -•- 144  =  4.34  cube  feet,  and  weight  of  a  cube  foot  of  such  steam 
=  .23  Ibs.,  and  58400  :  4.34  X  .23  ::  33000  :  .564,  which, x  60  minutes  =  33. 84,  or  34 
as  per  table. 

The  pressures  are  computed  on  premise  that  steam  is  maintained  at  a  uniform 
pressure  during  its  admission  to  cylinder,  and  that  expansion  is  operated  correctly 
to  termination  of  stroke. 

Column  10  is  quotient  of  work  in  foot-lbs.,  divided  by  Joule's  equivalent  772. 
Thus,  94  200-7-772  =  122. 

For  percentage  of  constituent  heat,  converted  from  102°  and  212°,  assume 
122  as  in  last  case : 

Then  122  x  9 -r-  ioo  =  10.98  per  cent  for  102°,  and  122  X  io-f-  ioo  =  12.2  per  cent, 
for  212°. 

"Wire-drawing"  will  cause  a  reduction  of  pressure  during  admission,  and  clear- 
ance will  vary  from  3  to  8  per  cent,  according  to  design  of  valve,  as  poppet,  long  or 
short  slide. 

In  practice,  wire-drawing  of  steam,  and  opening  of  exhaust  before  termination  of 
stroke,  involve  deviations  from  a  normal  condition,  for  which  deductions  must  be 
made,  added  to  which  there  is  the  back  pressure,  from  insufficient  condensation  in 
condensing  engines,  and  from  pressure  of  air  in  non-condensing  engines,  and  com- 
pression of  exhaust  steam  at  termination  of  stroke. 

To  Compute  Grain,  in.  Feed  "Water  at  High  Temperature. 

T  —  £-|-W  —  w  =  H.  T  and  t  representing  total  heat  in  steam  and  temperature  of 
feed  water,  W  and  w  temperature,  of  water  blown  off  and  fed  =  heat  lost  by  blowing 
off,  and  H  total  heat  required  from  fuel,  all  in  degrees. 

ILLUSTRATION.— Assume  steam  at  248°,  feed  water  100°  in  one  case  and  150°  in 
another,  and  density  — ,  and  total  heat  at  248°  =  1157°;  what  is  gain  ? 

1157  — 1004-248  —  ioo  =  1205°  =  total  heat  required  from  fuel. 


7  —  1504-248  —  150  =  1105°  = 


Then 


H  — H'_  1205  — 1105 
H  1205 


=  .083  =  8.3 per  cent, 


720 


STEAM. 


COMPOUND   EXPANSION. 


Compound  Expansion  is  effected  in  two  or  more  cylinders,  and  is 
tised  in  three  forms. 

ist.  When  steam  in  one  cylinder  is  exhausted  into  a  second,  pistons  of  the 
two  moving  in  unison  from  opposite  ends — that  is,  steam  from  top  or  for- 
ward-end of  first  cylinder  being  exhausted  into  bottom  or  after-end  of  the 
other,  and  contrariwise — this  is  known  as  the  Woolf  *  engine. 

2d.  Steam  from  the  ist  cylinder  is  exhausted  into  an  intermediate  vessel, 
or  "  receiver,"  the  pistons  being  connected  at  right  angles  to  each  other. 

3d.  Steam  from  receiver  is  exhausted  into  a  3d  cylinder  of  like  volume 
with  2d,  pistons  of  all  being  connected  at  angles  usually  of  120°. 
The  two  latter  types  are  those  of  the  compound  engine  of  the  present  time. 

Expansion  from  Receiver.  The  receiver  is  filled  with  steam  exhausted 
from  ist  cylinder,  which  is  then  admitted  to  2d,  or  2d  and  3d,  in  which  it  is 
cut  off  and  expanded  to  termination  of  stroke. 

Initial  pressure  in  ad,  or  2d  and  3d  cylinders,  is  assumed  to  be  equal  to  final  press- 
ure in  ist,  and  consequently  the  volume  cut  off  in  the  one  or  the  other  cylinders 
must  be  equal  in  volume  to  that  of  ist  cylinder,  for  its  full  volume  must  be  dis- 
charged therefrom. 

Inasmuch  as  3d  cylinder  is  but  a  division  of  the  ad,  with  addition  of  receiver, 
this  engine,  in  following  illustrations,  will,  for  simplification,  be  treated  as  having 
but  two  cylinders. 

In  illustration,  assume  ist  and  2d  cylinders  to  have  volumes  as  i  to  2,  with  like 
lengths  of  stroke,  and  that  steam  is  cut  off  at  .5  stroke,  and  equally  expanded  in 
both  cylinders,  the  ratio  of  expansion  in  each  cylinder  being  thus  equal  to  their 
volumes. 

Volume  received  into  2d  cylinder  would  be  equal  to  that  exhausted  from  ist,  as- 
suming there  would  not  be  any  diminution  of  pressure  from  loss  of  heat  by  inter- 
mediate radiation,  etc.  This  is  based  upon  assumption  that  expansion  occurs  only 
upon  a  moving  piston;  but  in  operation,  expansion  occurs  both  in  receiver  and  in 
intermediate  passages,  as  nozzles  and  clearances;  the  2d  cylinder,  therefore,  receives 
steam  at  a  reduced  pressure,  increased  volume,  and  reduction  of  ratio  of  expansion. 
To  meet  this,  and  attain  like  effects,  volume  of  2d  cylinder  must  be  increased  in 
proportion  to  increased  volume  of  steam  and  its  ratio  of  expansion.  Consequently, 
there  is  no  loss  of  effect  aside  from  increased  volume  and  weight  of  parts  by  inter- 
mediate expansion,  provided  primitive  ratio  of  expansion  is  maintained  by  giving 
relative  increased  volume  to  2d  cylinder. 

ILLUSTRATION.— Assume  cylinders  having  volumes  as  i  and  3,  initial  steam  of  ist 
cylinder  to  be  60  Ibs.  per  sq.  inch,  stroke  of  piston  6  feet,  cut  off  at  one  third,  and 
clearance  7  per  cent. 

Final  pressure,  as  per  rule,  page  711,  =22.62  Ibs.,  and  pressure  as  exhausted  into 
receiver,  reduced  one  fourth,  =  16.97  Ibs.,  assuming  there  is  no  intermediate  fall  of 
pressure.  The  steam,  therefore,  is  expanded  to  1.33  times  volume  of  cylinder;  a 
like  volume,  therefore,  must  be  given  to  2d  cylinder,  to  admit  of  this  at  a  like  press- 
ure. If,  therefore,  the  increased  terminal  volume  of  the  steam  in  the  ist  cylinder 
was  augmented,  including  a  clearance  of  7  per  cent.,  the  effect  would  be  as  follows: 

Volume  admitted  to  2d  cylinder  is  equal  to  volume  of  ist  added  to  its  clearance, 
or  to  .33  volume  of  2d  cylinder  added  to  its  clearance;  that  is,  to  .33  of  107  percent., 
or  35.66  per  cent.,  consisting  of  clearance,  and  35.66  —  7  =  28.66  per  cent,  stroke  of 
2d  cylinder.  The  steam  exhausted  into  2d  cylinder  thus  fills  less  than  .33  of  its  stroke 
by  4.67  (33.33  —  28.66).  As  steam  is  expanded  from  volume  of  ist  cylinder,  plus  its 
clearance,  to  2d  cylinder,  plus  its  clearance,  ratio  of  expansion  in  2d  cylinder  is  equal 
to  ratio  of  volume  of  both  cylinders,  which  is  3,  and 

joo  (representing  full  stroke)  -4-  j  22.62 

-E-£  =  3,  and  final  pressure =  7. 54  Ibs.  per  sq.  inch. 

28.66  +  7  3 

*  In  1825-28  James  P.  Allaire,  of  New  York,  adopted  this  design  of  engine  in  the  steamboats  Henrf 
Eckford,  Sun,  Commerce,  Swiflsure,  Post  Boy,  and  Pilot  Boy. 


STEAM,  721 

AaPimfng  volume  of  receiver,  or  augmented  terminal  volume,  for  expansion  in  2d 
cylinder,  to  have  proportions  of  i,  1.25, 1.33,  and  1.5  times  volume  of  ist  cylinder 
plus  Jts  clearance,  the  relations  would  be  as  follows: 


Augmented  terminal  volumes) 

E  *1_^ 

1.25 

i-33 

(times  volume  of 
•>      I      ist  cylinder. 

I.O7 

1.3-37 

T  4.27 

!do.        do. 
including     clear 

Fii»l  volumes  in  2d  cylinder) 
added  to  clearance  ) 
Ratio  of  expansion  in  2d  cyl'r.  . 
Intermediate    reductions    of  ) 

*•"/ 
321 

3 

0 

*'33i 

3-21 

2-4 
.2 

i-T-^/ 

3-21 
2.25 
.25 

ance. 
(  times  volume  of 
3-21    \     ist  cylinder. 

2 

(of  terminal  press- 
•33    |    ure  in  ist  cyl'r. 

o 

4-  52 

5.65 

11.31  Ibs.  per  sq.  inch. 

Pressures  in  receiver  and  ini-  1 
tial  pressure  in  2d  cylinder.  .  j 
Final  pressure  in  2d  cylinder  ... 

22.62 
7-54 

18.1 
7-54 

16.96 

7-54 

11.31       do.         do. 
7.54       do.         do. 

To   Compute   Expansion   in   a   Compound   Engine. 
RECEIVER  ENGINE. 

Ratio  of  Expansion.  In  ist  cylinder,  as  per  formula,  page  710.  In  2d  cylinder. 
n~I  r  =  ratio.  Of  Intermediate  Expansion.  — —  =  ratio,  n  representing  ratio 

of  intermediate  reduction  of  pressure  between  ist  and  zd  cylinder,  to  final  pressure  in 
ist  cylinder,  and  r  ratio  of  area  of  ist  cylinder  to  that  of^d. 

ILLUSTRATION.— Assume  n  ==  4,  and  r  =  3. 

Then  4~I  X  3  =  2.25  ratio,  and     4    =:  1.33  ratto. 

Total  or  Combined  Ratio  of  Expansion,  r  R'  =  product  of  ratio  ofist  and  zd  cyl- 
inders by  ratio  of  expansion  in  ist  cylinder.  As  when  r  =  3,  and  R'  =  2.653,  then 
2.653  X  3  =  7-959  total  ratio- 

Bence,  Combined  Ratio  of  Expansion  in  both  cylinders.  *-  —  r  R'=R".  R'  rep- 
resenting ratio  of  expansion  in  ist  cylinder,  and  R"  combined  ratio. 

ILLUSTRATION.— Assume  as  preceding,  and  R'  =  2.653. 

Then X  3  X  2.653  =  5-969  combined  ratio. 

To  Compute   KfFect  for  One    Stroke  and  a  Given   Ratio 
of  Expansion   in   ITirst    Cylinder. 

Without  Intermediate  Expansion.  RULE.  —  Multiply  actual  ratio  of  ex- 
pansion in  ist  cylinder  by  ratio  of  both  cylinders,  and  to  hyp.  log.  of  com- 
bined ratio  add  i;  multiply  sum  by  period  of  admission  to  ist  cylinder  plus 
clearance,  and  term  product  A. 

Divide  ratio  of  both  cylinders,  less  i,  by  ratio  of  expansion  in  ist  cyl- 
inder;  to  quotient  add  i ;  multiply  sum  by  clearance,  and  term  product  B. 

Subtract  B  from  A,  and  term  remainder  C.  Multiply  area  of  ist  cylinder 
in  sq.  ins.  by  total  initial  pressure  in  Ibs.  per  sq.  inch,  and  by  remainder  C. 
Product  is  net  effect  in  foot-lbs.  for  one  stroke. 

With  Intermediate  Expansion.  Add  effect  thereof  to  result  obtained  above, 
and  by  following  formula : 

Or,  I'  i  +  hyp.  log.  R"  —  c  ( i  -f  -|p- )  a  P  =  E.    a  representing  area  in  sq.  ins. , 

P  initial  pressure  in  Ibs.  per  sq.  inch  of  ist  cylinder,  I'  length  of  admission  or  point 
of  cutting  off  plus  clearance,  c  clearance  in  feet,  and  E  effect  in  foot-lbs. 

3? 


722 


STEAM. 


ILLUSTRATION.— Assume  areas  of  cylinders  i  and  3  sq.  ins.,  length  of  stroke  6  feet, 
pressure  of  steam  60  Ibs.  per  sq.  inch,  cut  off  at  2  feet,  clearance  7  per  cent,  and 
area  of  intermediate  space,  as  receiver,  one  third  volume  of  ist  cylinder. 

R"  =  ratio  of  expansion  in  2d  cylinder X  3  X  2.653  =  5-969  hyp.  log. 


2.653X2.25  +  1X2.42  —  3  —  i  -^-2.653  4-1  X-  42  X  i  X6o=  1.7865  +  i  X  2.42  — 
2^-2.6534-1  X-42  X  60  =  6.  743  —  .737  X  60  =  360.  36  foot-lbt. 

ist  Cylinder. 

Effect  on  piston  60  Ibs.  x  i  inch  x  2  feet  .......................  =120     foot-lbt. 

"      of  clearance  60  Ibs.  x  42  foot  ............................  =  25.2       *j  _ 

Total  initial  effect  =  6oX  2  X  .42  ..........................  =145.2  fooTlbs. 

Then  145.2  x  i  +  hyp.  log.  2.653  or  1.976  .......................  =286.91  foot-lbt. 

Less  effect  of  clearance  .......................................  =  25.2        " 

Net  effect  on  piston  above  vacuum  line  ....................  =  261.71  foot-lbt. 

Less  effect  of  back  pressure  60-1-2.653  =  22.61,  which,  x  3  sq.  )  ,  , 

ins.  and  2  feet  stroke  ......................................  J  —      >'00  _ 

Net  effect  on  piston  .......................................  =  126.05/00^0*. 

zd  Cylinder. 


145.2  X  i  +  hyp.  log.  2.25  or  1.81  ..............  =  262.81  foot-lbs. 

Effect  of  clearance  22.61  X  3  X.  42  .....  .  ......  =  28.49      "         =  234.32/00^6$. 

36°-  37  foot-lbs. 

Intermediate  reduction  of  pressure,  as  given  at  page  721,  =  .25  X  22.61  =5.65  Ibs. 
per  sq.  inch,  which,  x  3  sq.  ins.  and  by  2  per  foot  of  stroke,  =  33.9  foot-lbs. 

Hence  360.  36  +  33.9  =  394.  26  foot-lbs. 
Or,  by  sum  of  the  three  results,  viz.  : 
ist  cylinder.  .  .  ..................................................  126.05  foot-lbt. 

Intermediate  expansion  .......................  .  .................    33.9       " 

2d  cylinder  .....................................................  234.  32      *  ' 

394.27/00^6*.' 

WOOLF  ENGINE.     D.  K  Clark. 

Ratio  of  Expansion.—  In  ist  cylinder  a*  per  formula,  page  710.    In  id  cylinder, 
r       ^-aj^-i-f-aj  •=  ratio,    r  representing  ratio  of  area  ofist  cylinder  to  that  0/2^, 

I  and  I'  lengths  of  stroke  and  of  stroke  added  to  clearance,  in  ins.  or  feet,  and  x  ratit 
value  of  intermediate  volume. 

ILLUSTRATION  —Assume  I  =  6  feet,  V  —  7  per  cent.  =  .  42,  r  =  3,  and  x  =  .  333. 

3X6l2  +  '333 
Then  -  -^  -  =  2.353,  ratio  of  expansion  in  zd  cylinder. 

Total  Actual  Ratio  of  Expansion.    R'  (r  -p  -f  *)  =  ratio. 
ILLUSTRATION.—  Assume  preceding  elements,  R  =  2.653. 

Then  2.653  (3  X  g-^  +  .333)  =  2.653  X  3.137  =  8.322,  total  actual  ratio. 

Combined  Actual  Ratio  of  Expansion.    R'  (  r  -?  -f-  *J  -r-  1  -fa;  =  ratio. 
ILLUSTRATION.  —Assume  preceding  elements. 
2-653  (3  X  ^  --  1-  .333  -T-  i  -f-  333J  —    *322  =  6.242,  combined  actual  ratio. 


STEAM. 


723 


To  Attain  Combined  Ratio  of  Expansion  and  Final 

IPressvire    in    2d    Cylinder. 

Assuming  four  cases  as  taken  for  Receiver  Engine  with  a  clearance  of 
7  per  cent.    The  relations  would  be  as  follows: 


•  5 

•535 
1.605 


Intermediate  spaces  are o  '333 

Volume  of  ist  cylinder o  .357 

Add  to  these  1.07,  the  volume  of  iet ) 
cylinder  plus  its  clearance,  and....  ]    I>O7      M2? 

To  same  values  of  intermediate  space*! 

add  3,  the  volume  of  2d  cylinder,  I 

and  the  sums  are  the  final  volumes  [    3  3-357      3-535 

by  expansion  in  2d  cylinder J 

Ratios  ofexpansion  in  2dcyl'rarequo-  )        o 

tients  of  nnal  by  initial  volumes. .  }    2"  8°4    2.  352      2. 2O2 

Intermediate  falls  of  pressure  are,  in  ) 
parts  of  final  pressure  in  ist  cylinder  ) 


(  part  of  volume  of  ist  cylin- 
(     der  plus  its  clearance,  or, 


total  initial  volumes  for  ex- 
pansion in  2d  cylinder  or 
,     times  volume  of  ist  cyl'r. 


( times  volume  of  ist  cyl- 
4-°7    j     inder. 


1.07 
2.14 


o  5.94 

The  initial  pressures  for  expansion  in ) 

ad  cylinder  are  .........VT. J    1  '75 

23.75  17-81 

Etntt,  final  pressures  in  zd  eyVr  are. .  8. 47  7. 57 


•333 

7.92 
.66 

15-83 
7.19 


1.902  ratios  of  expansion. 

f  of  final    pressure  ;    or,  as- 
J     snming  initial  pressure  at 
I      63  Ibs.,  and  final  pressure 
I,    at  23.75  IDS->  &aey  are 
11.87  Ibt.  per  sq.  inch. 

(of  final  pressure  in  ist  cyl- 
>     \     inder,  or 

11.87  76*.  per  »q.  inch. 
6. 24  Ibs.  per  aj.  inch. 


Combined  Ratios  in  these  Four  Cases. 


ist.    ist  ratio  of  expansion  — 
2d       do.         do.     *  33.rP1 

to  2.  653           Combined  Ratio, 
to  2.804=2.653  X  2.804  =  7.44. 

ad.     ist      do. 
2d       do. 

3d.     ist      do. 
2d       do. 

4th.    ist      do. 
2d       do. 

do  

do.  OtfAJ 

to  2.  653 
tO  2.  352  =  2.  653  X  2.  352  =  6.  24. 

do  
do  

to  2.653 

tO  2.  202  =  2.653X2.  202  =  5.  84. 

do  
do  

to  2.653 
to  1.905  =  2.653  X  1-905  =  5.05. 

Initial  effect  of  st< 


pressure,  admitted  to  ist  cylinder,  for  2  feet,  or  one 
third  of  stroke  of  piston,  and  with  a  clearance  of  7  per  cent,  or  .42  feet,  is  as  follows: 
Effect  on  piston  .....  63  X  2  feet  =  126  foot-lbs.  f  Total  initial 

do.    in  clearance  .  .  63  X  .42  foot  =  26.  46  =  63  X  2.  42  =  1  52.  46  foot-lbs.  (     effect. 
This  sum  is  initial  effect,  on  which  effect  by  expansion  is  computed,  while  it  is 
26.46  foot-lbs.  in  excess  of  the  initial  effect  on  the  piston. 
The  total  effect,  then,  is  computed  as  follows: 

ist  case.       152.46  x  (i  +  hyp.  log.  7.44)  or  3.0069  =  458.27         Net  Effect. 
Less  effect  of  clearance  ..............  26.  46    431.  81  foot-lbs. 


2dcase. 
3d  case. 
4th  case. 


152.46  x  (i-fhyp-  log.  6.24)  or  2.831   =431.47 
Less  effect  of  clearance  ..............  26.46    405.01 

152.46  X  (i  4-  h7P-  log-  5-84)  or  2.7647  =  421.35 
Less  effect  of  clearance  ..............  26.  46    394.  89 


152.46  X  (i  -hhyp.  log.  5.05)  or  2.6294  =  399.29 
Less  effect  of  clearance  ..............  26.  46 


372.  83 


The  reductions  of  net  effect  in  2d,  sd,  and  4th  cases  are  6.2,  8.6,  and  13.7  per  cent 
of  effect  in  ist  case. 

To  Compute  Effect  for    One    Stroke    and  a  Q-iven   Com- 

bined   Actnal    Ptatio    of  Expansion. 

RULE.  —  To  hyp.  log.  of  combined  actual  ratio  of  expansion  (behind  both 
pistons)  add  i  ;  multiply  sum  by  period  of  admission  of  steam  to  ist  cylin- 
der, added  to  clearance,  and  from  product  subtract  clearance. 

Multiply  area  of  ist  cylinder  in  sq.  ins.  by  initial  pressure  of  steam  in  Ibs. 
per  sq.  inch  and  by  above  remainder.  Product  is  net  effect  in  foot-lbs.  for 
jone  stroke. 

' 


724  STEAM. 

EXAMPLE.— Assume  elements  of  ist  illustration  page  723. 
Hyp.  log.  6.24  +  1  =  2.831,  which,  x  2.42  =  6.85,  and  6.85  —  .42  and  remainder 
X  60  =  385.8/00^0$. 

Or,  V  (i  -f-  hyp.  log.  R')  —  C  x  a  P  =  E. 

Comparative    Effect   of*   Steam,    in    Receiver   and.   "Woolf 
Engines. 

The  effect  of  steam  in  a  compound  engine,  without  clearance  and  without  any 
intermediate  reduction  of  pressure,  is  the  same  whether  operated  in  a  receiver  or 
Woolf  engine. 

When,  however,  there  is  an  intermediate  space  between  the  two  cylinders,  as  a 
receiver,  there  is  an  intermediate  reduction  of  the  pressure  of  the  steam,  conse- 
quent upon  the  increase  of  its  volume  in  the  receiver;  the  reduction  of  pressure, 
therefore,  being  less  rapid  than  with  the  Woolf  engine,  the  effect  is  greater. 

In  illustration,  the  following  comparative  elements  of  the  effect  of  both  engines 
Is  furnished. 

RECEIVER.        (7  per  cent,  clearance. )          WOOLF. 


Ratio  of  Expansion.    Net  Effect. 

ist  case.... ..7.96 422.3  foot-lbs. 

»d  "'  5.97 421-55   " 

3d  "  5-3i 4I7-96 


Ratio  of  Expansion.    Net  Effect. 

ist  case 7.64 431.71  foot-lbs. 

2d  "  6.24 405.11   " 

3d,  "  5-84....... 394-99   " 


4th  " 3-98 402.78  Ma  "  5-05 372-93     " 

From  wliich  it  appears,  that  although  the  effect  of  a  receiver  engine  is  the  great- 
est, its  ratio  of  expansion  is  less  than  with  the  Woolf  engine. 

Also,  that  by  the  addition  of  clearance  to  the  pistons  of  each  engine,  the  actual 
ratios  of  expansion  are  sensibly  reduced,  as  compared  with  the  ratios  without 
clearance. 

INDICATOR. 
To   Compute    Mean.    IPressxire   "by   an   Indicator. 

;*  &  ff  ^  oooxo  RULE. — Divide  atmosphere  line,  o  o  in  fig- 
^  •  '  '  '  '  '  '  ure,  into  any  convenient  number  of  parts,  as 
feet  of  stroke  of  piston,  and  erect  perpendic- 
ulars at  each  point.  Measure  by  scale  of 
parts  (alike  to  that  of  diagram)  the  actual 
mean  pressure,  as  defined  between  the  two 
lines  at  top  and  bottom  of  diagram,  add  the 

results,  divide  sum  by  number  of  points,  and 

3   4    5   6    7    s   9  10    quotient  will  give  mean  pressure  in  Ibs.  per 
sq.  inch  upon  piston. 
EXAMPLE.— Pressures,  as  above  given,  are: 

35  +  35  +  35  +  34  +  3^  +  25  -f  16  -f  10 -f  8  -f  6  =  236,  which,  -=- 10,  =  23.6  Ibs. 
NOTE.— If  it  were  practicable  to  run  an  engine  without  any  load,  and  it  some- 
times is,  the  mean  pressure,  as  exhibited  by  an  indicator,  would  be  an  exact  meas- 
ure of  the  friction  of  the  engine. 

Conclusions   on    Actual   Efficiency   of  Steam. 

For  development  of  highest  efficiencies  of  steam,  as  used  in  an  engine,  means  for 
protecting  it  from  cooling  and  condensing  in  the  cylinder  must  be  employed.  Super- 
heating of  it  prior  to  its  introduction  into  a  cylinder  is  probably  most  efficient 
means  that  may  be  employed  for  this  purpose.  Application  to  cylinder  of  gases 
hotter  than  it  is  next  best  means;  and  next  is  the  steam-jacket. 

In  cases  of  locomotive  and  portable  engines,  consumption  of  steam  per  IIP  per 
hour  is  less  than  for  that  of  single-cylinder  condensing  engines  for  like  ratios  of  ex- 
pansion, which  is  due  to  effect  of  temperature  of  non-condensing  cylinders,  always 
exceeding  212°. 

It  is  dedncible  from  these  results  that  the  compound  engine  is  more  efficient  than 
the  single-cylinder,  and  that,  of  the  two  kinds  of  compound  engines,  the  receiver- 
engine  is  more  efficient  than  the  Woolf. 

Average  consumption  of  bituminous  coal  per  IIP  per  hour,  for  compound  engines 
in  long  voyages,  as  shown  by  Mr.  Bramwell,  ranged  from  i.  7  to  2. 8  Ibs.  (D.  K.  Clark,) 


STEAM. 


725 


To    Compute   Volume    of  "Water    Evaporated,    per   X^b. 
of  Coal. 

—  ^—  —  =  volume  of  water,  in  tts.    V  and  v  representing  volume  of  steam  and 

relative  volume  of  water,  in  cube  feet,  W  weight  of  cube  foot  of  water,  and  F  weight  of 
fuel  consumed,  both  in  Ibs.,  and  d  density  of  water,  in  degrees  of  saturation. 

ILLUSTRATION.  —  Take  case  at  foot  of  page.     ¥  =  449887  cube  feet,  u  =  8q8  cube 
feet,  W  =  64.3,  E  =  i,  and  F  =  4061  Ibs. 


Gain  in  Fuel,  and  Initial  Pressure  of  Steam  required,  ichen  Acting  Expan- 
sively, compared  with  Non-Expansion  or  Full  Stroke. 


Point  of 
Cutting  off. 

Gain  in 
Fuel. 

Cutting 
off. 

1  Point  of 
[Cutting  off. 

Gain  in 
Fuel. 

Cutting 
off. 

|   Point  of 
Cutting  off. 

Gain  in 
Fuel. 

Cutting 
off. 

Stroke. 

ft 

Per  Cent. 
22.4 
32 

Lbs. 
1.03 
1.09 

Stroke. 
1  -375 

Per  Cent. 
49.6 

Lbs. 
1.18 
1,32 

Stroke. 
•25 
1       -125 

PerCent. 
58.2 
67.6 

Lbs. 
1.67 

2.6 

ILLUSTRATION.— What  must  be  initial  pressure  of  steam  cut  off  at  .5,  to  be  equiv- 
alent to  ioo  Ibs.  per  sq.  inch  at  full  stroke  ? 

100  at  full  stroke  =  ioo,  and  ioo  X  1. 18  =  118  Ibs. 

To   Compute   Gain,   in   Fuel. 

RULB. — Divide  relative  effect  of  steam  by  number  of  times  the  steam  is 
expanded,  and  divide  i  by  quotient ;  result  is  the  initial  pressure  of  steam 
required  to  be  expanded  to  produce  a  like  effect  to  steam  at  full  stroke. 

Divide  this  pressure  by  number  of  times  the  steam  is  expanded,  and  sub- 
tract quotient  from  i,  remainder  will  give  gain  per  cent,  in  fuel. 

EXAMPLE.— When  steam  is  cutoff  at  .5,  what  is  gain  in  fuel,  and  what  mechanical 
effect  ? 

Relative  effect,  including  clearance  of  5  per  cent.,=  1.64;  number  of  times  of  ex- 
pansion =  2. 

1.64-5-  2  =  .82,  and  i  -5- .82  =  1.22  initial  pressure. 
x.22-5-2  =  .6i,  and  i  —.61  =.39  per  cent 

Mechanical  effects  of  steam  at  full  and  half  strokes  are  2  — 1.64  =  .36  difference. 

Hence,  1.64  :  .36  ::  50  (half  volume  of  steam  used) :  10.97  per  cent,  more  fuel  to 
produce  same  effect  at  half  stroke,  compared  with  steam  at  full  stroke. 

To    Compnte    Consumption    of*  3Txael    in    a   IP'irnaoe. 

When  Dimensions  of  Cylinder,  Pressure  of  Steam,  Point  of  Cut-off,  Revo- 
lutions, and  Evaporation  per  Lb.  of  Fuel  pei*  Minute  are  given. 

RULE. — Compute  volume  of  cylinder  to  point  of  cutting  off  steam,  in- 
cluding clearance.  Multiply  result  by  number  of  cylinders,  by  twice  number 
of  strokes  of  piston,  and  by  60  (minutes)  ;  divide  product  by  density  of  steam 
at  its  pressure  in  cylinder,  and  quotient  will  give  number  of  cube  feet  of 
water  expended  in  steam. 

Multiply  number  of  cube  feet  by  64.3  for  salt  water  (62.425  for  fresh), 
divide  product  by  evaporation  per  Ib.  of  fuel  consumed,  and  quotient  will 
give  consumption  in  Ibs.  per  hour. 

EXAMPLE. — Cylinder  of  a  marine  engine  is  95  ins.  in  diameter  by  10  feet  stroke 
of  piston;  pressure  of  steam  in  steam-chest  is  15.3  Ibs.  per  sq.  inch,  cut  off  at  .5 
stroke;  number  of  revolutions  14.5,  and  evaporation  estimated  at  ':  5  Ibs.  of  salt 
water  per  Ib.  of  coal;  what  is  consumption  of  coal  per  hour,  when  density  of  water 
is  maintained  at  2-32?  (See  Saturation,  page  726.) 

Volume  of  steam  at  above  pressure,  compared  with  water  (15.3-1-14.7),  =  838. 
Area  of  95  ins.  -{-2.5  per  cent,  for  clearance  -r- 144  =  50.45  cube  feet.  Point  of  cut- 
ting off  5  feet-}-  2.5  per  cent.  =  5  feet  1.5  ins.,  and  50.45  X  5  feet  1.5  ins,  x  14.5  X  a 
X  60  =  449  887  cube  feet  steam  per  hour. 


STEAM. 


Hence,  449  887  -=-  838  =  536.  86  cube  feet  water,  which,  X  64.3  =  34  520  Ibs.  ,  which 
-f-  8.  5  =  4061  Ibs.  coal  per  hour. 

NOTE.—  Elements  given  are  those  of  one  engine  of  steamer  Arctic,  and  consump- 
tion of  clean  fuel  (selected)  for  a  run  of  12  days  (one  engine)  was  3820  Ibs.  per  hour. 

Utilization  of  Coal  in  a  Marina  Boiler. 

Experiment  gives  from  .55  to  .8  per  cent,  of  the  heat  developed  in  the 
combustion  of  coal,  as  utilized  in  the  generation  of  steam.  Ordinarily  it 
may  be  safely  taken  at  ,65. 


SALINE    SATURATION   IN    BOILERS. 
Average  sea-water  contains  per  100  parts : 
Chloride  of  sodium  (com.  salt) .  .2.5;  Chloride  of  magnesium  .33.. 

Sulphuret  of  magnesium 53;  Sulphuret  of  lime ox.. 

Carbonate  of  lime  and  of  magnesia 


=  2.83 

=    -54 

.02 


Saline  matter. 
Water , 


Hence,  sea-water  contains  .0339111  part  of  its  weight  of  solid  matter  in  solution, 
and  is  saturated  when  it  contains  36.37  parts. 

Mean  quantity  of  salts,  or  solid  matter,  in  solution,  is  3.39  per  cent,  three  fourths 
of  which  is  common  salt. 

Removal   of  Incrustation   of  Scale   or    Sediment. 

Potatoes,  in  proportion  of  .033  weight  of  water.  Molasses,  in  proportion  of  1.6 
Ibs.  per  IP  of  boiler.  Oak,  suspended  in  the  water,  and  Mahogany  or  Oak  sawdust, 
and  Tanner's  and  Slippery  Elm  bark,  renewed  frequently,  according  to  volume  of  it, 
and  the  evaporation  of  the  water.  Muriate  of  Ammonia  and  Carbonate  of  Soda,  in 
quantity  to  be  determined  by  observation. 

BLOWING    OFF. 

To  Compute  Loss  of  Heat  l>y  Blowing  Off  of  Saturated 
Water   from   a   Steam-boiler. 

E"*"*  =  proportion  of  heat  lost,  S  —  T  x  E  =  heat  required  from  fuel  for 


water  evaporated  in  degrees,  and 


loss  of  heat  per  cent.    S  representing 


-  —  ,,  „  ,  . 

b  -  1    Jii  -j-  t 

sum  of  sensible  and  latent  heats  of  water  evaporated,  T  temperature  of  feed  water, 
t  difference  in  temperature  of  water  blown  off  and  that  supplied  to  boiler,  E  volume 
of  water  evaporated,  proportionate  to  that  blown  off,  the  latter  being  a  constant  quan- 
tity, and  represented  by  i. 

Values  of  E  at  following  degrees  of  saturation,  and  volumes  to  be  blown  off: 


... 

{j 

Is! 

32. 

| 

1  f 

1*1 

>  s 

32- 

Jrf 

1  ^ 

l-l 

^    S 

32. 

l« 

«     ?ta 

|a; 

>    S 

1-25 

'•35 

•25 

•35 
5 

4 
3 

2 

1.65 

I!  85 

•65 

•75 
85 

i-33 
1.18 

2 
2.25 
2-5 

I 

1.25 

i 
.8 
66 

2-75 
3 
2.25 

2 
2  25 

•57 
•5 
•45 

Thus,  when  water  in  a  boiler  is  maintained  at  a  density  of  —  ,  i  volume  of  it  U 

32 
evaporated,  and  an  equal  volume,  or  i,  is  to  be  blown  off. 

Hence  i  -j-  i  —  i  —  i  =.  ratio  of  volume  evaporated  to  volume  blown  off. 

ILLUSTRATION  i.—  Point  of  blowing  off  is  2  (32),  pressure  of  steam  is  15  3  Ibs.,  me* 
aurial  gauge,  and  density  of  feed  water  i  (32);  what  is  proportion  of  heat  lost? 

8  =  11578°      T=ioo°.      t=.  15.3  +  14.7  =250.4°  —  100°  =  150.4°.      E  =  i. 


STEAM.— STEAM-ENGINE.  727 

2.— Assume  point  of  blowing  off  1.75  (32);  what  would  be  loss  of  heat  per  cent,  in 
preceding  case? 

E  = .  75. -^^ ; =  15-9  per  cent,  lost  by  blowing  off. 

1157-8  —  ioo  X. 75  +  i5o-4 

3.— Assume  elements  of  preceding  case.  What  is  total  heat  required  from  fuel 
for  water  evaporated  ? 

1157.8  — ioo  X -75  =  793- 35°- 

To    Compute   Volume   of*  "Water    Blown.    Off   to    that 
Evaporated. 

When  Degree  of  Saturation  is  Given.  RULE. — Divide  i  by  proportionate 
volume  of  water  evaporated  to  that  blown  off,  or  value  of  E  as  above,  for 
degree  of  saturation  given,  and  quotient  will  give  number  of  volumes  blown 
off  to  that  evaporated. 

ILLUSTRATION. — Degree  of  saturation  in  a  marine  boiler  is  — -  ;  what  is  volume 
of  water  blown  off? 

E  =  i.25.    Then  1-^-1.25  =  .8  blown  off. 

Proportional  Volumes  of  Saline  Matter  in  Sea-water. 

Atlantic,  South. .  i  in  24 


Dead  Sea. 


North. .  i 


2-59 


Baltic i  in  152  I  British  Channel 1  in  28 

Black  Sea i  "    46    Mediterranean i  "  25 

Red  Sea i  "  131  |  Atlantic,  Equator. .  i  "  25 

When  saline  matter  at  temperature  of  its  boiling-point  is  in  proportion  of  10  per 
cent. ,  lime  will  be  deposited,  and  at  29. 5  per  cent  salt. 

Temperature  of  water  adds  much  to  extent  of  saline  deposits 


STEAM-ENGINE. 

The  range  of  proportions  here  given  is  to  meet  the  requirements  of 
variations  in  speed,  pressure,  length  of  stroke,  draught  of  water,  etc., 
in  the  varied  purposes  of  Marine,  River,  and  Land  practice. 

CONDENSING. 

For  a  Range  of  Pressures  of*  from   3O  to  SO  Itos.  (Mercu- 
rial   Grange)   per    Sq..  Inch.,  Ciit    Off   at    Half  Stroke. 

Piston-rod.  Cylinder  or  Air-pump  (Wrought  /row),  .1  to  .14  of  its  diam. ; 
(Steel\  .08  diam. ;  and  {Copper  or  Brass),  .11  and  .125. 

Condenser  (Jet).  .Volume,  .35  to  .6  of  cylinder.  (Surface.)  Brass  tubes, 
16  to  19  B  W  G,  .625  to  .75  in  diameter  by  from  5  to  10  feet  in  length,  and 
.75  to  1.25  in  pitch,  condensing  surfaces,  .55  to  .65  area  of  evaporating  sur- 
face of  boiler  with  a  natural  draught;  .7  to  .8  with  a  blower,  jet,  or  like 
draught.  Or,  for  a  temperature  of  water  of  60°,  1.5  to  3  sq.  feet  per  IIP. 

With  a  very  effective  and  sufficient  circulating  pump,  areas  may  be  reduced  to 
.5  and  .6. 

Effect  of  vertical  tube  surface,  compared  to  horizontal,  is  as  10  to  7. 

Air-pump  (Single  acting  and  direct  connection).  Volume  from  .15  to  .2 
steam  cylinder.  Or, 2-75>  For  Double  acting  put  4  for  2.75.  V  and  v 

representing  volumes  of  condensing  and  condensed  water  per  cube  foot,  and  n  strokes 
of  piston  per  minute. 

Foot  and  Delivery  Valves.    Area,  .25  to  .5  area  of  air-pump. 

Delivery  Valve  (Out-board).  With  a  Reservoir.  Area  from  .5  to  .8  Foot 
valve. 

NOTE.— Velocity  of  water  through  these  valves  should  not  exceed  12  feet  per 
second. 


728  STEAM-ENGINE. 


Steam  and  Exhaust  Valves.  —  (Popnet).^-^=ai'ea  for  steam,  asn     for 

"  24000  '   20000     J 

exhaust;  (Slide).  -    for  steam,  and  -    for  exhaust,     a  representing 

"   30000     J  22  750    J 

urea  of  steam  cylinder  in  sq.  ins.,  s  stroke  of  piston  in  ins.,  and  n  number  of  revolu- 
tions per  minute. 

Injection  Pipes.  —  One  each  Bottom  and  Side  to  each  condenser;  area  of 
each  equal  to  supply  70  times  volume  of  water  evaporated  when  engine  is 
ivorking  at  a  maximum  ;  and  in  Marine  and  River  engines  the  addition  of 
a  Bilge,  which  is  properly  a  branch  of  bottom  pipe. 

NOTE  i.—  Proportions  given  will  admit  of  a  sufficient  volume  of  water  when  en- 
gine is  in  operation  in  the  Gulf  Stream,  where  the  water  at  times  is  at  temperature 
of  84°,  and  volume  required  to  give  water  of  condensation  a  temperature  of  100°  is 
70  times  that  of  volume  evaporated. 

2.  Velocity  of  flow  of  water  through  cock  or  valve  20  feet  per  second  in  river  or  at 
shallow  draught,  and  30  feet  in  sea  or  deep  draught  service. 

Feed  Pump.*—  (Single  acting,  Marine),  Volume,  .006  to  .01  steam  cylinder. 
(River  and  Land),  or  when  fresh  water  alone  or  a  surface  condenser  is  used, 
.004  to  .007. 

NON-CONDENSING. 

If  or  a  Range  of  ^Pressures  of  from.  5O  to  ISO  Has.  GVIerou.- 
rial   Grange)  per   Sq..  Inch.,  Cnt  Off'  at  Half  Strolie. 

Piston-rod.  —  (Wrought  Iron),  Diam.,  .125  to  .2  steam  cylinder.  (Steel), 
i.  to  1.6  steam  cylinder. 

Steam  and  Exhaust  Valves.  —  Area  is  determined  by  rules  given  for  them 
in  a  condensing  engine,  using  for  divisors  30  ooo  and  22  750. 

A  decrease  in  volume  of  cylinder  is  not  attended  with  a  proportionate  decrease 
of  their  area,  it  being  greater  with  less  volume. 

Feed-pump.*  —  (Single  acting,  Marine),  Volume,  .008  to  .016  steam  cylin- 
der. (River  and  Land),  or  where  fresh  water  alone  is  used,  .005  to  .on". 

Greneral   Rules. 
Engines. 

Cylinder.     Thickness.—  (  Vertical),  ^£  =  t  ;  (Horizontal),  —  -  =  t  ;    (In- 

clined), divide  by  2000  in  a  ratio  inversely  as  sine  of  angle  of  inclination. 

D  representing  diameter  of  cylinder,  p  extreme  pressure  injbs  per  sq.  inch  that  it 
may  be  subjected  to,  and  t  thickness  in  ins. 

Shafts,  Gudgeons,  Journals,  etc.  To  resist  Torsion.   See  rules,  pp.  790,  796. 
Coupling  Bolts.     —    /-—  =  d.     n  representing  number  of  bolts,  D  diameter 

of  shaft,  d'  distance  of  centre  of  bolts  from  centre  of  shaft,  and  d  diameter  of  bolts, 
all  in  ins. 

Cross-head,  Wrought-iron.    (Cylinder),  ~±—  =  S,  and    /-r-  =  d,  or  —  =,  b. 

a  representing  area  of  cylinder  in  sq.  ins.,  I  length  of  cross-head  between  centres  of  its 
journals  in  feet,  and  S  product  of  square  of  depth  d,  and  breadth,  b,  of  section,  both 

in  ins.       (Air-pump),  —  =  S,  and  as  above  for  d  and  b. 


If  section  of  either  of  them  is  cylindrical,  for  S  put  -\Xs  X  1.7. 
Diam.  of  boss  twice,  and  of  end  journals  same  as  that  of  piston-rod.    Sec- 
tion at  ends  .5  that  of  centre. 

*  See  Formulas,  page  736. 


STEAM-ENGINE.  729 

Steam-pipe.— Its  area  should  exceed  that  of  steam-valve,  proportionate  to 
its  length  and  exposure  to  the  air. 

Connecting  -  rod.  —  Length,  2.25  times  stroke  of  piston  when  it  is  at  all 
practicable  to  afford  the  space ;  when,  however,  it  is  imperative  to  reduce 
this  proportion,  it  may  be  twice  the  stroke. 

Comparative  friction  of  long  and  short  connecting-rods  is,  for  length  of  stroke  ot 
piston,  12  per  cent,  additional;  twice,  3  per  cent. ;  and  for  thrice,  1.33. 

Neck.  —  Diam..  i  to  i.i  that  of  piston-rod.  Centre  of  body  (Horizontal), 
1.25  ins. ;  ( Vertical),  .06  inch  per  foot  of  length  of  rod. 

With  two  connecting-rods  or  links,  area  of  necks. 65  to. 75  area  of  attached 
piston-rod. 

Straps  of  Connecting-rods,  Links,  etc.  —  (Strap),  area  at  its  least  section 
.65  neck  of  attached  rod  ;  (Gib  and  Key),  .3  diam.  of  neck,  width,  1.25  times, 
(Slot)  1.35  times  (Draff)  of  keys  .6  to  .8  inch  per  foot.  Distance  of  Slot 
from  end  of  rod  .5  diam.  of  pin. 

/P I 
Pins  (Cranks,  Beams,  etc.).     3/— .  355  =  d.    P  representing  pressure  or  thrust 

of  rod  or  beam,  I  length  of  journal  in  ins.,  and  C,/or  Wrought  iron  =  640,  Cast,  560. 
Puddled  steel,  600,  and  Cast,  1200. 

Length,  1.3  to  1.5  times  their  diam.,  and  pressure  should  not  exceed  750 
Ibs.  per  sq.  inch  for  propeller  engine,  and  1000  for  side-wheel. 

Cranks  (Wrought-iron). — Hub,  compared  with  neck  of  shaft,  1.75  diam., 
and  i  depth.  Eye,  compared  with  pin,  2  diam.,  and  1.5  depth.  Web,  at  pe- 
riphery of  hub,  width,  .7  width,  and  in  depth  .5  depth  of  hub ;  and  at  periph- 
ery of  eye,  width,  .8  width,  and  in  depth,  .6  depth  of  eye. 

(Cast-iron.)  Diameters  of  Hub  and  Eye  respectively,  twice  diam.  of  neck 
of  shaft,  and  2.25  times  that  of  crank  pin. 

Radii  for  fillets  of  sides  of  web  .  5  width  of  web  at  end  for  which  fillet  is  designed ; 
for  fillets  at  back  of  web,  .5  that  at  sides  of  their  respective  ends. 

Beams,  Open  or  Trussed. — Length  from  centres  1.8  to  2  stroke  of  piston, 
and  depth  .5  length.  If  strapped,  Strap  at  its  least  dimensions  .9  area  of 
piston-rod,  its  depth  equal  to  .5  its  breadth.  End  centre  journals  each  i,  and 
main  centre  journals  2.5  times  area  of  piston  or  driving-rod. 

This  proportion  for  strap  is  when  depth  of  beam  is  .5  length,  as  above;  conse- 
quently, when  its  depth  is  less,  area  of  strap  must  be  increased;  and  when  depth  of 
strap  is  greater  or  less  than  .5  width,  its  area  is  determined  by  product  of  its  6  <Z2, 
being  same  as  if  its  depth  was  .5  its  width. 

(Cast-iron).  Area  of  Section  of  Centre.  -  ~=-=A.  p  representing 
extreme  pressure  upon  piston  in  Ibs.,  d  depth  in  ins.,  and  I  length  in  feet. 

Depth  at  centre  .5  to  .75  diam.  of  cylinder,  and,  when  of  uniform  thick- 
ness, a  thickness  of  not  less  than  .1  of  depth. 

Vibration  of  End  Centres. — 1-^-2  —  V(l-s-2)2  —  (s -~- 2)2  =  vibration  at  each 
end  ;  s  representing  stroke  of  piston,  in  feet. 

Plumber  Blocks  (Shaft).— Binder  d  ^—C  =  depth.    d  representing  diam. 

of  bolts  when  two  to  binder,  I  distance  between  bolts,  b  breadth  of  binder,  all  in  ins. , 
and  Cfor  wrought  iron  i,  steel  .85,  and  cast  iron  .2. 

Holding-down  Bolts.  P  -r-  3  C  =  area  at  base  of  thread  of  each  bolt.  C  for  mild 
steel  for  small  and  large  bolts  6000  and  7000,  for  wrought  iron  4500  and  6000,  if  but 
two  are  used. 

Binder  (Brass).    -  J^  =  depth. 


73O  STEAM-ENGINE. 

Cocks.— Angles  of  sides  of  plug  from  7°  to  8°  from  plane  of  it. 

Pumps. — Velocity  of  water  in  pump  openings  should  not  exceed  500  feet 
per  minute. 

Fly-wheels  and  Governors. — See  Rules,  pages  451  and  452. 
"Water- wheels. 

Water-wheels  (Arms). — Number  from  .75  to  .8  diam.  of  wheel  in  feet. 
(Blades)  Wood. — For  a  distance  of  from  5  to  5.5  feet  between  arms,  thick- 
ness from  .09  to  ,i  inch  for  each  foot  of  diam.  of  wheel. 

Area  of  blades,  compared  with  area  of  immersed  amidship  section  of  a 
vessel,  depends  upon  dip  of  wheels,  then*  distance  apart,  model  and  rig  of 
vessel. 

In  River  service,  area  of  a  single  line  of  blade  surface  varies  from  .3  to  .4 
that  of  immersed  section;  in  Bay  or  Sound  service,  it  varies  from  .15  to  .2; 
and  in  Sea  service,  it  varies  from  .07  to  .1. 

NOTE.— A  wrought-iron  blade  .625  inch  thick  bent  at  a  stress  withstood  by  an 
oak  blade  3.5  ins.  thick. 

Radial   and.   Feathering. 

Radial. — Loss  of  effect  is  sum  f>f  Joss  by  oblique  action  of  wheel  blades 
upon  the  water,  their  slip,  and  thrust  and  drag  of  arms  and  blades  as  they 
enter  and  leave  the  water. 

Loss  by  oblique  action  is  computed  by  taking  mean  of  square  of  sines  of 
angles  of  blades  when  fully  immersed  in  the  water. 

Loss  by  oblique  action  of  blades  of  wheel  of  steamer  Arctic,  when  her  wheels 
were  immersed  7  feet  9  ins.  and  5  feet  9  ins.,  was  25.5  and  18.5  per  cent.,  which 
was  the  loss  of  useful  eftect  of  the  portion  of  total  power  developed  by  engines, 
which  was  applied  to  wheels. 

Feathering. — Loss  of  effect  is  confined  to  thrust  and  drag  of  arms  and 
blades  as  they  enter  and  leave  the  water. 

Comparative  Effects.—  In  two  wheels  of  a  like  diameter  (26  feet,  and  6  feet  immer- 
sion), like  number  and  depth  of  blades,  etc.,  the  losses  are  as  follows  : 

Radial 26.6  per  cent.   |  Feathering 15.4  per  cent. 

Loss  of  effect  by  thrust  and  drag  in  a  feathering  wheel,  having  these  elements 
and  included  in  the  above  given  loss,  is  computed  at  2  per  cent. 

Relative  loss  of  effect  of  the  two  wheels  is,  approximately,  for  ordinary  immer- 
sions, 20  and  15  per  cent,  from  circumference  of  wheel. 

2  d* d'3 

Centre  of  Pressure,  —      _       —  d  =  c.    d  and  d'  representing  depths  of  blades 

below  surface  of  water,  and  c  centre  of  pressure,  all  in  like  dimensions,  from  bottom 
edge. 

In  the  cases  here  given,  centres  of  pressure  are  as  follows: 
Radial 6.4  ins.   |  Feathering 8.5  ias. 

Propellers. 

Propellers  (Screw).  —  Pitch  should  vary  with  area  of  circle  described  by 
screw  to  area  of  midship  section  of  vessel. 

AREA,  TWO-BLADED. 

Area  of  disk  of  propeller  to  mid- )  , 

ship  section  being  i  to } 

Ratio  of  pitch  to  the  diameter  of )  ~~T~ 

propeller  is  i  to }  '8      '-02     *•"      »-2     *-*7     '-3'      M      '-47 

For  Four-bladed  screws,  multiply  ratio  of  pitch  to  diam.  as  given  above, 
by  1.35.  Length,  .166  diam. 


STEAM-ENGINE. 


731 


Slip. — Slip  of  a  screw  propeller  is  directly  as  its  pitch,  and  economical 
effect  of  a  screw  is  inversely  as  its  pitch  j  greater  the  pitch  less  the  effect. 

An  expanding  pitch  has  less  slip  than  a  uniform  pitch,  and,  consequently, 
is  more  effective. 

To    Compute   Thrust   of  a   Propeller. 

IIP  —-  =  T.    S  representing  speed  of  vessel  in  knots  per  hour. 


SLIDE  VALVES. 
All  Dimensions  in  Inches. 

To  Compute  Lap  required  on  Steam  End,  to  Cut-off  at 
any   given    IPart   of*  Stroke   of  Piston. 

RULE. — From  length  of  stroke  subtract  length  of  stroke  that  is  to  be  made 
before  steam  is  cut  off;  divide  remainder  by  stroke,  and  extract  square 
root  of  quotient. 

Multiply  this  root  by  half  throw  of  valve,  from  product  subtract  half  lead, 
and  remainder  will  give  lap  required. 

EXAMPLE. — Having  stroke  of  piston  60  ins.,  stroke  of  valve  16  Ins.,  lap  upon  ex- 
haust side  .5  in.  =  one  thirty-second  of  valve  stroke,  lap  upon  steam  side  3.25  ins., 
lead  2  ins.,  steam  to  be  cut  off  at  five  sixths  stroke;  what  is  the  lap? 

60  —  |<  of  60  =  10.       /—  =  .408.    .408  x  *—  —  3.264,  and  3.264  —  -j  =  2.264  »«* 

To  Ascertain   Lap   required   on    Steam   End,  to  Cut-off 
at   various    Portions   of  Stroke. 

Distance  of  piston  from  end  of  its  stroke  when  steam  is  cut  off, 
in  parts  of  length  of  its  stroke. 


Valve 
without  Lead. 

Lap  in  parts  of) 
stroke j 


•354 


323 


.286 


.27 


•25 


A 

t 

1 

A 

A 

.228 

.204 

•"77 

•'44 

102 

ILLUSTRATION.—  Take  elements  of  preceding  case. 
Under  £  is  204,  and  .204  X  16  =  3.  264  ins.  lap. 

When  Valve  is  to  have  Lead.—  Subtract  half  proposed  lead  from  lap  as- 
certained by  table,  and  remainder  will  give  proper  lap  to  give  to  valve. 
If,  then,  as  last  case,  valve  was  to  have  2  ins.  lead,  then  3.264  —  2-7-  2  =  2.264  ins- 

To  Compute  at  what  Part  of*  Stroke  any  given  Lap  on 
Steam  Side  will  Cvat  off. 

RULE.  —  To  lap  on  steam  side,  as  determined  above,  add  lead  ;  divide  sum 
by  half  length  of  throw  of  valve.  From  a  table  of  natural  sines  (pages  390- 
402)  find  the  arc,  sine  of  which  is  equal  to  quotient  ;  to  this  arc  add  90°, 
and  from  their  sum  subtract  arc,  cosine  of  which  is  equal  to  lap  on  steam 
side,  divided  by  half  throw  of  valve.  Find  cosine  of  remaining  arc,  add  i 
to  it,  and  multiply  sum  by  half  stroke,  and  product  will  give  length  of  that 
part  of  stroke  that  will  be  made  by  piston  before  steam  is  cut  off. 

EXAMPLE.  —  Take  elements  of  preceding  case. 


Cos.  (sin.  *^i±-2  +9o°-cos.  i£)  +  ,  X  |=  <m  (3,°  ,3'+  9°°-73°  341 

=  48°  39',  and  cos.  48°  39'+  1  x  —  =  1.66  X  30  =  49.  8  ins. 

To   Ascertain    Breadth   of*  Ports. 

Half  throw  of  valve  should  be  at  least  equal  to  lap  on  steam  side,  added  to  breadth 
of  port.  If  this  breadth  does  not  give  required  area  of  port,  throw  of  valve  must  be 
increased  until  required  area  is  attained. 


732 


STEAM-ENGINE. 


Portion  of  Stroke  at  -which  Exhausting   Port  is  Closed 
and    Opened. 


Lap  on 

Exhaust 

Portion  of  Stroke  at  which  Steam 

Exhaust 

Portion  of  Stroke  at  which  Steam 

Side  of 

is  cat  off. 

Side  of 

ia  cut  off. 

Valve  in 

Valve  in 

Parts  of 
Us  Throw. 

J 

A: 

i 

A 

i 

1 

IL 

Parts  of 
its  Throw. 

i 

A 

j 

A 

i 

i 

A 

A 

B 

.125 

.178 

.161 

•  143 

.126 

.109 

•093 

.074 

•125 

•033 

.026 

.019 

.012 

.008 

.004 

.001 

.0625 

.us 

.1 

.085 

.071 

.058 

•043 

.0625 

.06 

.052 

.04 

•03 

.022 

•015 

.008 

.031  25 

•  113 

.101 

.085 

.069 

•053 

•043 

•033 

.031  25 

•073 

.066 

.051 

.042 

•033 

.023 

.013 

o 

.092 

.082 

.067 

•055 

.041 

•033 

.022 

0 

.092 

.082 

•055 

.044 

•033 

.022 

Units  in  columns  of  table  A  express  distance  of  piston,  in  parts  of  its  stroke,  from 
end  of  stroke  when  exhaust  port  in  advance  of  it  is  closed;  and  those  in  columns 
of  table  B  express  distance  of  piston,  in  parts  of  its  stroke,  from  end  of  its  stroke 
when  exhaust  port  behind  it  is  opened. 

ILLUSTRATION.— A  slide  valve  is  to  be  cut  off  at  one  sixth  from  end  of  stroke,  lap 
on  exhaust  side  is  one  thirty-second  of  stroke  of  valve  (16  ins. ),  and  stroke  of  piston 
is  60  ins.  At  what  point  of  stroke  of  piston  will  exhaust  port  in  advance  of  it  be 
closed  and  the  one  behind  it  open. 

Under  one  sixth  in  table  A,  opposite  to  one  thirty-second,  is  .053,  which  x  60, 
length  of  stroke  =  3. 18  ins. ;  and  under  one  sixth  in  table  B,  opposite  to  one  thirty- 
second,  is  .033,  which  x  60=  1.98  ins. 

If  lap  on  exhaust  side  of  this  valve  was  increased,  effect  would  be  to  cause  port  in 
advance  of  valve  to  be  closed  sooner  and  port  behind  it  opened  later.  And  if  lap  on 
exhaust  side  was  removed  entirely,  the  port  in  advance  of  piston  would  be  shut, 
and  the  one  behind  it  open,  at  same  time. 

Lap  on  steam  side  should  always  be  greater  than  that  on  exhaust  side,  and  differ- 
ence  greater  the  higher  the  velocity  of  piston. 

In  fast-running  engines,  alike  to  locomotives,  it  is  necessary  to  open  exhaust  valve 
before  end  of  stroke  of  piston,  in  order  to  give  more  time  for  escape  of  the  steam. 

To   Compute    Stroke   of  a    Slide   Valve. 

RULE.— To  twice  lap  add  twice  width  of  a  steam  port  in  ins.,  and  sum 
will  give  stroke  required. 

Expansion  by  lap,  with  a  slide  valve  operated  by  an  eccentric  alone,  cannot  be 
extended  beyond  one  third  of  stroke  of  a  piston  without  interfering  with  efficient 
operation  of  valve;  with  a  link  motion,  however,  this  distortion  of  the  valve  is 
somewhat  compensated.  When  lap  is  increased,  throw  of  eccentric  should  also  be 
increased. 

When  low  expansion  is  required,  a  cut-off  valve  should  be  resorted  to  in  addition 
to  main  valve. 

Xo    Com.pu.te   Distance    of    a   I?iston    from    End   of  its 
Stroke,  -when    Lead    produces   its    Effect. 

RULE.  —  Divide  lead  by  width  of  steam  port,  both  in  ins.,  and  term  the 
quotient  sine ;  multiply  its  corresponding  versed  sine  by  half  stroke,  and 
product  will  give  distance  of  piston  from  end  of  its  stroke,  when  steam  is  ad- 
mitted for  return  stroke  and  exhaustion  ceases. 

EXAMPLE.— Stroke  of  piston  is  48  ins.,  width  of  port  2.5  ins.,  and  lead  .5  inch; 
what  will  be  distance  of  piston  from  end  of  stroke  when  exhaustion  commences? 


=  «ine,  ver.  sin.  of  .2 


=  .0202,  and  .0202  x  —  =  -4848  ins. 


To   Compute    Lead,  when   Distance   of  a   Piston   from 

the    End    of  Stroke    is   given. 

RULE. — Divide  distance  in  ins.  by  half  stroke  in  ins.,  and  term  quotient 
versed  sine ;  multiply  corresponding  sine  by  width  of  steam  port,  and  prod- 
uct will  give  lead. 
EXAMPLE.— Assume  elements  of  preceding  case. 

4848  -4- 1-  = . oaoa  =  vcr.  sin. .  and  sine  of  ver.  sin.  0202  = .  2,  and .  2  X  t.  5  = .  5  inch, 


STEAM-ENGINE.  733 

To  Compute  13istari.ee  of*  a  Piston  from  Knd  of  its  Stroke, 
-wlieii   Steam   is   admitted  for  its   tteturn   Stroke. 

RULE. — Divide  width  of  steam  port,  and  also  that  width,  less  the  lead,  by 
.5  stroke  of  slide,  and  term  quotients  versed  sines  first  and  second.  Ascer- 
tain their  corresponding  arcs,  and  multiply  versed  sine  of  difference  between 
first  and  second  by  .5  stroke,  and  product  will  give  distance. 

EXAMPLE. — Assume  elements  of  preceding  case,  lap  =  .5  inch,  and  stroke  of 
Elide  6  ins. 

^~-  and  2'£~25  =  - 8333,  and  .667  and  ver.  sin.  80°  24'  <*,  7o°  33'  x  —  = . 3528  inch. 

To  Compute  Lap  and  Lead  of*  Locomotive  Valves. 
To  cut  off  at  .33,  .25,  and  .125  of  stroke  of  piston,  lap  ^=289,  .25,  and  .177  t,  outside 
lead  =  .07  t,  and  inside  lead  =  .3  t    t  representing  stroke  of  valve,  all  in  ins. 

HORSE-POWER 

Horse-power  is  designated  as  Nominal,  Indicated,  and  A  dual. 

Nominal,  is  adopted  and  referred  to  by  Manufacturers  of  steam-engines, 
in  order  to  express  capacity  of  an  engine,  elements  thereof  being  confined 
to  dimensions  of  steam  cylinder,  and  a  conventional  pressure  of  steam  and 
speed  of  piston. 

Indicated,  designates  full  capacity  in  the  cylinder,  as  developed  in  opera- 
tion, and  without  any  deductions  for  friction. 

Actual,  refers  to  its  actual  power  as  developed  by  its  operation,  involving 
elements  of  mean  pressure  upon  piston,  its  velocity,  and  a  just  deduction  for 
friction  of  operation  of  the  engine. 

To   Compute   Horse-power   of  an    Kiigine. 

Nominal . —  Non-condensing, ,  and  Condensing, =  IP  D  repre- 
senting diameter  of  cylinder  in  ins.,  and  v  velocity  of  piston  in  feet  per  minute. 

Non-condensing  is  based  upon  uniform  steam-pressure  of  60  Ibs.  per  sq. 
inch  (steam-gauge),  cut  off  at  .5  stroke,  deducting  one  sixth  for  friction  and 
losses,  with  a  mean  velocity  of  piston,  ranging  from  250  to  450  feet  per 
minute. 

Condensing  is  based  upon  uniform  steam-pressure  of  30  Ibs.  per  sq.  inch 
(steam-gauge),  cut  off  at  .5  stroke,  deducting  one  fifth  for  friction  and 
losses,  with  a  mean  velocity  of  piston  of  300  feet  per  minute  for  an  engine 
of  short  stroke,  and  of  400  feet  for  one  of  long  stroke. 

Actual. — Non-condensing.     -        — =H*.    A  representing 

area  of  cylinder  in  sq.  ins.,  P  mean  effective  pressure  upon  cylinder  piston,  inclusive 
of  atmosphere,  f  friction  of  engine  in  all  its  parts,  added  to  friction  of  load,  both  in 
[bs.  per  sq.  inch,  s  stroke  of  piston  in  feet,  and  r  number  of  revolutions  per  minute. 

Sum  of  these  resistances  is  from  12.5  to  20  per  cent.,  according  to  pressure  of 
eteam,  being  least  with  highest  pressure. 

*  This  value  is  best  obtained  by  an  Indicator;  when  one  is  not  used,  refer  to  rule 
and  table,  pp.  710-12.  In  estimating  value  of  P,  add  14.7  Ibs.,  for  atmospheric  press- 
ure,  to  that  indicated  by  steam  gauge  or  safety-valve.  Clearance  of  piston  at  each 
end  of  cylinder  is  included  in  this  estimate. 

t  This  value  may  be  safely  estimated  in  engines  of  magnitude  at  1.5  to  2  Ibs.  per 
sq.  inch,  for  friction  of  engine  in  all  its  parts,  and  friction  of  load  may  be  taken  at  5 
to  7.5  per  cent,  of  remaining  pressure.  Sum  of  these  resistances  in  ordinary  marine 
engines  is  from  10  to  20  per  cent,  according  to  pressure  of  steam,  exclusive  of  power 
required  to  deliver  water  of  condensation  at  level  of  discharge  or  load  line  of  a  ves- 
sel. For  pressure  representing  friction  for  different  designs  and  capacities  of  en- 
gines as  estimated  by  English  authority,  see  pp.  473-5  and  662. 

a  O 


734 


STEAM-ENGINE. 


ILLUSTRATION.—  Diameter  of  cylinder  of  anon-condensing  engine  is  loins.,  stroke 
of  piston  4  feet,  revolutions  45  per  minute,  and  mean  pressure  of  steam  (steam 
gauge)  60  Ibs.  per  sq.  inch. 

A  =  78-54  sq.  ins.   ¥=60+14.7  =  74.7^3.  /=2-5-f  (60+14.7—  2-5)X-  075  =  7.92  Ibs. 

Then  78.54  X  (60  +14.7  -  7.92  +  1^7)  X  2  X  4  X  45        4  6  jp 

33000 

NOTE  L—  Power  of  a  non-condensing  engine  is  sensibly  affected  by  character  of  its 
exhaust,  as  to  whether  it  is  into  a  heater,  or  through  a  contracted  pipe,  to  afford  a 
blast  to  combustion. 

2.—  If  an  indicator  is  not  used  to  determine  pressure  of  steam  in  a  cylinder,  a 
safe  estimate  of  it,  when  acting  expansively,  is  .9  of  full  pressure,  and  when  at  full 
stroke  from  .75  to  .8. 


Condensing. 


33000 

Power  required  to  work  the  air-pump  of  an  engine  varies  from  .7  to  .9  Ibs.  per  sq. 
inch  upon  cylinder  piston. 

ILLUSTRATION.  —  Diameter  of  cylinder  of  a  marine  steam  engine  ts  60  ins.,  stroke 
of  piston  10  feet,  revolutions  15  per  minute,  pressure  of  steam  50  Ibs.  per  sq.  inch, 
cut  off  at  25  stroke,  and  clearance  2  per  cent. 

A  =  2827.455.  ins.  P  (per  Ex.,  page  713)  =  28.62  Ibs.  f=  1.54-28.62  —  1.5  X  .05 
=  2.467  Ibs.  __  _ 

Then  2827.4  X  28.66-2.856  jOOOoJOS  =  66        ^ 

33000  ,)(,^ 

From  which  is  to  be  deducted  in  marine  engines  power  necessary  to  discharge 
water  of  condensation  at  level  of  load-line,  which  is  determined  by  pressure  due  to 
elevation  of  water,  area  of  air-pump  piston,  and  velocity  of  its  discharge  in  feet  per 
second. 

Indicated.     ^^  =  ff,  and  2f2°J£  =  A. 
33  ooo  P  2  s  r 

British   Admiralty   Rule.—  Nominal.     -  —  -or-^^IP. 

33000       6000 

French.—  (Force  de  Cheval.)     1.695  D2  L  r  =  W.    D  and  L  in  meters. 
ILLUSTRATION.—  Assume  a  diameter  of  cylinder  of  .254  meters,  with  a  stroke  of 
piston  of  .3  meters  and  250  revolutions  per  minute. 

1.695  X  .2542  X  -3  X  250  =  8.18  IP. 

A  Force  de  Cheval  =  4500  kilometers  per  minute  =  32  549  foot-lbs.  =  .987  57  IP. 
One  IP  =  1.0139  Force  de  Chevaux. 

Compound    Indicated.     ALr  (—  i  hyp.  log.  R"  —  6J  .000053  —  H? 

L  representing  length  of  stroke  in  feet,  R"  combined  ratio  of  both  cylinders,  and  b 
back  pressure. 

ILLUSTRATION.—  Assume  area  of  cylinder  3  sq.  ins.,  stroke  6  feet,  one  stroke  of 
piston,  and  steam  60  Ibs.  per  sq.  inch,  cut  off  at  .25. 

A  —  3  sq.  ins.,  L  =  6  feet,  n  =  *  stroke,  P  =  60  Ibs.,  R"  =  5.969,  6  =  3  Ibs. 
per  sq.  inch,  and  r  =  .  5,  and  i  -{-  hyp.  log.  R"  =  i  -f-  1.7865. 

Then3X6x  sxf  —  ~-  X  1  +  1.7865  —  3)  X  .000053  =  9  X  10.052X2.7865  —  3 
X  -000053  —  -0132  H>,  which,  X  2  for  i  revolution,  =  .0264  IP  per  revolution. 

To    Compvite    Volume    of  "Water    required    to    "be    Evapo- 
rated   in    an    Engine. 

RULE.  —  Multiply  volume  of  steam  expended  in  cylinder  and  steam-chests 
by  twice  number  of  revolutions,  and  multiply  product  by  density  of  steam 
at  given  pressure. 

*  f  For  reference  see  ist  and  zd  foot-note  on  previous  page. 


STEAM-ENGINE.  735 

EXAMPLE.— What  volume  of  water  will  an  engine  require  to  be  evaporated  per 
revolution,  diam.  of  cylinder  being  70  ins.,  stroke  of  piston  10  feet,  and  pressure  of 
steam  34  Ibs.  per  sq.  inch,  including  atmosphere,  cut  off  at  .5  of  stroke? 

Area  of  cylinder  =  3848. 5  ins.    10X12^-2  =  60  ins. ,  60  X  3848. 5  =  230910  cube  ins. 

Add,  for  clearance  at  one  end,  volume  of  nozzle,  steam-chest,  etc.,  17  317  cube  ins. 

Then  230910+  17  317-=- 1728  X  2  =  287.3  cube  feet,  which,  X  .001  336,  density  of 
steam  at  34  Ibs.  pressure  (see  Note  2),  =  .3838  cube  feet. 

NOTE  i.— This  refers  to  expenditure  of  steam  alone;  in  practice,  however,  a  large 
quantity  of  water  "foaming,"  differing  in  different  cases,  is  carried  into  cylinder  in 
combination  with  the  steam ;  to  which  is  to  be  added  loss  by  leaks,  gauges,  etc. 

2. — Volume  of  steam  is  readily  computed  by  aid  of  table,  pp.  708-9.  Thus,  den- 
sity or  weight  of  one  cube  foot  of  steam  at  above  pressure  =  .0835  Ibs.  Hence,  as 
62.5  Ibs. :  i  cube  foot ::  .0835  Ibs.  :  .001  336  cube  foot. 

To  Compute  "Volume  of  Circulating  "Water  required  toy 
an   £jiigine. 

"I4/"-3  —  _  v  T  representing  temperature  of  steam  upon  entering  the  con- 
denser, 1. 1',  and  t"  temperatures  of  feed  water,  of  water  of  condensation  discharged, 
and  of  circulating  water,  all  in  degrees. 

ILLUSTRATION. — Assume  exhaust  steam  at  8  Ibs.  per  sq.  inch,  temperatures  of  dis- 
charge 100°,  feed  water  120°,  and  sea- water  75°. 

Temperature  at  8  Ibs.  pressure  =  183°.     II14        —    3~I2°  =  41.95  times. 

100  —  75 

To  Compute  Volume  of  Flow  through  an  Injection  Pipe. 

RULE. — Multiply  square  root  of  product,  of  64.33  and  depth  of  centre  of 
opening  into  condenser,  from  surface  of  external  water,  added  to  height  of  a 
column  of  water  due  to  vacuum  in  condenser,  all  in  feet,  by  area  of  opening 
in  sq.  ins. ;  and  .6  product,  divided  by  2.4  (144  -j-  60)  will  give  volume  in 
cube  feet  per  minute. 

EXAMPLE. — Diameter  of  an  injection  pipe  is  5.375  ins.,  height  of  external  water 
above  condenser  6.13  feet,  and  vacuum  24.45  'ns-  >  what  is  volume  of  flow  per  min.? 

Area  of  5.375  ins.  =  22.69  ins->  c  =  .6.    Vacuum  24'45  1DS'  =  12  Ibs. :  12  X  2.24 

2.04 
feet  (sea- water)  =  26. 88  feet,  and  26. 88  -}-  6, 13  =  33.  i  feet. 

Then  V6*33  X  33^x  2ig£X^  =  g^I5  =  26l.73  ^tf^ 

To   Compute   Area  of*  an    Injection    Pipe. 

RULE.— Ascertain  volume  of  water  required  by  rule,  page  706,  in  cube  ins. 
per  second,  multiply  it  by  number  of  volumes  of  water  required  for  con- 
densation, by  rule,  page  707,  divide  it  by  velocity  due  to  flow  in  feet  per 
second,  and  again  by  12,  and  quotient  will  give  area  in  sq.  ins. 

EXAMPLE.— An  engine  having  a  cylinder  70  ins.  diam.,  stroke  of  piston  10  feet, 
revolutions  per  minute  15,  and  steam  19.3  Ibs.,  mercurial  gauge  cut  off  at  .5;  what 
should  be  area  of  its  injection  pipe  at  its  maximum  operation  ? 

Volume  of  cylinder  267.25  cube  feet,  cut  off  at  .5  =  133.625  ins. 

Density  of  steam  at  34  Ibs.  (19.3  -f  14-7)  =  -ooi  336.  Velocity  of  flow  of  injected 
water  (computed  from  vacuum  and  elevation  of  condensing  water)  33  feet  per  second. 

Thea  133.625  X  15  X  2  X  1728  -4-  60  ==  115452  cube  ins.  steam  per  second,  and 
115  452  X  .001  336  =  154.24  cube  ins.  water  per  second. 

Maximum  volume  of  water  required  to  condense  steam  is  about  70  times  volume 
of  that  evaporated,  which  only  occurs  in  the  Gulf  of  Mexico;  ordinary  requirement 
is  about  40  times. 

154.24  -f- 11.59  (=  7-5  Per  cent,  for  leakage  of  valves,  etc.)  =  165.83,  which,  X  70 
as  above,  =  n  608.  i  cube  ins.,  and  n  608. 1-^-33  X  12  =  29.31  sq.  ins. 


736 


STEAM-ENGINE. 


Coefficient  of  velocity  for  flow  under  like  conditions  =.  .6;  hence,  29.31-=-. 6  = 
48.8555.  ins. 

NOTE.— This  is  required  capacity  for  one  pipe.  It  is  proper  and  customary  that 
there  should  be  two  pipes,  to  meet  contingency  of  operation  of  one  being  arrested. 

To    Compute   Area   of*  a    Feed.   3?ump.    (Sea-water. ) 

RULE. — Divide  volume  of  water  required  in  cube  ins.  by  number  of  single 
strokes  of  piston,  both  per  minute,  and  divide  quotient  by  stroke  of  pump,  in 
ins. ;  multiply  this  quotient  by  6  (for  waste,  leaks,  "running  up,"  etc.),  and 
product  will  give  area  of  pump  in  sq.  ins. 

EXAMPLE.— Assume  volume  to  be  5  cube  feet  and  revolutions  of  engine  15  per 
minute,  with  a  stroke  of  pump  of  3.5  feet. 


5  X  1728 
15 


=  576,  which -7-3. 5  x  12  =  13.72,  and  13.72  X  6  =  82.32  sq.  ins. 


NOTB. — In  fresh  water,  this  proportion  may  be  reduced  one  half. 

STEAM-INJECTOR.       "Wm.     Sellers     <te     Co.,    Incorporated. 

Self-acting  Injector,  1887,  Class  N,  Improved. 
Volume    of    "Water    Discharged    per    Hour. 


No. 

I 

60 

ressure  of  g 
90 

>teaminLb8.                 II 

120          |           180         || 

I 

60 

'ressure  of  £ 
90 

team  in  Lb 

I2O 

a. 
180 

4-3 
5-4 
6-5 
7-5 

Cub.  feet. 

% 

129 
172 

Cub.  feet. 
67 
107 
'54 
206 

Cub.  feet. 
75 

121 

I76 
234' 

Cub.  feet. 
69 

137 
199 
265 

8-5 
9-5 
10.5 
n-5 

Cub.  feet. 
221 
276 
338 
405 

Cub.  feet. 
265 
332 
407 
446 

Cub.  feet. 
301 
376 
460 
55i 

Cub.  feet. 
340 
425 
520 
623 

Highest  admissible  temperature  of  water  supply  at  120  Ibs.  steam,  138°. 
Minimum  capacity,  36$  to  40$  of  maximum. 

To    Compute    Size   of*  Injector    required.. 

One  H3  per  hour  will  require  from  15  to  40  Ibs.  of  water  per  hour,  accord- 
ing to  character  of  engine. 

When  the  Ibs.  of  coal  burned  per  hour  can  be  ascertained,  divide  them  by 
7.5,  and  quotient  will  give  the  volume  of  water  in  cube  feet  per  hour. 

When  the  area  of  grate-surface  is  known,  multiply  it  by  1.6  for  IP. 

In  case  of  plain  cylindrical  boilers,  divide  the  number  of  sq.  feet  of  heat- 
ing-surface by  10  for  the  IP.  In  case  of  flue  boilers,  divide  by  12,  and  with 
multi-tubular  boilers,  by  15,  for  the  nominal  IP. 

To    Compute   "Volume   of*  Injection    "Water   required,   per 
IH*    per    Hour. 

OPERATION.— Assame  temperature  of  water  80°,  and  of  condensation  100°.  Vol- 
ume of  cylinder  per  IIP  as  per  formula,  page  716,  and  illustration,  page  717,  =  2.76 
feet  per  minute. 

Then,  as  per  rule  page  707,  "4    *  "V°°  =  52-3  cube  ins.  per  cube  foot  of  steam. 


nnnu. 
2.76X  52.3  X  62.5 
1728 


=  5.22  Ibs.,  which,  X  60,  =  313.2  Ibs. 


Xo  Compute   Net  "Volume   of*  Feed  Water   required   per 
IH»    per    Hour. 

OPERATION. —Assume  elements  of  formula,  page  716,  and  illustration,  page  717. 
Then.  1154X2. 76X60  =  19. ii  Ibs. 

Feed  Pipes.    —  Vv  =  diameter  for  small,  and  —  \/v,  for  large  pumps. 
d  representing  diameter  of  plunger  in  ins.,  and  v  its  velocity  in  feet  per  minute. 


STEAM-ENGINE. 


737 


Results   of  Operations    of  S team-engines.    (D.  K.  Clark.) 


CONDENSING  ENGINK. 

Actual 
Ratio  of 
Expan- 
sion. 

Steam 

iF. 

cut-oflF. 

Coal 

S. 

Initial 
Pressure 
at 
cat-off. 

Steam 
perlJP 
per  hour, 

SINGLE. 
Corliss  Saltaire  

c  2 

Lbs. 
14  51 

Lba. 

2   5 

Lbs. 
04.  e 

Lbs. 
17  A 

* 

O.O7 

14.27 

2  2 

1* 

18.7 

"         East  London  

•3  62 

12  Q2 

IO 

3.  o 

20.72 
19.6 

A     T-32 

fe 

18  62 

COMPOUND. 

J  Elder  &  Co                    J  Receiver  

1.8.5 

14-45 

i  61 

c6 

-  J.  fciaer  &  u>  j  Marine,  jacketed 
J  &E.Wood.  .,            .    (Receiver  

1.852 
4.01 

14.85 
10.94 

85.5 

*•  {stationary.  
n     ,,                                  (  Woolf,  stationary 

1.857 
2.486 

13-34 

13.18 

50-5 

S=-S 

America,  Woolf  {Shcylinder  

3.221 
2.31 

13-87 
actual 

9° 

22.51 
15-37 

"    Jacketed  {£t°»;; 

5-03 
3-77 

9IO 

20.71 



90 

14.1 

NON-CONDENSING. 

4.  8 

16  87 

76 

25  Q 

e 

14.  Q3 

70 

2Q  6 

A 

Si 

2.Q4 

21.  24. 

_ 

87 

21.24 

CYLINDERS. 

| 

= 
e.§ 

*   -O 

CYLINDERS. 

Most  Efficient 
Ratio  of  Ex- 
pansion. 

M 
ji 

CONDENSING. 

Single  cylinder,  jacketed.  .  . 

6 
4 
4 
6 

Lbs. 
19-5 

SM 

Compound,  jacketed,  Woolf 
Compound,  Woolf.  

10 

7 

4 
3 

Lbg. 
20.5 
23 

24 
21 

"           "        superheated 
Compound,  jacketed,  Re-) 

NON-CONDENSING. 

Single  cylinder,  t  jacketed.  . 
Single  cylinder.  t... 

*  From  boiler. 


f  70  Iba.  pressure. 


%  90  Ibs.  pressure. 


Standard  Operation  of  a  Portable  Engine. 


Grate 5.5   sq.  feet 

Heating  surface 220       "      ** 

Coal  per  IP  per  hour. ...      6.25  Ibs. 
"     u   sq.  foot  of  grate.      9       " 

"      "    hour 50       " 


Water  evaporated  from ) 

and  at  212°  per  hour.  } * 

44  u    per  IP  per  hour   62.5 

'»«*-*        u      <t   sq  foot  of  J      8,  Q 
grate f 


ItML 


Ratio  of  heating  surface  of  grate 40  to  i. 

MIXTURE  OF  AIR   AND  STEAM. 

Water  contains  a  portion  of  air  or  other  uncondensable  gaseous  matter,  and  when 
ft  is  converted  into  steam,  this  air  is  mixed  with  it,  and  when  steam  is  condensed 
it  is  left  in  a  gaseous  state.  If  means  were  not  taken  to  remove  this  air  or  gaseous 
matter  from  condenser  of  a  steam-engine,  it  would  fill  it  and  cylinder,  and  obstruct 
their  operation,  but,  notwithstanding  the  ordinany  means  of  removing  it  (by  air 
pump),  a  certain  quantity  of  it  always  remains  in  condenser. 

20  volumes  of  water  absorb  i  volume  of  air. 
3Q* 


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1 

STEAM-ENGINE. — BOILER.  739 

BOILER. 

Its  efficiency  is  determined  by  proportional  quantity  of  heat  of  com- 
bustion of  fuel  used,  which  it  applies  to  the  conversion  of  water  into 
steam,  or  it  may  be  determined  by  weight  of  water  evaporated  per  Ib. 
of  fuel. 

In  following  results  and  computations,  water  is  held  to  be  evaporated  from  stand- 
ard temperature  of  212°. 

Proportion  of  surplus  air,  in  operation  of  a  furnace,  in  excess  of  that  which  is 
chemically  required  for  combustion  of  the  fuel,  is  diminished  as  rate  of  combustion 
is  increased;  and  this  diminution  is  one  of  the  causes  why  the  temperature  in  a 
furnace  is  increased  with  rapidity  of  combustion. 

When  combustion  is  rapid,  some  air  should  be  introduced  in  a  furnace 
above  the  grates,  in  order  the  better  to  consume  the  gases  evolved. 

Naturcd  Draught. 

Grate  (Coat)  should  have  a  surface  area  of  i  sq.  foot  for  a  combustion  of 

15  Ibs.  of  coal  per  hour,  length  not  to  exceed  1.5  times  width  of  furnace,  and 
set  at  an  inclination  toward  bridge-wall  of  i  to  1.5  ins.  in  every  foot  of  length. 

When,  however,  rate  of  combustion  is  not  high,  in  consequence  of  low  ve- 
locity of  draught  of  furnace,  or  fuel  being  insufficient,  this  proportion  of  area 
must  be  increased  to  one  sq.  foot  for  every  12  Ibs.  of  fuel. 

Width  of  bars  the  least  practicable,  spaces  between  them  being  from  .5  to 
.75  of  an  inch,  according  to  fuel  used.  Anthracite  requiring  less  space  than 
bituminous.  Short  grates  are  most  economical  in  combustion,  but  generate 
steam  less  rapidly  than  long. 

Level  of  grate  under  a  plain  cylindrical  boiler  gives  best  effect  with  a  fire 
5  ins.  deep,  when  grate  is  but' 7.5  ins.  from  lowest  point. 

Depth,  Cast-iron,  .6  square  root  of  length  in  ins. 

(Wood),  their  area  should  be  1.25  to  1.4  that  for  coal. 

Automatic  (Vicar's).  —  Its  operation  effects  increased  rapidity  in  firing 
and  more  effective  evaporation. 

Ash-pit. — Transverse  area  of  it,  for  a  combustion  of  15  Ibs.  of  coal  per 
hour,  2  to  .25  area  of  grate  surface  for  bituminous  coal,  and  .25  to  .3  for 
anthracite.  Or  15  to  20  ins.  in  depth  for  a  width  of  furnace  of  42  ins. 

Furnace  or  Combustion  Chamber. — (Coal)  Volume  of  it  from  2.75  to  3  cube 
feet  per  sq.  foot  of  grate  surface.  (  Wood)  4.6  to  5  cube  feet. 

The  higher  the  rate  of  combustion  the  greater  the  volume,  bituminous 
coal  requiring  more  than  anthracite.  Velocity  of  current  of  air  entering 
an  ash-pit  may  be  estimated  at  12  feet  per  second. 

Volume  of  air  and  smoke  for  each  cube  foot  of  water  converted  into  steam  is, 
from  coal,  1780  to  1950  cube  feet,  and  for  wood,  3900. 

Rate  of  Combustion.  —  In  Ibs.  of  coal  per  sq.  foot  of  grate  per  hour. 
Cornish  Boilers,  slowest,  4 ;  ordinary,  10.  Stationary,  12  to  16.  Marine, 

16  to  24.     Quickest:  complete  combustion  of  dry  coal,  20  to  23;  of  caking 
coal.  24  to  27 ;  Blast  or  Fan  and  Locomotive,  40  to  120. 

Bridge-wall  (Calorimeter). — Cross-section  of  an  area  of  1.2  to  1.6  sq.  ins. 
for  each  Ib.  of  bituminous  coal  consumed  per  hour,  or  from  18  to  24  sq.  ins. 
for  each  sq.  foot  of  grate,  for  a  combustion  of  15  Ibs.  of  coal  per  hour. 

Temperature  of  a  furnace  is  assumed  to  range  from  1500°  to  2000°,  and 
volume  of  air  required  for  combustion  of  i  Ib.  of  bituminous  coal,  together 
with  products  of  combustion,  is  154.81  cube  feet,  which,  when  exposed  to 
above  temperatures,  makes  volume  of  heated  air  at  bridge- wall  from  600  to 
750  cube  feet  for  each  Ib.  of  coal  consumed  upon  grate. 


740 


STEAM-EM  GINE. — BOILER. 


Hence,  at  a  velocity  of  draught  of  about  12  feet  per  second,  area  at  bridge- 
wall,  required  to  admit  of  this  volume  being  passed  off  in  an  hour,  is  2  to  2.5 
sq.  ins.,  and  proportionately  for  increased  velocity,  but  in  practice  it  may  be 
1.2  to  1.6  ins. 

When  20  Ibs.  of  coal  per  hour  are  consumed  upon  a  sq.  foot  of  grate,  20  X  1.2  or 
1.6  =  24  or  32  S(l-  ms-  are  required,  and  in  a  like  proportion  for  other  quantities. 

Or,  When  area  of  flues  is  determined  upon,  and  area  over  bridge-wall  is 
required,  it  should  be  taken  at  from  .7  to  .8  area  of  lower  flues  for  a  natural 
draught,  and  from  .5  to  .6  for  a  blast. 

When  one  half  of  tubes  were  closed  in  a  fire-tubular  marine  boiler,  the  evapora- 
tion per  Ib.  of  coal  was  reduced  but  1.5  per  cent. 

Firing. — Coal  of  a  depth  up  to  12  ins.  is  more  effective  than  at  a  less 
depth.  Admission  of  air  above  the  grate  increases  evaporative  effect,  but 
diminishes  the  rapidity  of  it. 

Air  admitted  at  bridge-wall  effects  a  better  result  than  when  admitted  at 
door,  and  when  in  small  volumes,  and  in  streams  or  currents,  it  arrests  or  pre- 
vents smoke.  It  may  be  admitted  by  an  area  of  4  sq.  ins.  per  sq.  foot  of  grate. 

Combustion  is  the  most  complete  with  firings  or  charges  at  intervals  of 
from  15  to  20  minutes. 

With  a  fuel  economizer  (Green's)  an  increased  evaporative  effect  of  9  per 
cent,  has  been  obtained. 

When  external  flues  of  a  Lancashire  boiler  were  closed,  evaporative  power  was 
slightly  increased,  but  evaporative  efficiency  was  decreased;  and  when  25  per  cent, 
of  like  surface  in  setting  of  a  plain  cylindrical  boiler  was  cut  off.  evaporation  was 
reduced  but  1.5  per  cent.  When  temperature  at  base  of  chimney  was  630°,  with  a 
fire  12  ins.  in  depth,  it  was  decreased  to  556°  with  one  9  ins.  in  depth,  and  to  539° 
with  one  6  ins. 

High  wind  increases  evaporative  effect  of  a  furnace. 

Stationary  or  Land. — Set  at  an  inclination  downward  of  .5  inch  in  10  feet. 

Smoke  Preventing.— A.  test  of  C.  Wye  Williams's  design  of  preventing  smoke,  at 
Newcastle,  1857,  as  reported  by  Messrs.  Longridge,  Armstrong,  and  Richardson, 
gave  an  increased  evaporative  effect  with  the  "practical  prevention  of  smoke. " 
Hence  it  was  concluded,  "  That  by  an  easy  method  of  firing,  combined  with  a  due 
admission  of  air  in  front  of  furnace,  and  a  proper  arrangement  of  grate,  emission 
of  smoke  may  be  effectually  prevented  in  ordinary  marine  multi-tubular  boilers, 
with  suitable  coals.  2d.  That  prevention  of  smoke  increases  economic  value  of  fuel 
and  evaporative  power  of  boiler.  3d.  That  coals  from  the  Hartley  district  have  an 
evaporative  power  fully  equal  to  that  of  the  best  Welsh  steam-coals. " 

Heating    Surfaces. 

Marine  (Sea-water). — Grate  and  heating  surfaces  should  be  increased 
about  .07  over  that  for  fresh  water. 

Relative  Value  of  Heating  Surfaces. 

Horizontal  surface  above  the  flame  =  i     I  Horizontal  beneath  the  flame = .  i 

Vertical =  .5  |  Tubes  and  flues =.56 

Minimum  "Volumes  of*  Fuel  Consumed  per  Sq.  Foot  of 
GS-rate  per  Hour,  for  given  Stirface-ratios.    (D.  K.  Clark.) 


DESCRIPTION  OF 
BOILER. 

10 

15 

Si 
20 

irface-r 
30 

atios  of 
40 

Seating  J 
So 

Surface  t 

60 

o  Grate. 

75 

90 

100 

Stationary  

Lbs. 
•7 
•  7 

.2 

•  3 
•4 

Lbs. 

\:l 

•4 
•7 

i 

Lbs. 

is 

.8 
1.3 
1.8 

Lbs. 
6.8 

6-3 
1.8 

2.9 
4 

Lbs. 

12.  1 
II.  2 
3-2 
5-2 

7 

Lbs. 
18.9 
17-5 

II 

Lbs. 
26 
24 

11.7 

16 

Lbs. 

isT3 
25 

Lbs. 

I"3 
36 

Lba. 

32-5 
44 

Marine  
Portable  

Locomotive  (coal)  . 
"         (coke). 

At  extreme  consumption  of  fuel  (120  Ibs.)  coke  will  withstand  disturbing  effect 
of  a  blast  better  than  coal. 


STEAM-ENGINE. — BOILER.  74! 

A  scale  of  sediment  one  sixteenth  of  an  inch  thick  will  effect  a  loss  of  14.7  per 
sent,  of  fuel. 

One  sq.  foot  o{Jlre  surface  is  held  to  be  as  effective  as  three  of  heating. 
Relation   of  Grrate,  Heating    Surface,  and.    Fuel. 

When  Grate  and  Heating  Surface  are  constant,  greater  the  weight  of  fuel 
consumed  per  hour,  greater  the  volume  of  water  evaporated ;  but  the  volume 
is  in  a  decreased  proportion  to  fuel  consumed. 

In  treating  of  relations  of  grate,  surface,  and  fuel,  D.  K.  Clark,  in  his  valuable 
treatise,  submits,  that  in  1852  he  investigated  the  question  of  evaporative  perform- 
ance of  locomotive-boilers,  using  coke;  and  he  deduced  from  them,  that,  assuming 
a  constant  efficiency  of  fuel,  or  proportion  of  water  evaporated  to  fuel,  evaporative 
effect,  or  volume  of  water  which  a  boiler  evaporates  per  hour,  decreases  directly  as 
grate-area  is  increased;  that  is  to  say,  larger  the  grate,  less  the  evaporation  of  water, 
at  same  rate  of  efficiency  of  fuel,  even  with  same  heating  surface. 

2d.  That  evaporative  effect  increases  directly  as  square  of  heating  surface,  with 
same  area  of  grate  and  efficiency  of  fuel. 

3d.  Necessary  heating  surface  increases  directly  as  square  root  of  effect— viz.,  for 
four  times  effect,  with  same  efficiency,  twice  heating  surface  only  is  required. 

4th.  Necessary  heating  surface  increases  directly  as  square  root  of  grate,  with  same 
efficiency;  that  is,  for  instance,  if  grate  is  enlarged  to  four  times  its  first  area,  twice 
heating  surface  would  be  required,  and  would  be  sufficient,  to  evaporate  same  vol- 
ume of  water  per  hour  with  same  efficiency  of  fuel. 

Result  of  40  experiments  with  a  stationary  boiler  (fresh  water),  with  an 
evaporation  of  9  Ibs.  water  per  Ib.  of  fuel  consumed,  the  coefficient  .002  22 
was  deduced. 

Hence,  (— \  .00222  =  W.  W  representing  volume  of  water  in  cube  feet,  and  g 
and  h  areas  of  grate  and  heating  surfaces  in  sq.  feet. 

ILLUSTRATION.— Assume  a  heating  surface  of  90  feet,  and  a  grate  of  3;  what  will 
be  the  evaporation  ? 

Then  90-4-  3  x  .002  22  =  1.998  cube  feet. 

NOTE.— A  Galloway  stationary  boiler,  with  a  ratio  of  grate  area  of  34.3  and  a  con- 
sumption of  21.8  Ibs.  coal  per  hour,  evaporated  2.9  cube  feet  of  water  per  sq.  foot  of 
grate.  Hence  the  coefficient  in  this  case  would  be  .002  466. 

To  Compute  .A^reas  of  Grrate  and.  Heating  Surfaces, 
Volume  of  ^Water,  and.  AVeight  of  3Tuel. 

For  a  Temperature  0/281°,  or  Pressure  0/50  Ibs.  per  Sq.  Inch. 

To   Compute   \Veignt  of  Fuel. 
When  Water  per  Sq.  Foot  of  Grate  per  Hour  and  Surface  Ratio  are  Given. 

-^ —  =  F,  and  x  R2  =  (E  —  C)  F. 
ILLUSTRATION.—  Assume  elements  as  preceding. 

200 — .02  X  50 2  /200  \ 

— — —  — 15}  and  <02  x  502  _  i 10\  x  15  =  50. 

To  Compute  Ratio  of  Heating  Surface  to  Area  of  Grate, 
and.    to    Effect   a   <3-iven    Evaporation. 

When  Water  and  Fuel  per  Sq.  Foot  of  Grate  are  Given.     ^/W~CF  =  R. 

W  representing  water  evaporated  per  sq.  foot  of  grate,  and  F  fuel  consumed,  both 
in  Ibs.  per  hour.     C  and  x  specific  constants  for  each  type  of  boiler,  and  R  (h  ~-  g) 
ratio  of  heating  surface  to  grate. 
ILLUSTRATION.— Assume  W  =  200,  C  =  10,  F  =  15,  and  x  =  .02. 


/200-ioXis  200-.02X5Q2  d     /('3. 33-xo)  Xij_ 

V  -03  10  V  .02 


742 


STEAM-ENGINE. — BOILER. 


When  Efficiency  of  Fuel  and  Fuel  consumed  per  Sq.  Foot  of  Grate  per 
Hour  are  given.    •=•  =  E  or  efficiency  of  fuel  or  weight  of  water  evaporated  per  Ib. 

w    /^=*. 

To  Comptite  Fnel  that  Tnay  "be  consumed,  per  Sq..  Foot 
of  Grate  per  Hour,  corresponding  to  a  Gfiven  Effi- 
ciency, 

When  Efficiency  of  Fuel,  that  is,  Weight  of  Water  evaporated  per  Lb.  of 
Fuel,  and  the  Surface  Ratio,  are  given. 


- 

ILLUSTRATION.— Assume  elements  as  preceding. 


Combustion  of  Coal  per  sq.  foot  of  grate.—  Natural  Draught,  from  20  to  25  Ibs.  can 
be  consumed  per  hour.— Steam-jet,  30  Ibs.,  and  Exhaust-blast  65  to  80  Ibs. 

From  Results  of  Experiments  upon  Marine  Boilers,  see  Manual  of  D.  K.  Clark, 
page  808;  he  deduced  the  following  formula,  as  applicable  to  all  surface  ratios  in 
such  boilers. 

Newcastle  021 56  R2  +  9.71  F,  and  for  Wigan  .01  R2+ 10.75  F  =  W  in  Ibs. 

And  the  general  formulas  he  deduced  from  all  the  various  experiments  are  as 
follows. 

From  and  at  212°. 


Portable 008   R2  +  8.6   F  =  W      Marine 016  R2  +  10.25  F  =  W. 

Stationary. , .  .0222  R2  +  9-s6F  =  W,  |  Locomotive,  coal,  .009  R2  +  9.7    F  =  W. 
Locomotive,  coke 0178  R2-f- 7.94^  =  W. 

As  the  maximum  evaporative  power  of  fuel  is  a  fixed  quantity,  the  preceding 
formulas  are  not  fully  applicable  in  minimum  rates  of  its  consumption  and  evapo- 
rative quality. 

With  coal  and  coke  the  limits  of  evaporative  efficiency  may  be  taken  respectively 
at  12.5  and  12  Ibs.  water  from  and  at  212°. 

ILLUSTRATION  i.— Assume  a  marine  fire-tubular  boiler  with  a  surface  ratio  of  heat- 
ing surface  to  grate  of  30  and  a  consumption  of  coal  of  15  Ibs.  per  sq.  foot  of  grate 
per  hour,  what  will  be  its  evaporation  per  sq.  foot  of  grate? 

.016  X  30* -f- 10. 25  X  15  =  168. 15  Ibs, 

2 Assume  a  like  boiler,  using  fresh  water,  to  have  a  ratio  of  heating  surface  to 

grate  of  30  and  an  evaporation  of  165  Ibs.  water  per  sq.  foot  of  grate  per  hour,  what 
would  be  consumption  of  coal  per  sq.  foot  of  grate  per  hour? 

165 — .016  x  so2 

— £—  =  14.69  Ibs. 

10.25 

Tube  Surface  (Iron)  per  Ib.  of  coal  1.58,  per  sq.  foot  of  grate  32,  and  per  IBP  4.27 
sq.  feet. 

Locomotive  Boiler  has  from  60  to  90  sq.  feet  per  foot  of  grate,  and  consumes  65 
Ibs.  coal  per  sq.  foot  per  hour. 

Evaporative  Capacity  of  Tu/bes   of*  "Varying  Length. 

By  Temperatures.     (A  J.  Dutton,  Eng. -in- Chief,  R.  N.) 

Diameter,  external,  2.75  in*.     Length,  6  feet  8  ins.     Combustion  Chamber,  1644°. 
IN  TUBES. 


2  ins. . . 1426°  |  5  ins. . . 1398° 

3  "  ...1405°      6    "  ...1406° 

4  "...  1412°  '   7    "...  1400° 


8  ins... 1410° 
14  "  ...1368° 
20  "...  1295° 


32  ins... 1 198° 
44  "...1106° 
56  "  ...1015° 


68  ins 926° 

80   " 887° 

Connection.  782° 


STEAM-ENGINE. — BOILEE. 


743 


Results  of  Operation  of*  Boilers  under  "Varying  Propor- 
tions of  03- rate,  Area,  and.  Length  of  Heating  Surface, 
Dratignt  of  Furnace,  and.  Rate  of  Combustion. 


DESCRIPTION'. 

Area  of 
Grate. 

Heating 
Surface. 

Grate  to 
Heating 
Surface. 

Coal  per 
Sq.  Foot 
of  Grate 
per  Hour. 

Evapor 
Water  fi 
per  sq.  ft. 
of  grate. 

ation  of 
om  212° 
per  Ib. 
of  Coal. 

FUEL. 

Fire-tubular. 

Agricultural  and  Hoisting 

Sq.Feet. 
4-7 

(2|25 
|i6 
10.5 
10.6 

22 

18 
10.3 
10.3 
10.8 

Sq.  Feet. 
158 
220 
963-5 

818 
788 
1056 
748 
749 
9*5 
508 
151-2 
945 
767 

Ratio. 

Si 
75 

100 

& 

50 
49-3 
14 

30 
24.4 

Lbs. 

3.1 

30.86 
38 

45 
*57 

24-3 
23-6 
41.25 
27.63 
27.76 
28.87 
14 

Lbs. 
119 
151 
327 
375 
419 
1401 
265 
264 
468 
309.8 
205 

293-7 
141.4 

Lbs. 

9-33 
1.83 
0.6 
0.47 
1.04 
0.41 
0.7 

1.2 

I.36 

i-54 
7-39 
0.17 

O.I 

Welsh. 
f 

< 
t 

n 
Lanc'r 
Anth'e 
Welsh. 

English  } 

i 

2 

2* 

3  

"                 2 

3i-5 

i  New  Castle. 


3  Experimented  at  New  York. 


2  and  4  Wigan. 

*  Effect  of  reducing  the  tube-surfaces  was  tried  by  stopping  one  half  the  number  of  tubes  in  alter- 
nate  diagonal  rows,  so  that  the  tube  surface  was  reduced  206.5  sq.  feet.    The  results  with  fires  12  ins. 
deep  were  as  follows  t 

Tubes  open.        Tubes  half  closed. 
Coal  per  sq.  foot  of  grate  per  hour  ..................  25     Ibs.  24     Ibs. 

Water  from  212°  per  Ib.  of  coal  .....................  12.41"  12.23" 

Smoke  per  hour,  very  light.  ....  ....................    2.8  minutes.  8  minutes. 

Evaporative   Effects   of  Boilers   for   Different   Rates   of 
Comoustion,  and    Surface    Ratios.     (D.  K.  Clark.) 


Water  from  and  at  212°  per  Hour. 
Surface    Ratio   SO. 


STATIONARY. 

MARINE. 

PORTABLE. 

LOCOMOTIVB. 

Fuel  per 
Sq.  Foot 

Water 

Water 

Water 

Coal. 
Water 

Coke. 
Water 

of  Grate 
per  Hour. 

per 

Sq.  foot. 

per  Ib. 
of  Coal. 

per 

Sq.  foot. 

per  Ib. 
of  Coal. 

Sq.Pfoot. 

perlb. 
of  Coal. 

per 

Sq.  foot. 

per  Ib. 
of  Coal. 

per 

Sq.  foot. 

per  Ib. 

of  Coal. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

10 

116 

1.6 

117 

11.7 

93 

9-3 

105 

10.5 

95 

9-5 

15 

163 

0.9 

168 

XX.  2 

I36 

9 

154 

10.3 

135 

9 

20 

211 

0.6 

219 

IO.9 

179 

9 

202 

10.  I 

175 

8.7 

30 

307 

O.2 

322 

10.7 

265 

8.8 

209 

10 

254 

8.5 

Surface   Ratio  SO. 

i5 

l87 

2.5 

187.5 

12.5 

149 

9-9 

168 

II.  2 

!64 

10.9 

20 

247 

2.3 

248 

J2.5 

192 

9.6 

217 

10.9 

203 

IO.  2 

3° 

342 

1.4 

348 

ii.  6 

278 

9-3 

3i4 

10-4 

283 

9-4 

40 

438 

0.9 

450 

"•3 

364 

9.1 

411 

10.3 

362 

9 

50 

534 

0.7 

552 

ii 

450 

9 

508 

XO.I 

442 

8.8 

Surface   Ratio   ?5. 

Water. 

Fuel  per  Sq.  Foot  of  Grate  per  Hour  in  Lbs. 

30 

40 

So 

60 

75 

90 

IOO 

Lba. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs, 

LOCOMOTIVE,  coal.. 

Per  sq.  foot. 

342 

439 

536 

633 

778 

927 

IO1O 

ti                       U 

"          coke  . 

u   Ib.  coal. 
"    sq.  foot. 

11.4 

338 

ii 

418 

10.7 
497 

10.7 
576 

10.4 
695 

10.3 
8i5 

10.2 
894 

.. 

"   Ib.  coal. 

"•3 

10.4 

9.9 

9-6 

9-3 

9 

8.9 

When  a  heater  is  used,  and  temperature  of  feed-water  is  raised  above  that  ob- 
tained in  a  condensing  engine,  the  proportions  of  surfaces  may  be  correspondingly 
reduced. 


744 


STEAM-ENGINE. — BOILER, 


Results  of  Operation  of  varioxis  Designs  of  Boiler,  tin- 
der  varying  IPz-oportioiis  of  Grrate,  Calorimeter,  Area 
and.  Length  of  Heating  Surface,  Draught,  Firing,  and. 
Rate  of  Comtoustion.. 


STATIONARY. 

Area 
of 
Grate. 

Heat- 
ing 
Surface. 

Grate 
to 
Heating 
Surface. 

Circuit  of 
Heating 
Surface. 

§* 

li 

F 

£"3 

Coal  per 
Sq.  Foot 
of  Grate 
per  Hour. 

Water  E 

from  212° 
per  Ib.  of 
Coal. 

vaporated 

per  Sq.  Foot 
of  Grate 
per  Hour. 

Sq.  Ft. 

Sq.  Ft. 

Ratio. 

Feet. 

0 

Lbs. 

Lbs. 

Lbs. 

Lancashire  double) 

internal  and  ex-  > 

20.5 

612 

29.8 

79 

5" 

15-35 

8.33 

125  4 

ternalflued1...) 

«              "       2  

21 

767 

36.5 

80 

5°5 

21.5 

10.88 

204  4 

Galloway  vertical 
water-tubular2. 

21 

719 

22.8 

79 

5°5 

22.7 

10.77 

212  4 

"           "     2.  . 

31-5 

719 

34-3 

80 

630 

18.3 

10.17 

162  4 

Fairbairn1  1!'. 

20.  5 

-^87 

T  c  27 

8  67 

133 

20.1 
I4.2 

1017 

607 

377-5 

49'  5 
30-3 
26.8 

56 

o"/ 

510 

292 

»§••*/ 

16.42 

7-43 

«J.U/ 

8.12 

9,08 

133 

59  5 

Cylindrical  flued3.  .  . 

MARINE. 

At  Pressure  of  Atmosphere. 

Horizontal  fire-tub.2 

10.3 

508 

49-3 

— 

— 

27-5 

11.92 

328? 

U                          11           2 

10.3 

508 

49-3 

— 

— 

41.25 

11.36 

4698 

U                            (           2 

10.3 

302 

30 

— 

— 

24 

12.23 

2689 

(i                            I 

19-3 

749 

_ 

— 

21 

10 

182  «> 

11                           ( 

28.5 

749 

26.3 

— 

— 

21.15 

8.94 

164  »> 

(t                            ( 

28.5 

749 

26.3 

— 

— 

J9 

11.13 

3-35" 

U                          I 

(C.  Wye  William  ) 

iS-5 

749 

48.3 

- 

600 

37-4 

10.63 

398 

u                  ! 

22 

749 

34 

— 

600 

17.27 

11.7 

202" 

tt      .            ( 

«                  t 

42 

749 

17.6 

— 

— 

16 

9-65 

154  J3 

"                              4     3 

10.8 

150 

13-9 

8.5 



10.99 

8-95 

88  I4 

«                              4      3 

4-32 

147 

34 

8.5 

— 

27.58 

7.24 

40  i4 

1  Trial  in  France.    2  At  Wigan,  1866-68,  height  of  chimneys  100  feet.    3  Navy. 

yard,  Washington,  U.  S.,  chimney  61  feet.  4  At  pressure  of  atmosphere,  fires  12  ins. 
deep,  at  40  Ibs.  pressure,  evaporation  was  reduced  12  per  cent.  5  Bituminous  coals. 
6  Anthracite,  at  pressure  of  6.5  Ibs.  above  atmosphere.  7  Fires  14  ins.  deep,  air  ad- 
mitted through  furnace  -  doors.  8  Ditto  do.,  jet  blast.  9  Half  tubes  closed  up. 
i°  Air  through  grate  only.  "  Air  through  grate  and  door,  no  smoke.  I2  One  open- 
ing in  door,  temp.  625°,  with  two  633°,  with  four  638°,  and  with  six  600°.  J3  Long 
grates,  air  spaces  fully  open,  no  smoke.  J4  One  furnace,  anthracite  coal,  5  ins.  deep. 

Draught. 

Draught  of  Furnace.  —  Volume  of  gas  varies  directly  as  its  absolute  tem- 
perature, and  draught  is  best  when  absolute  temperature  of  gas  in  chimney 
is  to  that  of  external  air  as  25  to  12. 

T  -|-  4.61  2°         V 

—  ^  =  —  7  =  V".     V,  V,  and  V"  representing  absolute  temperatures  at  T 
32°  +  461.  2°      V 

or  temperature  given,  and  at  32°,  in  degrees  and  volume  of  furnace  gas  at  tempera- 
ture T  in  cube  feet. 

ILLUSTRATION.—  Assume  temperature  of  furnace  or  T  —  1500°,  and  12  Ibs.  air  per 
Ib.  of  fuel. 


volume  of  gas  per  Ib.  of  fuel  at  i 


I5°o-4-~f   *'o   ~  3'98' 
Ibs.  supply  of  air,  150  X  3-98  =  597  cube  feet. 

W  v  t 

—  —  ;-  =  C.     W  representing  weight  of  fuel  consumed  in  furnace  per  second  in  Ibs., 

v  volume  of  air  at  32°  supplied  per  Ib.  of  fuel  in  cube  feet,  t  absolute  temperature  oj 
gas  discharged  by  chimney  in  degrees,  a  area  of  chimney  in  sq.feet,  and  C  velocity  oj 
current  in  chimney  in  feet  per  second. 


STEAM-ENGINE. — BOILEB. 


745 


ILLUSTRATION. — Assume  W  =  .i6,    t>  =  i5o,    t=.  1000°,  and  a  =  5. 
.16  X  150  X  iooo_  24000 


5X493-2°  2466 

V 

—  .084  to  .087  =  D.    D  representing  weight  of  a  cube  foot  of  gas  discharged  by 

4Q3  2° 

thimney,  in  Ibs.      ILLUSTRATION.    2*- —  X  .086  =  .0424  Ib. 

*     r*2  /  /*z\ 

—  ( !  _|_  G  +  — )  =  H.    G  representing  a  coefficient  of  resistance  and  friction  of 
2  g  \  m/ 

«tr  through  grate  and  fuel,*  f  coefficient  of  friction  of  gas  through  flues  and  over 
sooty  surfaces,^  I  length  of  flues  and  chimney,  m  hydraulic  mean  depth,l  and  H  height 
of  chimney,  all  in  feet. 
ILLUSTRATION  i.— Assume  C  =  9.73,  I  —  60,  and  m  =  .72,  all  in  feet. 


=: 
64.33 

*y3g*  x*^'0  =  -'6 


04-33  V  -72 

2.— Assume  preceding  elements. 

When  H  is  given.    ^/Tn  2flf-j-i  +  G  +  ^j=C 

ILLUSTRATION.— Assume  preceding  elements.    V2o. 6  x  64. 33  -r- 14  =  9. 73  feet. 

.  192  x  pressure  in  Ibs.  per  sq.  foot  =  head  in  ins.  of  water. 

Temperature  at  base  of  smoke-pipe  or  chimney,  or  termination  of  flues  or 
tubes,  is  estimated  at  500° ;  and  base  of  chimney,  or  its  calorimeter,  should 
have  an  area  of  1.3  to  1.6  sq.  ins.  for  every  Ib.  of  coal  consumed  per  hour. 
With  tubes  of  small  diameter,  compared  to  their  length,  this  proportion  may 
be  reduced  to  i  and  1.2  ins. 

Admission  of  air  behind  a  bridge-wall  increases  temperature  of  the  gases, 
but  it  must  be  at  a  point  where  their  temperature  is  not  below  800°. 

Loss   of*  I*ressTire   "by   Flow   of  Air   in.   Pipes. 

Length  3280  Feet,  or  1000  Meters. 


Velocity  at 
Pi] 
Feet 
per  Second. 

Entrance  of 
>e. 
Meter 
per  Second. 

4       1        6 

Loss  c 

Diameter  of 
8 
f  Pressure  i 

Pipe  in  Int. 
10        |         12         |          14 

n  Lbs.  per  Sq.  Inch. 

3.28 
6.56 
9.84 
,3-12 
16.4 
19.68 

i 

2 

3 
4 

5 
6 

.114 

1.183 
2.06 

4.446 

.076 
•343 

1-374 
2.16 
2.964 

•057 
•25 
•SQ2 
1.03 
1.61 
2.223 

•057 
.21 

•477 
.84 
1.29 
1.778 

038 
172 
394 
687 
i  i 
1.482 

.038 
•153 

:l43 
,ll3 

At  Mount  Cenis  Tunnel,  the  loss  of  pressure  from  84  Ibs.  per  sq.  inch,  in  a  pipe 
7.625  ins.  in  diameter  and  i  mile  15  yards  in  length,  was  but  3.5  per  cent. 

Artificial   I3ranglit. 

In  production  of  draught  in  an  ordinary  marine  boiler,  from  20  to  33  per 
cent,  of  total  heat  of  combustion  of  fuel  is  expended. 

Blast.— By  experiments  of  D.  K.  Clark  and  others  it  was  deduced  that  the  vacuum 
in  back  connection  is  about  .7  of  blast  pressure,  and  in  the  furnace  from  .33  to  .5 
of  that  in  back  connection;  that  rate  of  evaporation  varies  nearly  as  square  root  of 
vacuum  in  back  connection;  that  best  proportions  of  chimney  and  passages  thereto 
are  those  which  enable  a  given  draught  to  be  produced  with  greatest  diameter  of 
blast  pipe;  for  the  manifest  reason,  that  the  greater  that  diameter,  the  less  the  back- 
pressure due  to  resistance  of  orifice,  and  that  these  proportions  are  best  at  all  rates 
of  expansion  and  speeds. 

*  Which,  in  furnaces  consuming  from  20  to  24  Ibs.  coal  per  sq.  foot  of  grate  per  hour,  is  assigned  by 
Peclet  at  12.  t  Estimated  by  tame  authority  at  .012. 

t  For  »  square  or  circular  flue  i«  .25  its  diameter. 


746      STEAM-ENGINE.  —  DRAUGHT.  —  SAFETY    VALVES. 

Velocity  of  Draught.    Locomotive.    36.5  VH  (T  —  t)  =  V.       H  representing 
height  of  chimney  or  pipe  in  feet,  T  awe?  <  temperatures  of  air  at  base  and  top  of  chim- 
ney, and  V  velocity  in  feet  per  second. 

Sectional  area  of  tubes  within  ferrules  .................  2    grate. 

"          "    of  smoke-pipe  ..........................  066    " 

Area  of  blast-pipe  (below  base  of  smoke-pipe)  ...........  015    " 

Volume  of  back  connection  ................  3  feet  X  area  of  grate. 

Height  of  smoke-  pipe  4  times  its  diameter. 

Steam-jet.  —  Rings  set  above  base  of  smoke-pipe,  and  should  equally  divide' 
the  area  ;  jets  .06  to  .1  inch  in  diameter,  3  ins.  apart  at  centres. 

A  Steam-jet,  involving  50  per  cent,  increased  combustion  of  coal,  produced 
45  per  cent,  more  evaporation  at  nearly  same  evaporation  per  Ib.  of  coal. 

Fan  Blowers.—  See  page  447. 

Comparative  Result  of  Experiments  with  a  Steam  -jet  in  a  Marine  Boiler, 
with  Bituminous  Coal.     (Nicoll  and  Lynn,  Eng.) 

Without  Jet.  With  Jet. 
Area  of  grate  ....................  sq.  feet  .........  .    10.3  10.3 

Coal  per  sq.  foot  of  grate  per  hour.  .  .  .  Ibs  ...........    27.  5  41.  25 

Water          "  "  u  ..........  293.1  419-37 

from  212°  per  Ib.  of  coal  "  ..........    11.9  11.36 


Comparative   EJffect  of  Draught  and.   Blasts. 
By  late  experiments  in  England,  with  boilers  of  two  steamers,  to  deter- 
mine relative  effects  of  the  different  methods  of  combustion,  the  results  were: 
Natural  draught  i,  Jet  1.25,  and  Blast  1.6. 


In  CylMM  Pipes. 


Flow  of  Air.    (Hawksley.) 


In  Conduits  of  Various  Sections.  796    /^  = 


IP-    I*  which  xmch  water  is 
taken  as  equivalent  to  a  pressure  of  5.2  Ibs.  per  sq.  inch  for  any  passage. 

V  representing  velocity  in  feet  per  second,  h  head  of  water  in  ins.  ,  d  diameter  of 
pipe,  I  length,  and  C  perimeter,  all  in  feet,  a  area  of  section  in  sq.feet,  Q  (V  a)  volume 
of  air  discharged  per  second  in  cube  feet,  and  IP  horse-power. 

Safety   "Valves. 

Up  to  a  pressure  of  100  Ibs.  per  sq.  inch,  area  in  sq.  ins.  equal  product  of 
weight  of  water  evaporated  in  Ibs.  per  hour  by  .006. 

Act  of  Congress  (U.  S.).—  For  boilers  having  flat  or  stayed  surfaces,  30  ins.  for 
every  500  sq.  feet  of  effective  heating  surface;  for  cylindrical  boilers,  or  cylindrical 
flued,  24  sq.  ins. 

Board  of  Trade,  Eng.—  Two  of  .5  inch  area  per  sq.  foot  of  grate.  Or,  /—  -  = 
diameter.  G  representing  area  of  grate  in  sq.  ins. 

Locked  Safety-valves.—  Effective  heating  surface,  less  than  700  sq.  feet,  valve  2  ins. 
in  diameter;  less  than  1500,  3  ins.  in  diameter;  less  than  2000,  4  ins.  in  diameter; 
less  than  2500,  5  ins.  in  diameter;  and  above  2500,  6  ins.  in  diameter. 

Or,  (.05  G  -}-  .005  S)  /T5r  —  area  °feach  of  two  valves.  G  representing  sq.  t'ncft, 
per  sq.  foot  of  grate,  and  S  sq.  inch,  per  sq.foot  of  heating  surface. 


STEAM-ENGINE. — FLUES    AND    TUBES.  747 

ILLUSTRATION.— Assume  G  =  50  sq.  feet,  S  =  1600  sq.  feet,  and  P  =  80  Ibs.  (m.  g.) 
Then,  (.05X50 +  .005X1600)  X  V  100-^-80  =^-2. 5 +  8  X  1.118=11.73  sq.  ins. 

Pipes. 

Area.  .25  G  -f-  .01  S  /— .  G  representing  area  of  grate  and  S  area  of  heat- 
ing surface,  both  in  sq.feet,  and  P  pressure  per  mercurial  gauge  in  Ibs. 

(Copper),  Thickness.  Steam,  .  125  -f  ^^ ;  Feed, .  125  -f-  g^ ;  Blow  (Bottom 
and  Surface),  .  125  -^-  ;  Supply,  .i-\ ;  Discharge, . i  -f  — -  ;  Feed,  Suction, 

9000  300  200 

and  Bilge  discharge,  .09  +^,  and  Steam  Blow-off,  -05  +  —  .    d  representing 
internal  diam.  of  pipe,  and  p  internal  pressure  per  sq.  inch  in  Ibs. 

Flanges.  —  Of  brass,  thickness  4  times  that  of  pipe ;  breadth,  2.25  times 
diam.  of  bolt ;  bolts,  diam.  equal  to  and  pitch  5  times  thickness  of  flange. 
For  lower  pressure  or  stress,  pitch  of  bolts  6  times. 

;  ITln.es    and.   Tubes. 

Flues  and  Tubes.— Cross  section,  for  15  Ibs.  of  coal  consumed  per  hour, 
an  area  of  from  .18  to  .2  area  of  grate,  area  being  measurably  inverse  to 
diameter,  and  directly  increased  with  length.  Thus,  in  Horizontal  Tubular 
Boilers,  area  .18  to  .2  area  per  sq.  foot  of  grate,  and  in  Vertical  Tubular  .22 
to  .25,  area,  decreasing  with  their  length,  but  not  in  proportion  to  reduction 
of  temperature  of  the  heated  air,  area  at  their  termination  being  from  .7 
to  .8  that  of  calorimeter  or  area  immediately  at  bridge-wall. 

Large  flues  absorb  more  heat  than  small,  as  both  volume  and  intensity  of  heat  is 
greater  with  equal  surfaces. 

Tubes. — Surface  i  sq.  foot,  if  brass,  and  1.33,  if  iron,  for  each  Ib.  of  coal 
consumed  per  hour ;  or  20  of  brass  and  27  of  iron  for  each  sq.  foot  of  grate, 
and  2.6  sq.  feet  of  brass  and  3.7  of  iron  per  IIP. 

Set  in  vertical  rows,  and  spaces  between  them  increased  in  width  with 
number  of  the  rows. 

Temperature  of  base  of  Chimney  or  Smoke-pipe,  or  termination  of  the 
flues  or  tubes,  is  estimated  at  500° ;  and  base  of  chimney,  or  its  calorimeter, 
with  natural  draught,  should  have  an  area  of  1.33  sq.  ins.  for  every  Ib.  of 
coal  consumed  per  hour.  With  tubes  of  small  diameter,  compared  to  their 
length,  this  proportion  may  be  reduced  to  i  and  1.2  ins. 

When  combustion  in  a  furnace  is  very  complete,  the  flues  and  tubes  may 
be  shorter  than  when  it  is  incomplete. 

Evaporation. 

i  sq.  foot  of  grate  surface,  at  a  combustion  of  15  Ibs.  coal  per  hour,  will 
evaporate  2.3  cube  feet  of  salt  water  per  hour. 

A  sq.  foot  of  heating  surface,  at  a  like  combustion  of  fuel,  will  evaporate 
from  5  to  6.2  Ibs.  of  salt  water  per  hour ;  and  at  a  combustion  of  40  Ibs.  coal 
per  hour  (as  upon  Western  rivers  of  U.  S.),  from  10  to  n  Ibs.  fresh  water, 
exclusive  of  that  lost  by  being  blown  out  from  boilers. 

13.8  to  17.2  sq.  feet  of  surface  will  evaporate  i  cube  foot  of  salt  water  per 
hour,  at  a  combustion  of  15  Ibs.  coal  per  hour  per  sq.  foot  of  grate. 

Relative  evaporating  powers  of  Iron,  Brass,  and  Copper  are  as  i,  i  32,  and  1.56. 

NOTE. — Boilers  of  Steamer  Arctic,  of  N.  Y.,  vertical  tubular,  having  a  surface  of 
33.5  to  i  of  grate,  consuming  13  Ibs.  of  coal  per  sq.  foot  of  grate  per  hour,  evapo- 
rated 8. 56  Ibs.  of  salt  water  per  Ib.  of  coal,  including  that  lost  by  blowing  out  of 
saturated  water. 


743      STEAM-ENGINE. SMOKE-PIPES  AND  CHIMNEYS. 

Water  Surface. 

At  low  evaporations,  3  sq.  feet  are  required  for  each  sq.  foot  of  grate  siu> 
face,  and  at  high  evaporation  4  to  5  sq.  feet. 

Steam    Room. 

From  15  to  18  times  volume  that  there  are  cube  feet  of  steam  expended 
for  each  single  stroke  of  piston  for  25  revolutions  per  minute,  increasing 
directly  with  their  number.  Or,  .8  cube  feet  per  IIP  for  a  side-wheel  engine, 
and  .65  for  an  ordinary  and  .55  for  a  fast-running  screw-propeller. 

Space  is  required  proportionate  to  volume  of  steam  per  stroke  of  piston 
Tli us,  with  like  boilers,  the  space  may  be  inversely  as  the  pressures. 

Steam-drums  and  steam-chimneys,  by  their  height,  add  to  the  effect  of 
their  volume,  by  furnishing  space  for  water  that  is  drawn  up  mechanically 
by  the  current  of  steam,  to  gravitate  before  reaching  the  steam-pipe. 

Grate.  —  Area  in  sq.  feet  per  Ib.  of  coal  per  hour  for  following  boilers. 
Width,  1.5  diameter  of  furnace: 
Cornish  and  Lancashire,  slow  I  Portable,  moderate  forced  . .    03  sq.  foot 

combustion .2  sq.  foot.    Locomotive  and  like,  strong 

Marine, tubular 0510.066"     "     |     blast... 01  "     *' 

Thickness  of  Tubes  per  B  W  G. 

External  diameter  in  ins 2    2.25   2.5   2.75   3   3.25   3.5   3.75  4 

Thickness  for  pressure  of  50  Ibs.,  number..  12  12      n     n      n  10      10    10       9 
"         "         "      ioo    u         "      ..it   10        99        98        88        7 

Smoke-pipes   and.   diimneys. 

Area  at  their  base  should  exceed  that  of  extremity  of  flue  or  flues,  to 
which  they  are  connected. 

In  Marine  service  smoke-pipe  should  be  from  .16  to  .2  area  of  grate.  In 
Locomotive,  it  should  be  .1  to  .083. 

Intensity  of  their  draught  is  as  square  root  of  their  height.  Hence,  rela- 
tive volumes  of  their  draught  is  determined  by  formula: 

yh  .  i  a  =  volume  in  sq.  feet,  h  representing  height  of  pipe  or  chimney  in  feet,  and 
a  its  area  in  sq.  feet. 

When  wood  is  consumed  their  area  should  be  1.6  times  that  of  coal. 

Chimneys  (Masonry).— Diameter  at  their  base  should  not  be  less  than  from 
.1  to  .n  of  their  height. 

Batter  or  inclination  of  their  external  surface  .35  inch  to  a  foot,  which  is 
about  equal  to  i  brick  (.5  brick  each  side)  in  25  feet. 

Diameter  of  base  should  be  determined  by  internal  diameter  at  top,  and 
necessary  batter  due  to  height. 

Thickness  of  walls  should  be  determined  by  internal  diameter  at  top ; 
thus,  for  a  diameter  of  4  feet  and  les»,  thickness  may  be  i  brick,  but  for  a 
diameter  in  excess  of  that  1.5  bricks. 

Area.    ~r  =  <*•    C  representing  weight  of  coal  consumed  per  hour  in  Ibs..  and 
T/h 

a  area  of  ditto  at  top,  in  sq.  ins. 

(Brick  masonry.)— 25  tons  weight  per  sq.  foot  of  brickwork  in  height  is 
safe  if  laid  in  hydraulic  mortar. 

Less  the  height  of  a  smoke-pipe  or  chimney,  the  higher  the  temperature  ol 
its  gases  is  required. 


STEAM-ENGINE. PUMPS. — PLATES    AND   BOLTS.       749 

Velocities  of  Current  of  Heated  Air  in  a  Chimney  100  Feet  in  Height. 
In  Feet  per  Second. 


External  Air. 

Air 

150° 

at  Base 
250° 

of  Chirnr 
350° 

ey. 
45oe 

External  Air. 

Air 
150° 

at  Base 

250' 

of  Chimr 
JSP" 

ey. 
450* 

Feet 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

10° 

24 

30 

33 

35 

60° 

*9 

26 

29 

33 

32° 

22 

28 

3i 

34 

700 

18 

25 

29 

32 

50° 

20 

27 

30 

33 

80° 

17 

24 

28 

32 

When  Height  of  Chimney  is  less  than  100  feet. — Multiply  velocity  as  ob- 
tained for  temperature  by  .1  square  root  of  height  of  chimney  in  feet. 

Draught  consequent  upon  a  steam -jet  in  a  smoke-pipe  or  chimney  is 
nearly  equal  to  that  of  a  moderate  blast. 

The  most  effective  draught  is  when  absolute  temperature  of  heated  air  or 
gas  is  to  that  of  external  air  as  25  to  12,  or  nearly  equal  to  temperature  of 
melting  lead. 

In  chimneys  of  gas  retorts,  ovens,  and  like  furnaces,  the  draught  is  more 
intense  for  a  like  height  of  chimney  than  in  ordinary  furnaces,  in  con- 
sequence of  the  great  mass  of  brick  masonry,  which,  becoming  heated,  adds 
to  intensity  of  draught. 

Chimneys.    Lawrence  Manufacturing  Co.,  Mass.     Octagonal. 
Height  above  ground  211  feet.     Diameters  15,  and  10  feet  1.5  ins.    Wall  at  base 
93.5,  and  at  top  11.5  ins.    Shell  at  base  15  ins.,  at  top  3.75  ins. 
Foundation  22  feet  deep. 

England.—  Square.—  Height 190  feet    Diameter  at  base. ..,    ...20  feet 

"     300    "  »         «       29    " 

Round.         "     312    "  "         "       30    " 

«     3oo    "  "         "      20   u 

Diameter  at  base  usually  i  of  height  above  ground. 

Vacuum  at  base  of  chimney  ranges  from  .375  to  43  his.  of  water. 

Circulating  3?u.mps. 

Single-acting.  —  .6  volume  of  single-acting  air-pump  and  .32  of  double* 
acting. 
Double-acting.  —  .53  volume  of  double-acting  air-pumpb 

Volume  of  Pump  compared  to  Steam  Cylinder  or  Cylinders. 

Engine.  Pump  Volume. 

Expansive,  1.5  to  5  times Single-acting 08   to  .045. 

Compound do.          04510.035. 

Expansive,  1.5  to  5  times Double-acting .045  to  .025. 

Compound do 02510.02. 

Valves. — Area  such  as  to  restrict  the  mean  velocity  of  the  flow  to  450  feet 
per  minute. 

PLATES  AND  BOLTS. 

Wrouglit-irori.— Tensile  strength  ranges  from  45500  to  70000  Ibs. 
per  sq.  inch  for  plates,  and  60000  to  65000  Ibs.  for  bolts,  being  increased 
when  subjected  to  a  moderate  temperature. 

English  plates  range  from  45000  to  56000  Ibs.,  and  bolts  from  55000  to 
59  ooo  Ibs. 

D  K.  Clark  gives  best  quality  of  Yorkshire  56  ooo  Ibs.,  of  Staffordshire  44  800  Ibs. 

Test  of  IPlates.  (U.  S.)  —  All  plates  to  be  stamped  at  diagonal  corners  at 
about  four  ins.  from  edge,  and  also  in  or  near  to  their  centre,  with  name  of  manu- 
facturer, his  location,  and  tensile  stress  they  will  bear. 

Plates  subjected  to  a  tensile  stress  under  45  ooo  Ibs.  per  sq.  inch,  should  contract 
in  area  of  section  12  per  cent.,  45000  and  under  50000, 15,  and  50000  and  over,  25, 
at  point  of  rupture. 


750 


STEAM-ENGINE. — PLATES. 


Brands.  (C  No.  i)  Charcoal  No.  i.—  Plates,  will  sustain  a  stress  of  40000  Ibs.  per 
sq.  inch;  hard  and  unsuited  for  flanging  or  bending. 

(C  No.  i  R  H)  Reheated,  hard  and  durable,  suited  for  furnaces,  unsuited  for  con- 
tinued bending. 

(C  H  No.  i  S)  Shell,  will  sustain  a  stress  of  50000  to  54000  Ibs.  in  direction  of  fibre, 
and  34000  to  44000  across  it:  hard  and  unsuited  for  flanging  or  even  bending  with 
a  short  radius. 

(C  H  No.  i  F)  Flange,  will  sustain  a  stress  of  50000  to  54000  Ibs.,  soft  and  suited 
for  flanging. 

(C  H  No.  i  F  B)  Furnace  and  (C  H  No.  i  F  F  B)  Flange  Furnace.  The  first  is 
hard,  but  capable  of  being  flanged,  the  other  is  hard,  and  suited  for  flanging. 

The  especial  brands  are  Sligo,  Eureka,  Pine,  etc. 

The  best  English  plates  known  are  the  Yorkshire,  as  Low  Moor,  Bowling,  Farnley, 
Monk  Bridge,  Cooper  &  Co.,  etc.  (See  Steam-boilers,  W.  H.  Shock,  U.  S.  N.,  1880.) 

Steel. — Tensile  strength  ranges  from  75000  to  96000  Ibs.  Mr.  Kirkaldy 
gives  85  966  Ibs.  as  a  mean. 

When  used  in  construction  of  boiler-plates  should  be  mild  in  quality,  containing 
but  about  .25  to  .33  per  cent,  of  carbon;  for  when  it  contains  a  greater  proportion, 
although  of  greater  tensile  strength,  it  is  unsuited  for  boilers,  from  its  hardness  and 
consequent  shortness  in  its  resistance  to  bending. 

Crucible  steel  may  be  used,  but  that  obtained  by  the  Bessemer  or  Siemens-Martin 
process  is  best  adapted  for  boiler-plates.  Its  strength  becomes  impaired  by  the 
processes  of  punching  and  shearing,  rendering  it  proper  thereafter  to  submit  it  to 
annealing. 

Steel  rivets,  when  of  a  very  mild  character  and  uniformly  heated  to  a  bright  red, 
are  superior  to  iron  in  their  resistance  to  concussion  and  stress. 

Copper. — Tensile  strength  is  33  ooo  Ibs.,  being  reduced  when  subjected 
to  a  temperature  exceeding  120°.  At  212°  being  32  ooo,  and  at  550°  25  ooo  Ibs. 

"Wrouglit-iron    Shell   IPlates. 

Pressure   and.   Thickness.     ((7.  S.  Law.) 

Based  upon  a  Standard  of  One  Sixth  of  Tensile  Strength  of  Plates.    Iron  or  Steel 


Results  with  a  Tensile  Strength  0/50000  Lbs. 

Thick- 

Diameters in  Ins. 

ness. 

36 

38 

40 

42 

44 

46 

48 

54 

60 

66 

72 

78 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Tbs7 

Lbs. 

Lbs. 

Lba. 

.25 

116 

no 

104 

99 

95 

91 

87 

77 

69 

63 

58 

53 

•3125 

145 

137 

130 

124 

118 

IJ3 

109 

96 

87 

79 

72 

67 

•375 

'74 

165 

156 

149 

142 

136 

130 

116 

104 

87 

80 

•5 

232 

220 

208 

198 

190 

182 

174 

*54 

138 

126 

116 

106 

84 

90 

96 

IO2 

108 

"4 

120 

126 

132 

i35 

140 

144 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 
•4375 

69 
80 

3 

61 

58 
68 

52 
61 

49 
57 

47 

55 

46 
53 

44 

43 
50 

•5625 

99 
in 

92 
103 

98 

81 

87 

I 

69 
78 

65 
73 

63 

61 

69 

i 

57 
64 

148 

138 

130 

121 

115 

109 

103 

97 

94 

88 

85 

•875 

172 

160 

152 

142 

136 

128 

122 

114 

no 

106 

102 

TOO 

i 

198 

184 

l62 

154 

146 

138 

130 

126 

122 

118 

114 

To  which  20  per  cent,  is  to  be  added  for  double  riveting  and  drilled  holes. 

Iron  plates  .375  inch  in  thickness  will  bear,  with  stay  bolts  at  4,  5,  and  6  ins. 
apart  from  centres,  respectively  170,  150,  and  120  Ibs.  per  sq.  inch. 

Iron  plates,  as  tested  by  Mr.  Phillips  at  Plymouth  Dockyard,  .4375  inch  in  thick- 
ness, with  screw  stay  bolts  1.375  ins.  in  diameter  riveted  over  heads,  15. 75  and  15.25 
ins.  from  centre  =  240  sq.  ins.  of  surface  for  each  bolt;  bulged  between  bolts  and 
drew  from  bolts  at  a  pressure  of  105  Ibs.  per  sq.  inch  of  plate. 

Iron  plates  .5  inch  in  thickness,  under  like  conditions  with  preceding  case,  bulged 
and  drew  from  bolts  at  a  pressure  of  140  Ibs.  per  sq.  inch  of  plate.  Hence,  it  ap- 
pears, resistances  of  plates  are  as  squares  of  their  thickness. 

When  nuts  were  applied  to  ends  of  bolt  through  .4375  inch  plate,  its  resistance  in- 
creased to  165  Ibs.  per  sq.  inch  of  ulate. 


STEAM-ENGINE. — SHELLS. — PLATES.  75 1 

Cylindrical   Shells.    (&•  S.  Law.) 

Xo  Compute  Pressure  for  a  Griven  Thickness  and 
Diameter,  or  Thickness  for  a  Griven.  Pressure  and 
Diameter. 

For  Pressure.  RULE. — Multiply  thickness  of  plate  in  ins.  by  one  sixth 
of  tensile  strength  of  metal,  and  divide  product  by  radius  or  half  diameter 
of  shell  in  ins. 

When  rivet-holes  are  drilled,  and  longitudinal  courses  are  double  riveted, 
add  one  fifth  to  result  as  above  attained. 

EXAMPLE.  —Assume  boiler  8  feet  in  diam.,  and  plates  .5  inch  thick;  what  work- 
ing pressure  will  it  sustain,  tensile  strength  of  plates  equal  to  a  stress  of  60000  Ibs.? 


8  X  12 


•  5  X  60000-:- one  sixth-: =  ^-^-  =  104.16  Ibs. 

2  48 

For  Thickness.    RULE. — Multiply  pressure  by  radius  of  shell,  and  divide 
product  by  one  sixth  of  tensile  strength  of  metal. 
EXAMPLE.— Assume  pressure,  radius,  and  tensile  strength  as  preceding. 

104.16x96-^2   =  5000  _ 
60 ooo-r- one  sixth      10000 
For  Evaporation  of  Salt  Water.— Add  one  sixth  to  thickness  of  plates  and  sec- 
tional  area  of  stay  bolts. 

For  Freight  and.  River  Steaniboats. 

Standard.  —  150  Ibs.  pressure  for  a  boiler  42  ins.  in  diameter  and  plates 
.25  inch  thick. 

For  Pressure.  RULE.— Multiply  thickness  of  plate  by  12  600,  and  divide 
result  by  radius  of  boiler  in  ins. 

EXAMPLE.— Assume  a  boiler  42  ins.  in  diameter,  and  plates  .25  inch  in  thickness; 
what  working  pressure  will  it  sustain? 

.25  X  12  600 -7-42 -r- 2  =  150  Ibs. 

Proof.— All  boilers  by  U.  S.  Law  to  be  tested  to  a  hydrostatic  pressure  of  50  per 
cent,  above  that  of  their  working  pressure. 

Relative   Mean.    Strength    of  Riveted   Joints    compared 
to   that   of  IPlates. 

Allowances  being  made  for  Imperfections  of  Rivets,  etc. 

Plates,  100 ;  Triple,  .72  to  .75;  Double  or  Square,  .68  to  .72;  Double 
with  double  abut  straps,  .7  to  .75 ;  Staggered,  .65 ;  Single,  .56  to  .6. 

Board  of  Trade,  England. 

Coefficient  or  Factor  of  Safety.  —  When  shells  are  of  best  material  and 
workmanship,  rivet-holes  drilled  when  plates  are  in  place,  abut  strapped, 
plates  at  least  .625  inch  in  thickness  and  double  riveted,  with  rivets  com- 
puted at  a  resistance  not  to  exceed  75  per  cent,  over  the  single  shear,*  the 
coefficient  is  taken  at  5.  Boilers  must  be  tested  by  hydrostatic  pressure  to 
twice  that  of  working  pressure. 

Tensile  strengths  of  plates  are  taken,  with  fibre  47  ooo  Jbs.  per  sq.  inch, 
across  it  40000  Ibs.,  and  when  in  superheaters  from  30000  to  22  400  Ibs. 

47  — -  =  P,  and  —  — —  =  t.  P  representing  pressure  that  shell  will  sus- 
tain per  sq.  inch  in  Ibs.,  B  least  per  cent,  of  strength  of  rivet  or  plate  (whichever  it 
least)  at  lap,  D  diam.  of  shell  and  t  thickness  of  plate,  both  in  ins.,  and  C  coefficient 
of  safety. 

*  Shearing  or  detruaive  resistance  of  wrought  iron  U  from  70  to  80  per  cent,  of  its  tensile  strength* 


752 


STEAM-ENGINE. — SHELLS. — PLATES. 


«ox  6x5X5  =  .5  inch. 
50  ooo  X- 75X2 


Pd      _      2<C      _        . Pd 
— -  =  C,     — —  =P,  and  — -  = 
at  d  2  C 


ILLUSTRATION.— Assume  T  =  50000  Ibs.  tensile  strength  of  plate,  B  =  75  per  cent., 
D=  120  ins.,  C  =  s,  and  t=z.$.  What  pressure  will  shell  sustain,  and  what  should 
be  thickness  of  plates  for  such  pressure  and  diameter? 

SooooX.75X.5Xa 

120X5 

Tor  all  practicable  deficiencies  in  drilling,  punching,  and  riveting  in  trans- 
verse courses,  if  existing,  this  coefficient  is  increased  up  to  6.75,  and  in  lon- 
gitudinal courses  to  8.75,  and  when  courses  are  not  properly  broken,  an 
addition  is  made  to  above  of  .4. 
Diameter  of  rivets  should  not  be  less  than  thickness  of  plates. 

Molesworth. 

d  representing  diameter  and  t  thickness  of 

metal,  both  in  ins.,  P  working  pressure  in  Ibs.  per  sq.  inch,  and  C  as  follows  : 

Single  riveted.  Double  riveted. 

Best  Yorkshire  plates )  .  fh    f  tpn<silA   (C  =  62oo      and      7800 

"    Staffordshire  plates. ...}   one  ninth  of  tensile  >u=  ,.       6200 

Ordinary  plates. )  ("  =  3000        »        3700 

Working  stress  not  to  exceed  .2  tensile  strength  of  joint  or  riveted  plate. 
Then  for  a  pressure  of  no  Ibs.,  and  a  diameter  of  42  ins.,  as  given  for  a  standard 
U.  S.  boiler. 

Taking  C  as  above  for  best  single-riveted  plate  at  6200,     "°x  42  =  .372  -f  ins. 

2  X  O2OO 

in  thickness,  or  .  122  inch  in  excess  of  U.  S.  Law  for  a  plain  cylindrical  boiler,  single 
riveted. 

Lloyd's. 
Thickness  of  shells  to  be  computed  from  strength  of  longitudinal  joints. 

/JC      ,  ;     PD  t  JC  p  —  d  .  na 

•  =  P,  — —  =  t,  =  D,  — =  x,  and  —  =  *.  t  representing  thick- 
ness of  plate,  D  diameter  of  shell,  p  pitch  and  d  diameter  of  rivets,  all  in  ins.  ;  J  per 
cent,  of  strength  of  joint  or  rivets,  the  least  to  be  taken;  C  a  constant  as  per  table  ; 
P  working  pressure  in  Ibs.  per  sq.  inch  ;  n  number  and  a  area  of  rivet ;  x  per  cent, 
of  strength  of  plate  at  joint  compared  with  solid  plate,  and  zper  cent,  of  strength  of 
tivets  compared  with  solid  plate. 

When  plates  are  drilled,  take  .9  of  z,  and  when  rivets  are  in  double  shear, 
put  1.75  a  for  a. 


Constants. 

IRON  PLATES. 

STEEL  PLATES. 

JOINT. 

.5  inch 
and 
under. 

.75  inch 
and 
under. 

Above 
.7Sinch. 

•3sr 

under. 

inland 
under. 

.75  inch 
and 
under. 

Above 
•75inch. 

T        (  punched  holes  .. 

155 
170 

165 
1  80 

170 

IQO 

200 

215 

230 

240 

Lap  i  drilled       do   ....... 

Double  abut  }  punched  holes 
strap          (  drihed       do. 

170 

1  80 

1  80 
190 

•Jrf 

IQO 
200 

215 

230 

250 

260 

When  plates,  as  in  steam-chimneys,  superheaters,  etc.,  are  exposed  to  direct  ac- 
tion of  the  flame,  these  constants  are  to  be  reduced  .33. 

ILLUSTRATIONS.  —  Assume  pitch  4  ins.,  diam.  of  rivet  1.375  ins.,  and  thickness  of 
plate  i  inch,  both  single  and  double  riveted.     Area  1.375  =  1.48  sq.  ins. 

4~''375  =  .  656  per  cent,  strength  of  joint  compared  to  solid  plate.    ?-^4^-  =  .  37 
4  4X1 

d  r'75        —  =  .647 
4X 


per  cent,  strength  of  rivet  to  solid  plate  when  single  riveted,  an 

per  cent,  when  do 
by  go  with  drilled. 


per  cent,  when  double  riveted.    Rivets  at  Joint.    ™  X  100  with  punched  holes  and 


STEAM-ENGINE.  —  PLATES.  —  ABUT   STRAPS,  ETC.      753 

Flates. 

To  Co**ipnte  Thickness  of  I>lates  for  a  Griven  Pressure 
and  /i>itcn,  and.  I*ressu.re  and  IPitch  for  GJ-iven  Thicli- 
ness« 

-  rrcp.       /-——p,  and      /—£-  =  t-     t  representing  thickness  of  metal  in 

p2  V      "  V       vy 

sixteeH'is  of  an  inch,  p  pitch  of  stays  or  distance  apart  at  centres  in  ins.,  P  working 
pressure  in  Ibs.  per  sq.  inch,  and  C  a  constant,  as  follows  : 

For  a  Tensile  Strength  of  Metal  of  50  ooo  Lbs.  per  Sq.  Inch. 

Screw  Stay-bolts  with  Riveted  Heads.—  Plates  up  to  .4375  inch  in  thickness  C  =  90, 
and  above  that  100. 

Screiv  Stay-bolts  with  Nuts.  —  Plates  up  to  .4375  inch  in  thickness  C  =  uo,  and 
above  that  120. 

Screw  Stay-bolts  with  Double  Nuts  and  Washers.  —  Up  to  4.375  ins.  in  thickness 
C  as  140,  and  above  that  160. 

When  stay-bolts  are  not  exposed  to  corrosion,  these  constants  may  be  reduced  .2. 

Resistance  of  a  flat  surface  decreases  in  a  higher  ratio  than  space  between 
stays.  Hence,  C  must  be  decreased  in  proportion  to  increase  of  pitch  above 
that  of  ordinary  boiler-plates. 

ILLUSTRATION  i.—  Assume  pressure  no  Ibs.  per  sq.  inch,  and  pitch  of  stays  5  ins.  ; 
what  should  be  thickness  of  plate  for  screw-bolts  and  riveted  heads? 

C  =  95.    Then  J1-^^-  =  y/—  =  5-  38-  sixteenth, 

2.  —  Assume  thickness  of  metal  5  sixteenths  inch  thick,  stay-bolts  screwed  and 
riveted  over  its  threads,  and  working  pressure  of  steam  80  Ibs.  per  sq.  inch, 

C  =  95.    Then       S**9S  =  5-45  ins.  pitch. 


A-font   Straps. 

Double  Abuts  should  be  at  least  .625  thickness  of  plate  covered.  Single, 
.125  thicker  than  plate  covered,  and  Double,  .625. 

Stays. 

Direct.  —  Tensile  stress  should  not  exceed  5000  Ibs.  per  sq.  inch  for  Iron, 
and  7000  for  Steel. 

Diagonal  or  Oblique.  —  Ascertain  area  of  direct  stay  required  to  sustain 
the  surface  ;  multiply  it  by  length  of  diagonal  stay,  and  divide  product  by 
length  of  a  line  drawn  at  a  right  angle  to  surface  stayed,  to  end  of  diagonal 
8ta}%  and  quotient  will  give  area  of  stay  increased  to  that  which  is  required. 

Stress  upon  an  oblique  stay  is  also  equal  to  stress  which  a  perpendicular 
stay  supporting  a  like  surface  would  sustain,  divided  by  cosine  of  angle 
which  it  forms  with  perpendicular  to  surface  to  be  supported. 

ILLUSTRATION.  —  Assume  pressure  no  Ibs.  per  sq.  inch,  area  of  supported  surface 
36  sq.  ins.,  and  angle  of  stay  45°;  what  would  be  pressure  or  stress  upon  stay? 

Cosine  45°  =  .  707  1  1.    Then  1  10  x  36  -5-  .  707  1  1  =  5600  tot. 


754  STEAM-EN  GLNE. — GIKDEKS. — FLUES,  ETC. 


I»roporti< 

DIMENSIONS. 

5ns  of*  Eyes   of*  Stays,  Rod 

No.  i.                               No.  2. 
FORGED  AND  WELDED. 

s,  etc. 

No.  3- 
DRILLED  FROM  BAI 

SL 

c 

C 

No.  i.  a  and  a  =  x      inch. 

rrt 

b=   .9     « 

*>(  ©    ) 

~" 

t>(  @  ) 

«=   -75    " 

\    *•_./    1 

— 

V     ^-S    J 

No.  2.  a  and  a  =  i        " 

T^ 

/• 

\ 

Nk 

6=   .6     " 

«=   -75    " 

No.  3.  a  and  a  =  i        " 

a 

% 

a 

^ 

n 

a 

i 

&  =    -75    " 

""" 

«  =    .875  " 

m 

•>-~ 

bss 

.  - 

6® 

L- 

When  drilled  from  upset  bar,  dimensions  same  as  for  No.  i.  Pins  when  of  steel 
.66  neck  of  rod. 

Stay-bolts.— /row,  are  not  to  be  subjected  to  a  greater  stress  than 
6000  Ibs.  per  sq.  inch  of  section  ;  Steel,  8000  Ibs.,  both  areas  computed  from 
weakest  part  of  rod,  and  when  of  steel  they  are  not  to  be  welded. 

To   Compute    Diameter    and.    IPitch    of*  Stay  -  "bolts,  and 
Resistance   they   will    Snstain. 


Screwed. 


=  d,     ^=p,     and  (^-)2  =  P- 


Socket. 


=|7,    and  (^—  )  =P.    d  representing  diameter  in  ins. 
\  P  / 


95 


ILLUSTRATION.—  Assume  pitch  of  stay  bolts  6  ins.,  and  working  pressure  100  Ibs. 
per  sq.  inch;  what  should  be  diameters  of  bolts,  both  screw  and  socket? 


6  X  -v/ioo  ,  6  X  -v/IO( 

— ^-^ =  .857  inch  Screwed,  and 4 — 

70  95 

GJ-irders.     (Lloydjs.) 
P(L— p)DL_  /P(L  — 


=  .63-j-tncA  Socket. 


Cd*t 


(L-p)DL"'  OS*-"?  V 0* =  *  L^^ew^ 

2en^A  of  girder,  d  its  depth,  t  its  thickness  at  centre  or  sum  of  its  thicknesses,  D  its 
distance  apart  from  centre  to  centre,  and  p  pitch  of  stays,  all  in  ins.,  and  C  a  constant 
as  per  following : 

One  stay  to  each  girder,  C  =  6000.     If  two  or  three  =  9000.     If  four  —  10  200. 

ILLUSTRATION. —  Assume  triple  stayed  girder,  24  ins.  in  length,  3  ins.  in  depth,  j 
inch  thick,  and  stayed  at  intervals  of  6  ins. ;  what  working  pressure  will  it  sustain? 

_  9000  X  62  X  i         324  ooo 

C  =  9000.    Then  -^ —  —  =  222L  —  =  125  Ibs. 

(24  — 6)X6X24        2592 

Flues,  Arched   or   Circular   innrnaces.     U.  S.  Law. 

.3125  inch  for  each  16  ins.  of  diameter.  English  iron,  being  harder  than 
American,  is  better  constructed  to  resist  compression,  and  consequently  a 
less  thickness  of  metal  is  required  for  like  stress. 


Lloyd's. 


/PLD 


-  =  L-  ^representing 
external  diameter  of  flue  or  furnace,  and  t  thickness  of  plate,  both  in  ins.,  L  length 
of  flue  or  furnace  between  its  ends  or  between  its  rings,  in  feet,  and  P  working  press- 
ure in  Ibs.  per  sq.  inch. 

ILLUSTRATION.  —  Assume  diameter  of  flue  16  ins.,  length  6  feet,  and  working  press 
use  of  steam  80  Ibs.  per  sq.  inch. 


_,  /8o  x  6  x  16 

Then     /  —  —  —  —  -  = 


J 
=  .29  inch.    Furnace.  —  P  not  to  exceed  • 


STEAM-ENGINE.  —  RIVETING.  75  5 

ILLUSTRATION.  —  Assume  diameter  of  a  circular  furnace  or  width  of  a  semicircular 
one  48  ins.,  working  pressure  of  steam  80  IDS.,  and  length  6  feet. 


Then  -  =  ^-257  =  .507  inch  thickness. 

RIVETING. 

Plates.  —  The  strength  of  a  joint  is  determined  by  ascertaining  which 
of  the  two,  the  plate  or  the  rivets,  has  the  least  resistance  ;  the  stress  on  the 
first  being  tensile  and  the  latter  detrusive. 

The  tensile  strength  is  to  be  taken  from  that  of  the  article  under  consider- 
ation, making  due  allowances  for  construction  and  location  of  the  joint,  and 
the  consequent  variation  of  stress,  as  with  or  across  the  fibre  of  the  metal, 
or  exposed  to  high  heat  as  in  a  superheater. 

With  or  Across  the  Fibre.  —  From  experiments  of  Mr.  D.  Kirkaldy  and 
others,  the  difference  in  strength  of  Iron  plates  is  ascertained  to  be  from  6.5 
to  18  per  cent.,  the  average  10  per  cent. 

Steel  Plates.  —  The  relative  strength  of  plates  with  or  across  the  fibre,  as 
determined  by  Mr.  Kirkaldy,  for  "Fagersta"  is  9  per  cent.,  and  for  "Siemens" 
it  is  without  material  difference. 

Holes.  —  The  relative  strength  of  plates  when  subjected  to  drilled  or 
punched  holes,  as  determined  by  the  experiments  of  Mr.  Kirkaldy,  is  shown 
to  be  15  per  cent. 

In  Riveted  Joint  exposed  to  a  tensile  stress,  area  of  rivets  should  be  equal 
to  area  of  section  of  plates  through  line  of  rivets,  running  a  little  in  excess 
up  to  .5625  inch  diameter  of  rivet,  and  somewhat  less  beyond  that,  area  be- 
ing determined  by  relative  shearing  and  tensile  resistances  of  rivet  and 
plate. 

NOTE.—  For  Riveting  of  Hulls  of  Vessels,  see  pp.  828-30. 

Essentially  by  Nelson  Foley. 

Single    Lap    Riveting. 
=b  for  plate,        ^  =  V  for  rivets,        -A-=p,        ptV  =  a,      and 

'  -  1  =  d  .  p  representing  pitch,  t  thickness  of  plate,  and  d  diameter  oj  rivets, 
i  —  o 

all  in  ins.,  a  sectional  area  of  rivets  in  sq.  ins.  ,  n  number  oj  rivets,  and  b  and  b  per 
cent,  of  plate  between  holes  and  of  section  of  rivets  to  solid  plate,  i.  e.  plate  before 
being  punched. 

ILLUSTRATION,—  Assume  p  =  3  ins.  ,    d  =  i  inch,    a  =  .  7854  inch,  and  t  =  5  inch. 

•3  ""'  =  .  66  per  cent,  strength  of  lap,      '7  54  =  .  523  per  cent,  of  rivet  to  solid  plate, 
3  3X5 


3  X  .5  X  .523-f-  =  .7 


When  Shearing  Strength  of  Rivet  is  not  Equal  to  Tensile  Strength  of  Plate. 
—  Then  diameter  of  rivet  must  be  increased  in  ratio  of  excess  of  strength  of 
plate  over  rivet. 

Or,  ^^  —  T-  t  =  d.    T  and  S  representing  tensile  and  shearing  strengths,  which  may 

i  —  o  S 
be  takzn  at  5  and  4  for  Iron  and  ^  and  6  for  Steel. 

When  full  value  of  rivet  sectiom  is  not  allowed  as  by  Lloyd's  rules  for  drilled 
holes,  b'=:&'x.9- 


756 


STEAM-ENGINE. — RIVETING. 


Pitches    as   Determined   by   Diameter  of  Rivets. 


Plate 
between 
Edges 
of  Holes. 

Pitch  = 
Diam.  of 
Rivet  X 

Plate 
between 
Edges 
of  Holes. 

Pitch  = 
Diam.  of 
Rivet  X 

Plate 
between 
Edges 
of  Holes. 

Pitch  = 
Diam.  of 
Rivet  X 

Plate 
between 
Edges 
of  Holes. 

Pitch  = 
Diam.  of 
Rivet  X 

Per  Cent. 
50 
52 
55 

2 
2.08 
2.22 

Per  Cent. 
62 

2.38 
2-5 

2.63 

Per  Cent. 
65 
68 
70 

2.86 
3-13 
3-33 

Per  Cent. 
72 
75 
78 

3-57 
4 
4-55 

OPERATION.— If  distance  between  edges  of  holes,  or  p  —  d,  =65  per  cent,  of  solic 
plate,  and  diam.  of  rivet  i  inch,  then  2.86  X  i  =  2.86  ins.  pitch. 

When  Plate  and  Rivets  are  of  equal  strength  in  ultimate  tension,  &'  =  6,  =  B. 
Hence,  *'^  t  =  a.  In  illustration  of  B,  assume  p  =  3,  d  =  1. 1,  and  t  =  .5. 
Then  3  —  i.i  =  1.9,  and  ^=  .633  =  Z>,  or  per  cent,  of  strength  of  punched  to 

tolid  plate.    Area  1. 1  =  .95,  and    '9     =  .633  =  &',  or  per  cent,  of  section  of  rivet  to 
tolid  plate.    Hence,  B  =  .  633. 
ILLUSTRATION.— -Assume  as  shown,  6=1.633. 


Diameter  of  Rivets   as   Determined   toy-   Plate. 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thick  ness 
of  Plate  X 

Per  Cent. 

T  =  S. 

.9  per  cent, 
of  Section 

Per  Cent. 

T  =  S. 

.9  per  cent. 
ofSection 

Per  Cent. 

T  =  S. 

.9  per  cent, 
ofSection 

of  Rivet. 

of  Rivet. 

of  Rivet. 

52 

53 

1-38 
1.44 

i-53 
i-59 

55 
S^ 

It 

I:P 

f 
60 

1.76 
1.91 

i-95 

2.12 

54 

i-5 

1.66 

57 

1.69 

i.87 

62 

2.08 

2.31 

OPERATION.—  If  thickness  of  plate  =  .5  inch  and  plate  and  rivet  have  equal  resist- 
ance, or  B  =  62  per  cent.,  then  .5  X  2.08  =  1.04  ins.  diameter. 


9  =  .44l8amt()/d, 


Donble    T-jap    Riveting. 

Preceding  formulas  for  single  lap  riveting  apply  to  this,  with  substitution 
of  2d  for  a  and  .64  for  1.27. 

ILLUSTRATION.  —  Assume  p  =  3  ins.  ,  t  =  .5  inch,  and  b'  =  .589. 
3X.5X.589      .l  .         ., 

i—  -75  3 


Diameter  of  Rivets   as   Determined    by    Plate. 


B 

•r  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

Per  Cent. 

T  =  S. 

.0  per  cent. 
ofSection 

Per  Cent. 

T  =  S. 

.9  per  cent, 
of  Section 

Per  Cent. 

T  =  S. 

.9  per  cent, 
of  Section 

of  Rivet. 

of  Rivet. 

of  Rivet. 

68 

1-35 

1-5 

71 

1.56 

1-73 

74 

1.81 

2 

69 

1.42 

i-57 

72 

1.64 

1.82 

75 

1.91 

2.12 

70 

1.48 

1.65 

73 

1.72 

1.91 

76 

2 

2.25 

OPERATION.  —  Assume  t  — .  5  inch  and  B  —  70  per  cent. ,  tensile  strength  compared 
to  shearing  being  as  7  to  6.     What  should  be  diameter  of  the  rivets? 

.5  X  1.48  x  —  =  .863  inch.    When  rivets  are  in  double  shear,  put  1.9  a  for  a. 


STEAM-ENGINE. — DUTY. — EVAPOEATION. 


757 


Triple   Lap    Riveting. 

Preceding  formulas  for  single  lap  riveting  apply  to  this,  with  substitution 
>f  3  a  for  a  and  .42  for  1.27. 
ILLUSTRATION.—  Assume p  =  3  in*.,  t  .5  inch,  and  b'  =  .883. 


-  =  •75 1, 


and 


i=.8836'. 


3X-5 
Diameter   of*  Rivets   as   Determined   by   Plate. 


B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =  Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  =Thlckne»i 
of  Plate  X 

Per  Cent. 

t  =  S. 

.9  per  cent. 
ofSection 

Per  Cent. 

T  =  S. 

.9  per  cent. 
ofSection 

Per  Cent. 

T  =  S, 

.9  per  cent 
ofSection 

of  Riret. 

of  Rivet. 

of  Rivet. 

70 

•99 

I.I 

73 

I.I5 

1.27 

76 

i-34 

1.49 

7i 

1.04 

'•IS 

74 

I.  21 

'•34 

77 

1.42 

1.58 

72 

1.09 

1.  21 

75 

1.27 

1.41 

78 

i-5 

1.67 

OPERATION.— As  shown  by  preceding  tables. 

Q-eneral    Formulas   and. 

1.27  BT 


Rivets  in  Single  Shear. 
Rivets  in  Double  Shear. 
Rivets  in  Triple  Shear. 


Illxist  rations. 

'=*-»•  FTf=6' 


i  (i—  B)S 

i^S.-*- 

i.27BT     , 
fcS(.-B)S<  =  <l'"ld 


a  2.5  S 

J9«T     = 


Zigzag  Riveting.  Strength  of  plate  between  holes  diagonally  is 
equal  to  that  horizontally  between  holes,  when  diagonal  pitch  =  .6  and  hor- 
izontal =  diameter  of  rivet  -f  .4. 

Thus,  .6  p  -f-  .4  p  —  diagonal  pitch. 

Dnty   of*  Steam-engines. 

The  conventional  duty  of  an  engine  is  the  number  of  Ibs.  raised  by  it  i 
foot  in  height  by  a  bushel  of  bituminous  coal  (112  Ibs.). 

Cornish  Engine.—  Average  duty,  70  ooo  ooo  Ibs.  ;  the  highest  duty  ranging 
from  47000000  to  101900000  Ibs. 

A  condensing  marine  engine,  working  with  steam  at  .75  Ibs.  (mercurial 
gauge),  cut  off  at  .5  stroke,  will  require  from  1.75  to  2  Ibs.  bituminous  coal 
per  IP  per  hour. 

Relative    Cost   of*  Steam-engines   for   Eqxial   Effects. 

In  Lbs,  of  Coal  per  E*  per  Hour.  Lba 

A  theoretically  perfect  engine  ...........................................  66 

A  Cornish  condensing  engine  ..........................................  2.  38 

A  marine  condensing  engine  ....................................  1.75  to  3 

Evaporative   I?o-wer   of  Boilers. 

The  Evaporative  power  of  a  boiler,  in  Ibs.  of  water  per  Ib.  of  fuel  consumed, 
is  ascertained  approximately  by  formula 

i-833  (    0  .  T.  )  «  =  Ibs.     S  representing  total  heating  surface  in  sq.  feet  F  fuel 

\2  b  -f-  1  1 

consumed  in  Ibs.  per  hour,  and  e  theoretical  evaporative  power  of  the  fuel. 

ILLUSTRATION.  —  Assume  evaporative  power  of  the  fuel  at  15,  consumption  pel 
hour  800  Ibs.,  and  heating  surface  1600. 

Then  I'833  (,6ooxT+8oo)  X  IS  =  i 

3S 


758 


STEAM-ENGINE. — WEIGHTS. 


iUr.    ..833 


The  eva 


45 


vaporative  power  of  different  fuels,  from  and  at  212°,  is,  for  coals,  from  14 
to  16.8  IDS.,  the  average  of  Newcastle  being  15.3,  for  patent  fuels  15.66,  Lignite  13.5, 
Coke  13.3,  Peat  10.3,  and  Woods,  when  dry,  8.1.    See  A.  E.  Seaton,  London,  1883. 

Notes   on.   Hoi*se-po~wer. 

A  Lancashire  boiler  with  a  heating  surface  of  610  sq.  feet  and  a  grate-area  of  25 
will  evaporate  in  ordinary  operation  50  cube  feet  of  water  per  hour  ;  3.12  sq.  feet  of 
horizontal  section  per  cube  foot  of  water,  and  .5  sq.  foot  of  grate-area  per  cube  foot. 

Nominal.    Flue  Boilers.—  Usually  computed  at  5.  5  to  6  sq.  feet  of  horizontal 
section,  15  sq.  feet  of  heating  surface,  and  i  sq.  foot  of  grate-area. 
The  IIP  of  such  boilers  will  range  from  3  to  4  times  that  of  the  nominal. 
Multitubular  Boilers.—  .75  sq.  foot  of  grate-area  and  2.5  ofheating  surface. 

^Weights   of  Steam-engines. 
Side-wheels.—  American  Marine  (Condensing). 


ENGINE. 

Frame. 

Water- 
wheels. 

c 

No. 

ylinders. 
Volume. 

Weight  per 
Cube  Foot. 

SERVICE. 

Wood.* 
Wood.* 
Wood.* 
Wood.* 
Wood.* 
Iron. 
Iron. 
Iron. 

Wood. 
Wood. 
Wood. 
Wood. 
Iron. 
Iron. 
Iron. 
Iron. 

I 

2 

I 
2 
I 
2 
2 
2 

Cube  Feet. 
63 
216 
430 
253 
725 
540 
1502 
«tt«> 

Lbs. 
1  100 

I04of 
1225 
1480$ 
1089! 
850 
55o§ 

1  100 

River. 
Coast. 
Coast. 
Coast. 
Sea. 
Sea. 
Sea. 
Sea. 

t( 

u 

H 

Oscillating  

Inclined... 

*  Without  frame. 


t  With  frame  1109. 


t  Including  boilers. 


§  Single  frame. 


Screw  Propellers.— American  Marine  (Condensing). 


ENGINE. 

Cyli 
No. 

nders. 
Volume. 

Engine. 

WEIGHTS. 
Boilers. 

Per  C.  Ft. 
Cylinder. 

SEH- 

VICX. 

Vertical  direct,  Jet  Condens'g  .  . 
"           "     Surface  Cond'g  . 
"           "     Jet             "      . 

i<           «       <(               n 

it                     U             ((                            U 

Horizontal  back-action  

Cube  Feet. 
4 
12.5 
12.5 

$ 

68 

67 
4.8 

24-3 
4235.6 
35.86 

2.77 

Lbs. 
22040 
59000 
48130 
120450 
i  523  060 
289  680 

201000 
24705 
94I96 
I  O22  400 
30534 
172028 
I44IO 
14759 

Lbs. 

12  100 
32OOO 
35000 
98000 
985600 
200800 
200593 
26372 
88050 
840000 
27301 
100065 
22481 
22417 

Lbs. 
8535 
7280 
6650 
6620 
4958 
7212 
6009 
10641 
75oo 
4380 
16066 
7774 
i9834 
13421 

Sea. 
Sea. 
Sea. 
Coast 
Sea. 
Sea. 
Sea. 
Coast. 
Sea. 
Sea. 
Coast. 
Sea. 
River. 
Coast. 

Vertical  compound  (     *> 

;;         ;;     JJ| 

"      direct.  .             1  3  a 

«     «:::::::::  i^a 

••         "    Non-Condensing. 

«                 ««             it                  U 

English  Marine  (Condensing). 


DESCRIPTION. 

Cy 

No. 

linders. 
Volume. 

Engines. 

WKJ 

Propeller 
and 

Shafting. 

GHTS. 

Boilers 
and 
Water. 

Total. 

Per 
IIP 

Per 

Cube  Ft 
Cylinder. 

Trunk  

2 
2 

2 
2 

2 

6 

2 
2 

Cube  Ft. 
230 
382 

393 
440 
24 
707 
52 
'43 

Tons. 

121 
223 

105 
117 
4-2S 

497 
55 
130 

Tons. 
47 
85 
48 

43 

W75 

15 

27 

Tons. 
257 
303 
144 
135 
7-25 
656 
no 

162 

Tons. 
425 
6n 
357 
295 
12.25 
1320 
1  80 
319 

Lbs. 
465 
338 
781 
560 
60 
368 
35* 
309 

Tons. 
1.85 
z.  6 
•9 
•  7 
•52 
1.87 

3-44 
2.23 

Horizontal  direct      .  . 

Vertical  direct  

Oscillating 

Vertical  compound  
Horizontal  compound.  .  . 

STEAM-ENGINE. WEIGHT   OF   BOILERS. 


Land-engines.— (Non-condensing. ) 


759 


ENGINE. 

Volume 
of 
Cyl'r. 

Engine. 

Spur-wheel 
and 
Connections. 

Sugar-Mill 
Complete. 

Boilers, 
Grates,  etc. 

Engine  per 
Cube  Foot 
of  Cylinder. 

Vertical)  18  ins.  X4  feet 
beam  f  30  ins.  X  5  feet 
Horizon'l,  14  ins.  X  2  feet 
"        22  ins.  X  4  feet 

7 
24-5 

2.2 

10.6 

LbB. 

67200 
105000 
10914 
56000 

Lbs. 
37800 
137  '79 

Lbs. 
89600 
265  879 

Lbs. 
26880 
75000 
8200 
30140 

Lba. 
9600 
4290 
5100 
5600 

To  Compxite  Weight  of*  a  Vertical  Beam  and  Side-wheel 
Jet    Condensing   Engine.    (T.  F.  Rowland,  A.S.C.E.) 

Including  all  Metals,  Boiler  and  A  ttackments.  Smoke-pipe,  Grates,  Iron  Floors, 

and  Iron  in  Wooden  Water-wheels,  omitting  Coal-bunkers. 

For  a  Pressure  per  Mercurial  Gauge  of  40  Ibs.per  Sq.  Inch. 

For  surface  condenser  add  10  to  15  per  cent 

RULE. — Multiply  volume  of  cylinder  in  cube  feet  by  Coefficient  in  follow- 
ing table  corresponding  to  length  of  stroke,  and  product  will  give  rough 
weight  in  Ibs.  For  finished  weight  deduct  6  per  cent. 

Stroke.       Coefficient.  I      Stroke.       Coefficient.       Stroke.       Coefficient.       Stroke.       Coefficient. 


Feet. 

,4 


2467 
2340 


2213 

2OOO 


Feet. 
9 


1865 
1730 


1619 
1546 


EXAMPLE  i.— What  are  the  rough  and  finished  weights  of  a  vertical  beam  engine, 
cylinder  80  ins.  in  diameter  and  12  feet  stroke  of  piston  ? 

Area  of  80  ins.  =  5026. 56,  which  x  12  feet  =  419  cube  feet,  and  x  1546  for  12  feet, 
stroke  =  647  774  Ibs.  rough  weight. 

Then  647  774  X  .06  =  38  866,  and  647  774  —  38  866  =  608  908  Ibs.  finished  weight. 


WEIGHTS  OF  BOILERS. 

Weights  of  Iron  Boilers  (including  Doors  and  Plates,  and  exclusive  of  Smoke- 
pipes  and  Grates)  per  Sq.  Foot  of  Heating  Surface. 

Surface  Measured  from  Grates  to  Base  of  Smoke-pipe  or  Top  of  Steam  Chimney. 

BOILER.    For  a  Working  Pressure  of  40  Lb».  Weight. 


Single  return,  Flue  * water  bottom. . 

"          "         "      Multi-flue'*!*.!!  '.".".water  bottom!! 


Horizontal  return,  Tubularf water  bottom . . 

"       t — 


Vertical 


Horizontal  direct,  Tubular*. 


t water  bottom. . . 


Lbs. 

25.6  to  32.9 

24  to  30 
27     to  45 

25  to  43 
22.5  to  35 
21     to  33 

17.7  to  26.7 
18.5  to  26. 5 

19.8  to  23.8 

17       tO  21 

23. 5  to  24 
18.1  to  18.6 
16.3  to  17.3 
24     to  26 


Cylindrical,  external  furnace, t  36  ins.  in  diam.,  .25  inch  thick. . 
"          Flue  "       $361042       "       .25    "       "     .. 

Horizontal  direct,  Tubular Locomotive 

Vertical  Cylinder  direct,  Tubular — 

Weight  of  Cylindrical  Furnace  and  Shell  Boilers,  all  complete  for  Sea  Service  and 
for  a  pressure  of  60  Ibs.  steam,  200  Ibs.  per  IIP. 

*  Section  of  furnace  square.     Shell  cylindrical.  t  Section  of  furnace  and  shell  square. 

t  Wrought-iron  heads,  .375  inch  thick,  flues,  .25  inch,  and  surface  computed  to  half  diameter  of  shell. 

NOTES.— i,  The  range  in  the  units  of  weight  arises  from  peculiarities  of  construc- 
tion, consequent  upon  proportionate  number  of  furnaces,  thicknesses  of  metal,  vol« 
ume  of  shell  compared  with  heating  surface,  character  of  staying,  etc. 

2.  If  pressure  is  increased  the  above  units  must  be  proportionately  increased. 


760      STEAM-ENGINE. — BOILEB-POWER,  COMBUSTION. 

Boiler-power. 

The  power  of  a  boiler  is  the  volume  or  weight  of  steam  alone  (indepen- 
dent of  any  water  that  it  may  hold  in  suspension)  that  it  will  generate  at  its 
operating  pressure  in  a  unit  of  time,, 

Marine  boilers  of  the  ordinary  type  and  proportions,  with  natural  draught,  burn- 
ing anthracite  coal,  produce  3.5  to  5.5  IIP  per  sq.  foot  of  grate  per  hour;  with  a 
free  burning  or  a  semi-bituminous  coal,  5  to  7.5  IIP;  and  with  a  foroad  draught, 
with  25  to  30  Ibs.  best  coal  per  sq.  foot  of  grate  per  hour,  8  to  10  IIP. 

Marine  engines,  operating  with  a  steam-pressure  of  35  Ibs.  (m.  g.),  and  with  mod- 
erate expansion,  consume  30  Ibs.  steam  per  IIP  per  hour,  and  with  a  high  rate  of 
expansion,  under  a  pressure  of  70  Ibs. ,  20  Ibs.  steam. 

With  a  blast  draught  and  consuming  30  to  40  Ibs.  of  a  fair  quality  of  coal  per  sq. 
foot  of  grate  per  hour,  7  to  10  IP  per  hour  can  be  attained. 

In  locomotive  boilers,  having  from  50  to  90  sq.  feet  of  heating  surface  per  sq.  foot 
of  grate,  and  at  a  rate  of  combustion  of  from  45  to  125  Ibs.  of  coke,  an  average  evap- 
oration of  9  Ibs.  of  water  per  Ib.  of  coke  has  been  attained  at  ordinary  temperatures 
and  pressure. 

To  Compute  Volume   of*  Air   and  Q-as   in   a   Furnace. 

When  Volume  at  a  Given  Temperature  is  known.  RULE. — Multiply  given 
volume  by  its  absolute  temperature,  and  divide  product  by  the  given  abso- 
lute temperature. 

NOTE.— Absolute  temperature  is  obtained  by  adding  461°  to  given  or  acquired 
temperature. 

EXAMPLE. — Assume  volume  of  air  entering  a  furnace  at  i  cube  foot,  its  tempera- 
ture^0, and  temperature  of  furnace  1623°;  what  would  be  the  increase  of  volume? 

*  X  16230  + 461°  _  2084 
600  +  461°       -  521  ~ 

"Volume   of  Furnace   Q-as   per   Lib.  of  Coal.     (Rankine.) 


Tempera- 
ture. 

12  Lbs. 

A.ir  Supplied 
18  Lbs. 

24  Lbs. 

Tempera- 
ture. 

12  Lb8. 

Air  Supplied 
18  Lbs. 

24  Lbs. 

If 

ISO 
161 

225 
241 

300 
322 

752° 

III2 

369 

479 

553 
718 

738 
957 

104 

172 

258 

344 

1472 

588 

882 

1176 

212 

205 

3<>7 

409 

1832 

697 

1046 

1395 

572 

3H 

471 

628 

2500 

906 

1357 

1812 

Temperature  of  ordinary  boiler  furnaces  ranges  from  1500°  to  2500°. 
The  opening  of  a  furnace  door  to  clean  the  fire  involves  a  loss  of  from  4  to  7  per 
cent,  of  fuel. 
For  other  illustrations,  see  ante,  page  744-6. 

Rate   of  ConVbustion. 

The  rate  of  combustion  in  a  furnace  is  computed  by  the  Ibs.  of  fuel  consumed  per 
sq.  foot  of  grate  per  hour. 

In  general  practice  the  rate  for  a  natural  draught  is,  for  anthracite  coal  from  7  to 
16  Ibs.,  for  bituminous,  from  10  to  25  Ibs.,  and  with  artificial  or  forced  draught,  as  by 
a  blower,  exhaust-blast,  or  steam -jet,  the  rate  may  be  increased  from  30  to  120  Ibs. 

The  dimensions  or  size  of  coal  must  be  reduced  and  the  depth  of  the  fire  increased 
directly,  as  the  intensity  of  the  draught  is  increased. 

Temperature  of  gases  at  base  of  chimney  or  pipe  should  be  600°,  and  frictional 
resistance  of  surface  of  chimney  is  as  square  of  velocity  of  current  of  gases. 

Ordinarily  from  20  to  32  per  cent,  of  total  heat  of  combustion  is  expended  in  tlie 
production  of  the  chimney  draught  in  a  marine  boiler,  to  which  is  to  be  added  the 
Josses  by  incomplete  combustion  of  the  gaseous  portion  of  the  fuel  and  the  dilution 
of  the  gases  by  an  excess  of  air,  making  a  total  of  fully  60  per  cent.  (Steam-boilers. 
Wm,  H.  Mock,  U.  S.  N.y  1881.) 


STRENGTH    OF   MATERIALS. — ELASTICITY.  761 

STRENGTH  OF  MATERIALS. 

Strength  of  a  material  is  measured  by  its  resistance  to  alteration  of 
form,  when  subjected  to  stress  and  to  rupture,  which  is  designated  as 
Crushing,  Detrusive,  Tensile,  Torsion,  and  Transverse,  although  trans- 
verse is  a  combination  of  tensile  and  crushing,  and  detrusive  is  a  form 
of  torsion  at  short  lengths  of  application. 

ELASTICITY   AND    STRENGTH. 

Strength  of  a  material  is  resistance  which  a  body  opposes  to  a  per- 
manent separation  of  its  parts,  and  is  measured  by  its  resistance  to 
alteration  of  form,  or  to  stress. 

Cohesion  is  force  with  which  component  parts  of  a  rigid  body  adhere  to 
each  other. 

Elasticity  is  resistance  which  a  body  opposes  to  a  change  of  form. 

Elasticity  and  Strength,  according  to  manner  in  which  a  force  is  exerted 
upon  a  body,  are  distinguished  as  Crushing  Strength,  or  Resistance  to  Com- 
pression ;  Detrusive  Strength,  or  Resistance  to  Shearing ;  Tensile  Strength, 
or  Absolute  Resistance ;  Torsional  Strength,  or  Resistance  to  Torsion ;  and 
Transverse  Strength,  or  Resistance  to  Flexure. 

Limit  of  Stiffness  is  flexure,  and  limit  of  Resistance  is  fracture. 

Neutral  Axis,  or  Line  of  Equilibrium,  is  the  line  at  which  extension  ter- 
minates and  compression  begins. 

Resilience,  or  toughness  of  bodies,  is  strength  and  flexibility  combined ; 
hence,  any  material  or  body  which  bears  greatest  load,  and  bends  most  at 
time  of  fracture,  is  toughest. 

Stiffest  bar  or  beam  that  can  be  cut  out  of  a  cylinder  is  that  of  which 
depth  is  to  breadth  as  square  root  of  3  to  i ;  strongest,  as  square  root  of  2  to 
i ;  and  most  resilient,  that  which  has  breadth  and  depth  equal. 

Stress  expresses  condition  of  a  material  when  it  is  loaded,  or  extended  in 
excess  of  its  elastic  limit. 

General  law  regarding  deflection  is,  that  it  increases,  cceteris  paribus,  di- 
rectly as  cube  of  length  of  beam,  bar,  etc.,  and  inversely  as  breadth  and  cube 
of  depth. 

Resistance  of  Flexure  of  a  body  at  its  cross-section  is  very  nearly  .9  of  its 
tensile  resistance. 

Coefficient   of*  Elasticity. 

Elasticity  of  any  material  subjected  to  a  tensile  or  compressive  force, 
within  its  limits,  is  measured  by  a  fraction  of  the  length,  per  unit  of  force 
per  unit  of  sectional  area,  termed  a  constant,  and  coefficient  of  elasticity  is 
usually  defined  as  the  weight  which  would  stretch  a  perfectly  elastic  bar  of 
uniform  section  to  double  its  length. 

Unit  of  force  and  area  is  usually  taken  at  one  Ib.  per  sq.  inch.  E  represent- 
ing denominator  of  fraction. 

EXAMPLE.— If  a  bar  of  iron  is  extended  one  i2ooooooth  part  of  its  length  per  Ib. 
of  stress  per  sq.  inch  of  section,  t  r 

12000000       E ' 

The  bar  would,  therefore,  be  stretched  to  double  its  normal  length  by  a  force  ot 
12000000  Ibs.  per  sq.  inch,  if  the  material  were  perfectly  elastic. 


762 


STRENGTH    OF    MATERIALS. ELASTICITY. 


The  same  method  of  expressing  coefficient  of  elasticity  is  applied  to  re- 
sistance to  compression.  That  is,  coefficient,  in  weight,  is  expressed  by  de- 
nominator of  fraction  of  its  length  by  which  a  bar  is  compressed  per  unit  of 
weight  per  sq.  inch  of  section. 

Ultimate  extension  of  cast  iron  is  sooth  part  of  its  length. 

Extension  of  Cast-iron  Bars,  when  suspended  Vertically. 

i  Inch  Square  and  10  Feet  in  Length.     Weight  applied  at  one  End. 
Weight.    Extension.        Set.          Weight.    Extension.        Set.          Weight.     Extension.        Set. 


Lba. 
529 


Ins. 
.0044 

.0092 


Lb3. 

2117 
4234 


Ins. 
.0190 
.0397 


Ins. 

.000059 
.00265 


Lbs. 
8468 
14820 


Ins. 

.0871 


Ins. 

.00855 
•Q25S5 


"Woods. — MM.  Chevaudier  and  Wertheim  deduced  that  there  was  no 
limit  of  elasticity  in  woods,  there  being  a  permanent  set  for  every  extension. 
They,  however,  adopted  a  set  of  .00005  of  length  as  limit  of  elasticity. 
This  is  empirical. 

MODULUS    OF    ELASTICITY. 

Modulus  or  Coefficient  of  Elasticity  of  any  material  is  measure  of  its 
elastic  reaction  or  force,  and  is  height  of  a  column  of  the  material, 
pressing  on  its  base,  which  is  to  the  weight  causing  a  certain  degree  of 
compression  as  length  of  material  is  to  the  diminution  of  its  length. 

It  is  computed  by  this  analogy :  As  extension  or  diminution  of  length 
of  any  given  material  is  to  its  length  in  inches,  so  is  the  force  that  pro- 
duced that  extension  or  diminution  to  the  modulus  of  its  elasticity. 

P I 

Or,  x  :  P  : :  I :  w  =  — .   x  representing  length  a  substance  i  inch  square  and  i  foot 

in  length  would  be  extended  or  diminished  by  force  P,  and  w  weight  of  modulus  in  Ibs. 

To   Compute   "Weight   of"  M.odulns    of  Elasticity. 
RULE. — As  extension  or  compression  of  length  of  any  material  i  inch 
square,  is  to  its  length,  so  is  the  weight  that  produced  that  extension  or  com- 
pression, to  modulus  of  elasticity  in  Ibs. 

EXAMPLE.— If  a  bar  of  cast  iron,  i  inch  square  and  10  feet  in  length,  is  extended 
.008  inch,  with  a  weight  of  1000  Ibs.,  what  is  the  weight  of  its  modulus  of  elasticity? 

.008  :  120  (10  X  12)  ::  1000  :  15000000  Ibs. 
To    Compete    Modnlns    of  Elasticity. 

When  a  Bar  or  Beam  is  Supported  at  Both  Ends  and  Loaded  in  Centre. 
RULE.— Multiply  weight  or  stress  per  sq.  inch  in  Ibs.  by  length  of  material 
in  ins.,  and  divide  product  by  modulus  of  weight. 

Or,  —  =  E;     —  =  M;     —  W.    I  representing  length  in  ins.,  M  modulus, 

W  weight  in  Ibs.  per  sq.  inch,  and  E  compression  or  extension. 

EXAMPLE  i.— If  a  wrought-iron  rod,  60  feet  in  length  and  .2  inch  in  diameter,  is 
subjected  to  a  stress  of  150  Ibs.,  what  will  it  be  extended? 

Modulus  of  elasticity  of  iron  wire  is  28  230  500  Ibs.  (see  following  table),  and  area 
ofit.22X  .7854  =  . 31416. 

=  477.46  Ibs.  per  sq.  inch,  and  60  X  12  =  720  ins. 


.31416 


Then  477.46  X 


720      = 


=  OI2  18  inch 


"  28  230  500      28  230  500 
a.— Take  elements  of  preceding  case  under  rule  for  weight  of  modulua 
120X1000          ..,      . 008  x  15  ooo  ooo  _ 


15000000 


-  =  .008  inch. 


-  =  1000  Ibs. 


STRENGTH    OP    MATERIALS. COHESION, 


763 


Modulus  of  3 

SUBSTANCES. 

Slastici 
Height. 

ty  and  "> 

Weight. 

TVeight  of  Var: 

SUBSTANCES. 

LOU*  1M£ 
Height. 

iterials. 

Weight. 

Ash 

Feet. 
4970000 
4600000 

2  460000 
4  II2000 

4800000 
5680000 
8330000 
4440000 

2  79OOOO 
5000000 
6OOOOOO 
5750000 
7550000 
8  377  OOO 

Lbs. 
•  i  656  670 
1345000 
8464  ooo 
14  632  720 
18240000 
1499500 

2Ol6  DOO 
5550000 

8  844  300 
170000 
2370000 
17  968  500 
25  820  ooo 

28  230  qoo 

Larch  

Feet. 
4415000 
146000 
i  850000 
2400000 
6570000 

2  150000 
4750000 
8700000 
8970000 
8530000 
9OOOOOO 
1672000 
1053000 
4480000 

Lbs. 
1074000 
720000 
1080400 
3  300000 
2071  ooo 
2508000 
i  710000 
2430000 
i  830000 
26650000 
28689000 
1718800 
3510000 

1  3  44O  OOO 

Beech             

Brass,  yellow  
"      wire 

Lignum-vitae  
Limestone  

Copper  cast 

Mahogany        .   .  . 

Elm  

Marble,  white  
Oak     

Fir  red 

Glass 

Pine,  pitch  

Gun-metal 

"    white  

Hempen  fibres.  .  .  . 
Ice 

Steel,  cast  

"     wire  

Iron  cast  

Stone,  Portland  .  .  . 
Tin  cast 

"    wrought  
"    wire... 

Zinc  — 

"Weight    a    Material    will    "bear   per    Sq..  Inch,  -without 
Permanent    Alteration    of  its    Length. 


MATERIAL. 

Lbs. 

MATERIAL. 

Lbs. 

MATERIAL. 

Lbs. 

Metals. 
Brass 

Stones,  etc. 
Marble  

Woods. 
Beech 

Gun  metal  

IOOOO 

Limestone*  

Elm     .  . 

3 

15  coo 

Portland  

1500 

Fir  red  

42QO 

'  *    wrought 

17  800 

Larch     .   . 

060 

Lead       

I  500 

Woods. 

Mahogany  

•JOOO 

Steel  .  . 

4^000 

Ash  ... 

TUO 

Oak  .  . 

JUOU 

3060 

*  Tensile  strength  2800. 
Comparative    Resilience   of  "Woods. 


Ash i 

Beech 86 

Cedar 66 


Chestnut 

Elm 

Fir 


Larch 84 

Oak 63 

Pitch  Pine 57 


Spruce 64 


Teak. 


•59 


Yellow  Pine. ..  .64 


MODULUS   OF   COHESION. 

To  Compute  Length  of  a  Prism  of  a  Material  -which  would. 
t>e   Severed  toy  its  own  Weight  when    Sxaspended. 

RULE. — Divide  tensile  resistance  of  material  per  sq.  inch  by  weight  of  a 
foot  of  it  in  length,  and  quotient  will  give  length  in  feet. 

ILLUSTRATION.— Assume  tensile  resistance  of  a  wrought- iron  rod  to  be  60000  Ibs. 
per  sq.  inch.  Weight  of  i  foot  =  3.4  Ibs. 

Then  60000-4-3.4  =  17647.06/66*. 

length  in  Feet  required  to  Tear  Asunder  the  following  Substances: 

Rawhide 15  375  feet.  |  Hemp  twine. . .  75  ooo  feet.  |  Catgut 25  ooo  feet. 

Elasticity  of  Ivory  as  compared  with  Glass  is  as  .95  to  i. 

When  Height  is  given.  RULE.— Multiply  weight  of  i  foot  in  length  and 
i  inch  square  of  material  by  height  of  its  modulus  in  feet,  and  product  will 
give  weight. 

To    Compute    Height   of  Modulus   of  Elasticity. 
RULE.— Divide  weight  of  modulus  of  elasticity  of  material  by  weight  of 
i  foot  of  it,  and  quotient  will  give  height  in  feet. 

EXAMPLE.  —Take  elements  of  preceding  case  (page  762),  weight  of  i  foot  being 
3  Ibs. ;  what  is  height  of  its  modulus  of  elasticity  ? 

15000000-:- 3  =  5000  ooo  feet 


764  STRENGTH    OF   MATERIALS. — CRUSHING. 

From  a  series  of  elaborate  experiments  by  Mr.  E.  Hodgkinson,  for  the 
Railway  Structure  Commission  of  England,  he  deduced  following  formulas 
for  extension  and  compression  of  Cast  Iron : 

Extension :  13  934  040 290  743  200  -^  =  W. 

Compression :  12  931 560  - —  522  979  200  -^  =  W.  e  and  c  representing  extension 
and  compression,  and  I  length  in  ins. 

ILLUSTRATION.— What  weight  will  extend  a  bar  of  cast  iron,  4  inc.  square  and  10 
feet  in  length,  to  extent  of  .2  inch? 

.2  2*~ 

13  934  040  X  —  —  290  743  200  —  =  23  223.4  —  807.62  =  22  415. 78,  which  X  4  ins. 
=  89666.12  Ibs. 

CRUSHING   STRENGTH. 

Crushing  Strength  of  any  body  is  in  proportion  to  area  of  its  section, 
and  inversely  as  its  height. 

In  tapered  columns,  it  is  determined  by  the  least  diameter. 

When  height  of  a  column  is  not  5  times  its  side  or  diameter,  crushing 
strength  is  at  its  maximum. 

Cast  Iron. — Experiments  upon  bars  give  a  mean  crushing  strength  of 
100 ooo  Ibs.  per  sq.  inch  of  section,  and  5000  Ibs.  per  sq.  inch  as  just  sufficient 
to  overcome  elasticity  of  metal ;  and  when  height  exceeds  3  times  diameter, 
the  iron  yields  by  flexure.  When  it  is  10  times,  it  is  reduced  as  i  to  1.75 ; 
when  it  is  15  times,  as  i  to  2 ;  when  it  is  20  times,  as  i  to  3 ;  when  it  is  30 
times,  as  i  to  4 ;  and  when  it  is  40  times,  as  i  to  6. 

Experiments  of  Mr.  Hodgkinson  have  determined  that  an  increase  of 
strength  of  about  one  eighth  of  destructive  weight  is  obtained  by  enlarging 
diameter  of  a  column  in  its  middle. 

In  columns  of  same  thickness,  strength  is  inversely  proportional  to  the 
I>63  power  of  length  nearly. 

A  hollow  column,  having  a  greater  diameter  at  one  end  than  the  other, 
has  not  any  additional  strength  over  that  of  an  uniform  cylinder. 

Wrought  Iron.— Experiments  give  a  mean  crushing  stress  of  47  ooo  Ibs. 
per  sq.  inch,  and  it  will  yield  to  any  extent  with  27  ooo  Ibs.  per  sq.  inch, 
while  cast  iron  will  bear  80  ooo  Ibs.  to  produce  same  effect. 

Effects. — A  wrought  bar  will  bear  a  compression  of  -g^-g  of  its  length,  with- 
out its  utility  being  destroyed. 

With  cast  iron,  a  pressure  beyond  27  ooo  Ibs.  per  sq.  inch  is  of  little,  if 
any,  use  in  practice. 

Glass  and  hard  Stones  have  a  crushing  strength  from  7  to  9  times  greater 
than  tensile ;  hence  an  approximate  value  of  their  crushing  strength  may  be 
obtained  from  their  tensile,  and  contrariwise. 

Various  experiments  show  that  the  capacity  of  stones,  etc.,  to  resist  effects 
of  freezing  is  a  fair  exponent  of  that  to  resist  compression. 

Seasoning. — Seasoned  woods  have  nearly  twice  crushing  strength  of  un- 
seasoned. 

Elastic    Limit   compared    to    Crxxsliing    Resistance. 

Wrought- iron  Commerce 545 

Bessemer  steel 615 

Cast  steel 473 


Cast  steel 692 

Fagersta  steel {  '25 


STRENGTH   OF   MATERIALS. — CRUSHING. 


765 


Crushing  Strength,  of  various  Materials,  deduced  from 
Experiments  of  Maj.  \Vade,  Hodglzinson,  Capt.  IVIeigs, 
TJ»  S.  .A..,  Stevens  Institute,  and  by  Gr.  JL*.  Vose. 

Meduced  to  a  Uniform  Measure  of  One  Sq.  Inch. 
CAST  IRON. 


FlGURBS  AND  M  ATSRIAL. 

Crushing 
Weight. 

FIGURES  AND  MATERIAL. 

Crushing 

Weight. 

Gun-  metal  American  •••••••• 

Lbs. 

Lbs. 

««                 "        J 

85000 

Stirling,  mean  of  all,  English  .. 

12*  395 

"           mean  

125000 
IOOOOO 

'*       extreme,  English  

134400 
53760 

Low  Moor,  No.  i,  English  
"         No.  2.      *'      ...... 

62450 

O2  33O 

Average  (Hodgkinson),  English 

153200 
84  240 

Clyde.        No,  *      " 

zoo  oqo 

IOQ7OO 

WEOUGHT  IEOH. 


American,  extreme.. . 


47040 

VARIOUS  METALS. 


averaga. 


65200 
40000 
37850 


Aluminium  bronze,  95  cop.  .... 

I2QQ2O 

p     eoooo 

164800 

"              "           soft  

66 

I     "     tempered    • 

Steel,  cast  ( 

I0500O 

"     Siemens  

335000 

I 
"     Fagersta  

250000 
IS4SOO 

Lead  

IS500 

77^0 

Elastic  Crushing  Strength  of  Wrought  Iron  and  Crucible  Steel  is  equal  to  its  tea 
eile,  of  Bessemer  Steel,  50  per  cent  of  its  transverse  strength, 

WOODS. 


Ash  

6663 

6061 

Birch  { 

3300 

Box  

7900 

IO  SI3 

Cedar  red  .... 

6 

"     seasoned  

6  noo 

5  35° 

Elm      

6831 

4  '  seasoned  

802=; 

Larch     .  { 

3200 

Locust.  .  . 

5500 

O  III 

Mahogany,  Spanish  

8  1  08 

Maple  { 

8100 

Oak,  American  white.          .... 

IOOOO 

'  '    Canadian  white  

'    Q 

"           "        live 

6  850 

"    English  { 

95oo 

Pine,  pitch  

6484 

"     white  

"     yellow  

8000 

Spruce,  white  

Teak  

Walnut.  .. 

66^s 

Chestnut.. 


....    900 

Hemlock 600 

Pine,  white 800 


Crosswise  of  Fibre. 

Pine,  Yellow- South...  1400 

"     Oregon 1200 

' '     Northern 1000 


Redwood 800 

Spruce 700 

White  Oak 2000 


Increase  in  Strength  of  Oabes  of  Sandstone,  per  Sq.  Inch  (under  Blocks 
of  Wood\  as  Area  of  Surface  is  increased.     (Gen'l  Gillmore,  U.  S.  A.) 


STONE. 

•5 

I 

i-5 

INC 

2 

HKS. 
2.25 

2-75 

3 

Yellow  Berea  sandstone  .  . 
Blue         "            " 

Lbs. 
6080 

Lbs. 
6990 
9500 

Lbs. 
8226 
10730 

Lbs. 
8955 
12000 

Lbs. 
9130 
12500 

Lbs. 
9838 
13200 

Lbs. 
10125 

STRENGTH    OF   MATERIALS. — CRUSHING. 


Stones,  Ce 

FIGURES  AND  MATERIAL. 

jment 

Crushing 
Weight. 

s,  etc.    (Per  Sq.  Inch.) 

FIGURES  AND  MATERIAL. 

Crushing 
Weight. 

Basalt,  Scotch  

Lbs. 
8300 
16800 
800 
I  400 
6  222 
I02I9 

14  216* 

3630 
800 

4000 

1440 
1650 
7200 
2250 
5600" 

808 
2228 

1543 
17000 
32000 
1280 
600 
3800 
2464 
5980 
2330 

2650 

1800 
342 
750 
3270 
1280 
460 
775 
3522 

33J9 
3069 
2991 
31  ooo 
19600 
10760 

6339 
10450 
12850 

B,  N.  J. 
3atent  Offic 
Eleilly,  Ord 

Lbs. 
5340 
457° 
15583 
15000 
18800 
4000 
9000 
7800 
3  'So 
3600 
14000 
8057 
18061 

'39J7 
18941 
17440 
12624 
9630 
22702 
8950 

3  360 
10382 
u  156 
18248 
10124 
500 
800 
760 
240 
460 
595 

120 

3850 
12000 

5340 
7850 
11789 
5825 
3136 
2554° 
10762 

5710 
13890 
23744 
5000 
8300 

v  York. 
stitute. 

"       Welsh  

Beton,  N.  Y.  S.  ConcretiDg  Co.  j 

11       Quincy  Mass    .  . 

Greenstone,  Irish  

Limestone  J 

"      hard  burned  

"         compact,  Eng  

"      coniiDon                        < 

'  {      yellow-faced  burned,  Eng. 
"      Stourbridge  fire-clay,  '» 
"      Staffordshire  blue,      u 
"      stock  English  

"         Anglesea    "  
"         Irish          "  

Marble  Baltimore  Md             { 

"       East  Chester,  N.Y.f... 
"       Hastings  N.  Y  

*'      Fareham,  English  

"      red,  English  

"       Irish  

"      Sydney  N.  S  

"       Italian  

"           "       white  

Cement,  Hydraulic,  pure,  Eng.  | 
"       Portland,  sand  i  

'       Lee  Mass 

1       Montgomery  Co.  ,  Pa.  ... 

"          "           sand  3 

'       Stockbridge,  Mass.*.... 
'       Symington,  large  
"     fine  crystal  
"     strata  horizontal 

Masonry,  brick,  common  { 

"           "      in  cement  
Mortar  good    

"          "     3  mos  

"         u     i  sand,  3  mos  
"          **     9  mos  

*'          "     i  sand,  9  mos.  ..  . 
"          "    12   inch   cubes,  ) 
12  mos. 
i  sand  and  gravel  ) 
"  3 
"       Roman  

"       lime  and  sand  

"           u         u        beaten.  .  . 
"       common  

"       pure,  Eng  
u       Rosendale  

Oolite  Portland  

'  '       Sheppey  Eng 

Pottery-  pipe  Chelsea  . 

Concrete,  lime  i,  gravel  3.  ...  { 
Freestone  Belleville  N  J 

Sandstone  Aquia  Creek  §  

"          Connecticut  II  

"         Connecticut 

'         Craigleth,  Eng  

"         Dorchester,  Mass  
"          Little  Falls,  N.  Y.... 
Glass  crown 

'         Derby  grit  "  ..  . 

1         Holyh'd  quartz,  Eng. 
'         Seneca  IT  .  .   . 

Gneiss  

4         Yorkshire,  Eng.  
Slate  Irish  { 

Granite,  Aberdeen,  Eng  

u        Cornish,        "   ....;... 
"        Dublin          " 

Terra  Cotta          .             . 

"       Newry,         " 

Whinstone  Scotch  

*  Tested  by  author  at  Stevens  Institut 
§  Capitol,  Treasury  Department,  and  '. 
11  Cn.mwell,  Conn.    Tested  by  J.  W,  ] 

t  Post-office,  Wash.         \  City  Hall,  Ne\ 
e,  Washington,  D.  C. 
nance  Dept.,  U.S.A.        f  Smithsonian  Ir 

Safe  .Load   of*  Hollow,  Cylindrical,  and  Solid  Columns, 
A-rches,  Chords,  etc.,  of  Cast    Iron. 

Hollow  Columns.    Per  Sq.  Inch.    (F.  W.  Shields,  M.  I.  C.  E.) 


Length. 

Thick- 
ness. 

Load. 

Length. 

Thick- 
ness. 

Load. 

Length. 

Thick- 
ness. 

Load. 

Length. 

Thick- 
ness. 

Load. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

20  to  24 
diam's. 

•375 
•5 

2800 
336o 

20  tO  24 

diam's. 

.625 
•75 

3920 
4480 

25  to  30 
diam's. 

•375 
•5 

2240 
2800 

25  to  3° 
diam's. 

•625 
•75 

336o 
3920 

Solid  Colnmns,  etc.— 3360  IDS.  per  sq.  inch.    (Brunei.) 
Arclies.— 5600  Ibs.  per  sq.  inch. 


STRENGTH  OF  MATERIALS. — CRUSHING. 


767 


Chords  and  Posts.— i  inch  diameter  and  not  more  than  15  diameters  in 
length  .  2  of  breaking  weight  of  metal. 

.625  inch  diameter  and  not  more  than  25  diameters  in  length  .5  of  breaking  weight 
of  metal,  and  when  more  than  25  diameters  in  length  from  .1  to  .025  of  breaking 
weight  of  metal  (Baltimore  Bridge  Co.) 

Wrought-iron    Cylinders   and    Rectangular   TiVbes. 


LENGTH. 

External 
Diameter. 

Internal 
Diameter. 

Thickness, 

Area. 

Crushing  Weight 
per  Sq.  Inch. 

CYLINDERS. 

Ins. 

Ins. 

Ins. 

Sq.  Ins. 

Lbs. 

10  feet 

1-495 

1.292 

.1 

•444 

14661 

10     " 

2.49 

2.275 

.107 

.804 

29779 

10     " 

6.366 

6.106 

2-547 

35886 

RECTANGULAR  TUBES. 

10    feet 

4.1         X     4.1 

•03 

•504 

10980 

5      " 

4-1         X     4-1 

•03 

•504 

li  514 

TO 

•d 

4.1         X     4.1 

.06 

1.02 

19261 

to       " 

"5 

4.25      X     4-25 

•134 

2-395 

21585 

7-5    " 

4.25      X     4-25 

•134 

2-395 

23203 

10         " 

i 

8.4        X     4-25 

(.26 
(.126 

6.89 

29981 

10         " 

8.  i        X     8.  i 

.06 

2.07 

13276 

7.66" 

8.1        X     8.1 

.06 

2.07 

13300 

10         " 

)     Internal 

8.1        X     8.1 

.0637 

3-551 

5       "    . 

j  diaphrag's 

8.1        X     8.x 

.0637 

3-55' 

23208 

Strength,  per  Sq..  Inch,  of  3 -Inch   Cntoes  under   Bloclts 
of  Wood.    (Gertl  Gillmore,  U.  S.  A.) 

Surfaces  Worked  to  a  Clear  Bed. 


GRANITE. 

Lbs. 

22250 
15000 
17750 
14750 
18250 
16187 
12500 
21250 
14100 
12423 
19500 
14937 
*5937 
13370 
17750 
12875 
17500 
20750 
24040 

"475 
25000 
20700 
13900 
18500 
12600 
16900 
18000 
25000 

21  500 

LIMESTONE. 
Bardstown,  Ky.  ,  dark  
Cooper  Co.,  Mo.,  dark  drab.  .... 
Erie  Co.  ,  N.  Y.  ,  blue  

Lbi. 
16250 
6650 
12250 
3650 

13504 

13062 
7612 
9687 
20025 

9850 
u  700 
6950 
4350 
17725 
7250 
10250 
8850 
6250 
7450 
9687 
6800 
13500 
10700 
6250 
12000 

?i5o 
750 

5000 

"       light 

Westchester  Co    N  Y  

MARBLE. 
East  Chester,  N.  Y.  

Millstone  Point,  Conn  
New  London,  Conn  

"  gray  

Dorset,  Vt  
Mill  Creek,  111.,  drab  

North  Bay  Wis    drab        .  .     . 

Westerly,  R.  I.,  gray  
Fall  River  Mass    gray  

SANDSTONE. 
Little  Falls,  N.  Y.  ,  brown  
Belleville  N  J  ,  gray  

Garrisons,  Hudson  River,  gray.  . 
Duluth  Minn    dark  

Middletown,  Conn.,  brown  

Keene,  N.  H.,  bluish  gray  
Used  in  Central  Park,  N.  Y.,  red 
Jersey  City,  N.  J.,  soap  

Medina  N  Y    pink     

LIMESTONE. 
Glen's  Falls,  N.  Y  

Vermillion,  O.  ,  drab  

Fond  du  Lac,  Wis.,  purple  
Marquette,  Mich.,        u     
Seneca  0    red  brown  

Lake  Champlain,  N.  Y.  

Cana.job.arie  N  Y      

Cleveland,  0.  ,  olive  green  
Albion,  N.  Y,  brown  
Kasota,  Minn.  ,  pink  
Fontenac,  Minn.,  light  buff.  .... 
Craigleth  Edinburgh     

Garrisons          '*             

Marblehead,  0.  ,  white  
Joliet  111    white  

Lime  Island,  Mich.  ,  drab  .  .  .  .  | 
Sturgeon  Bay,  Wis.  ,  bluish  drab 

Dorchester,  N.  B.,  freestone.  .  .  . 
Massillon,  O.,  yellow  drab  
Warrensburg,  Mo.,  bluish  drab. 

768  STRENGTH    OF    MATERIALS. CRUSHING. 

To   Compute    Crushing   "Weight   of  Columns, 
Deduced  by  Mr.  L.  D.  B.  Gordon  from  Results  of  Experiments  of  various  Authors 

METALS. 

Cast    Iron.    (Hodgkinson.) 
Solid  or  Hollow. 

Round,  — r=  W.     Rectangxalar,  — =  W. 

'  400  500 

For  L,  T,  U,  T,  etc.,  put    '9  a 2      (Unwin). 

"Wr ought    Iron.    (Stoney.)  ~1 

Solid  or  Hollow. 
Round, ^-  =  W.      Rectangular, —  =  W. 

2400  7000 

Steel.    (Baker.) 

Solid.  —  Strong  steel. 
51  a                                                  51  a 
Round,  —  =  W.       Rectangular, —  i=W. 

i-\ i-f— — 

900  1600 

Solid.— MM  steel. 

30  a                                                      30  a 
Round, ^~  =  w-       Rectangular, —  =  W. 

1400  "~  2480 

a  representing  area  of  section  of  metal  in  sq.  ins.,  r  ratio  of  length  to  least  external 

diameter  or  side  in  like  terms,  and  W  crushing  weight  in  tons. 
ILLUSTRATION.— What  is  the  crushing  weight  of  a  hollow  cylindrical  column  of 

cast  iron,  10  ins.  in  diameter,  20  feet  in  length,  and  i  inch  in  thickness? 

a  =.  area  of  10  ins. — area  of  10  —  i  X  2  =  28.28  ins.        r  =  —    —  =  24,  and  24* 

36X28.28       1018.08 
=  576.       Then, — -  —  — =  417. 25  tons  =  934  640  Ibs. 

Safe   Loads.— Cast  Iron,  one  fifth.    Wrought  Iron  or  Steel,  one  fourth. 
WOODS.     (C.  Shaler  Smith. ) 

C  a —  \V         C  representing  coefficient  of  material,  a  area  of  section 

(~yr^ —  '     in  sq.  ins. ,  I  length,  and  d  diameter  or  least  side,  both  in 

— )  X  -004  like  terms,  and  W  crushing  weight  in  Ibs. 

Coefficients.*    For  Crushing  Stress  per  Sq.  Inch  of  Section. 

Hemlock 3100  I  White  Pine 3500  I  Georgia  Pine 5000 

Spruce 3500  I  Yellow  Pine 5000  |  Oak,  White 6000 

(Hodgkinson.) 

Ash 9000  I  Beech 7050  I  Elm 7000 

"    Canadian 7000  |  Cedar. 5100  |     "    rock 10000 

ILLUSTRATION. — Assume  a  Yellow-pine  column  10  ins.  square  and  ia  ft.  in  length. 
5000  X  io2  500000 

—  273  373  »*• 


I+("/<^)2X.oo4 
Safe    Load.*    One  fifth.        (Department  of  Buildings,  City  of  New  York.) 


STRENGTH    OF   MATERIALS. — CRUSHING. 


769 


To   Compute    Safe   Load,   of  Columns, 

it  Iron. 
)  80000  a 


Cai 

Round,  or 
Rectangular, 

Solid  or  Hollow. 


"Wrought   Iron. 

40000  a 


=  W. 


For  Mild  Steel  put  48  ooo,  and  for  Strong  or  Hard  put  60000. 

a  representing  area  of  section  in  sq.  ins. ,  I  length  of  column  in  ins. ,  r  radius  oj 

Gyration  =/— ,  I  moment  of  Inertia  (see  p.  819),  C  coefficient,  and  W  safe  load  in  Ibs. 


Coefficients. 

Rou 

Solid. 

ND. 

Hollow. 

RECTANC 

Solid. 

ULAR. 

Hollow. 

Cast  Iron  

.000164 
.000047 
—  .000022 
—  .oooo« 

.000272 
.000059 
—  .000035 
—  .000087 

.000189 
.000049 
—  .000033 
—  .  ooo  006 

.000267 
.000047 
—  .000081 

.OOO  TCC 

Wrought  Iron    

Steel,  Mild  

Do.    Strong..., 

ILLUSTRATION.—  What  is  the  safe  load  for  a  Cylindrical  and  Hollow  Cast-iron  col- 
umn, 10  ins.  in  external  diameter,  8  ins.  internal,  and  20  feet  in  length  ? 

Area  =  28.  28  sq.  ins.     I  =  5  4  —  44  X  .  7854  =  289.  8.     r  =  /^f  =  3.  22. 

V     20.20 
80  000  X  28.  28  2  262  400 

•  -  —  --  — 


.  00027 


»  +  SSSoX.ooo,7] 


-=181  zoo 


2.  Assume  a  solid  column  of  Strong  Steel  of  like  diameter  and  15  feet  in  length. 

Area  =  78. 54  sq.  ins. ,  and  r  =  2. 5. 
78.54  X  60000 4712400 4712400 fi       , 

4  X[i<"  5184  X  — -000053]       4X>725  Ibs. 


4  X   i-f  I—      -)X  —  -ooo 

For  Relative  Value  of  various  Woods  and  Comparison  of  Long  and  Short  Col- 
umns, see  page  976. 

Weight  TDorne  with  Safety-  t>y  Solid  Cast-iron  Columns. 

In  looo  Lbs. — (New  Jersey  Steel  and  Iron  Co.) 


Length. 
Feet. 

2 

Ins. 

& 

Ins. 

In5, 

6 
Ins. 

m7, 

iJlJLl 

8 
Ins. 

1KTKK 

In9s. 

10 
Ins. 

ii 
Ins. 

12 

Ins. 

13 

Ins. 

14 
Ins. 

15 
Ins. 

5 

12.4 

44 

I  O2 

184 

288 

414 

560 

728 

916 

1126 

1354 

— 

_ 

— 

6 

9.4 

36 

88 

264 

S32 

884 

1082 

1320 

157° 

—  - 

— 

7 

7.2 

30 

76 

146 

242 

360 

502 

660 

850 

1056 

1282 

'530 

1798 

2086 

8 

24 

66 

130 

218 

332 

470 

630 

812 

1016 

1240 

1486 

1754 

2040 

9 

— 

20 

56 

114 

198 

306 

44° 

596 

774 

974 

1196 

1440 

1706 

1992 

0 

— 

18 

48 

102 

1  80 

282 

410 

56o 

739 

932 

1152 

1392 

1656 

1940 

2 

— 

— 

38 

80 

136 

238 

354 

494 

658 

846 

1056 

1292 

1550 

1828 

4 
6 

— 

- 

28 

64 
52 

122 
100 

2OO 
170 

304 
262 

432 
378 

586 
520 

® 

966 
878 

1192 
1094 

1440 
1332 

1712 

8 

— 

— 

— 

44 

84 

144 

226 

332 

462 

616 

796 

1000 

1:228 

1482 

20 



— 

— 

72 

124 

196 

292 

410 

552 

720 

912 

1130 

1372 

For    Tn"bes    or   Hollow    Columns. 

Subtract  weight  that  may  be  borne  by  a  column,  of  diameter  of  internal 
diameter  of  tube  from  external  diameter,  and  remainder  will  give  weight 
that  may  be  borne.  Thickness  of  metal  should  not  be  less  than  one  twelfth 
diameter  of  column. 

ILLUSTRATION.— Required  the  safe  load  of  a  solid  cast-iron  column  6  ins.  in  diam- 
eter and  20  feet  in  length. 

Under  6  and  in  a  line  with  20  is  72,  which  x  1000  =  72  ooo  Ibs. 

NOTE.— This  is  about  one  sixth  of  destructive  weight. 

3  A 


77O          STRENGTH    OF   MATERIALS. — DEFLECTION. 

DEFLECTION. 
Deflection   of  Bars,  Beams,  Grirders,  etc. 

Experiments  of  Barlow  upon  deflection  of  wood  battens  determined, 
that  deflection  of  a  beam  from  a  transverse  strain,  varied  directly  as  cube 
of  length  and  inversely  as  breadth  and  cube  of  depth,  and  that  with 
like  beams  and  within  limits  of  elasticity  it  was  directly  as  the  weight. 

In  bars,  beams,  etc.,  of  an  elastic  material,  and  having  great  length  com- 
pared to  their  depth,  deductions  of  Barlow  will  apply  with  sufficient  accu- 
racy for  all  practical  purposes ;  but  in  consequence  of  varied  proportions  of 
depth  to  length,  of  varied  character  of  materials,  of  irregular  resistance  of 
beams  constructed  with  scarphs,  trusses,  or  riveted  plates,  and  of  unequal 
deflection  at  initial  and  ultimate  strains,  it  is  impracticable  to  deduce  any 
exact  laws  regarding  degrees  of  deflection  of  different  and  dissimilar  figures 
and  proportions. 

From  an  experiment  of  Mr.  Tredgeld  it  was  shown  that  deflection  of  cast 
iron  is  exactly  proportionate  to  load  until  stress  reaches  a  certain  magnitude, 
when  it  becomes  irregular. 

In  experiments  of  Hodgkinson,  it  was  further  shown  that  sets  from  de- 
flections were  very  nearly  as  squares  of  deflections. 

In  a  rectangular  bar,  beam,  etc.,  position  of  neutral  axis  is  in  its  centre, 
and  it  is  not  sensibly  altered  by  variations  in  amount  of  strain  applied.  In 
bars,  beams,  etc.,  of  cast  and  wrought  iron,  position  of  neutral  axis  varies  in 
same  beam,  and  is  only  fixed  while  elasticity  of  beam  is  perfect.  When  a 
bar,  beam,  etc.,  is  bent  so  as  to  injure  its  elasticity,  neutral  line  changes,  and 
continues  to  change  during  loading  of  beam,  until  its  elasticity  is  destroyed. 

When  bars,  beams,  etc.,  are  of  same  length,  deflection  of  one,  weight  being 
suspended  from  one  end,  compared  with  that  of  a  beam  Uniformly  Loaded, 
is  as  8  to  3 ;  and  when  bars,  etc.,  are  supported  at  both  ends,  deflection  in  like 
case  is  as  5  to  8.  Whence,  if  a  bar,  etc.,  is  in  first  case  supported  in  middle, 
and  ends  permitted  to  deflect,  and  in  second,  ends  supported,  and  middle 
permitted  to  descend,  deflection  in  the  two  cases  is  as  3  to  5. 

Of  three  equal  and  similar  bars  or  beams,  one  inclined  upward,  one  down- 
ward, at  same  angle,  and  the  other  horizontal,  that  which  has  its  angle  up- 
ward is  weakest,  the  one  which  declines  is  strongest,  and  the  one  horizontal 
is  a  mean  between  the»two. 

When  a  bar,  beam,  etc.,  is  Uniformly  Loaded,  deflection  is  as  weight,  and 
approximately  as  cube  of  length*or  as  square  of  length ;  and  element  of  de- 
flection and  strain  upon  beam,  weight  being  the  same,  will  be  but  half  of  that 
when  weight  is  suspended  from  one  end. 

Deflection  of  a  bar,  beam,  etc.,  Fixed  at  one  End,  and  Loaded  at  other, 
compared  to  that  of  a  beam  of  twice  length,  Supported  at  both  Ends,  and 
Loaded  in  Middle,  strain  being  same,  is  as  2  to  i ;  and  when  length  and 
loads  are  same,  deflection  will  be  as  16  to  i,  for  strain  will  be  four  times 
greater  on  beam  fixed  at  one  end  than  on  one  supported  at  both  ends ;  there- 
fore, ah1  other  things  being  same,  element  of  deflection  will  be  four  times 
greater ;  also,  as  deflection  is  as  element  of  deflection  into  square  of  length, 
then,  as  lengths  at  which  weights  are  borne  in  their  cases  are  as  i  to  2,  de- 
flection is  as  i :  22  x  4=  i  to  16. 

Deflection  of  a  bar,  beam,  etc.,  having  section  of  a  triangle,  and  supported 
at  its  ends,  is  .33  greater  when  edge  of  angle  is  up  than  when  it  is  down. 

In  order  to  counteract  deflection  of  a  beam,  etc.,  under  stress  of  its  load, 
where  a  horizontal  surface  is  required,  it  should  be  cambered  on  its  upper 
surface,  equal  to  computed  deflection. 


STRENGTH    OF   MATERIALS. — DEFLECTION.  77! 

Safe  Deflection. — One  fortieth  of  an  inch  for  each  foot  of  span,  with  a 
factor  of  safety  for  load  of  .33  of  destructive  weight  =  y^u,  but  for  ordinary 
loads  and  purposes, 

Cast  Iron,  y^^  to  -^^ ;  and  Wrought  Iron,  y^j-  to  ^rinF  or  TTDTF» 
after  beam,  etc.,  has  become  set. 

When  Length  is  uniform,  with  same  weight,  deflection  is  inversely  as 
breadth  and  square  of  depth  into  element  of  deflection,  which  is  inversely  as 
depth.  Hence,  other  things  being  equal,  deflection  will  vary  inversely  as 
breadth  and  cube  of  depth. 

ILLUSTRATION.— Deflections  of  two  pine  battens,  of  uniform  breadth  and  depth,  and 
equally  loaded,  but  of  lengths  of  3  and  6  feet,  were  as  i  to  7.8. 

Deflection  of  different  bars,  beams,  etc.,  arising  from  their  own  weight, 
having  their  several  dimensions  proportional,  will  be  as  square  of  either  of 
their  like  dimensions. 

NOTE.— In  construction  of  models  on  a  scale  intended  to  be  executed  in  full  di- 
mensions, this  result  should  be  kept  in  view. 

When  a  continuous  girder,  uniformly  loaded,  is  supported  at  three  points 
by  two  equal  spans,  middle  portion  is  deflected  downwards  over  middle  bear- 
ing, and  it  sustains  by  suspension  the  extreme  portions,  which  also  have  a 
bearing  on  outer  bearings.  Middle  portion  is,  by  deflection,  convex  up- 
wards, and  outer  portions  are  concave  upwards ;  and  there  is  a  point  of 
"contrary  flexure,"  where  curvature  is  reversed,  being  at  junction  of  con- 
vex and  concave  curves,  at  each  side  of  middle  bearing.  This  point  is  dis- 
tant from  middle  bearing,  on  each  side,  one  fourth  of  span.  Of  remaining 
three  fourths  of  each  span,  a  half  is  borne  by  suspension  by  middle  portion, 
and  a  half  is  supported  by  abutment.  Hence,  distribution  of  load  on  bear- 
ings is  easily  computed,  as  given  above.  Deflection  of  each  span  is  to  that 
of  an  independent  beam  of  same  length  of  span  as  2  to  5. 

In  a  beam  of  three  equal  spans,  deflection  at  middle  of  either  of  side  spans 
is  to  that  of  an  independent  beam  as  13  to  25. 

In  a  long  continuous  beam,  supported  at  regular  intervals,  deflection  of 
each  span  is  to  that  of  an  independent  beam  of  one  span  as  i  to  5. 

Cylinder.— If  a  bar  or  beam  is  cylindrical,  Barlow  gives  the  deflection  1.7 
times  that  of  a  square  beam,  other  things  being  equal ;  D.  K.  Clark  puts  it 
at  1.47. 

Formulas  fbr  Deflection   of  Beams  of  Rectangular   Sec- 
tion, etc. 

Loaded  at  One  End.    ^^  n  =  D.     Loaded  Uniformly.      *b  ^     =  D. 


Both  i 
s.) 


Loaded  in  Middle.  =  D     Looted  Uniformly. 

Ends. 


Supported  at  Both  Ends. 


5  = 


Loaded  in  Middle.     -  =  D.      Loaded  Uniformly.  =  D. 

16  6  di  C  8  X  16  6  d3  C 

m2  n2  W 
Loaded  at  any  one  Point.      ^  =  D. 

Supported  in  Middle. 

Ends  Loaded  Uniformly.      -  3  f  ,Wl7-  =  D. 
5  X  ID  o  uj  O 

I  representing  length  in  feet,  b  breadth,  and  d  depth,  both  in  ins.,  W  weight  or  stress 
in  Ibs.,  m  and  n  distances  of  weight  between  supports,  C  a  constant,  and  D  deflection 
in  ins. 


7/2  STRENGTH    OF   MATERIALS. — DEFLECTION. 

Deflection    of  Beams    or    Bars    of   Rectangular 

iSection. 
To   Compute   Deflection  of  a   Rectangular  Beam  or   Bar. 

Supported  at  Both  Ends.     Loaded  in  Middle. 

CAST   IRON. 
73  vy 

Rectangular  Beams.     —. 7-77  =  D-    Cylindrical.    For  36  ooo  put  24  ooo. 

36  ooo  b  d* 

I  representing  length  in  feet,  b  and  d  in  inches. 

ILLUSTRATION Assume  a  rectangular  bar  of  cast  iron,  i  inch  square  and  loaded 

with  224  Ibs.,  4.5  feet  between  its  supports. 

224  X  4-  5 —  _  20412  _         inc^ 
36  ooo  X  i  X  i 3      36  ooo 
By  actual  experiment  of  Mr.  Hodgkinson  the  deflection  was  .561  inch. 

WROUGHT   IRON. 
£3  W 

Rectangular  Beams. =-T. -  —  D.     Cylindrical.     For  60000  put  42000. 

60  ooo  &  d3 

WOODS. 

73   TIT 

^  =  D.     I  representing  length  in  inches,  and  W  weight  in  tons. 
Mean  of  LasleWs,  Barlow,  etc. 


Ash  Canadian.... 

c 

.  .  .  1476 

c 
Iron-  wood  4228 

Oak,  French  .... 

C 

...  2656 

"    Eng 

Larch            ...     .     2100 

Beech  

2418 

Mahogany  Honduras  2118 

Pitch  pine  

...  2968 

Blue  Gum 

"         Mexican    3608 

Elm 

1227 

u         Spanish..  3360 

Fir  Dantzic 

Oak  Baltimore            2761 

Spruce   

.  .  .    3300 

"    Riga 

"    Canadian     ....  3445 

Greenheart.  .  . 

...  1888 

"    Eng.  ..            ..  1848 

Yellow  pine  — 

...  2084 

ILLUSTRATION. — What  is  the  deflection  of  a  floor  beam  of  Yellow  pine,  3  by  12  ins., 
12  feet  between  its  supports,  under  a  uniformly  distributed  load  of  3000  Ibs.? 


8  X  3  X  i23  x  2084  X  .216       18668415 

• • zr: ;=  1.25  tons. 

5X2985984  14929920 

*  %  for  being  uniformly  distributed. 
By  a  test  of  a  like  beam,  the  deflection  was  .2125. 

Z3  W 

For  Cylindrical  Beams  deduct  one-third  from  these  constants,  or -~. _  =  D. 

3.14^4  C 

For    Torsional    Deflection    of  Iron    Shaft.     (D.  K.  Clark.) 

Cast  Iron,  - — —  =  D.  Wrought  Iron,  • ^=  D. 

044  a4  I37°  «4 

W  in  tons  and  r  radius  or  distance  of  applied  power. 

Deflection    of   Continuous    Girders    or    Beams. 

Beams  of  Uniform  Dimensions,  Supported  at  Three  or  More  Bearinga. 
(D.  K.  Clark.) 

2.  Three  Equal  Spans  or  4  Bearings. 
Weight  on  ist  and  4th  bearing  —   .4  W  I 
"        "  2d     "    sd        "       =  i.i  Wl 

3.  Four  Equal  Spans  or  5  Bearings. 

Weight  on  ist  and  sth  bearing  =  .39  W  I  \  Weight  on  2d  and  4th  bearing  =  1.14  W I 
Weight  on  sd  bearing  =  .93  W  I. 


i.  Two  Equal  Spans  or  3  Bearings. 
Weight  on  ist  and  3d  bearing  —  .  375  W  I 


"  2d  bearing 


STRENGTH    OF   MATERIALS. — DEFLECTION. 


773 


To  Compxite   IVTaximnm    Load,  that   may  "be  "borne  "by  a 
Rectangular    Beam. 

Deflection  not  to  exceed  Assigned  Limit  of  one  hundred  and  twentieth  of  an 

Inch  for  each  Foot  of  Span. 
Supported  at  Both  Ends.    Loaded  in  Middle. 

— j  =  W.    b  and  d  representing  breadth  and  depth  in  ins.,  I  length  in  feet,  C  con- 
stant, and  W  weight  or  load  in  Ibs. 

Constants. 


Cast  Iron           .         0003 

Oak  white  027 

Oak  red  

oto 

Wrought  Iron     0021 

Hickory                        018 

Pine  pitch        ...   .      033 

Pine  white.  "  

O3Q 

Teak                              024 

u     vellow  .  .006 

Chestnut,  horse  .  .  . 

borne  by  a  beam  o 
ed  in  its  middle? 

=  ^m.^  Ibs. 

..    .051 

r  white 

ILLUSTRATION.—  What  is  maximum  load  that  may  be 
pine,  3  by  12  ins.,  20  feet  between  its  supports,  and  load 

3  X  i23        5184 
C  —  .0^0.        Then  r-=-P  —  ^—4  = 

WROUGHT  IRON. 
Deflection   of  "Wrovight-iron    Bars. 

Supported  at  Both  Ends.      Weight  applied  in  Middle. 


i. 

A 

Weight  and  Deflection 

all  ' 

No.                    FORM. 

•s 

1 

by  Actual 
Observation. 

at  one  sixth 
of  Destruc- 
tive Weight. 

at     1    thof 
an  Inch  for 
each  Foot  of 
Span. 

Constant 
Reduced  W 
and  Deflec 
W/3 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lba. 

Ins. 

C 

i.  American.  K 

1.83 

I 

i 

600 

.06 

266 

.027 

148 

.015 

X 

2.  English...    " 

2-75 

2 

2 

4480 

.08 

1310 

.022 

1310 

.022 

1.29 

3.       "      ....    § 

2.75 

1-5 

2.5 

8960 

.104 

2128 

.025 

1873 

.022 

1.25 

4.       «      ....    " 

2.75 

i-5 

3 

8960 

.088 

3800 

.037 

2259 

.022 

.88 

To  Compute  Deflection  of*,  and  'Weight  that  may  "be  toorne 
Toy,  a  .Rectangular  Bar  or  Beam  of  Wrought  Iron. 


-  —  C. 


6oooo6d3D~~  60000  b'd*'C 

ILLUSTRATION.— What  weight  will  a  beam  2  ins.  in  breadth,  5  ins.  in  depth,  and 
15  feet  between  its  supports,  bear  with  safe  deflection  of  -j-W  of  an  inch  for  each 
foot  of  space,  or  -j-^  of  its  length  ? 

C  from  table  =  .88.    D  =  y^  of  15  = .  12  inch. 
6ooooX2X53X.88x.i2 
-- 


D.  K.  Clark  gives  for  Elastic  deflection,  47  ooo  for  Rectangular  bars,  and  32  ooo  for 
Cylindrical. 

NOTE. — Deflection  of  ^-^  to  ^-J-^  of  the  length  may  be  allowed  under  special  cir- 
cumstances ;  but  under  ordinary  loads  the  deflection  should  not  exceed  one  fourth 
of  these,  as  ^-^  to  -^far. 

Practice  in  U.  S.  is  to  allow  i21OQ  after  girder  has  taken  its  permanent  set. 

In  small  bridges  there  is  a  slight  increase  in  deflection  from  high  speeds,  about 
.166  or  .  144  of  the  normal  deflection,  with  the  same  load  moving  at  slow  speed. 

In  large  girders  there  is  no  perceptible  difference  between  the  deflection  at  high 
and  low  speeds. 


774 


STRENGTH    OF   MATERIALS. DEFLECTION. 


Deflection   of  Wronght-iron    Rolled.    Beai 

Supported  at  Both  Ends.      Weight  applied  in  Middle. 


70000  d2  (4  a  +  1.155  a')  D 

. 

Flanges. 

Weight  and  Deflection 

No.     FORM. 

I 

Width. 

Mean 
Thick- 
ness. 

Web. 

Depth. 

by  Actual 
Observation. 

at  one  sixth 
of  Destructive 
Weight. 

C 

I 

Feet. 

10 

IDS. 
3 

Inch. 
.485 

Inch. 

•5 

Ins. 
7 

Lbs. 
12000 

•4 

Lbs. 
3800 

Inch. 
.127 

1.05 

2.             u 

3-          " 

20 
2O 

4.6 
5-7 

.8 

•643 

•  5 
.6 

9-85 
11.75 

l6oOO 
20000 

1.15 

.85 

6300 
8000 

•453 
•34 

.92 
.98 

To  Compute  Deflection  of,  and.  "Weight  that  may  "be  "borne 
"by,a"Wrovight-iron.  Rolled  Beam  of  TJnifbrxn  and  Sym- 
metrical Section. 

Supported  at  Both  Ends.      Weight  applied  in  Middle.     (D.  K.  Clark.) 


70000  d2  (4  a  +  i .  155  a')  D  _ 


-=W. 


70000  d2  (4  a-f- 1.155  «') 
I  representing  span  in  feet,  d  reputed  depth,  or  depth  less  thickness  of  lower  flange 
in  ins.,  a  area  of  section  of  lower  flange,  a'  area  of  section  of  web  for  reputed  depth 
of  beam,  both  in  sq.  ins.,  and  W  weight  or  stress  in  Ibs. 

ILLUSTRATION. — What  is  deflection  of  a  wrought-iron  rolled  beam  of  New  Jersey 
Steel  and  Iron  Co.,  10.5  ins.  in  depth,  flanges  5  by  .5  ins.,  and  width  of  web  .47 
inch,  when  loaded  in  its  middle  with  8000  Ibs.,  and  supported  over  a  span  of  20  feet? 

d  =  10.5  —  .5  =  10  ins.,     a  —  5  X  -5  =  2. 5  sq.  ins.,  and  a'  =  10  X  -47  —  4-7  sq.  ins. 

8oOO  X  2O3  64OOOOOO 

Then  —  -  =  — '         —  =  .59  inch. 

70000  X  io2  X  (4  X  2.5  +  1.155  X  4-7)      I07999  5°° 

If  weight  is  uniformly  distributed,  divide  by  112  500  instead  of  70000. 

A  like  beam  6  ins.  in  depth,  loaded  with  2608  Ibs.,  and  supported  over  a  span  of 
12  feet,  gave  by  actual  test  a  deflection  of  .3  inch,  and  by  above  formula  it  is  also 
.3  in«h. 

NOTE.— Deflection  for  such  a  beam,  for  a  statical  weight  or  stress  of  17  100  Ibs., 
uniformly  distributed,  by  rules  of  N.  J.  Steel  and  Iron  Co. ,  would  be  .  54  inch,  which, 
with  difference  in  weights,  will  make  deflections  alike. 

Deflection    of  \Vronght-iron    Riveted    13 earns. 
Supported  at  Both  Ends.     Weight  applied  in  Middle. 

-J4- — T, =  C  at  Reduced  Weight  and  Deflection. 


No.  FORM. 

Length. 

Flanges. 

Angles. 

Web. 

Feet. 

Ins. 

Ins. 
2.125X2 

Inch. 
I 

'I 

7 

— 

X-28 
2.125X2 
X.29 

}, 

4-5X 

2X2 

'I 

n.66 

•5 
4-5X 
•375 

X-3I25 
2X2 
X-3'25 

•25 

4-5X 

2X2 

3-      " 

28-5 

•5 
7    X 

X-375 
3X3 

•375 

•5 

X-4375 

16.5 


Weight  and 

bv  Actual 
Observation. 

Deflection 
at  one  sixth 
of  Destructive 
Weight. 

C 

Lbs. 

Inch. 

Lbs. 

Inch. 

4216 

.1 

4062 

.096 

•63 

77280 

.46 

12880 

•075 

1.96 

"5584 

.875 

19265 

.148 

3-86 

STBENGTH    OF   MATERIALS. — DEFLECTION. 


775 


To  Compnte  Deflection  of,  and  "Weight  that  may  "be  "borne 
by,  a  Riveted  Beam  of  "Wrought  Iron. 

—r--, -7K =  D.  ,68000  (^±-^-  +  —  WcD 


W. 


a,  a',  and  a"  representing  areas  of  upper  and  lower  flanges  with  their  angle  pieces, 
and  of  web  for  its  entire  depth,  all  in  sq.  ins. 

NOTE. — If  there  are  not  any  flanges,  as  in  No.  i,  angle  pieces  alone  are  to  be  computed  for  flange 
Area. 

ILLUSTRATION.— What  weight  will  a  riveted  and  flanged  beam  of  following  dimen- 
sions sustain,  at  a  distance  between  its  supports  of  25  feet,  and  at  a  safe  deflection 
of .  2  inch  or  ^^  of  its  length  ? 

Top  flange 6  X -5  ins.      \     Web 5  in*. 

Bottom  flange 6X-5   "       |     Depth 17      " 

Angles * 2.25  X  2.25  X  .5  ins. 

a  and  a'  each  =  6  X  •  5  =  3  +  2.25  +  2.25  — .5  x  .5  X  2  =  7  sq.  ins. 
a"  — .  5  x  17  =  8. 5  sq.  ins.    C,  as  per  No.  2,=  .43,  but  inasmuch  as  flanges  in  thii 
case  are  much  heavier,  assume  .5. 


168000  ( 


Then- 


2X.5 


25s 


Strength  of  a  Riveted  beam  compared  to  a  Solid  beam  is  as  i  to  1.5,  while  for 
«qual  weights  its  deflection  is  1.5  to  i. 

Tu.~bular   GJ-irders.    "Wrought   Iron. 


Supported  at  Both  Ends.     Weight  applied  in  Middle. 

c. 

•g  £«g 

1 

til 

j 

Depth. 

^ 

1 

111  8 

No.                  SECTION. 

tL-.'C 

•o 

*& 

? 

S  °  1 

* 

Inter- 

Ex- 

'« 

q~ 

C  00  "SCO 

^ 

J      M 

« 

nal. 

ternal. 

1 

Q 

ojj 

Feet. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Inch. 

( 

t.           Thickness  .03  inch 

3-75 

1.9 

2-94 

3 

448 

.z 

.03 

28! 

a.     "            "        .525  " 

30 

15-5 

22.95 

24 

33685 

-56 

.24 

47: 

top          .372  "  ) 

3.     '         bottom    .244  " 

30 

16 

23.28 

24 

32538 

i.  ii 

.24 

22. 

sides        .125  "  ) 

4.     "    Thickness  .75    " 

45 

24 

34-25 

35-75 

128850 

1.85 

.36 

362 

5.  fj  Thickness  .0375" 

'7 

12 

11.925 

12 

2755 

.65 

.136 

6s 

6.  f\           "        .0416" 
7-  U           "        -'43  " 

17 
17 

9.25 

,  9-25 

13-535 
14.714 

13.62 
15 

2262 
16800 

.62* 
1.39* 

.136 
.136 

4' 
"S 

*  Destructive  weight. 

To   Compnte    Deflection    of,  and   "Weight    that    may  "be 
loorne    by,  a  \Vroxight-iron    Txabxilar   G-irder. 

16  &  d3  C  D  W  I3 

ILLUSTRATION.— What  weight  may  be  safely  borne  by  a  wrought-iron  tube,  alike 
>  No.  3  in  preceding  table,  for  a  length  of  30  feet,  and  a  deflection  of  .32  inch  ? 


to  N 


<  243  x  224  Xv24  _  190253629  _         ^ 
—3  27000         ' 


Flanged  Rails. 

Deflection  of  Iron  and  Steel  Flanged  Rails  within  their  elastic  limit,  comparad 
with  their  transverse  strength,  is  as  17  to  20,  and  with  double-headed  it  is  as  n  to  23. 


7/6         STRENGTH   OF  MATERIALS, — DEFLECTION. 

RAILS. 

Supported  at  Both  Ends.     Weight  applied  in  Middle. 


Iron. 


No.   FORM. 

rl 

Head. 

Bottom. 

Weight 
Y?rd. 

I 

Ares. 

Observed 
Weight  and 
Deflection. 

Destructive 
Weight 
and 

Deflection. 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

S<i.  Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

•     I 

2-75 

2.25X1 

2.25X1 

60 

4-5 

6.166 

13440 

•034 

26680 

.065 

3. 

3- 

4-5 
5 

2.3  Xi 
2.3  Xi 

2.3  Xi 
2.3  Xi 

82 

4-5 
5-4 

6.68 
8.25 

II  200 
25760 

.11 

.2 

24640 
51520 

.204 
.378 

<  T 

2-75 

3-5  X  .8 

2.25X1 

60 

4 

6.7 

II  200 

•035 

26680 

.065 

*  JL 

2.58 

2.23XZ 

3-5  X  .6 

57 

3-5 

5-85 

II  2OO 

.097 

20160 

.128 

Steel. 

No.  FORM. 

r  J 

1 

Bottom. 

N 

Web. 

Dept 

Centres 
of 
Heads. 

h. 

1 

Area. 

Observed 
Weight  and 
Deflection. 

Destructive 

Weight 
and 
Deflection. 

Feet. 

Ilia. 

Ins. 

Lbs. 

Inch. 

Ins. 

Ins. 

S.Ius 

Lbs. 

In. 

Lbs. 

Inch. 

•p 

5 

— 

- 

78 

•75 

4.2 

5-4 

7.67 

36086 

•25 

80192 

•55 

7-     " 
Bessemer. 

3.62 

~ 

- 

86 

- 

- 

5-5 

8-43 

22400 

.14 

26680 

.165 

8X 

5 

2-5 

6-375X;37 

84 

•65 

3-37 

4-5 

8.24 

27290 

.24 

27290 

.24 

To  Compute  Deflection  of*  Doxi"ble-headed.  Rails  -within 
Elastic   Limit.     (D.  K.  Clark.) 

Supported  at  Both  Ends.     Weight  applied  at  Middle. 


Wl* 


IRON. 

r  =  D.    a  representing  area  of  one  head,  less  portion  per* 

57ooo(4ad/2-f  1.155  td*) 

taining  to  web,  d  whole  depth  of  rail,  d'  vertical  distance  between  centres  of  heads, 
t  thickness  of  web,  all  in  ins.,  I  length  in  feet,  and  W  weight  in  Ibs. 

STEEL. 

For  57  ooo  put  67  400. 

ILLUSTRATION. — Take  case  No.  3  (Iron),  in  preceding  table,  with  a  weight  of  26000 
Ibs. ;  what  will  be  its  deflection  between  bearings  5  feet  apart? 


Then 


a  =  1.911.    $'  =  4.2.    $  =  5.4.    £  =  .82. 
26000X5'  3250000 


57000  (4  X  i-9«  X  4-22  + 1.155  X  .82  X  5-43)      57oooX  284 


= .  2  inch. 


To  Compute  Deflection  of  Iron  and   Steel  Rails  of  TJn- 
symmetrical    Section   -within    Elastic    Limits. 

Elastic  Deflection  of  Steel  Flanged  Rails  of  Metropolitan  Railway  of  London,  as 
determined  by  Mr.  Kirkaldy,  at  a  span  of  5  feet,  and  loaded  in  middle,  was  .02  inch 
per  ton.  (See  Manual  of  D.  K.  ClarJc,  pp.  667-670.) 


STRENGTH    OF   MATERIALS. — DEFLECTION. 


777 


CAST  IRON. 

Reflection  of*  Rectangular  Bars  and  Beams  of  various 
Sections,  etc.,  by  U.  S.  Ordnance  Corps,  Barlow, 
liodgliinson,  and  Cubitt. 

Supported  at  Both  Ends.     Weight  applied  in  Middle. 


A 

5  .  c 

i 

Weight  and  Deflection. 

«S8   \ 

No.                   FOEM. 

« 

*0  M 

r 

Breadth. 

i 

By  Actual 
Observation. 

At  one  sixth 
of  Breaking 
Weight. 

At     *    thof 
an  inch  for 
each  foot  of 

Constant  t 
Bduced  We 
nd  Deflecti 
W/3 

•3 

span. 

PJ  * 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

C 

i.American  —  1| 

1.66 

2 

2 

5000 

.036 

1666 

.012 

1805 

.013 

3.8! 

2.  English  " 

it 

I 

I 

212 

I008 

•32 
•4 

80 
5333 

.12 
2.  II 

22 
3370 

•033 
i«33 

4.11 

4.       "                      1 

4-5 

3 

I 

1  120 

1.42 

215 

.27 

30 

•037 

2.37 

4                 1 

5.^  n 

4-5 

i 

{2'-l 

223I 

•5i 

422 

.1 

156 

•037 

2-33 

To    Compute    Deflection    of*,   and    "Weight    that    may    be 
borne  by,  a  Rectangular  Bar  or  Beam  of  Cast  Iron. 
WP  ^          10400  6  ds  CD 


-  =  C. 


10400  b  d^D  10400  6  ri3  C 

ILLUSTRATION.— What  weight  will  a  beam  2  ins.  in  breadth,  5  ins.  in  depth,  and 
16  feet  between  its  supports,  bear  with  safe  deflection  of  T^  of  an  inch  for  each 
foot  of  span,  or  j-fa  of  its  length  ? 
C  from  table  =  3. 89.     D  —  T^TJ  of  16  =  .  133  ins. 

10400X2  X  53  X  3-89  X  .  133  _  '  345. '62  _  „ 

"IP""  =  ~^96~: 

Clark  gives  C  uniform  for  Rectangular  bars  of  2.69,  and  1.85  for  Cylindrical. 

FLANGED  BEAMS.      Cast   Iron. 
Supported  at  Both  Ends.     Weight  applied  in  Middle. 

To  Compute  Deflection  of,  and  "Weight  that  may  be 
borne  by,  a  Flanged  Beam  of  Cast  Iron  of  Uniform 
And  Symmetrical  Section. 

WW 

f=D. 


27  OOP  d2  (4  a  +  1. 155  a'2)  D  = 


27000  d2  (4  a-j-  1.155  a'2 

ILLUSTRATION.— What  is  deflection  of  a  cast-iron  beam  (Hodgkinson's)  7.15  ins., 
flanges  2.6  x  .86  ins.  and  5  X  1.6  ins.,  and  width  of  web  i  inch,  when  loaded  in  its 
middle  with  n  200  Ibs.,  over  a  span  of  15  feet? 

d  =  7-i5  — 1.6  =  5.55  *«*•»  a  =  5  X  x.6  =  8  in*.,  and  a' =  7. 15  — 1.6  =  5.55  ins. 

-1200X153 37800090  _        M 

"27000X30-8(32  +  35.57) 


Then- 


27oooX  S-552  (4  X  8  +  1.155  X  S-552) 

NOTE  i. — The  observed  deflection  of  this  beam  was  1.28  ins.,  at  one  sixth  of  its  de- 
structive weight  it  was  .3,  and  at  y^  of  an  inch  for  each  foot  of  span  it  wa§ 
.125  inch. 

2.— The  mean  ratio  of  elastic  to  destructive  stress  is  73  per  cent. 

Formulas  for  value  of  deflection  signify  that  deflection  varies  directly  as  weight, 
and  as  cube  of  length ;  and  inversely  as  breadth,  cube  of  depth,  and  coefficient  of 
elasticity. 


773 


STRENGTH    OF    MATERIALS. DEFLECTION. 


Elastic  Strength  of  Beams  of  Unsymmetrical  Section. — Elastic  strength  is 
approximately  deducible  from  ultimate  strength,  according  to  ordinary  ratio 
of  one  to  the  other,  ascertained  experimentally.  Elastic  strength  and  de- 
flection of  a  homogeneous  beam  of  any  section  is  same,  whether  in  its  nor- 
mal position  or  turned  upside  down. 

Comparative    Strength,   and.    Deflection    of  Cast-iron 
Flanged.    Beams. 


Description  of  Beam. 

Comp. 
Strength. 

Description  of  Beam. 

Comp. 
Strength. 

lieain  of  equal  flanges 

.58 

Beam  with  flanges  as  i  to  4  5 

78 

"     with  only  bottom  flange. 

.72 

i  to  5.5.. 

:e 

"        "            "        i  to  4.... 

.63 
•73 

"         "          "         i  to  6.  73. 

.92 

SHAFTS. 

To   Compute    Deflection    and    Distributed   "Weight  for 

Limit   of  Deflection. 

"Wrought    Iron. 

Deflection.  Weight. 


Round. 


Square. 


Round. 


Square. 


Round.     -7T7T- 


Square.     •—£-• 


•ported  at  Enc 

U. 

and 
and 

and 
and 

and 
and 

Fixed  at  Ends.           Supported  at  Ends.           Fixed  at  End 
Wl«                            664  d*      mmjt       1330  d4 

it 
W. 

W. 

W. 
W 

W. 
W 

66  400  d4 

~~ 

"V  J3 

I*                I* 

97554      and      I95°54 

97  500  ** 

195000  S4 

Cast   Iron. 

394  d*      and       79°d* 

39400  d* 

79000  d4 

WJ3 

ia                                        £2        — 

580  s4                   n6os4 

58000  s* 
78^00  d* 

116000  s4 
Steel. 

Wf3      -p. 

158000  a4 

fa                         i» 
788  d4      -«H      J576  d* 

J«                          «a 

1160  &4           .      2320  64 
—  .  and     -^—  —  — 

116000  s* 


232000  5*  " 


<f  representing  diameter  and  s  side  of  shaft,  in  ins.,  I  length  between  centres  of  bear- 
ings,  in  feet,  and  W  weight  in  Ibs. 

Deflection  of  a  Cylindrical  Shaft  from   its  Weight  alone, 
when  Supported  at  Both  Knds. 

•007318  ^FC  =  D-     l  representing  length  in  feet,  d  diameter  in  ins.,  and  C  con- 
stant, ranging  from  475  to  550. 
The  greatest  admissible  deflection  for  any  diameter  is  .00167  —  =  D. 


Admissible   Distances   between    Bearings.     ^.9128  d  C  =  L 

Diam. 
of  Shaft. 

Disti 
Wrought 
Iron. 

ince. 
"    Steel. 

Diam. 
of  Shaft. 

Dist 

Wrought 
Iron. 

ance. 
Steel. 

Diam. 
of  Shaft. 

Disti 
Wrought 
Iron. 

ince. 
Steel. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

i 

2 

12.27 
15.46 

12.  6l 

15.84 

I 

20.99 
22.3 

21-57 
22.92 

9 

10 

25-53 
26.44 

26.24 
27.18 

3 

17.7 

18.19 

7 

23-48 

24.13 

ii 

27-3 

28.05 

4 

19.48 

20.02 

8 

24-55 

25.23 

12 

28.1 

28.88 

When  Ends  of  Shaft  are  rigidly  connected  at  Ends. 

Barlow  gives  D  —  .66  of  results  obtained  by  above  formula;  but  when  deflectior 
of  attached  length  is  considerable,  Navier  gives  D  =  .25  of  above. 


STRENGTH    OF    MATERIALS. DEFLECTION.  779 

Deflection   of  Mill   and    Factory   Shafts. 

19  W 

=  D.    I  representing  length  between  supports  in  ins.,  W  weight  at  middle 


6*rd*C 
infos.,d  diameter  of  shaft  in  ins.,  and  C  <w-  follows: 

Bessemer  steel , 3800000   |   Wrought  iron 3500000 

To    Compnte    Deflection   of  a   Cylindrical    Shaft. 

RULE.— Divide  square  of  three  times  length  in  feet  by  product  of  follow- 
ing Constants  and  square  of  diameter  in  ins,,  and  quotient  will  give  deflection. 

Cast  iron,  cylindrical 1500  |  Wrought  iron,  cylindrical 1980 

"       «*     square 2560!  *     square 3360 

EXAMPLE.— Length  of  a  cast-iron  cylindrical  shaft  is  30  feet,  and  ite  diameter  in 
centre  15  ins. ;  what  is  its  deflection? 

8100 

:  .024  ins. 


337500 

SPRINGS. 
Flexure  of  a  spring  is  proportional  to  its  load  and  to  cube  of  its  length, 

Deflection  of  a  Carriage  Spring. 

A  railway-carriage  spring,  consisting  of  10  plates  .3125  inch  thick,  and  a 
of  .375  inch,  length  2  feet  8  ins.,  width  3  ins.,  and  camber  or  spring  6  ins., 
deflected  as  follows,  without  any  permanent  set : 
.5  ton 5  inch.  I  1.5  tons 1.5  ins.  I  3  tons 3  ins, 

1          " I  "         |2  " 2  !«      J4       K     *     "   . 

Compression  of  an  India-rubber  Buffer  of  3  Ins.  Stroke. 

i    ton 1.3   ins.  I  2  tons. 2        ins.  I    5  tons 2.75  ina 

i-Stons. i-7S  "    |3    "  2.375    "   1 10    "  3 

GJ-eneral   Deductions. 

Defection  depends  essentially  upon  form  of  Girder,  Beam,  etc. 

A  continuous  weight,  equal  to  that  a  beam,  etc.,  is  suited  to  sustain,  will 
not  cause  deflection  of  it  to  increase  unless  it  is  subjected  to  considerable 
changes  of  temperature. 

Heaviest  load  on  a  railway  girder  should  not  exceed  .16  of  that  of  de- 
structive weight  of  girder  when  laid  on  at  rest. 

Semi-girders  or  Beams. — Deflection  of  a  beam,  etc.,  fixed  at  one  end  and 
loaded  at  other,  is  32  times  that  of  same  beam  supported  at  both  ends  and 
loaded  in  middle. 

Deflection  consequent  upon  Velocity  of  Load. — Deflection  is  very  much  in- 
creased by  instantaneous  loading;  by  some  authorities  it  is  estimated  to  be 
doubled. 

Momentum  of  a  railway  train  in  deflecting  girders,  etc.,  is  greater  than 
effect  from  dead  weight  of  it,  and  deflection  increases  with  velocity. 

When  motion  is  given  to  load  on  a  beam,  etc.,  point  of  greatest  deflection 
does  not  remain  in  centre  of  beam,  etc.,  as  beams  broken  by  a  travelling  load 
are  always  fractured  at  points  beyond  their  centres,  and  often  into  several 
pieces. 

Heaviest  running  weight  that  a  bridge  is  subjected  to  is  that  of  a  loco- 
motive and  tender,  which  is  equal  to  2  tons  per  lineal  foot. 

Girders  should  not,  under  any  circumstances,  be  deflected  to  exceed  .025 
inch  to  a  foot  in  length. 


STRENGTH    OF   MATERIALS. — DEFLECTION. 

A  carriage  was  moved  at  a  velocity  of  10  miles  per  hour ;  deflection  was 
3  inch,  and  when  at  a  velocity  of  30  miles  deflection  was  1.5  ins. 

In  this  case,  4150  Ibs.  would  have  been  destructive  weight  of  bars  if  ap- 
plied in  their  middle,  but  1778  Ibs.  would  have  broken  them  if  passed  over 
them  with  a  velocity  of  30  miles  per  hour. 

Relative  Elasticity  of  various  Materials.     (Trumbull) 

Ash 2.9  I  Cast  Iron i     I  Pine,  white 2.4  I  Pine, pitch. ...  2.9 

Beech 2.1  |  Elm  and  Oak  ..  2.9  |     "     yellow...  2.6  |  Wrought  Iron.    .86 

Cast  Iron. — Permanent  deflection  is  from  .33  to  .5  of  its  breaking  weight, 
and  deflection  should  never  exceed  .125  of  ultimate  deflection,  and  it  is  not 
permanently  affected  but  by  a  stress  approaching  its  destructive  weight. 

By  experiments  of  U.  S.  Ordnance  Corps  (Report,  1852),  set  or  permanent  deflec- 
tion was  .38  of  its  breaking  weight,  ultimate  deflection  .133  ins.  Deflection  for 
T^  of  span  =  .013,  or  .  i  of  ultimate  deflection. 

By  experiments  of  Mr.  Hodgkinson  (See  Rep.  of  Commas  on  Railway  Structures, 
London,  1849),  set  for  English  iron  bore  a  much  greater  proportion  to  its  breaking 
weight. 

A  beam,  etc.,  will  bend  to  .33  of  its  ultimate  deflection  with  less  than  .33 
of  its  breaking  weight,  if  it  is  laid  on  gradually,  and  but  .16  if  laid  on 
rapidly. 

Chilled  bars  deflect  more  readily  than  unchilled. 

Results  of"  Experiments  on.  the   Subjection  of"  Cast-iron 
Bars   to   continued.    Strains. 

(Rep.  of  Commas  on  Railway  Structures,  London,  1849.) 

Cast-iron  bars  subjected  to  a  regular  depression,  equal  to  deflection  due  to 
a  load  of  .33  of  their  statical  breaking  weight,  bore  10000  successive  de- 
pressions, and  when  broken  by  statical  weight,  gave  as  great  a  resistance  as 
like  bars  subjected  to  a  like  deflection  by  statical  weight. 

Of  two  bars  subjected  to  a  deflection  equal  to  that  carried  by  half  of  their 
statical  breaking  weight,  one  broke  with  28  602  depressions,  and  the  other 
bore  30  ooo,  and  did  not  appear  weakened  to  resist  statical  pressure. 

Hence,  Cast-iron  bars  will  not  bear  continual  applications  of  .33  of  their 
breaking  weight.  . 

Mr.  Tredgold,  in  his  experiments  upon  Cast  Iron,  has  shown  that  a  load  of  300 
Ibs.,  suspended  from  middle  of  a  bar  i  inch  square  and  34  ins.  between  its  sup- 
ports, gave  a  deflection  of  .16  of  an  inch,  while  elasticity  of  metal  remained  unim- 
paired. Hence  a  bar  i  inch  square  and  i  foot  in  length  will  sustain  850  Ibs.,  and 
retain  its  elasticity. 

Wrought  Iron. — All  rectangular  bars,  having  same  bearing,  length,  and 
loaded  in  their  centre  to  full  extent  of  their  elastic  power,  will  be  so  deflect- 
ed that  their  deflection,  being  multiplied  by  their  depth,  product  will  be  a 
constant  quantity,  whatever  may  be  their  breadth  or  other  dimensions,  pro- 
vided their  lengths  are  same. 

A  bar  of  Wrought  Iron,  2  ins.  square  and  9  feet  in  length  between  its  sup- 
ports, was  subjected  to  100000  vibratory  depressions,  each  equal  to  deflec- 
tion due  to  a  load  of  .55  of  that  which  permanently  injured  a  similar  bar, 
and  their  depressions  only  produced  a  permanent  set  of  .015  inch. 

Greatest  deflection  which  did  not  produce  any  permanent  set  was  due  to 
rather  more  than  .5  statical  weight,  which  permanently  injured  it. 

A  wrought-iron  box  girder,  6x6  ins.  and  9  feet  in  length,  was  subjected 
to  vibratory  depressions,  and  a  strain  corresponding  to  3762  Ibs.,  repeated 
43  37°  timea,  did  not  produce  any  appreciable  effect  on  the  rivets. 


STRENGTH   OF   MATERIALS. — DEFLECTION.          781 

Deflection  of  Solid  rolled  beams  compared  to  Riveted  beams  is  as  i  to  1.5. 

Wrought-iron  Girders  of  ordinary  construction  are  not  safe  when  sub- 
jected to  violent  impacts  or  disturbances,  with  a  load  equal  to  .33  of  their 
destructive  weight. 

Wood. — La  consequence  of  wood  not  being  subjected  to  weakening  by  the 
effect  of  impact,  a  factor  of  safety  of  5  for  single  pieces  is  held  to  be  suffi- 
cient, but  for  structures,  in  consequence  of  loss  of  strength  in  its  connections, 
a  factor  of  from  8  to  10  becomes  necessary. 

\Vorlring    Strengtli   or    Factors    of  Safety.* 

Elastic  strength  of  materials  is,  in  general  terms,  half  of  its  ultimate  de- 
structive or  breaking  strength.  If  a  working  load  of  .5  elastic  strength,  or 
.25  of  ultimate  strength,  be  accepted,  equal  range  for  fluctuation  within 
elastic  limit  is  provided.  But,  as  bodies  of  same  material  are  not  all  uni- 
form in  strength,  it  is  necessary  to  observe  a  lower  limit  than  .25  where 
material  is  exposed  to  great  or  to  sudden  variations  of  load  or  stress. 

Cast  Iron.— Mr.  Stoney  recommends  .25  of  ultimate  tensile  strength,  for 
dead  weights ;  .16  for  bridge  girders ;  and  .125  for  crane  posts  and  machin- 
ery. In  compression,  free  from  flexure,  cast  iron  will  bear  8  tons  (17920 
Ibs.)  per  sq.  inch ;  for  arches,  3  tons  (6720  Ibs.)  per  sq.  inch ;  for  pillars, 
supporting  dead  loads,  .16  of  ultimate  strength;  for  pillars  subject  to 
vibration  from  machinery,  .125 ;  and  for  pillars  subject  to  shocks  from 
heavy-loaded  wagons  and  like,  .1,  or  even  less,  where  strength  is  exerted  in 
resistance  to  flexure. 

Wrought  Iron. — For  bars  and  plates,  5  tons  (n  200  Ibs.)  per  sq.  inch  of 
net  section  is  taken  as  safe  working  tensile  stress ;  for  bar  iron  of  extra 
quality,  6  tons  (13440  Ibs.).  In  compression,  where  flexure  is  prevented, 
4  tons  (8960  Ibs.)  is  safe  limit ;  in  small  sizes,  3  tons  (6720  Ibs.).  For  col- 
umns subject  to  shocks,  Mr.  Stoney  allows  .16  of  calculated  breaking  weight ; 
with  quiescent  loads,  .25.  For  machinery,  .125  to  .1  is  usually  practised; 
and  for  steam-boilers,  .25  to  .125. 

Mr.  Roebling  claims  that  long  experience  has  proved,  beyond  shadow  of 
a  doubt,  that  good  iron,  exposed  to  a  tensile  strain  not  above  .2  of  its  ulti- 
mate strength,  and  not  subject  to  strong  vibration  or  torsion,  may  be  de- 
pended upon  for  a  thousand  years. 

Steel. — A  committee  of  British  Association  recommended  a  maximum 
working  tensile  stress  of  9  tons  (20 160  Ibs.)  per  sq.  inch.  Mr.  Stoney  rec- 
ommends, for  mild  steel^  .25  of  ultimate  strength,  or  8  tons  (17920  Ibs.)  per 
sq.  inch.  Limit  for  compression  must  be  regulated  very  much  by  nature  of 
steel,  and  whether  it  be  annealed  or  unannealed.  Probably  a  limit  of  9  tons 
(20 160  Ibs.)  per  sq.  inch,  same  as  limit  for  tension,  would  be  safe  max- 
imum for  general  purposes.  In  absence  of  experience,  Mr.  Stoney  further 
recommends  that,  for  steel  pillars,  an  addition  not  exceeding  50  per  cent, 
should  be  made  to  safe  load  for  wrought-iron  pillars  of  same  dimensions. 

Wood. — One  tenth  of  ultimate  stress  is  an*  accepted  limit.  Piles  have,  in 
some  situations,  borne  permanently  .2  of  their  ultimate  compressive  strength. 

Foundations. — According  to  Professor  Rankine,  maximum  pressure  on 
foundations  in  firm  earth  is  from  17  to  23  Ibs.  per  sq.  inch;  and,  on  rock,  it 
should  not  exceed  .125  of  its  crushing  load. 

Masonry.  —  Mr.  Stoney  asserts  that  working  load  on  rubble  masonry, 
brick-work,  or  concrete  rarely  exceeds  .16  of  crushing  weight  of  aggregate 
mass ;  and  that  this  seems  to  be  a  safe  limit.  In  an  arch,  calculated  pressure 
should  not  exceed  .05  of  crushing  pressure  of  stone. 

*  Essentially  from  Manual  of  D.  K.  Clark,  London,  1877. 


782  STRENGTH    OF   MATERIALS. — DETRUSIVE. 

Ropes. — For  round,  working  load  should  not  exceed  .14  of  ultimate  strength, 
and  for  flat  .11. 

Dead  Load.        Live  Load. 

Perfect  material  and  workmanship 2  4 

Dr.  Rankine  gives    (  Good  ordinary  material  (  w^d .  t30  s  to  jo 

following  factors :  )     and  workmanship . . .  J  Masonry  |  "y[-  V  [        4  8 

A  Dead  Load  is  one  that  is  laid  on  very  gradually  and  remains  fixed. 
A  Live  Load  is  one  that  is  laid  on  suddenly,  as  a  loaded  vehicle  or  train 
passing  swiftly  over  a  bridge. 

DETRUSIVE    OR    SHEARING    STRENGTH. 

Detrusive  or  Shearing  Strength  of  any  body  is  directly  as  its  strength, 
or  thickness,  or  area  of  shearing  surface. 

Results    of  Experiments    upon    Detrusive    Strength,   of 
JMetals   with    a    launch. 


METALS. 

Diameter 
of 
Punch. 

Thickness 
of 
Metal. 

Power 
exerted. 

Power  requ 
Surface  of  M 
Sq.I. 

ired  for  a 
etal  of  One 
ich. 

Brass  

Ins. 

i 

Ins. 

.045 

Lbs. 
5448 

Lbs. 
37000 

f  Si-0  2. 

30000 

^  -•  """^ 

•5 

.08 

3983 

30000 

S,H| 

Steel  

i 
.5 

•3 
.25 

21  250 
3472O 

90000 

CD   g>    1 

"    Bessemer.  

.875 

•75 

(  103  600 

51800 

•5 

i 

•17 

(184800 
II  950 
82870 

92400 
43900 

Pi 

2 

1.06 

297400 

44300 

??^ 

To  Compute    IPower  to   Punch,   Iron,  Brass,  or  Copper. 

RULE.— Multiply  product  of  diameter  of  punch  and  thickness  of  metal  by 
150000  if  for  wrought  iron,  by  128000  if  for  brass,  and  by  96000  if  for 
cast  iron  or  copper,  and  product  will  give  power  required,  in  Ibs. 

EXAMPLE.— What  power  is  required  to  punch  a  hole  .5  inch  in  diameter  in  a  plato 
of  brass  25  inch  thick?  >g  x  -25  x  I28ooo  =  l6ooo  lbs, 

Comparison,   "between   IDetrusive    and.   Transverse 

Strengths. 

Assuming  compression  and  abrasion  of  metal  in  application  of  a  punch  of 
one  inch  in  diameter  to  extend  to  .125  of  an  inch  beyond  diameter  of  punch, 
comparative  resistance  of  wrought  iron  to  detruswe  and  transverse  strain, 
latter  estimated  at  600  Ibs.  per  sq.  inch,  for  a  bar  i  foot  in  length,  is  as  3  to  i. 


WOODS. 
Detrusive    Strength   of*  Woods. 


Lbs. 

Spruce 470 

Pine,  white 490 


Lba. 

Pine,  pitch...  510 
Hemlock 540 


Ash 


Chestnut 690 


Per  Sq.  Inch. 

Lbs.  Lbs. 

650     Oak 780 

Locust 1180 


To  Compute  Length  of  Surface  of  Resistance  of  "Wood 
to  Horizontal  Thrust. 

RULE.— Divide  4  times  horizontal  thrust  in  Ibs.  by  product  of  breadth  of 
wood  in  ins.,  and  detrusive  resistance  per  sq.  inch  in  Ibs.  in  direction  of  fibre, 
and  quotient  will  give  length  required. 

EXAMPLE.— Thrust  of  a  rafter  is  5600  Ibs.,  breadth  of  tie  beam,  of  pitch  or  Georgia 
pine,  is  6  ins. ;  what  should  be  length  of  beyond  score  for  rafter  ? 

Assume  strength  510  as  above.       Then  ^ — ^-^  =  224°°  =  7.32  ins. 

^  6  X  510        3060 


STRENGTH    OF   MATERIALS. — DETRUSIVE. 


783 


Shearing. 
"Wrought    Iron. 

Resistance  to  shearing  of  American  is  about  75  per  cent.,  and  of  English 
80  per  cent.,  of  its  tensile  strength. 

Resistance  to  shearing  of  plates  and  bolts  is  not  in  a  direct  ratio.  It  ap- 
proximates to  that  of  square  of  depth  of  former,  and  to  square  of  diameter 
of  latter. 

Results    of*  Experiments    upon.    Shearing    Strength,   of 

Various    Metals    t>y   Parallel    Cutters. 

Wrought  Iron.  —Thickness  from  .5  to  i  inch,  50000  Ibs.  per  sq.  inch. 

Made  try  Inclined  Cutters,  angle  =  7°. 


PLATES.              ;    ; 

Thickness. 

Power. 

BOLTS. 

Diam. 

Power. 

Brass 

Ins. 
05 

Lbs. 
54O 

Brass  

Ins. 
I.  II 

Lbs. 
29700 

6 

•775 

ii  310 

Steel 

24. 

14  030 

Steel  

•775 

28  720 

Wrought  iron  ] 

•51 

39x5o 

Wrought  iron  i 

•32 

3093 

I 

44800 

\ 

1.142 

354"> 

Result  of  Experiments  in  Shearing,  made  at  tJ.  S.  Navy 
Yard,  Washington,  on   "Wrought-iron    Bolts. 


Diam. 

Minimum. 

Stress. 
Maximum. 

PerSq.Inch. 

Diam. 

Minimum. 

Stress. 
Maximum. 

PerSq.Inch. 

Inch. 
•5 
•75 

Lbs. 
8900 
18400 

Lbs. 
9400 
19650 

Lbs. 
44149 
39553 

Mean  4 

Inch. 
.875 

i 

L  033  Ibs. 

Lbs. 
25500 
32900 

Lbs. 
27600 
35800 

Lbs. 
41503 
40708 

Result   of  Experiments   on    .87"£5    Inch   "Wrought-iron. 
Bolts.    (E.  Clark.) 


Lbs. 

Single  shear 54096 

Double   "    46904 


Tons. 
24.15 


Double  shear  of  two  .625-inch  plates 
riveted  together  (one  section) ....  45  696 


Tensile  strength 50 176  Ibs. 

Riveted   Joints. 

Experiments  on  strength  of  riveted  joints  showed  that  while  the  plates 
were  destroyed  with  a  stress  of  43  546  Ibs.,  the  rivets  were  strained  by  a 
stress  of  39088  Ibs. 

Cast   Iron. 

Resistance  to  shearing  is  very  nearly  equal  to  its  tensile  strength.  An 
average  of  English  being  24000  Ibs.  per  sq.  inch. 

Steel. 

Shearing  strength  of  steel  of  all  kinds  (including  Fagersta)  is  about  72  per 
cent,  of  its  tensile  strength. 

Treenails. 

Oak  treenails,  i  to  1.75  ins.  in  diameter,  have  an  average  shearing  strength 
of  1.8  tons  per  sq.  inch,  and  in  order  to  fully  develop  their  strength,  the  planks 
into  which  they  are  driven  should  be  3  times  their  diameter. 

Woods. 

When  a  beam  or  any  piece  of  wood  is  let  in  (pot  mortised)  at  an  inclina- 
tion to  another  piece,  so  that  thrust  will  bear  in  direction  of  fibres  of  beam 
that  is  cut,  depth  of  cut  at  right  angles  to  fbres  should  not  be  more  than  .a 
of  length  of  piece,  fibres  of  which,  by  their  cohesion,  resist  thrust. 


784  STRENGTH    OF   MATERIALS.  —  PENSILE. 

TENSILE    STRENGTH. 

Tensile  Strength  is  resistance  of  the  fibres  or  particles  of  a  body  to 
separation.  It  is  therefore  proportional  to  their  number,  or  to  area  of 
its  transverse  section,  and  in  metals  it  varies  with  their  temperature, 
generally  decreasing  as  temperature  is  increased.  In  silver,  tenacity 
decreases  more  rapidly  than  temperature  ;  and  in  copper,  gold,  and  plat- 
inum less  rapidly. 

Cast   Iron- 

Experiments  on  Cast-iron  bars  give  a  tensile  strength  of  from  4000  to 
5000  Ibs.  per  sq.  inch  of  its  section,  as  just  sufficient  to  balance  elasticity  of 
the  metal;  and  as  a  bar  of  it  is  extended  the  12300^1  part  of  its  length  for 
every  1000  Ibs.  of  direct  strain,  or  one  sixteenth  of  an  inch  in  64.06  feet  per 
sq.  inch  of  its  section,  it  is  deduced  that  its  elasticity  is  fully  excited  when 
it  is  extended  less  than  the  24<Doth  part  of  its  length,  and  extension  of  it  at 
its  limit  of  elasticity,  which  is  about  .5  of  its  destructive  weight,  is  esti- 
mated at  isooth  part  of  its  length. 

Average  ultimate  extension  is  5ooth  part  of  its  length. 

A  bar  will  contract  or  expand  .000006173  inch,  or  the  i62oooth  of  its 
length,  for  each  degree  of  heat  ;  and  assuming  extreme  of  moderate  range 
of  temperature  in  this  country  140°  (—  20°  -f  120°),  it  will  contract  or  ex- 
pand with  this  change  .0008642  inch,  or  the  usyth  part  of  its  length. 


It  follows,  then,  that  as  1000  Ibs.  will  extend  a  bar  the  i23ooth  part  of 
its  length,  contraction  or  extension  for  iisyth  part  will  be  equivalent  to  a 
force  of  10648  Ibs.  (4.75  tons)  per  sq.  inch  of  section.  It  shrinks  in  cooling 
from  one  eighty-fifth  to  one  ninety-eighth  of  its  length. 

Mean  tensile  strength  of  American,  as  determined  by  Maj.  Wade  for  U.  S. 
Ordnance  Corps,  is  31  829  Ibs.  (14.21  tons)  per  sq.  inch  of  section;  mean  of 
English,  as  determined  by  Mr.  E.  Hodgkinson  for  Commission  on  Applica- 
tion of  Iron  to  Railway  Structures,  1849,  is  19484  Ibs.  (8.7  tons)  ;  and  by  Col, 
Wilmot,  at  Woolwich,  in  1858,  for  gun-metal,  is  23  257  Ibs.  (10.35  tons). 
varying  from  12320  Ibs.  (5.5  tons)  to  25  520  Ibs.  (10.5  tons). 

Mean  ultimate  extension  of  four  descriptions  of  English,  as  determined 
for  Commission  above  referred  to,  was,  for  lengths  of  10  feet,  .1997  inch, 
being  6ooth  part  of  its  length  ;  and  this  weight  would  compress  a  bar  the 
775th  part  of  its  length. 

Tensile  strength  of  strongest  piece  ever  tested  —  45970  Ibs.  (20.52  tons). 
This  was  a  mixture  of  grades  i,  2,  and  3  from  furnace  of  Robert  P.  Parrott 
at  Greenwood,  N.  Y.,  and  at  3d  fusion. 

At  2.5  tons  per  sq.  inch  it  will  extend  same  as  wrought  iron  at  5.6  tons. 
From  experiments  of  Maj.  Wade  he  deduced  the  following  mean  results  : 


Density. 
7.225 


Tensile.        I      Transverse.      I        Torsion.        I       Crushing.       I        Hardness. 
31829         |  8182  8614  I        144916  22.34 


Tensile  per  sq.  inch  of  section ;  Transverse  per  sq.  inch,  one  end  fixed, 
load  applied  at  other  end  at  a  distance  of  i  foot ;  and  Torsion  per  sq.  inch, 
stress  applied  at  end  of  a  lever  i  foot  in  length. 

Green  sand  castings  are  6  per  cent,  stronger  than  dry,  and  30  per  cent, 
stronger  than  chilled ;  but  when  castings  are  chilled  and  annealed,  a  gain  of 
115  per  cent,  is  attained  over  those  made  in  green  sand. 

Resistance  to  crushing  and  tensile  stress  is  for  American  as  4.55  to  i,  and 
for  English  as  5.6  to  7  to  i.  Strength  increasing  with  density. 


STRENGTH    OF    MATERIALS. TENSILE.  78$ 

Remelting. — Strength,  as  well  as  density,  are  increased  by  repeated  re- 
meltings.  The  increase  is  the  result  of  the  gradual  abstraction  of  the  con- 
stituent carbon  of  the  iron,  and  the  consequent  approximation  of  the  metal 
to  wrought  iron. 

Result  of  the  4th  melting  of  pig  iron,  as  determined  by  Major  Wade,  was 
to  increase  its  strength  from  12880  Ibs.  (5.75  tons)  to  27888  Ibs.  (12.45 
tons),  and  its  specific  gravity  from  6.9  to  7.4. 

Three  successive  meltings  of  Greenwood  iron,  N.  Y.,  gave  tensile  strength 
of  21  300,  30  100,  and  35  700  Ibs. 

Result  of  5th  melting  by  Mr.  Bramwell  was  to  increase  strength  oOva4ian 
iron  from  16  800  Ibs.  (7.5  tons)  to  41  440  Ibs.  (18.5  tons). 

Remelting  increases  its  resistance  to  a  crushing  stress  from  70  to  &*>  tons 
(14  per  cent.)  per  sq.  inch  of  section. 

Hot   and    Cold   Blast. 

Mr.  Hodgkinson  deduced  from  experiments  that  relative  strength  o*  1.2 
and  3  ins.  square  was  as  100,  80,  and  77,  and  that  hot  blast  had  less  tensile 
strength  than  cold  blast,  but  greater  resistance  to  a  crushing  stress. 

Captain  James  ascertained  that  tensile  strength  of  .75  inch  bars,  cut  out 
of  2  and  3  inch  bars,  had  only  half  strength  of  a  bar  cast  i  inch  square. 

Mr.  Robert  Stephenson  concluded,  from  experiments  of  recent  date,  that 
average  strength  of  hot  blast  was  not  much  less  than  that  of  cold  blast ;  but 
that  cold  blast,  or  mixtures  of  cold  blast,  were  more  regular,  and  that  mixt- 
ures of  cold  blast  and  hot  blast  were  better  than  either  separate. 

Stirling's    M.ixed   or  Toxighened   Iron. 

By  mixture  of  a  portion  of  malleable  iron  with  cast  iron,  carefully  fused 
in  a  crucible,  a  tensile  strain  of  25  764  Ibs.  has  been  attained.  This  mixt- 
ure, when  judiciously  managed  and  duly  proportioned,  increases  resistance 
of  cast  iron  about  one  third ;  greatest  effect  being  obtained  witfc  a  propor- 
tion of  about  30  per  cent,  of  malleable  Iron. 

Malleable    Cast   Iron. 

Tensile  strength  of  annealed  malleable  is  guaranteed  by  some  Manufact- 
urers of  it  at  56000  Ibs. ;  it  is  capable  of  sustaining  22400  Ibs.  without  per- 
manent set. 

"Wrought   Iron. 

Experiments  on  English  bars  gave  a  tensile  strength  of  from  22000  Ibs. 
to  26  400  Ibs.  per  sq.  inch  of  its  section,  as  just  sufficient  to  balance  elasticity 
of  the  metal;  and  as  a  bar  of  it  is  extended  the  28oooth  part  of  its  length 
for  everv  1000  Ibs.  of  direct  strain,  or  one  sixteenth  of  an  inch  in  116.66  teet 
per  sq.  inch  of  its  section,  it  is  deduced  that  its  elasticity  is  fully  excited 
when  it  is  extended  the  loooth  part  of  its  length,  and  extension  of  it  at  its 
limit  of  elasticity,  which  is  from  .45  to  .5  of  its  destructive  weight,  is  esti- 
mated at  i52oth"part  of  its  length. 

A  bar  will  expand  or  contract  .000006614  inch,  or  151  200  part  of  its  length 
for  each  degree  of  heat;  and  assuming,  as  before  stated  for  cast  iron,  that 
extreme  range  of  temperature  in  air  in  this  country  is  140°,  it  will  contract 
or  expand  with  this  change  .000926,  or  io8oth  of  its  length,  which  is  equiva- 
lent to  a  force  of  20  740  Ibs.  (9.25  tons)  per  sq.  inch  of  section. 

Mean  tensile  strength  of  American  bars  and  plates  (45000  to  76000), 
60  500  Ibs.  (27  tons)  per  sq.  inch  of  section ;  as  determined  by  Prof.  Johnson 
in  1836,  is  55  900  Ibs. ;  and  mean  of  English,  as  determined  by  Capt.  Brown, 
Barlow,  Brunei,  and  Fairbairn,  is  53  900  Ibs. ;  and  by  Mr.  Kirkaldy,  bars  and 
plates  (47040  to  55910)  51 475  Ibs.  (22.97  tons). 

a  U* 


;86 


STRENGTH    OF   MATERIALS. — TENSILE. 


Greatest  strength  observed  73449  Ibs.  (32-79  tons). 

Ultimate  strength,  as  given  by  Mr.  D.  K.  Clark,  59  732  Ibs.  (26.66  tons). 

Average  ultimate  extension  is  6ooth  part  of  its  length. 

Strength  of  plates,  as  determined  by  Sir  William  Fairbairn,  is  fully  9  per 
cent,  greater  with  fibre  than  across  it. 

Resistance  of  wrought  iron  to  crushing  and  tensile  strains  is,  as  a  mean, 
as  1.5  to  i  for  American;  and  for  English  1.2  to  i. 

Reheating.— Experiments  to  determine  results  from  repeated  heating  and 
laminating,  furnished  following : 

From  i  to  6  reheatings  and  rollings,  tensile  stress  increased  from  43904 
Ibs.  to  61  824  Ibs.,  and  from  6  to  12  it  was  reduced  to  43904  again. 

Effect  of  Temperature.— Tensile  strength  at  different  temperatures  is  as 
follows:  60°,  i;  114°,  1.14;  212°,  1.2;  250°,  1.32;  270°,  1.35;  325°,  1.41; 
435°,  1-4- 

Experiments  of  Franklin  Institute  gave  at 

80° 56000  Ibs.  I    720° 55  ooo  Ibs.  I  1240° 22000  Ibs. 

570° 66500   "     |  1050° 32000   "    |  1317° 9000   " 

Annealing.— Tensile  strength  is  reduced  fully  i  ton  per  sq.  inch  by  an- 
nealing. 

Cold  Rolling. — Bars  are  materially  stronger  than  when  hot  rolled,  strength 
being  increased  from  one  fifth  to  one  half,  and  elongation  reduced  from  21 
to  8  per  cent. 

Hammering  increases  strength  in  some  cases  to  one  fifth. 

Welding. — Strength  is  reduced  from  a  range  of  3  to  44  per  cent.  20  per 
cent.,  or  one  fifth,  is  held  to  be  a  fair  mean. 

Temperature. — From  o°  to  400°  strength  is  not  essentially  affected,  but  at 
high  temperature  it  is  reduced.  When  heated  to  redness  its  strength  is  re- 
duced fully  25  per  cent. 

Tensile  strength  at  23°  was  found  to  be  .024  per  cent,  less  than  at  64°, 

Cutting  Screw  Threads  reduces  strength  from  n  to  33  per  cent. 

Hardening  in  water,  oil,  etc.,  reduces  elongation,  but  does  not  essentially 
increase  the  strength. 

Case  Hardening  reduces  strength  fully  10  per  cent. 

Galvanizing  does  not  affect  strength  of  plates. 

Angled  Bars,  etc. — Their  strength  is  fully  10  per  cent,  less  than  for  bolts 
and  plates. 

Elements  connected  with  Tensile  Resistance  of  various 
Substances. 


af| 

«  M 

If  £ 

gf£ 

1? 

If  rf 

SUBSTANCES. 

^1 

*$$ 

SUBSTANCES. 

S-g^ 

25| 

•*~ 
NS 

w.S0 

||l 

as 

li 

ii1 

Lbs. 

Lbs. 

Beech  

3  355 

.•5 

Wrought  iron  ordinary. 

Cast  iron  English 

•  J 

"            "     Swedish  .... 

' 

,3 

**       "     American  
Oak  

5000 
2856 

.2 

"           "     English  

24  400 
(18850 

•34 
•35 

Steel  plates,  .  5  inch  

52000 

'.11 

"            "      American.  .  . 

15000 

.26 

75  700 

' 

"wire,  No.  9,  unannealed 

Yellow  pine.  .  . 

"    "        "     annealed  .  . 

47  53Z 
ID  1OO 

!'*< 

Turning. — Removing  outer  surface  does  not  reduce  the  strength  of  bolts. 


STRENGTH    OF   MATERIALS. — TENSILE. 


787 


TIE-BODS. 

Results  of  Experiments  on.  Tensile  Strength.  of  Wrough.t~ 
iron    Tie-rods. 


Common  English  Iron,  1. 1875  Ins.  in  Diameter, 

DESCRIPTION  OF  CONNECTION. 


I  Breaking  Weight. 


Semicircular  hook  fitted  to  a  circular  and  welded  eye 

Two  semicircular  hooks  hooked  together 

Right-angled  hook  or  goose-neck  fitted  into  a  cylindrical  eye 

Two  links  or  welded  eyes  connected  together 

Straight  rod  without  any  connective  articulation 


Lbs. 
14000 
16  220 
29120 
48160 
56000 


Ratio   of  Ductility   and.   IMallea'bility-  of  Metals. 


In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

Gold. 
Silver. 
Platinum. 

Iron. 
Copper. 
Zinc. 

Tin. 
Lead. 
Nickel. 

Gold. 
Silver. 
Copper. 

Tin. 
Platinum. 
Lead. 

Zinc. 
Iron. 
Nickel. 

Relative  resistance  of  Wrought  Iron  and  Copper  to  tension  and  compres- 
sion is  as  100  to  54.5. 

Steel. 

Experiments  of  Mr.  Kirkaldy,  1858-61,  give  an  average  tensile  strength 
for  bars  of  134  400  Ibs.  (60  tons)  per  sq.  inch  for  tool-steel,  and  62  720  Ibs. 
(28  tons)  for  puddled.  Greatest  observed  strength  being  148  288  Ibs.  (66.2 
tons).  Plates,  mean,  86800  Ibs.  (32  to  45.5  tons)  with  fibre,  and  81  760  Ibs. 
(36.5  tons)  across  it. 

Its  resistance  to  crushing  compared  to  tension  is  as  2.1  to  i. 

Hardening. — Its  strength  is  very  materially  increased  by  being  cooled  in 
oil,  ranging  from  12  to  55  per  cent. 

Crucible. — Experiments  by  the  Steel  Committee  of  Society  of  Civil 
Engineers,  England,  1868-70,  give  a  tensile  strength  of  91  571  Ibs.  per  sq. 
inch  (40.88  tons),  with  an  elongation  of  .163  per  cent.,  or  i  part  in  613,  and 
an  elastic  extension  of  .000034  7th  part  for  every  1000  Ibs.  per  sq.  inch,  or 
i  part  in  28818. 

Bessemer.— Experiments  by  same  Committee  give  a  tensile  strength 
of  76653  Ibs.  per  sq.  inch  (34.22  tons)  with  an  elongation  of  .144  per  cent., 
or  i  part  in  695,  and  an  elastic  extension  of  .oooo3482d  part  for  every  IOOG 
Ibs.  per  sq.  inch,  or  i  part  in  28  719. 

Result  of  Experiments  by  Committe  of  Society  of  Civil 
Engineers  of  England,  1868-T'O,  and  Mr.  Daniel  Kir- 
Isaldy,  IS-rS. 

Per  Sq.  Inch. 


STIEL. 

Elastic  Strength. 

Elastic  £ 
in  Parts 
of 
Length. 

xtension 

per 

1000  Lbs. 

Ratio  of 
Elastic 
to 
Ultimate 
Strength. 

Destrnctire 
Weight. 

Crucible  

Lbs. 

49840 
44800 
48608 
39200 
32080 
28784 

Tons. 
22.25 
20 
21.7 

'7-5 
14.56 
12.85 

Per  Cent. 
.225 
.204 

In  Length. 
.0005 
.00045 

Per  Cent. 
58.2 
59 
59-2 
5i-5 
46.4 

44-4 

Lbs. 

86464 

75757 
78176 
72576 
69888 
64512 

Tom. 
38.6 
33-82 
34-9 
32.4 
31.2 
28.8 

Bessemer  

Fagersta,  unannealed  . 
"       annealed.... 
Siemens,  unannealed.  . 
"       annealed.... 

788 


STRENGTH    OF   MATERIALS. TENSILE. 


Average   Tensile    Elasticity   of   Steel   Bars    and.   Plates. 

(jCom.  of  Civil  Engineers,  1870. ) 


DESCRIPTION. 

Elasticity  per 
Sq.  Inch. 

Elastic  Exten- 
sion in  Parts  of 
Length. 

Ratio  of  Elas- 
tic to  Destruc- 
tive Strength. 

Bars. 

Lbs. 
5°  557 

Parts. 

i  in  485 

Per  Cent. 
58.2 

43814 

i  in  675 

55 

56  560 

64.8 

34048 

55-6 

"         hciiiitrH'rcd  and  roll6d  

55  574 

64.7 

*'                  "           *4    annealed 

AO8t:8 

54 

"         plates  unannealed-        

•ao  710 

i  in  980 

SO-  2 

26  O4O 

i  in  1020 

56.5 

32  500 

46.4 

28780 

44.4 

40  174 

58.8 

Kruno's  shaft.  .. 

4.2  112 

i  in  181; 

Tensile  strength  of  steel  increases  by  reheating  and  rolling  up  to  second 
operation,  but  decreases  after  that. 

Tensile  Strength,  of  Various  Materials,  deduced  from 
Experiments  of  TJ.  S.  Ordnance  department,  ITair- 
'bairn,  liodgkinson,  KLirkaldy,  and  Iby  the  Author. 

Power  or  Weight  required  to  tear  asunder  One  Sq.  Inch,  in  Lbs. 


METALS.  Lbs. 

Antimony,  cast 1 053 

Bismuth,  cast 3  248 

Cast  Iron,  Greenwood 45  970 

"         mean,  Major  Wade. . .  31  829 

"          gun- metal,  mean 37232 

"         malleable,  annealed..  56000 

"         Eng. ,  strong 29  ooo 

"            "     weak 13400 


•    \       21  280 

"     gun-metal 23  257 

"            *'     mean* 19484 

"            «*     Low  Moor,  No.  2  14076 

"            "     Clyde,  No.  i....  16125 

«*             "          "      No.  3 —  23468 

"            "     Stirling,  mean..  25764 

Copper,  wrought 34  ooo 

"        rolled. 36  ooo 

"       cast 24250 

"       bolt 36800 

"       wire 61200 

Gold 20  384 

Lead,  cast 1 800 

"     pipe ••••  2240 

"       "    encased 3759 

"     rolled  sheet 3320 

Platinum  wire 53000 

Silver,  cast 40000 

Steel,  cast,  maximum 142  ooo 

"        **    mean 88560 

41     puddled,  maximum 173  817 

"     Amer.  Tool  Co 179  980 

(I        wi-e  f    210000 

^ \    300000 

1     plates,  lengthwise 96  300 

u      crosswise ....    93  700 

"     Chrome  bar 180  ooo 


METALS.  Lbs. 

Steel,  Pittsburgh,  mean 94  450 

«     Bessemer,  rolled [-'#*£ 

"  "         hammered.. 

Eng.,  cast 

"    plates,  mean. 

plates 

puddled  plates 


152900 
134000 
93500 
86800 
62720 


crucible 91570 

homogeneous 96  280 

blistered,  bars 104  ooo 

Fagersta  bars 

"        plates... 

Whitworth's 


"      Siemens's  plates. . 

"        "      Krupp's  shaft 

Tin,  cast 

"    Banca 2100 

Wire  rope,  per  Ib.  w't  per  fathom  4  480 

"       "     galvanized  steel,    "  6720 

Wrought  Iron,  boiler  plates. . .  {  ^  500 

rivets 63  ooo 

bolts,  mean 60  500 

"     inferior 30000 

hammered 54  ooo 

shaft 44  750 

wire 73  600 

'    No.  9 100000 

u    No.  20 120000 

"    diam.  .0069  inch  301 168 

"   galv'ized.osS  "  64960 

Eng.,  heavy  forging.  33600 

a  plates,  lengthw'e  53800 
"       4<      crosswise   48800 


By  Comru's  on  application  of  Iron  to  Railway  Structure. 


STRENGTH    OF 

METALS. 
Wrought  Iron,  Eng.,  mean  

MAT 

Lba. 

51000 
57600 
48800 
65920 
40000 

3130° 
56000 
63000 
51760 
595oo 
49000 
72000 
48900 

7000 
16000 

85120 

IIOOO 

71600 

96320 

3670 
18000 

49000 

5091  5 
34464 
23500 
33000 
38080 
36000 
29000 
18000 
33600 
42040 
7  ooo 
48700 

14000 
9500 
16000 
6300 
14000 
ii  500 
15000 
7000 
23000 
19000 
ii  400 
7500 
ii  600 
12500 

10000 

6000 
12400 
27000 
6000 
13000 
18000 

15860 

12  000 
II  OOO 

16000 

17350 

23000 

ERIALS.  —  TENSILE. 

WOODS. 
Larch  | 

789 

LbB. 

4200 
9500 
11800 
16000 
20500 

21  OOO 
8000 
I2OOO 
20333 
2522Z 
16500 
16380 
IO25O 
9500 
4500 

4200 
9860 
19200 
I4OOO 
13000 
IlSoO 
13000 
13300 
7OOO 
10833 
IO29O 
12400 
9600 
13000 
15000 
21000 
7800 

££ 

13000 
8000 

2300 
550 

1469 
300 
500 

77 
750 

100 

290 

860* 

393 
7i3 
948 
1152 

201 

319 
310 
214 

284 
104 
102 

700 

Lignum  vitas  

ti          «    Thames  

Locust 

"          "    armor-plates  .... 

Mahogany,  Honduras  

"    bar  { 
"      "  charcoal  
"          "    rivet,  scrap  
41       Russian,  bar,  best  

"          Spanish  { 

Oak,  Pa.  seasoned  

"     Va.,        "      

"     white  

"       Swedish,   "    best.... 

u                u             u 

u     live,  Ala.  

"     red  

"     African  

"    sheet  { 

"     English  { 

ALLOYS  OR  COMPOSITIONS. 
Alloy,  Cop.  60,  Iron  2,  Zinc  35,  Tin  2  . 

"     Dantzic  

Pear  

Pine,  Va  

"     Riga  

Aluminium  Cop  90  

'  '     yellow.  

"          maximum  

"     white  

Bell-  metal 

"     red  

Brass  cast     

Poon  

"     wire 

Poplar  

Bronze  Phosphor   extreme  

Redwood,  Cal  

"           mean..;  

Spruce,  white...                      ..  j 

"       ordinary  
Cop.  10,  Tin  i  

I 
Sycamore  \ 

«  f  «'•'  ;•'•-:•::::: 

Teak,  India  

*'      2,  Zinc  i  

Gun-metal  ordinary 

Walnut,  Eng  

"         mean      

'  '         bars  

"        Mich  

Speculum  metal      .              .... 

Willow  

Yellow  metal  

Yew  

WOODS. 
Ash,  white  

Across  Fibre. 
Oak   

Pine 

MISCELLANEOUS. 
Basalt,  Scotch  

"    English  

Bamboo  

Bay 

Beton,  N.  Y.  Stone  Con'g  Co  { 

Beech  English  

Birch  

"     Amer    black  

Brick  extreme  

"      inferior  < 

Bullet  
Cedar,  Lebanon  

Cement  Portland  7  days  | 

"     West  Indian  
"     American  

u                   pure,  i  mo  

Chestnut 

sand  2,  320  days.  . 

pure,         " 
"                   sand  i,  in  water  ) 
i  mo.  j 

"                      "    3,  i  year.... 
;<    5,i     "    
"                        "    7,1     "    ..-. 

Cypress       

Ebony               

Elm     { 

"     Alabama     

Hickory  

"        Rosedale,  Ulst.  Co.  ,  7  days 
"              "        sand  i,  30    " 
(i              it                                ( 

Holly        

1 

I 

790 


STRENGTH   OF  MATERIALS. — TORSION. 


MISCELLANEOUS. 
Cement,  Roman,  in  water  7  days 

Lbs. 
90 

MISCELLANEOUS.             Lbs. 

"              "       i  mo.. 
"             "       i  year 

"5 

286 

"      hydraulic  ..,               ..  f        8s 

"       sand  i,  42  days. 

284 

"^UI                           *'  t      130 

"           u    3       " 

199 

IDO 

Flax                                 

25  ooo 

Glue 

4OOO 

Granite                •  

578 

"        fine  green  1260 

3  5°° 

((               Ai-KrrtotVi                                                  5^3 

12000 

Arbroatn  1261 

16000 

"       Caithness  x^3 

Leather  belting  

33° 

**            PnrflanH                                             "57 

i 
Limestone  \ 

070 

rtland  looo 

( 

Marble  statuary             •       

Silk  fibre  52  ooo 

"    'Italian  

5200 

Slate..,                             9£°° 

Marble,  white  
"      Irish... 

9000 
17600 

Whalebone  7  ooo 

TOKSIONAL    STRENGTH. 
SHAFTS  AND  GUDGEONS. 

Shafts  are  divided  into  Shafts  and  Spindles,  according  to  their  mag- 
nitude, and  are  subjected  to  Torsion  and  Lateral  Stress  combined,  or  to 
Lateral  Stress  alone. 

A  Gudgeon  is  the  metal  journal  or  Arbor  upon  which  a  wooden  shaft 
revolves. 

Lateral  Stiffness  and  Strength. — Shafts  of  equal  length  have  lateral  stiff- 
ness as  their  breadth  and  cube  of  their  depth,  and  have  lateral  strength  as 
their  breadth  and  square  of  their  depths. 

Shafts  of  different  lengths  have  lateral  stiffness  directly  as  their  breadth 
and  cube  of  their  depth,  and  inversely  as  cube  of  their  length ;  and  have 
lateral  strength  directly  as  their  breadth  and  as  square  of  their  depth,  and 
inversely  as  their  length. 

Hollow  Shafts  having  equal  lengths  and  equal  quantities  of  material  have 
lateral  stiffness  as  square  of  their  diameter,  and  have  lateral  strength  as  their 
diameters.  Hence,  in  hollow  shafts,  one  having  twice  the  diameter  of  an- 
other will  have  four  times  the  stiffness,  and  but  double  the  strength ;  and 
when  having  equal  lengths,  by  an  increase  in  diameter  they  increase  in  stiff- 
ness in  a  greater  proportion  than  in  strength. 

When  a  solid  shaft  is  subjected  to  torsional  stress,  its  centre  is  a  neutral 
axis,  about  which  both  intensity  and  leverage  of  resistance  increase  as  radius 
or  side ;  and  the  two  in  combination,  or  moment  of  resistance  per  sq.  inch, 
increase  as  square  of  radius  or  side. 

Round  Shaft. — Radius  of  ring  of  resistance  is  radius  of  gyration  of  sec- 
tion, being  alike  to  that  of  a  circular  plate  revolving  on  its  axis,  viz.,  .7071 
radius.  The  ultimate  moment  of  resistance  then  is  expressed  by  product 
of  sectional  area  of  shaft,  by  ultimate  shearing  resistance  per  sq.  inch  of 
material  by  radius,  and  by  .7071. 

Or,  .7854  d2  r  S  X  .7071  =  .278  d*  S  =  R  W.      (D.  K.  Clark.) 

d  representing  diameter  of  shaft  and  r  radius,  S  ultimate  shearing  stress  of  mate- 
rial in  Ibs.  per  sq.  inch,  R  radius  through  which  stress  is  applied,  in  in*.,  and  W 
moment  of  load  or  destructive  stress,  in  Ibs. 


STKENGTH   OF   MATERIALS. — TORSION.  791 

Round  Shaft. — Strength,  compared  to  a  square  of  equal  sectional  area, 
is  about  as  i  to  .85.  Diameter  of  a  round  section,  compared  to  side  of 
square  section  of  equal  resistance,  is  as  i  to  .96. 

Square  Shaft. — Moment  of  torsional  resistance  of  a  square  shaft  exceeds 
that  of  a  round  of  same  sectional  area,  in  consequence  of  projection  of  cor- 
ners of  square ;  but  inasmuch  as  material  is  less  disposed  to  resist  torsional 
stress,  the  resistance  of  a  square  shaft,  compared  to  a  round  one  of  like  area 
of  section,  is  as  i  to  1.18,  and  of  like  side  and  diameter,  as  1.08  to  i. 

Hence,  •!l*8x*ol"'8  =  W.     Hott™  Round  ShajU.    -^  (d«-d'«)  S  =  w 

When  Section  is  comparatively  Thin.         7    =  W.     *  representing  side, 

d  and  d'  external  and  internal  diameters,  and  t  thickness  of  metal  in  ins. 

Torsional  Angle  of  a  bar,  etc.,  under  equal  stress,  will  vary  as  its  length. 
Hence,  torsional  strength  of  bars  of  like  diameters  is  inversely  as  their 
lengths. 

Stress  upon  a  shaft  from  a  weight  upon  it  is  proportional  to  product  of  the  parts 
of  shaft  multiplied  into  each  other.  Thus,  if  a  shaft  is  10  feet  in  length,  and  a  weight 
upon  centre  of  gravity  of  the  stress  is  at  a  point  2  feet  from  one  end,  the  parts  2 
and  8,  multiplied  together,  are  equal  to  16 ;  but  if  weight  or  stress  were  applied  in 
middle  of  the  shaft,  parts  5  and  5,  multiplied  together,  would  produce  25. 

When  load  upon  a  shaft  is  uniformly  distributed  over  any  part  of  it,  it  is  consid- 
ered as  united  in  middle  of  that  part;  and  if  load  is  not  uniformly  distributed,  it  is 
considered  as  united  at  its  centre  of  gravity. 

Deflection  of  a  shaft  produced  by  a  load  which  is  uniformly  distributed  over  its 
length  is  same  as  when  .625  of  load  is  applied  at  middle  of  its  length. 

Resistance  of  body  of  a  shaft  to  lateral  stress  is  as  its  breadth  and  square 
of  its  depth ;  hence  diameter  will  be  as  product  of  length,  of  it,  and  length 
of  it  on  one  side  of  a  given  point,  less  square  of  that  length. 

ILLUSTRATION. — Length  of  a  shaft  between  centres  of  its  journals  is  10  feet;  what 
should  be  relative  cubes  of  its  diameters  when  load  is  applied  at  i,  2,  and  5  feet 
from  one  end?  and  what  when  load  is  uniformly  distributed  over  length  of  it? 
I  x  I1  —  I*=d3;  and  when  uniformly  distributed,  d3^-2  =  d1. 

ioX  1  =  10—  i*  =  g=zcube  of  diameter  at  i  foot;  ioX  2  =  20— 22  =  i6  =  cufc« 
of  diameter  at  2  feet ;  10  X  5  =  50  —  s2  =  25  =  cube  of  diameter  at  5  feet. 

When  a  load  is  uniformly  distributed,  stress  is  greatest  at  middle  of  length,  and 
is  equal  to  half  of  it ;  25  -r-  2  =  12. 5  =  cube  of  diameter  at  5  feet. 

Torsional  Strength  of  any  square  bar  or  beam  is  as  cube  of  its  side,  and 
of  a  cylinder  as  cube  of  its  diameter.  Hollow  cylinders  or  shafts  have  great- 
er torsional  strength  than  solid  ones  containing  same  volume  of  material. 

To   Compute    IDiameter    of*  a    Solid    Shaft    of  Cast    or 
"Wrought    Iron    to    Resist    Lateral    Stress    alone. 

When  Stress  is  in  or  near  Middle.  RULE.— Multiply  weight  by  length  of 
shaft  in  feet ;  divide  product  by  500  for  cast  iron  and  560  for  wrought  iron, 
and  cube  root  of  quotient  will  give  diameter  in  ins. 

EXAMPLE. — Weight  of  a  water-wheel  upon  a  cast-iron  shaft  is  50000  Ibs.,  its  length 
30  feet,  and  centre  of  stress  of  wheel  7  feet  from  one  end;  what  should  be  diameter 
of  its  body? 

3/  /50QOO  X  3<A  _  I4  42  inf ?  if  wigM  was  in  middle  of  its  length. 

Hence  diameter  at  7  feet  from  one  end  will  be,  as  by  preceding  Rule,  30  x  7  — 
72  =  161  =  relative  cube  of  diameter  at  7  feet;  30  X  15  —  is2  =  22 5  =  relative  cube 
of  diameter  at  is  feet,  or  at  middle  of  its  length. 

Then,  as  -^225  :  14.42  ::  ^161  :  12.89  in*.,  diameter  of  shaft  at  7  feet  from  one  end. 


7Q2  STRENGTH    OF   MATERIALS.  —  TORSION. 

For  Bronze,  420;  Cast  steel,  1000  to  1500;  and  Puddled  steel,  500. 

When  Stress  is  uniformly  laid  along  Length  of  Shaft.  RULE.  —  Divide 
cube  root  of  product  of  weight  and  length  by  9.3  for  Cast  iron  and  10.6 
for  Wrought  iron,  and  quotient  will  give  diameter  in  ins. 

EXAMPLE.  —  Apply  rule  to  preceding  case.     -  —  =  12.31  ins. 

For  Bronze,  8.5  ;  Cast  steel,  18.6  to  27.9  ;  and  Puddled  steel,  9.3. 

When  Diameter  for  Stress  applied  in  Middle  is  given.  RULE.  —  Take  cube 
root  of  .625  of  cube  of  diameter,  and  this  root  will  give  diameter  required. 

EXAMPLE.—  Diameter  of  a  shaft  when  stress  is  uniformly  applied  along  its  length 
is  14.42  ins.  ;  what  should  be  its  diameter,  stress  being  applied  in  middle? 

^.625  X  14.423  =  ^.625  X  3000  =  12.33  ins. 

To  Compute  Diameter  of  a  Solid   Shaft  of  Cast  Iron  to 

Resist   its    "Weight   alone. 

RULE.  —  Multiply  cube  of  its  length  by  .0x57,  and  square  root  of  product 
will  give  diameter  in  ins. 

EXAMPLE.—  Length  of  a  shaft  is  30  feet;  what  should  be  its  diameter  in  body? 
V(3Q3  X  -007)  =  Vl89  =  '3-75  ins 

HOLLOW  SHAFTS. 

To  Compute  Diameter  of  a  Hollow  Shaft   of  Cast  Iron 
to    Sustain    its    Load    in    Addition    to   its    Weight. 

When  Stress  is  in  or  near  Middle.  BULK.  —  Divide  continued  product  of 
.012  times  cube  of  length,  and  number  of  times  weight  of  shaft  in  Ibs.,  by 
square  of  internal  diameter  added  to  i,  and  twice  square  root  of  quotient 
added  to  internal  diameter  will  give  whole  diameter  in  ins. 

EXAMPLE.  —  Weight  of  a  water-wheel  upon  a  hollow  shaft  30  feet  in  length  is  2.5 
times  its  own  weight,  and  internal  diameter  is  9  ins.  ;  what  should  be  whole  diam- 
eter of-  shaft  ? 


To   Compute    Diameter   of  a  Round  or  Square   Shaft    to 

Resist  Combined  Stress  of  Torsion  and  \Veight. 
RULE.  —  Multiply  extreme  of  pressure  upon  crank-pin,  or  at  pitch-line  of 
pinion,  or  at  centre  of  effect  upon  the  blades  of  a  water-wheel,  etc.,  that  a 
shaft  may  at  any  time  be  subjected  to  ;  by  length  of  crank  or  radius  of 
wheel,  etc.,  in  feet  ;  divide  the  product  by  Coefficient  in  following  Table,  and 
cube  root  of  quotient  will  give  diameter  of  shaft  or  its  journal  in  ins. 


EXAMPLE. — What  should  be  diameter  for  journal  of  a  wrought-iron  water-wheel 
shaft,  extreme  pressure  upon  crank-pin  being  59  400  Ibs.,  and  crank  5  feet  in  length? 

C  =  120.       3 1 =  ^5/2475  =  13-53  in*- 

When  Two  Shafts  are  used,  as  in  Steam-vessels,  etc.,  ivith  One  Engine. 
RULE. — Divide  three  times  cube  of  diameter  for  one  shaft  by  four,'  and 
cube  root  of  quotient  will  give  diameter  of  shaft  in  ins. 

'    v      4 

EXAMPLE. — Area  of  journal  of  a  shaft  is  113  ins. ;  what  should  be  diameter,  two 
shafts  being  used  ? 

-,  y  I23 

Diameter  for  area  of  113  =  12.       Then  —  1296,  and  ^I296  =  io-9  to*. 


STRENGTH    OP   MATERIALS. — TOESION. 


793 


Torsional    Strength    of  Various    Metals. 

(Maj.  Wm.  Wade,  U.  S.  Ordnance  Corps,  1851,  Steel  Committee  [England,  1868],  and 

Stevens  Institute,  N.  J.,  1878.) 

Reduced  to  a  Uniform  Measure  of  One  Inch  in  Diameter  or  Side. 
Stress  applied  at  One  Foot  from  Axis  of  Body  and  at  Face  of  Axis. 


BARS  AND  METALS. 

Tensile 
Strength. 

Destruct 

at 
25  Ins. 

ive  Stress 

Computed 
at 
12  Ins. 

Torsional 
Strength 

—  ?  =  T. 

C 

2 

Ins. 

oefficU 

nt  — 

10 
Ins. 

2=i 

15 

Ins. 

V. 

20 

Ins. 

CAST  IRON. 

Lbs. 

Lbs. 

Lbs. 

Oo'H'^j 
Area  i  sq.  inch  ) 

45000 

520 

1082 

492 

100 

95 

90 

85 

80 

..  Diam.  {3.|5  ins..  I 
Area  2.  97  sq.  ins.  ) 

« 

3800 

79°4 

230 

45 

40 

35 

30 

25 

,&».    Diam.  )  Least  .  .  . 
=  i.gf  Mean  ... 

9000 

1550 

3664 

isoj 

^^     ins.     )  Greatest. 

31  029 
45000 

2145 
2840 

59°7 

850) 

HH    Side  i  inch...  .  ) 

liH     Area  i  sq.  inch  ) 

350 

728 

728 

125 

120 

"5 

no 

105 

WROUGHT  IRON. 

^^  Diam.   (  Least  .  .  . 

38027 

1250 

2600 

376) 

mm    =,.9    Mean.... 

56300 

1375 

2860 

416 

1  20 

"5 

no 

105 

100 

^^     ins.     (  Greatest. 
Area  2.  83  sq.  ins. 

74592 

1500 

3120 

452) 

BRONZE. 

tt  Diam.=  (Least  — 

17698 

500 

1040 

152 

3° 

28 

26 

__ 

_ 

1.9  ins.  (Greatest. 

56786 

650 

1352 

197 

38 

36 

34 





Area  2.  83  sq.  ins. 

CAST  STEEL. 

«,  Diam.=  (Least  
1.9  ins.  (Greatest. 
Area  2.  83  sq.  ins. 

42000 
128000 

2600 
7760 

5408 
16140 

788 
2353 

1  60 
475 

470 

465 

~" 

BESSEMER  STEEL. 

u  Diam.  =  i.382  ins.  ) 
Area  i.  5  sq.  ins.  ) 

36960 

1568 

3261 

1236 

245 

240 

235 

230 

225 

To   Compute   Diameter   of  Shafts   of  Oak;   and.   Pine. 

Multiply  diameter  ascertained  for  Cast  Iron  as  follows:  Oak  by  1.83, 
Yellow  Pine  by  1.716. 

Metals   and    Woods. 

Ultimate  Torsional  Strength. — Of  Cast  Iron  may  be  taken  as  equal  to  its 
transverse  strength  for  American  and  .9  for  English,  or  as  .26  of  its  tensile 
strength  for  American  and  .23  for  English.  Of  Wrought  Iron,  as  .7  to  .8  of 
its  transverse  strength  for  American  and  .7  to  i  for  English,  and  of  Steel,  as 
.72  of  its  tensile  strength. 

Elastic  Torsional  Strength. — Of  Cast  Iron  may  be  taken  as  equal  to  its 
transverse  strength,  of  Wrought  Iron  40  per  cent,  of  its  ultimate  torsional 
strength,  of  Steel  44  per  cent,  of  its  tensile  strength,  and  45  per  cent,  of  its 
ultimate  torsional  strength. 

Bessemer  Steel. — Has  a  torsional  strength  of  6670  Ibs.  per  sq.  inch  at  a  ra- 
dius of  one  foot,  being  somewhat  less  than  that  of  Cast  Iron,  Fagersta  has  50 
per  cent,  of  its  ultimate  transverse  strength,  and  Siemens  44.5  per  cent,  of 
its  ultimate  tensile. 


794 


STRENGTH    OF   MATERIALS. — TORSION. 


NOTE. — Examples  here  given  are  deduced  from  instances  of  successful  practice-, 
where  diameter  has  been  less,  fracture  has  almost  universally  taken  place,  stresi 
being  increased  beyond  ordinary  limit. 

2. — When  shafts  of  less  diameter  than  12  ins.  are  required,  Coefficients  here  given 
may  be  slightly  reduced  or  increased,  according  to  quality  of  the  metal  and  diame- 
ter of  shaft;  but  when  they  exceed  this  diameter,  Coefficients  may  not  be  increased, 
as  strength  of  a  shaft  decreases  very  materially  as  its  diameter  increinSes. 

Order  of  shafts,  with  reference  to  degree  of  torsional  stress  to  which  they 
may  be  subjected,  is  as  follows : 

i.  Fly-wheel.      |    2.  Water-wheel.     |    3.  Secondary  shaft.    |    4.  Tertiary,  etc. 
Hence,  diameters  of  their  journals  may  be  reduced  in  this  order. 

To  Compute  Diameter  of  a  "Wrought-iron  Centre  Shaft 
for  connecting  T\vo  Engines  at  a  Right  Angle. 

Conditions  of  such  a  shaft  are  as  follows : 

Greatest  stress  that  it  is  subjected  to  is  when  leading  engine  is  at  .75  of 
its  stroke,  and  following  engine  .25  of  its  stroke ;  hence,  position  of  each 
crank  is  as  sin.  22°  30'  X  2  =  .7071  of  length  of  crank  or  radius  of  power. 

Consequently,  3  /—  ^Z —  _  &    p  representing  extreme  pressure  on  piston. 

NOTE.—  In  computing  P  it  is  necessary  to  take  very  extreme  pressure  that  piston 
may  be  subjected  to,  however  short  the  period  of  time.  Average  pressure  does  not 
meet  requirement  of  case. 

ILLUSTRATION.— Extreme  pressure  upon  each  piston  of  two  engines  connected  at 
a  right  angle  was  m  592  Ibs.,  and  stroke  of  pistons  10  feet;  what  should  have  been 
diameter  of  centre  shaft  ?  and  what  of  each  wheel  or  driving  shaft  ? 

3  /  /HI  592  X  2  X  .707  W       ,  7788955 

V    \ —     — T* /  =  \/    "2      =       4*  T6      ft' 

For  ordinary  mill  purposes,  driving  shafts  should  be  as  cube  root  of  .75  cube  of 
centre  shaft.  ,g  g3 

Thus  3  / —  ^3  16.70  ins. 

To  Compute  Torsional  Strength   of  Hollow  Shafts   and 
Cylinders. 

RULE.— From  fourth  power  of  exterior  diameter  subtract  fourth  power  of 
interior  diameter,  and  multiply  remainder  by  Coefficient  of  material ;  divide 
this  product  by  product  of  exterior  diameter  and  length  or  distance  from  axis 
at  which  stress  is  applied  in  feet,  and  quotient  will  give  resistance  in  Ibs. 

(^4  — $'4)  0 

Or,  i '—.  =  R. 

EXAMPLE.— What  torsional  stress  may  be  borne  by  a  hollow  cast-iron  shaft,  hav- 
ing diameters  of  3  and  2  ins.,  power  being  applied  at  one  foot  from  its  axis? 

C=:i3o.     s«  —  2*  X  130  =  8450,  which  -r-3>;  i  =—  =  2816.6  Ibs. 

To  Compute  Torsional  Strength  of  Ronnd  and  Sqnai-e 
Shafts. 

RULE.— Multiply  Coefficient  in  preceding  Table  by  cube  of  side  or  of 
diameter  of  shaft,  etc.,  and  divide  product  by  distance  from  axis  at  which 
stress  is  applied  in  feet ;  quotient  will  give  resistance  in  Ibs. 

ILLUSTRATION.— What  torsional  stress  may  be  borne  by  a  cast-iron  shaft  of  best 
material,  2  ins.  in  diameter,  power  applied  at  2  feet  from  its  axis. 

C  from  table  =  130.     I3°X23  =  ^i?  =  S20  Ibs 

2  2 

For  steamers,  when  from  heeling  of  vessel  or  roughness  of  sea  the  stress  may  be 
confined  to  one  wheel  alone,  diameter  of  journal  of  its  shaft  should  be  equal  to 
hat  of  centre  shaft. 


STRENGTH    OF   MATERIALS.  —  TORSION.  /Q5 

GUDGEONS. 
To   Compute   Diameter   of  a   Single   Q-udgeon   of  Cast 

Iron,  to    Support    a  given.   \Veight   or    Stress. 
RULE.  —  Divide  square  root  of  weight  in  Ibs.  by  25  for  Cast  iron,  and  2tS 
for  Wrought  iron,  and  quotient  will  give  diameter  in  ins. 

EXAMPLE.—  Weight  upon  a  gudgeon  of  a  cast-iron  water-wheel  shaft  is  62  500  Ibs.  ; 
what  should  be  its  diameter  ? 


25  25 

To  Compute  Diameter  of  Two  Q-udgeons  of  Cast  Iron, 

to   Support   a  given.    Stress   or  "Weight. 
RULE.  —  Proceed  as  for  two  shafts,  page  792. 

To  Compute  "Ultimate  Torsional  Strength,  of  Round,  and 
Square    Shafts.     (D.  K.  Clark.) 

Cast  Iron.    Round.    ^M1?  =  W;   x.534^r  =  *;  and  ^  =  S. 


Sguare.     •***  =  W,  and  ..363/^5  =  *         Hollow.     ^  ^^  S  =  W. 

S  representing  ultimate  shearing  strength,  and  W  moment  of  load,  both  in  Ibs.,  s  side 
of  square  shaft,  and  R  radius  of  stress,  both  in  ins. 

ILLUSTRATION.—  What  is  ultimate  torsional  strength  of  a  round  cast-iron  shaft 
4  ins.  in  diameter,  stress  applied  at  5  feet  from  its  axis  ? 

Assume  S  =  20  ooo  Ibs.        Then  '  2yB  X  4*  X  2°  °°°  =  5930  Ibs. 

5  X  12 

By  experiments  of  Major  Wade,  ordinary  foundry  iron  has  a  torsional  strength 
of  7725  Ibs.,  or  644  Ibs.  per  sq.  inch  at  radius  of  one  foot. 

Thus,  take  preceding  illustration.      Then  ^^  —  —  =  8240  Ibs. 

"Wrought   Iron.    Round.    -^^  -  =  W.      Square.    -^-^  —  =  W. 

When  Torsional  Strength  per  sq.  inch  for  radius  of  i  inch  is  ascertained, 
substitute  C  for  .278,  .4,  .2224,  or  .32. 

Stress  which  will  give  a  bar  a  permanent  set  of  .5°  is  about  .7  of  that 
which  will  break  it,  and  this  proportion  is  quite  uniform,  even  when  strength 
of  material  may  vary  essentially. 

Wrought  Iron,  compared  with  Cast  Iron,  has  equal  strength  under  a  stress 
which  does  not  produce  a  permanent  set,  but  this  set  commences  under  a  less 
force  in  wrought  iron  than  cast,  and  progresses  more  rapidly  thereafter. 
Strongest  bar  of  wrought  iron  acquired  a  permanent  set  under  a  less  strain 
than  a  cast-iron  bar  of  lowest  grade. 

Strongest  bars  give  longest  fractures. 

cj,  ,  »  ,  .g  d^  S  When  S  is  not  known,  substitute  for 

'na'  ~1T~  '  872  5  =72  per  cent,  of  tensile  strength. 

Torsional  Strength  of  Cast  Steel  is  from  2  to  3  times  that  of  Cast  Iron. 

Following  rules  are  purposed  to  apply  in  all  instances  to  diameters  of 
journals  of  shafts,  or  to  diameter  or  side  of  bearings  of  beams,  etc.,  where 
length  of  journal  or  distance  upon  which  strain  bears  does  not  greatly  ex- 
ceed diameter  of  journal  or  side  of  beam,  etc.  ;  hence,  when  length  or  distance 
is  greatly  increased,  diameter  or  side  must  be  correspondingly  increased. 

Coefficients  for  torsional  breaking  stress  of  Iron,  Bronze,  and  Steel,  as  de- 
termined by  Major  Wade,  are:  Wrought  Iron,  640;  Cast  Iron,  560;  Bronze, 
460  ;  Cast  Steel,  1120  to  1680.  Puddled  Steel  does  not  differ  essentially  from 
that  of  cast  iron 


796 


STRENGTH    OF   MATERIALS. — TORSION. 


Formulas  for  Minimum  and  Maximum  Diam.  of  Wrought-iron  Shafts. 
(A.  E.  Seaton,  London,  1883,  and  Board  of  Trade,  Eng.) 

Compound  Engines.  ^J  — —— —  S  =  diameter.  D  and  d  representing  diam- 
eter of  low  and  high  pressure  cylinders,  and  S  half  stroke,  all  in  ins.,  p  pressure  of 
steam  in  boiler,  in  Ibs.  per  sq.  inch,  and  C  a  coefficient,  as  follows : 


Aofle 
Crank. 

Sha 
Crank. 

ts. 
Pro- 
peller. 

Angle 
of 
Crank. 

Sba 
Crank. 

ts. 
Pro- 
peller. 

Angle 
of 
Crank. 

Shaf 
Crank. 

ts. 
Pro- 
peller. 

An?,e 
Crank. 

Sha 
Crank. 

'ts. 
Pro- 
peller. 

900 

(2468 
(4000 

2880 
5400 

100° 

(2279 
(4000 

2659 
5400 

110° 

(2131 
Uooo 

2487 
5400 

120° 

(2016 
(4000 

2352 
5400 

I/  -  C  =  diameter.    A.  E.  Seaton,  London,  1883. 

Side-wheel  Engines,  Sea  Service.  —  One  cylinder  crank  journal,  C  =  8o;  outboard 
loo  ;  Two  cylinder  crank  journal  50;  outboard  65;  and  centre  shaft  58. 

Propeller  Engines.—  One  cylinder  crank  journal  150;  Tunnel  130;  Two  cylinder 
compound  crank  130;  Tunnel  no;  Two  cranks,  crank  100;  Tunnel  85;  Three  cranks, 
crank  90;  and  Tunnel  78. 

River  Service.—  C  may  be  reduced  one  fifth. 

ILLUSTRATION.—  With  a  compound  propeller  engine,  steam  cylinders  20  and  40 
ins.  in  diameter,  by  40  ins.  stroke,  operating  under  a  pressure  of  80  Ibs.  steam 
(mercurial  gauge),  what  should  be  the  diameter  of  the  shafts  of  wrought  iron? 


=8.24  ins.  crank 


and 


A60 
V    54< 


-  x  40  =  7.46  ins.  propeller  shaft. 


Journals   of  Shafts,  etc. 

Journals  or  bearings  of  shafts  should  be  proportioned  with  reference  to 
pressure  or  load  to  be  sustained  by  the  journal.  Simplest  measure  of  bear- 
ing capacity  of  a  journal  is  product  of  its  length  by  its  diameter,  in  sq.  ins.  ; 
and  axial  area  or  section  thus  obtained,  multiplied  by  a  coefficient  of  pressure 
.  per  sq.  inch,  will  give  bearing  capacity. 

Sir  William  Fairbairn  and  Mr.  Box  give  instances  of  weights  on  bearings  of 
shafts,  etc.,  from  which  following  deductions  are  made,  showing  pressure  per  sq. 
inch  of  axial  section  of  journal  : 

Crank  pins,  687  to  1150  Ibs.  per  sq.  inch. 

Link  bearings,  456  to  690  Ibs.  per  sq.  inch. 

Pressure  on  bearings,  as  a  general  rule,  should  not  exceed  750  Ibs.  per  sq.  inch  of 
axial  area. 

Length  of  Journals  should  be  1.12  to  1.5  times  diameter. 

Journals  of  Locomotives  or  Like  Axles  are  usually  made  twice  diameter,  and  to 
sustain  a  pressure  of  300  Ibs.  per  sq.  inch  of  axial  area,  or  10  sq.  ins.  per  ton  of  load. 

Solid.    Cylindrical    Couplings    or    Sleeves. 
^-f-\/5-5  d=D;    $d  =  L',    .8d  =  J;    .25  d-f~-i2  —  k.     d  representing  diameter, 
and  L  length  of  sleeve,  I  length  of  lap  or  scarf  of  shaft,  k  breadth  of  key,  its  depth  be- 
ing half  its  breadth,  and  D  diameter  of  coupling  or  sleeve,  all  in  ins. 


Flanged   Couplings. 

d  +  VS-Sd^D;  3d4-i  —  F;  .3d-j-.4  =  ?;  d+i  =  L;  Z-^-4  —  s. 
senting  diameter  of  body  of  coupling,  F  diameter  of  flanges,  I  thickness  of  b 
L  length  of  each  coupling,  s  projection  of  end  of  one  shaft  and  retrocessi 
from  centre  of  coupling,  and  d  diameter  of  shaft,  all  in  ins. 

Supports   for    Shafts.     (Molesworth.) 
5  tyd*  =  L.    L  representing  distance  of  supports  apart,  in  feet. 


.     D  repre- 
oth flanges, 
retrocession  of  other 


STRENGTH    OF  MATERIALS. TOBSION. 


797 


To  Resi*t  Lateral  Stress.     fg-  =  D.    W  representing  weight  or  pressure 
sit  centre  of  length  in  Ibs.,  and  D  diameter  or  side,  if  square,  in  ins. 

Value  o/C.— Wrought  Iron,  560;  Cast  Iron,  500;  Cast  Steel,  1000  to  1500;  Bronze, 
420 ;  and  Wood,  40.    When  Weight  is  distributed  put  2  C. 

Values  of  C  for  Shafting  of  Various  Metals,  as  observed  by  different 

Authorities,  and  deduced  from  Formulas  of  Navier.    -^r^r  =  c- 

Ultimate  Resistance. 


METAL. 

c 

METAL. 

C 

METAL. 

C 

WROUGHT  IRON. 
American,  Pemb«,Me. 
"        Ulster.... 
"        mean  
English,  refined  

Swedish 

61673 
61815 
66436 
49148 
54585 
61909 

and 

, 

CAST  IRON. 
American,  mean  j 

44        1  8  trials 
English,  mean..  | 

Factory   Sliai 

36846 
38300 
42821 

44957 
22132 
38217 

ts.     (. 

STEEL. 
American,  Conn  .  . 
"         Spindle 
"    Nash.  I.  Co. 
English,  Shear.  .  ,  . 

Bessemer  j 

82926 
102131 
95213 

III  191 

73060 

79662 

Cylindrical 

r                              I 

T.  B.  Francis.) 
Square. 

mean 35000 

"      Eng.  30000 


i6WR 


Mean  value  of  T. 

Wrought  Iron....  {  W000  I  Steel f    76000 

(  94000  i  }  iiiooo 

;<    mean 50000!     u  mean 86000 

"       "    Eng.  45000 1     *'     "   Bessemer   78000 

ILLUSTRATION.— What  is  the  ultimate  or  destructive  weights  that  may  be  borne 
by  a  Round  Cast-iron  shaft  2  ins  in  diameter,  and  by  a  Square  shaft  1.75  ius.  side 
stress  applied  at  25  ins  from  axis  ?  Assume  T  =  36  ooo. 

Round  Square. 

6000       ,      _      /,.75»x  36000^.,  l^,/7=i8i9, 


\ 
3  J 


3    -f- 


*  Ibs. 


Their  lengths  should  be  reduced,  and  diameter  increased,  in  following  cases  : 
ist.  At  high  velocities,  to  admit  of  increased  diameter  of  journals,  thereby 
rendering  them  less  liable  to  heating,    ad.  As  they  approach  extremity  of  a 
line  of  shafting.    3d.  Attachment  of  intermediate  pulleys  or  gearing. 


Prime  Movers  of  Power. 
3/P 


=  *,  and  .ox  n  d*  =  IH>. 


Transmitters  of  Power. 

^  and  .02  n  d,  =  IH>. 


--5nIH*  =  d,  and  .016  n  d*  =  Iff  .  3/3I'2^IIP  =  d,  and  .032  n  d»  =  IH». 


Iron. 
Steel 

Cast 
Iron. 

IIP  representing  horse-power  transmitted,  n  number  of  revolutions,  and  d  diameter 
of  shaft  in  ins. 

ILLUSTRATION  i.— What  should  be  diameter  of  a  wrought-iron  shaft,  to  simply 
transmit  128  H?  at  100  revolutions  per  minute? 

3  /5°  X  I2   =  3  /_i£2  =  4  ins. 
V      10°        V  I0° 

2.— What  H?  will  a  steel  shaft  of  4  ins.  diameter  transmit  at  100  revolutions  per 
minute? 

.032  X  ioo  X  43  =  204.8  horses. 
3X* 


STRENGTH    OF    MATERIALS.  —  TRANSVERSE. 


TRANSVERSE    STRENGTH. 

Transverse  or  Lateral  Strength  of  any  Bar,  Beam,  Rod,  etc.,  is  in  proper- 
tion  to  product  of  its  breadth  and  square  of  its  depth;  in  like-sided  barss 
beams,  etc.,  it  is  as  cube  of  side,  and  in  cylinders  as  cube  of  diameter  ol 
section. 

When  One  End  is  Fixed  and  the  Other  Projecting,  strength  is  inversely  a* 
distance  of  weight  from  section  acted  upon  ;  and  stress  upon  any  section  ii 
directly  as  distance  of  weight  from  that  section. 

When  Both  Ends  are  Supported  only,  strength  is  4  times  greater  for  an 
equal  length,  when  weight  is  applied  in  middle  between  supports,  than  if  one 
end  only  is  fixed. 

When  Both  Ends  are  Fixed,  strength  is  6  times  greater  for  an  equal  length, 
when  weight  is  applied  in  middle,  than  if  one  end  only  is  fixed. 

When  Ends  Rest  merely  upon  Two  Supports,  compared  to  one  When  Ends  are 
Fixedj  strength  of  any  bar,  beam,  etc.,  to  support  a  weight  in  centre  of  it,  is 
as  2  to  3. 

When  Weight  or  Stress  is  Uniformly  Distributed,  weight  or  stress  that  can 
be  supported,  compared  with  that  when  weight  or  stress  is  applied  at  one  end 
or  in  middle  between  supports,  is  as  2  to  i. 

Metals. 

In  Metals,  less  dimension  of  side  of  a  beam,  etc.,  or  diameter  of  a  cylinder, 
greater  its  proportionate  transverse  strength,  in  consequence  of  their  having 
a  greater  proportion  of  chilled  or  hammered  surface,  compared  to  their  ele- 
ments of  strength,  resulting  from  dimensions  alone. 

Strength  of  a  Cylinder,  compared  to  a  Square  of  like  diameter  or  sides,  is 
as  5.5  to  8.  Strength  of  a  Hollow  Cylinder  to  that  of  a  Solid  Cylinder,  of 
same  area  of  section,  is  about  as  1.65  to  i,  depending  essentially  upOE  the 
proportionate  thickness  of  metal  compared  to  diameter. 

Strength  of  an  Equilateral  Triangular  Beam,  Fixed  at  One  End  and 
Loaded  at  the  Other,  having  an  edge  up,  com  oared  to  a  Square  of  the  same 
area,  is  as  22  to  27  ;  and  strength  of  one,  having  an  edge  down,  compared  to 
one  with  an  edge  up,  is  as  10  to  7. 

NOTE.  —  In  Barlow  and  other  authors  the  comparison  in  this  case  is  made  when 
the  beam,  etc.,  rested  upoa  supports.  Hence  the  stress  is  contrariwise. 

Strongest  rectangular  bar  or  beam  that  can  be  cut  out  of  a  cylinder  is  one 
of  which  the  squares  of  breadth  and  depth  of  it,  and  diameter  of  the  cylinder, 
are  as  i,  2,  and  3  respectively. 

Cast   Iron. 

Mean  transverse  strength  of  American,  as  determined  by  Major  Wade,  U 
681  Ibs.  per  sq.  inch,  suspended  from  a  bar  fixed  at  one  end  and  loaded  at 
the  other  ;  and  mean  of  English,  as  determined  by  Fairbairn,  Barlow,  and 
others,  is  500  Ibs. 

Experiments  upon  bars  of  cast  iron,  i,  2,  and  3  ins.  square,  give  a  result 
of  transverse  strength  of  447,  348,  and  338  Ibs.  respectively  ;  being  in  the 
ratio  of  i,  .78,  and  .756. 

^Woods. 

Beams  of  wood,  when  laid  with  their  annular  layers  vertical,  are  stronger 
than  when  they  are  laid  horizontal,  in  the  proportion  of  8  to  7. 

Relative  Stiffness  of  Materials   to  Resist   a  Transverse 

Stress. 
Ash  ...........  080  I  Cast  Iron  ----  i        I  Oak  ...........  095  J  Wrought  iron  i.  3 

Beecb  .........  073  I  Elm  ..........  073  |  White  pine...  .1      |  Yellow  pine..    .087 


STRENGTH    OF   MATERIALS. — TRANSVERSE. 


799 


Strength  of  a  Rectangular  Beam  in  an  Inclined  position,  to  resist  a  vertical 
stress,  is  to  its  strength  in  a  horizontal  position,  as  square  of  radius  to  square 
of  cosine  of  elevation ;  that  is,  as  square  of  length  of  beam  to  square  of  dis- 
tance between  its  points  of  support,  measured  upon  a  horizontal  plane. 

WOODS. 


California  Red  Pine. 
California  Spruce. . . 
Canadian  Red  Pine.. 
Cedar 


5000 


Ultimate  Resistan 
Chestnut  

ce. 

Lbs. 

5000 
7000 
35oo 
6000 

Oregon  Pine 

Lbe. 
6500 
4000 
6000 

AOOO 

Georgia  Pine  

Spruce  

Hemlock      

White  Oak 

Northern  Pine.  .  . 

White  Pine.  .  . 

Transverse    Strength    of*  "Various    Materials. 

(U.  S.  Ordnance  Department,  Hodgkinson,  Fairbaim,  Kirkaldy,  by  the  Author,  and 
Digest  of  Physical  Tests.) 

Power  reduced  to  uniform  Measure  of  One  Inch  Square,  and  One  Foot  in  Length; 
Weight  suspended  from  one  End. 

Safe  Stress. 


METALS. 

Brass 260 

Cast  Iron,  mean  (Maj.  Wade) 681 

ordinary 575 

extreme,  West  P't  F'dry  980 

gun -metal,*     "        "  740 

Eng. ,  Low  Moor,  cold  blast.  472 

Ronkey,  581 

Ystalyfera,        "  770 

mean,  65  kinds.. ....  500 

"  1 5  kinds,  cold  blast  641 

planed  bar. 518 

Copper 244 

Steel,  hammered,  mean 1500 

cast,  soft 1540 

"    hard 4200 

hematite,  hammered 1620 

Krupp's  shaft 2096 

Fagersta,  hammered 1200 

Wrought  Iron,  mean 600 

"          "     English 475 

"         «     Swedisht 665 

WOODS. 

Ash 220 

"  English 160 

"  Canada. 120 

Balsam,  Canada 87 

Beech 130 

"     white 112 

Birch 137 

Cedar,  white 160 

"     Cuba,  mean 84 

Chestnut 160 

Elm. 125 

"  Canada,  red 170 

Fir,  Baltic,  mean 153 

"  Canada,  yellow J  ^ 

"        "       red 120 

Ureenheart,  Guiana 160 


WOODS  (Continued). 

Gum,  blue 136 

Hackmatack 102 

Hemlock 100 

Hickory 210 

Larch,  Russian ng 

Lignumvitse 162 

Locust 295 

Mahogany u2 

Maple 202 

Oak,  white 150 

4    live 160 

'    African 207 

'    English 130 

'    French 160 

'    Canada. 146 

'    Spanish 105 

Pine,  white 125 

*     Pitch 137 

'     yellow 130 

4     Georgia 200 

Poon 184 

Spruce,  Canada 125 

"      black 87 

Sycamore 125 

Tamarack 100 

Teak 165 

Walnut 112 

STONES,  BRICKS,  ETC. 
Brick,  common,  mean 


« 
Cement,  mean 


Portland 


"    hydraulic,  Portland. . 

"    Roman 

"       Puzzuolana 

"       Portland,  i  year 

Concrete,  Eng.,  fire-brick  beam,) 
cement f 


20 

40 
15 

10.2 

37-5 
5 

2 

4-5 


*  This  was  with  a  tensile  strength  of  27  ooo  Ibs. 

t  With  840  Ibs.  the  deflection  was  i  inch,  and  the  elasticity  of  the  metal  destroyed. 


8oo 


STRENGTH    OF    MATERIALS. TRANSVERSE. 


STONES,  BRICKS,  ETC. 
Concrete,  Eng. ,  fire-brick,  sand  3, ) 

lime  i J       '7 

"         Eng.,  clay  and  chalk. ...    5.4 

Flagging,  blue,  New  York 3*-2S 

freestone,  Conn 13 

Dorchester 10. 8 

New  Jersey,  mean 19 

New  York 24 

Eng. ,  Craigleth 10. 7 

"      Darby,  Victoria. ..    1.3 

"      Park  Spring 4. 3 

Glass,  flooring 42. 5 

Granite,  blue,  coarse 18 

"        Quincy 26 

"        mean 25 

"        Eng.,  Cornish 22 

Limestone n  to  15 

"         English it 

Marble,  Vermont,  mean 92 


STONES,  BRICKS,  ETC. 

Marble,  Adelaide 4. 5 

"      Italian,  white IX<6 

Mortar,  lime,  60  days 2.5 

"       i  lime,  i  sand , . .    2 

i     "      2     "    i-75 

i  4          1-25 

Oolite,  English,  Portland 21.2 

Paving,  Scotch,  Caithness 68 

' '       Ireland,  Valentia 68. 5 

"       Welsh 157 

"       English,  Yorkshire,  blue..  10.4 

"  "        Arbroath ?7 

Slate 81 

"    Bangor 90 

"    English,  Llangollen 43 

Stones,  English,  Bath 5.2 

"         Kentish,  Rag 35.8 

"        Yorkshire,  landing  22.5 
"      Caen 12.5 


Elastic  Transverse  Strength  of  Woods,  compared  with  their  Breaking  Weight, 
is  as  follows : 


PerCent.  I 

Ash 29        Norwa; 

Beech 25 


Elm 32 

Larch 38 


Per  Cent. 

iy  Spruce  ____     30 
Oak,  Dantzic  ......     36 


English  ......     33 

Pitch  Pine  .........     24 


Per  Cent. 

Red  Pine 29 

Riga  Fir 30 

Teak 32 

Yellow  Pine 30 


Increase    in.    Strength   of  several   "Woods   "by    Seasoning. 

Per  Cent 
Ash 44.7  |  Beech 61.9  |  Elm 12.3  |  Oak 26.1  |  White  pine.... 9 

Concretes,  Cements,  etc. 


MATERIALS. 

Breaking 
Weight. 

MATERIALS. 

Breaking 
Weight. 

CONCRETES  (English). 
Fire-brick  beam  Portl'd  cement 

Lbs. 

BRICKS  (English). 
Best  stock  

Lbs. 

u      sand  3  parts,  lime  i  part 

.7 

Fire-brick  

CEMENTS  (English). 

New  brick  

5-4 

Old  brick  

Q  I 

Portland                                    t 

37-5 

Stock-brick,  well  burned  

*i 

Sheppev.  .  . 

IO.2 

inferior,  burned.  .  . 

2-5 

Transverse    Strength,    of  "Variovis    Figures    of  Cast    Iron. 

Reduced  to  Uniform,  Measure  of  Sectional  Area,  of  One,  Inch  Square  and  One  Foot  in 
Length.     Fixed  at  one  End  ;   Weight  suspended  from  the  other. 


Form  of  Bar  or  Beam. 

Breaking 
Weight. 

Form  of  Bar  or  Beam. 

Breaking 
Weight. 

Wi  Square  

Lbs. 
673 

1    Rectangular  prism. 

Lbs. 

^^  Square,  diagonal  vertical.  .  . 

568 

i        2  X  .  5    ins.  in  depth  .  .  . 
**        3  X  .33    "    in  depth.  .  . 
"        4  X  .25    "    in  depth.  .  . 
A  Equilateral  triangle,  an) 
edge  up  j 

1456 
2392 
2652 

560 

•  Cylinder  
j 

573 

WW  Equilateral  triangle,  an  ) 

958 

O  Hollow  cylinder;  greater) 
diameter  twice  that  of  \ 
lesser.  .  .                     .  .  ) 

794 

T2  ins.  in  depth  X  2  X  ) 
.268  inch  in  width.  ..  } 
\     2  ins.  in  depth  x  2  x  ) 
JL       .268  inch  in  width  .  .  j 

2068 
555 

STRENGTH    OF   MATERIALS. — TRANSVERSE. 


80 1 


Solid    and.    Hollow    Cylinders   of  various    IVIaterialSo 

One  Foot  in  Length.    Fixed  at  one  End  ;  Weight  suspended  from  the  other. 


MATERIALS. 

External 
Diam. 

Internal 
Diam. 

Breaking 
Weight. 

MATERIALS. 

External 
Diam. 

Internal 
Diam. 

Breaking 
Weight. 

WOODS. 

Ash  

Ins. 
2 

Inch. 

Lbs. 
685 

METAL. 

Cast  iron    cold) 

las. 

Ina. 

Lbs. 

60  A. 

blast                ) 

3 

— 

12000 

Fir*  

2 

772 

STONE  -WARE 

White  pine.  . 

I 
2 

— 

610 

Rolled  pipe  of  ) 
fine  clay  j 

2.87 

1.928 

IQO 

*  An  inch-square  batten,  from  same  plank  as  this  specimen,  broke  at  139  Ibs. 

Formulas    for   Transverse    Stress   of  Rectangular   Bars, 
Beams,  Cylinders,  etc. 

Fixed  at  One  End.    Loaded  at  the  Other. 


Bars,  Beams,  etc.    —  =  £ 

/z  w 
and  Cylinder  3/—=b  and  d. 

Fixed  at  Both  Ends.    Loaded  in  Middle. 

IW  6S6d2  6  S  6  d2 

Bars,  Beams,  etc.     ^^  =  8;        — ^—  =  W;       -^-= 

/— — -  =  d ;    and  Cylinder  3  /— _  =  b  and  d. 
V  6  S  b  V  6  b 

Fixed  at  Both  Ends.    Loaded  at  any  Other  Point  than  in  Middle. 
Bars,  Beams, etc.    2MnW     •-    3'6<J2S     "     2"'MW     '         2TOnW 


llbd* 


Supported  at  Both  Ends.    Loaded  in  Middle. 

IW 
Bars,  Beams,  etc. 


4S6  d2 
L__  =  W; 


i      =  d;    .Dd  Cylinder  = 


IW 

4Sd2 


Supported  at  Both  Ends.    Loaded  at  any  Other  Point  than  in  Middle. 


Bars,  Beams,  etc.      ,  .    J0  r^  S ; 
I  b  d2 


mnW 


=  6; 


H  W  /IW 

In  Square  Beams,  etc.,  for  b  and  d  put  |/-g--  =  ^/g^  =  d.    In  Cylinders,  for 

6  d2  put  d3  as  above. 

When  weight  is  uniformly  distributed,  same  formulas  will  apply,  W  repre- 
senting only  half  required  or  given  weight. 

S  representing  stress  in  a  Bar,  Ream,  or  Cylinder,  one  foot  in  length,  and  one  inch 
square,  side,  or  in  diameter;  and  W  weight,  in  Ibs.;  b  breadth,  and  d  depth  in  ins.; 
I  length,  m  distance  of  weight  from  one  end,  and  nfrom  the  other,  all  in  feet. 

B  r  i  clr-wor  k . 

A  brick  arch,  having  a  rise  of  2  feet,  and  a  span  of  15  feet  9  ins.,  and  2 
feet  in  width,  with  a  depth  at  its  crown  of  4  ins.,  bore  358400  Ibs.  laid  along 
its  centre. 


8O2         STBENGTH    OF   MATERIALS.  —  TRANSVERSE. 

Coefficient   or   Factor   of  Safety. 

Coefficient  or  factor  of  safety  of  different  materials  must  be  taken  in  view 
of  importance  of  structure,  or  instrument,  probable  or  required  period  of  du- 
ration of  it,  and  if  it  is  to  bear  a  quiescent,  vibratory,  gradual,  or  percussive 
stress,  and  to  meet  these  varied  conditions,  it  will  range  from  .125  to  .3  of 
the  maximum  or  ultimate  strength  here  given  or  ascertained. 

To  Compute  Transverse  Strength  of  a  Rectangular  Bar 

or    Beam. 

When  a  Bar  or  Beam  is  Fixed  at  One  End,  and  Loaded  at  the  Other. 
RULE.  —  Multiply  Coefficient  of  material  in  preceding  Tables,  or,  as  may  be 
ascertained,  by  breadth  and  square  of  depth  in  ins.,  and  divide  product  by 
length  in  feet. 

NOTE.  —When  abeam,  etc.,  is  loaded  uniformly  throughout  its  length,  result  must 
be  doubled. 

EXAMPLE.—  What  weight  will  a  cast-iron  bar,  2  ins.  square  and  projecting  30  ins. 
in  length,  bear  without  permanent  injury  ? 

Assume  strength  of  material  at  660,  and  its  elasticity  at  one  fifth  or  .2  of  its 
strength. 

,       66oX.2X2X22      1056 
Then  -  =  —  —  =  422.  4  Ibs. 
2.5  2.5 

If  Dimensions  of  a  Beam  or  Bar  are  Required  to  Support  a  Given  Weight 
at  its  End.  RULE.  —  Divide  product  of  weight  and  length  in  feet  by  Coeffi- 
cient of  material,  and  quotient  will  give  product  of  breadth  and  square  of 
depth. 

EXAMPLE.—  What  is  the  depth  of  a  wrought-iron  beam,  2  ins.  broad,  necessary  to 
support  576  Ibs.  suspended  at  30  ins.  from  fixed  end? 

Assume  strength  of  iron  at  150. 

Then  2'5iXQ576  =  9.6,  and     /^  =  2.  19  ins.  depth. 

When  a  Beam  or  Bar  is  Fixed  at  Both  Ends,  and  Loaded  in  the  Middle. 
RULE.—  Multiply  Coefficient  of  material  by  6  times  breadth  and  square  of 
depth  in  ins.,  and  divide  product  by  length  in  feet. 

NOTE.  —  When  beam  is  loaded  uniformly  throughout  its  length,  result  must  be 
doubled. 

EXAMPLE.—  What  weight  will  a  bar  of  cast  iron,  2  ins.  square  and  5  feet  in  length, 
support  in  middle,  without  permanent  injury? 

Assume  strength  of  material  as  in  a  previous  case  at  .2  of  660. 

Then 


If  Dimensions  of  a  Beam  or  Bar  are  Required  to  Support  a  Given  Weight 
in  Middie,  between  Fixed  Ends.  RULE.  —  Divide  product  of  weight  and 
length  in  feet  by  6  times  Coefficient  of  material,  and  quotient  will  give  prod- 
uct of  breadth  and  square  of  depth. 

EXAMPLE.  —  What  dimensions  will  a  square  cast-iron  bar,  5  feet  in  length,  require 
to  support  without  permanent  injury  a  stress  of  2160  Ibs.  ? 

Assume  strength  of  material  at  .2  of  660  or  132,  as  preceding. 

Then  -^  —  —  —  =  -  —  —  -  =  13.64,  which,  divided  by  2  for  assumed  breadth  =  6.82, 
and  ^/6.  82  =  2.61  ins.  depth. 

When  Breadth  or  Depth  is  Required.  RULE.  —  Divide  product  obtained 
by  preceding  rules  by  square  of  depth,  and  quotient  is  breadth;  or  by 
breadth,  and  square  root  of  quotient  is  depth. 

EXAMPLE.—  If  128  is  the  product,  and  depth  is  8;  then  i28---82  =  2,  breadth 
Also,  128  -s-  2  —  64,  and  ^64  =  8,  depth. 


STRENGTH   OF   MATERIALS.  —  TRANSVERSE.          803 

When  Weight  w  not  in  Middle  between  Ends.  RULE.  —  Multiply  Coefficient 
of  material  by  3  times  length  in  feet,  and  breadth  and  square  of  depth  in 
ins.,  and  divide  product  by  twice  product  of  distances  of  weight,  or  stress 
from  either  end. 

EXAMPLE.—  What  weight  will  a  cast-iron  bar,  fixed  at  both  ends,  2  ins.  square  and 
5  feet  in  length,  bear  without  permanent  injury,  2  feet  from  one  end? 

Assume  strength  of  material  at  .2  of  660  or  132,  as  preceding. 
Tnen  »3«  X  3  X  5  X  a  X  »•  =  15840  =          ^ 

2X(2X3)  12 

When  a  Beam  or  Bar  is  Supported  at  Both  Ends,  and  Loaded  in  Middle. 
RULE.  —  Multiply  Coefficient  of  material  by  4  times  breadth  and  square  of 
depth  in  ins.,  and  divide  product  by  length  in  feet. 

NOTE.—  When  beam  is  loaded  uniformly  throughout  its  length,  result  must  be 
doubled. 

EXAMPLE.  —  What  weight  will  a  cast-iron  bar,  5  feet  between  the  supports,  and  2 
ins.  square,  bear  in  middle,  without  permanent  injury? 
Assume  strength  of  iron  at  132,  as  preceding. 

Then  132X2X4X22  =  4224  -r-  5  =  844.  8  Ibs. 

If  Dimensions  are  Required  to  Support  a  Given  Weight.  RULE.  —  Divide 
product  of  weight  and  length  in  feet  by  4  times  Coefficient  of  material,  and 
quotient  will  give  product  of  breadth,  and  square  of  depth. 

When  Weight  is  not  in  Middle  between  Supports.  RULE.  —  Multiply  Coef- 
ficient of  material  by  length  in  feet,  and  breadth  and  square  of  depth  "in  ins., 
and  divide  product  by  product  of  distances  of  weight,  or  stress  from  either 
support. 

EXAMPLE.  —  What  weight  will  a  cast-iron  bar,  2  ins.  square  and  5  feet  in  length, 
support  without  permanent  injury,  at  a  distance  of  2  feet  from  one  end,  or  support? 
Assume  strength  of  iron  at  132.  as  preceding. 
' 


Then  BSo  Ibs. 

2  X  (5  —  2)  6 

To  Compute  IPressure   -upon  Ends    or   -upon    Supports. 

RULE  i.  —  Divide  product  of  weight  and  its  distance  from  nearest  end  or 
support,  by  whole  length,  and  quotient  will  give  pressure  upon  end  or  sup- 
port farthest  from  weight. 

2.  —  Divide  product  of  weight  and  its  distance  from  farthest  end,  or  sup- 
port, by  whole  length,  and  quotient  will  give  pressure  upon  end  or  support 
nearest  weight. 

EXAMPLE.—  What  is  pressure  upon  supports  in  case  of  preceding  example? 

80  X  2  =  352  Ibs.  upon  support  farthest  from  the  weight  ;  88oX3  =  528  Ibs.  upon 
support  nearest  to  weight. 

When  a  Bar  or  Beam,  Fixed  or  Supported  at  Both  Enr's,  bears  Two 
Weights  at  Unequal  Distances  from  Ends. 

f^=  --  \-  -™-  =  pressure  at  w  end,  and  ^-^  -}-  —  -—  =  pressure  at  T7  end. 
L  Li  Li  Li 

m  and  n  representing  distances  of  greatest  and  least  weights  from  their  nearest 
end,  W  and  w  greatest  and  least  weights,  L  whole  length,  I  distance  from  least  weight 
to  farthest  end,  and  I'  distance  of  greatest  weight  from  farthest  end. 

ILLUSTRATION.  —  A  beam  10  feet  in  length,  having  both  ends  fixed  in  a  wall,  bears 
two  weights—  viz.,  one  of  1000  Ibs.,  at  4  feet  from  one  of  its  ends,  and  the  other  of 
2000  Ibs.  ,  at  4  feet  from  the  other  end  ;  what  is  pressure  upon  each  end  ? 


8O4          STRENGTH    OF    MATERIALS. TRANSVERSE. 

When  Plane  of  Bar  or  Beam  Projects  Obliquely  Upward  or  Downward, 
When  Fixed  at  One  End  and  Loaded  at  the  Other.    RULE. — Multiply  Co* 
efficient  of  material  by  breadth  and  square  of  depth  in  ins.,  and  divide  product 
by  product  of  length  in  feet  and  cosine  of  angle  of  elevation  or  depression. 
NOTE.— When  beam  is  loaded  uniformly  along  its  length,  result  must  be  doubled. 
EXAMPLE.— What  is  weight  an  ash  beam,  5  feet  in  length,  3  ins.  square,  and  pro- 
jecting upward  at  an  angle  of  7°  15',  will  bear  without  permanent  injury? 

Assume  breaking  weight  of  ash  at  160,  and  its  elasticity  at  .25  of  its  strength,  aa4 
cosine  of  7°  15'  =  .992. 

Then =  — -  =  217.74  '&*• 

5X-992  4-96 

To  Compute  Transverse  Strength  of  an  Equilateral  Tri 

angle   or   T   Beam. 

RULE. — Proceed  as  for  a  rectangular  beam,  taking  following  proportions 
of  Coefficient  of  material : 

Fixed  at  One  or  (  E(luilateral  triangle,  edge  up 6  X  d2  X  .2    C 

nZkvT       \  Equilateral  triangle,  edge  down b  x  d2  X  .- 

Both  Ends.      { j  beam  flange  up 


/  (  Equilateral  triangle,  edge  up b  X  d2  x  . 34 

D'  ji    f  j        \  Equilateral  triangle,  edge  down 6  X  d2  X  . 2 

Both  Ends.      (7  beam,  flange  up.. 6Xd2x.42 

To  Compute  Transverse  Strength   of  a   Solid   Cylinder. 

RULE. — Proceed  as  for  a  rectangular  beam,  and  take  .6  of  Coefficient  or 
of  product. 

A  mean  of  18  results  with  cold  blast  gun  metal,  gave  a  coefficient  for  740  Ibs. 

When  Fixed  at  One  End,  and  Loaded  at  the  Other.  RULE. — Multiply 
weight  to  be  supported  in  Ibs.  by  length  of  cylinder  in  feet ,  divide  product 
by  .6  of  Coefficient  of  material,  and  cube  root  of  quotient  will  give  diameter. 

NOTE. — When  cylinder  is  loaded  uniformly  throughout  its  length,  cube  root  of 
half  quotient  will  give  diameter 

EXAMPLE.— What  should  be  diameter  of  a  cast-iron  cylindrical  beam  of  gun-metal, 
8  ins.  in  length,  to  break  at  15000  Ibs.  ? 


/i 

V 


=2.8a  ins. 


6X740  V    444 

When  Fixed  at  Both  Ends,  and  Loaded  in  Middle.  RULE.—  Multiply 
weight  to  be  supported  in  Ibs.  by  length  of  cylinder  between  supports  in 
feet  ;  divide  product  by  .6  of  Coefficient  of  material,  and  cube  root  of  one 
sixth  of  quotient  will  give  diameter. 

NOTE.—  When  cylinder  is  loaded  uniformly  along  its  length,  cube  root  of  half  the 
quotient  will  give  diameter. 

EXAMPLE.—  What  is  the  diameter  of  a  cast-iron  cylinder  of  gun-metal,  2  feet  be- 
tween supports,  that  will  break  at  35  964  Ibs.  ? 


Mean  results  of  cylinder  and  square  bars  gave  444  and  740  Ibs.    Hence,  strength 
of  a  cylinder  compared  to  a  square  is  as  444  to  740  or  .6  to  i. 


Then  =  4795,  ft,. 

To  Conap-ute   Diameter   of  a    Solid  Cylinder  to  Support 

a   given    Weight. 

When  Supported  at  Both  Ends,  and  Loaded  in  Middle.  RULE.—  Multiply 
weight  to  be  supported  in  Ibs.  by  length  of  cylinder  between  supports  in 
feet  ;  divide  product  by  .6  of  Coefficient  of  material,  and  cube  root  of  one 
fourth  of  quotient  will  give  diameter. 

I 


STRENGTH   OF   MATERIALS. — TRANSVERSE.          80$ 

NOTE.— When  cylinder  is  loaded  uniformly  along  its  length,  cube  root  of  half  the 
juotient  will  give  diameter. 

EXAMPLE. — What  is  diameter  of  a  cast-iron  gun-metal  cylinder,  i  foot  between  its 
Supports,  that  will  break  at  48000  Ibs.  ? 


Rectangular.    (D.  K.  Clark.) 

{x)  Loaded  at  Middle.    — ~  =  W.      (2)  Loaded  at  One  End. 
Cylindrical. 

(3)  Loaded  at  Middle.    5'5f  d2  =  W.    (4)  Loaded  at  One  End.    - 

I  W 

W  representing  ultimate  stress  in  tons. 
Above  Coefficients  are  for  iron  of  a  tensile  strength  of  7  tons  per  sq.  inch. 

d)      (2)     (3)     (4)  d)     (2)      (3)     (4) 

For  12  tons  put  13.8    3.4     9.4    2. 


Hence,  for  8  tons  put    9.2    2.3    6.3    z.6 
9       u        10.4    2.6    7.1    1.8 


xi. 5    2.9 
12.7    3.2 


It 


13  14.5      3.6      10.2 

14  16       4        ii        2.8 

15  "        17-2    4-3    «-8 


To  Compute  .Destructive  "Weiglit,  or  Loads  that  may  "be 
Tborne  t»y  "Wrouglit-iron  Rolled.  Beams  and  Q-irders, 
or  Riveted.  Tubes  of  various  Figures  and  Sections. 

Supported  at  Both  Ends.    Load  applied  in  Middle. 

When  Section  of  Beam  or  Girder  is  that  of  any  of  the  Figures  in  follow- 
ing Table.    RULE. — Divide  product  of  area  of  section,  depth,  and  Coefficient 
for  girder,  etc.,  from  following  Table,  by  length  between  supports  in  feet, 
and  quotient  will  give  destructive  weight  in  Ibs. 
NOTE.— The  Coefficients  given  are  based  upon  experiments  with  English  iron. 

Solid    Beams. 

ILLUSTRATION. — What  load  will  destroy  a  wrought- iron  grooved  beam  of  following 
dimensions,  10  feet  in  length  between  supports,  and  loaded  in  its  middle? 

Flanges,  5.7  X  .6  inch;  Web,  .6  inch;  Depth,  11.75  ins. ;  Area,  13.34  S(l-  jns« 
Assume  Coefficient  4638  as  for  like  case  (n)  in  following  table,  page  806. 

13.34X11.75X4638      726983 

-"To—       -  =  -^  =  72698.30*. 

Ultimate  stress  for  such  a  beam  by  experiment  was  estimated  at  97997  Iba 

Formulas  of  Various  Authors  give  following  Restdts: 
D.  K.  CLARK.     g,(4  <*+^555  a)  =  W.    a  representing  area  of  section  of  lower 

flange,  a'  area  of  section  of  web,  less  one  flange,  d  depth  of  beam,  less  average  depth 

if  one  flange,  all  in  ins.,  I  length  in  feet,  and  W  ultimate  destructive  weight  in  tons. 

This  formula  is  based  upon  the  assumption  that  the  beam  has  lateral  support. 


...75-6  (4  X  5-7  X  .6  +  I..55  X  ...75-6  X  .«)  =  iap  =  3>78>  which  ._ 
=  89 107  Ibs. 
MOLESWORTH.     4  °  *  **'  =  W.     C  =  7616  Hw. ,  and  /or  b  d2  put  bd'  —  ab'  d'3. 

6  =  5.7  :  fe/=::5-7— •  -6-T-2  =  2.55  :  ^  =  11.75  :  d' =  11.75  —  -6 X  2=10.55  : 
and  5.7  x  u-752  —  2  X  2.55  x  10. 55^  =  786.94  —  567.63  =  219.31. 
Then,  4X7616x219.3x^6781060^  ^ 

IO  X  12  I2O 

6  and  d  representing  exterior  and  V  and  d'  interior  dimensions,  and  I  length  all  in  ins. 
Fairbairn's  formula  would  give  a  result  less  than  half  of  the  first,  and  Hodgkin- 
son's  alike  to  that  of  Molesworth. 


8o6 


PLATE    AND    BOX    GIRDERS. 


Steel    Plate    Grirders.  Steel    Box    Grirders. 

Safe  Loath  in  Tons  of  2000  Ibs.     Uniformly  Distributed. 

Tensile  Strength,  15000  Ibs. 
Plate.  Carnegie  Steel  Co.  33 ox. 


SOX.  5  Ins.         33X.G  Ins. 

SOX.  £3  Ins.         33X.C5  Ins. 

\Vet>  Plate. 

Welo  Plates. 

Flanges,  12  X  .375  ins.    Anglet,  5  X  .5 

Flanges,  16  X  .375     Flanges,  20  X  .4375 
ins.                             ins. 

X  3.5  ins. 

Angles,  3.5  X  3-5  X  -5  ins. 

c" 

LOAD. 

c* 

LOAD. 

LOAD. 

(=  S. 

LOAD. 

ill 

5  • 

{!| 

ill 

^  „• 

||| 

§2  i 

'o    . 

||1 

*"• 

fii 

ijj 

%*i 

S|| 

111 

s'J! 

||| 

^fl 

flf 

lj| 

WJ 

ill 

H-,  0  o 

'-'  °  2 

Feet. 

Tons. 

Tons. 

Feet. 

Tons. 

Tons. 

Feet. 

Tons. 

Tons. 

Feet. 

Tons. 

Tons. 

20 

93.67 

4.62 

20 

105.82 

5  08 

20 

II2.6 

6.61 

2O 

150.2 

9.17 

21 

89.22 

4.38 

21 

100.78 

4.85 

21 

107.24 

6-3 

21 

1  43-  1 

8-75 

22 

85-15 

4.19 

22 

96.2 

4.62 

22 

102.37 

6 

22 

136-5 

8-33 

23 

81.46 

4 

23 

92.01 

4.42 

23 

97.92 

5-75 

23 

130.6 

7-96 

24 

78.07 

3-83 

24 

88.  1  8 

4-23 

24 

93.83 

5-52 

24 

125.2 

7-64 

25 

74-94 

3-68 

25 

84-65 

4.06 

25 

90.08 

5-3 

25 

1  20.  i 

7-33 

26 

72.06 

3-54 

26 

81.39 

3-91 

26 

86.62 

5-09 

26 

"5-5 

7.06 

3 

69.4 
66.91 

3-42 
3-29 

27 
28 

78.38 
75.58 

3-76 
3-63 

27 
28 

83.41 
80.42 

4.9 
4-73 

27 

28 

III.  2 
107.3 

6.8 
6-54 

29 

64.6 

29 

72.98 

3-5 

29 

77-65 

4-57 

29 

103.6 

6.32 

30 

62.45 

3-°7 

30 

3-39 

30 

75.07 

4.41 

30 

100.2 

6.1 

31 

60.44 

2-97 

31 

68.'  26 

3-29 

31 

72.65 

4.27 

31 

96.9 

592 

32 

58.55 

2.88 

32 

66.14 

32 

70.38 

4-13 

32 

93-9 

5-73 

33 

56.77 

2-79 

33 

64.13 

3*o8 

33 

68.24 

4.02 

33 

91 

5.56 

55-" 

34 

62.24 

2-99 

34 

66.23 

3-9 

34 

88.4 

5-39 

36 

52.04 

2!  56 

36 

58.79 

2-83 

36 

62.56 

3-67 

36 

83-4 

5-og 

38 

49-3 

2.42 

38 

55-7 

2.67 

59.26 

3-48 

38 

79 

4.82 

40 

46.83 

2.31 

40 

52-9 

2-55 

40 

56-31 

3-3 

4° 

4-58 

36X.G  Ins.        4rSX.6QO  Ins. 

36X.S  Ins.          4SX.S  Ins. 

"W"el>. 

\Vebs. 

Flanges,    12  X.  37  5 

Flanges,    14  X  -625 

Flanges,  24  X  .5625 

'Flanges,  30  X  .6875 

ins.     Angles,  5  X 

ins.     Angles,  6  X 

ins.     Angles,  4  X 

ins.     Angles,  5  X 

3.5  X.  5  ™* 

6  X  -625  ins. 

3.5  X.5*f»*. 

3.5X.5*™?. 

20 

"8.35 

5-54 

20 

176.01 

7-74 

20 

213-3 

12.22 

20 

329-2 

18.29 

21 

112.7 

5.28 

21 

167.63 

7-37 

21 

203.3 

11.65 

21 

3I3-5 

17.41 

22 

107.57 

5-04 

22 

1  60.  02 

7-03 

22 

194.1 

II.  12 

22 

299.2 

16.63 

23 

102.9 

4.82 

23 

153.06 

6-73 

23 

185-5 

10.64 

23 

286.2 

15-9 

24 

98.61 

4-63 

24 

146.68 

6-44 

24 

177.9 

10.2 

24 

274-3 

15.24 

25 

94.66 

4-44 

25 

140.82 

6.18 

25 

170.8 

9.78 

25 

263.3 

14.63 

26 

91.02 

4.27 

26 

135.39 

5-95 

26 

164.3 

9-44 

26 

253-2 

14.07 

27 

87.65 

4  ii 

27 

130.38 

5-73 

27 

158-1 

9.06 

27 

243.8 

'3-55 

28 

84-53 

3.96 

28 

125-73 

5-52 

28 

152-4 

8-73 

28 

235.2 

13.06 

29 

81.61 

3.82 

29 

121.38 

5-34 

29 

147.2 

8-43 

29 

227 

12.61 

30 

78.89 

3-7 

30 

"7.35 

5-17 

30 

142.3 

8-15 

3° 

219.5 

12.18 

31 

76-34 

3-58 

31 

113-56 

4.98 

31 

137-7 

7.88 

31 

212.3 

11.79 

32 

73-96 

3.46 

32 

III.OI 

4-85 

32 

133-4 

7-65 

32 

205.7 

11.42 

33 

71.72 

3.36 

33 

106.67 

4-7 

33 

129.3 

7.42 

33 

199.5 

10.08 

34 

69.61 

3-27 

34 

103-55 

4-55 

34 

125-5 

7-2 

34 

193.6 

10.75 

36 

65-75 

3-07 

36 

97.78 

4-3 

36 

118.6 

6.81 

36 

182.9 

10.  15 

38 

62.28 

2.91 

38 

92.64 

4-07 

38 

112.4 

6-44 

38 

173.2 

9.62 

40 

59-14 

2-77 

40 

88 

3-87 

40 

106.7 

6.12 

40 

164.5 

9.14 

BUCKLING.— To  arrest  the  buckling  of  these  girders,  strips  of  plate,  termed  Fillers,  are  set  vertical 
on  the  outer  sides  only  of  a  web  plate,  together  with  other  vertical  angles,  termed  Sti/eners,  both  of 
which  are  riveted  to  the  web  plate,  and  both  of  these  additions  are  set  at  intervals,  dependent  upon  the 
length  of  the  girder  and  the  character  of  its  stress. 


STRENGTH    OF    MATERIALS. TRANSVERSE.          8O/ 

Rolled    Steel    Beams. 

Safe  Load  for  One  Foot,  Uniformly  Distributed  and  Supported  Sidewise. 
Carnegie    Steel    Co.,    P»ittst>u.rg,    !>a. 


Index. 

Depth. 

Designation. 

Wi 

Flange. 

ith. 
Web. 

Area. 

Section. 

Weight 
Foot. 

Loa 
Tensile 
perSq 
12500 

ds. 
Strength 
.  Inch. 
16000 

Ins. 

Ins. 

Ins. 

Sq.  Ins. 

Lbs. 

Lbe. 

Lbs. 

B77 

3 

Light 

2.423 

.263 

1.91 

6-5 

15000 

19  loo 

Heavy 

2.521 

.361 

2.21 

7-5 

16200 

20700 

Standard 

2-33 

•'7 

1.63 

5-5 

13800 

17600 

623 

4 

Light 

2-733 

.263 

2-5 

8.5 

26  500 

33900 

Heavy 

2.88 

.41 

3.09 

10.5 

29800 

38100 

Standard 

2.66 

.19 

2.21 

7-5 

24900 

31800 

B  21 

5 

Light 

3-147 

•357 

3.6 

12.25 

45400 

58100 

Heavy 

3-294 

•504 

4-34 

'4-75 

50500 

64600 

Standard 

3 

.21 

2.87 

9-75 

40300 

51  600 

B  19 

6 

Light 

3-452 

•352 

4-34 

14-75 

66600 

86300 

Heavy 

3-575 

•475 

5-07 

17.25 

72800 

93100 

Standard 

3-33 

.23 

3-6i 

12.25 

60500 

77500 

Bi7 

7 

Light 
Heavy 

3-763 
3.868 

•353 
•458 

5-15 
5-88 

17-5 

20 

93300 
100400 

119400 
128600 

Standard 

3-66 

.25 

4.42 

15 

86300 

110400 

Bis 

8 

Light 

4.087 

•357 

6.03 

20.5 

126200 

161600 

Heavy 

4.271 

•541 

7-5 

25-5 

142600 

182  500 

Standard 

4 

.27 

5-33 

18 

118500 

15170° 

Bi3 

9 

Light 

4.446 

.406 

7-35 

25 

170300 

217000 

Heavy 

4.772 

•732 

10.29 

35 

207  ooo 

265000 

Standard 

4-33 

.29 

6-31 

21 

157300 

20  1  300 

B  ii 

10 

Light 

4-805 

•455 

8.82 

30 

223600 

286300 

Heavy 

5.099 

•749 

11.76 

40 

264  500 

338500 

Standard 

4.66 

7-37 

25 

203500 

260500 

B    9 

12 

Light 

5.086 

^436 

10.29 

35 

317000 

405800 

Standard 

5 

•35 

9.26 

299700 

383700 

B    8 

12 

Light 

5.366 

.576 

13.24 

45 

396800 

5o79oo 

Heavy 

5.612 

.822 

16.18 

55 

445800 

57o6oo 

Standard 

5-25 

.46 

11.84 

4° 

373500 

478  loo 

B    7 

15 

Light 

5.55 

.46 

13-24 

45 

506400 

648200 

Heavy 

5-746 

•656 

16.18 

55 

567  800 

726800 

Standard 

5-5 

.41 

12.48 

42 

490800 

628300 

B    5 

15 

Light 

6.096 

.686 

19.12 

65 

706700 

904600 

Heavy 

6.292 

.882 

22.06 

75 

768000 

983000 

Standard 

6 

•59 

17.67 

60 

676600 

866100 

B    4 

15 

Light 

6-479 

.889 

25 

85 

908600 

i  163000 

Heavv 

6-774 

1.184 

29.41 

IOO 

I  000600 

i  280  700 

Standard 

6.4 

.81 

23.81 

80 

883900 

i  131  300 

B8o 

18 

Light 

6.095 

•555 

17-65 

60 

779600 

997700 

Heavy 

6.259 

.719 

20.59 

70 

853000 

1091  900 

Standard 

6 

.46 

!5-93 

55 

736700 

943000 

B    3 

20 

Light 

6-325 

-575 

20.59 

70 

I  016600 

I  301  200 

Heavy 

6-399 

•649 

22.06 

75 

1  057  400 

i  353  5oo 

Standard 

19.08 

65 

974700 

i  247600 

B      2 

20 

Light 

7-063 

.663 

25 

85 

I  257  200 

1609300 

Heavy 

7.284 

.884 

29.41 

IOO 

1379800 

i  766  loo 

Standard 

7 

.6 

23-73 

80 

I  222  IOO 

i  564  300 

B    i 

24 

Light 

7.07 

•57 

25 

85 

I  505  900 

i  927  600 

Heavy 

7.254 

•754 

29.41 

IOO 

I  653  300 

2115900 

Standard 

7 

•5 

23.32 

80 

1449900 

1855900 

Index  refers  to  Illustration  in  Catalogue. 

For  safe  load  of  IRON  deduct  25  per  cent. 

For  permanent  stress,  absolutely  free  from  vibration,  a  greater  strain  would  be 
allowable,  and,  contrariwise,  if  the  stress  is  vibrative  or  mainly  that  of  a  live  load, 
the  loads  here  given  should  be  relatively  reduced. 


8o8 


STRENGTH    OF    MATERIALS. — TRANSVERSE. 


A  difference  of  25  per  cent.  In  either  direction  should  be  made,  according  to  char- 
acter of  load  to  be  supported  or  stress  to  be  borne. 

Elastic  Transverse  Strength  of  Wrought-iron  Bars  is  about  45  per  cent,  of  their 
transverse  strength,  of  Solid  rolled  beams,  50  per  cent. ;  and  of  double  -  headed 
rails,  46  per  cent,  of  their  transverse  strength ;  of  Fagersta  Steel,  56  per  cent,  of 
its  transverse  strength ;  of  double-headed  Steel  rails,  47  per  cent. ;  of  Bessemer 
Steel,  37.5  to  48  per  cent. ;  and  of  Steel  flanged,  68  per  cent. 

Transverse  strength  of  Solid  Cast-iron  Beams  or  Girders  is  about  50  per  cent,  of 
ultimate  strength. 

NOTE.— Actual  breaking  weight  of  a  10.5  ins.  beam  of  New  Jersey  Steel  and  Iron 
Co.,  weight  35  Ibs.  per  foot,  for  a  length  of  span  of  20  feet,  is  60000  Ibs. 

Rolled.    Steel    Deck    Beams. 

Safe  Load  for  One  Foot,  Uniformly  Distributed,  Supported  Sidewise. 


Depth. 

Web. 

Flange. 

Area. 

Add  to  Web 
for  each 
Ib.  increase. 

Weight. 

Loa 
Tensile 
perSq 

I2OOO 

ds. 
Strength 

16000 

Ins. 

Ins. 

Ins. 

Sq.  Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

6 

2.8 

4.38 

4.1 

.049 

14.1 

48800 

65  100 

6 

4-3 

4-53 

5 

17.16 

57600 

76800 

7 

3-1 

4.87 

5-3 

.042 

i8.ii 

77300 

103000 

7 

5-4 

6.9 

23.46 

93400 

124600 

8 

3.1 

5 

5-9 

•037 

20.15 

97400 

1  29  800 

8 
9 

4-7 
4-4 

4-94 

•033 

3*. 

112600 
141  800 

150  100 

189  100 

9 

5-7 

5-07 

8.8 

30 

156400 

208  500 

10 

3-8 

5-25 

8 

.029 

27.23 

169600 

226  100 

10 

6-3 

5-5 

10.5 

— 

35-7 

205  600 

274100 

"•5 

4.2 

9-5 

.026 

32-2 

221  000 

294700 

"•5 

5-5 

5-3 

10.9 

— 

37 

244800 

326500 

Steel    Bulb    Angles. 

5 

.31 

2-5 

2.94 

— 

10 

32500 

43300 

6 

.31 

3 

3-62 

— 

12.3 

45300 

60400 

6 

•38 

3 

4.04 

— 

13-75 

52  800 

70400 

6 
7 

.50 
•34 

3 
3 

4.71 

— 

17.2 
16 

60400 
69600 

80500 

92  800 

7 

•44 

3 

5-37 

— 

18.25 

76700 

IO2  3OO 

8 

.41 

3-5 

5.66 

— 

19.23 

93600 

124800 

9 

•44 

3-5 

6.41 

— 

21.8 

115700 

154200 

10 

.48 

3-5 

7.8 

— 

26.5 

158800 

211  700 

10 

.63 

3-5 

9.41 

— 

32 

172500 

23OOOO 

Operation    of  Tables. 
To   Compute  Depth  of  a  Beam  to  Support  a  Uniformly 

Distributed.    Load. 

RULE. — Multiply  load  in  Ibs.  by  length  of  span  in  feet,  and  take  from 
table  the  beam,  load  of  which  is  nearest  and  in  excess  of  product  obtained. 

ILLUSTRATION. — What  should  be  depth  of  a  steel  beam  to  sustain  with  safety  a 
uniformly  distributed  load  of  30000  Ibs.,  over  a  span  of  15  feet? 

30000  X  15  =  450000,  which  is  load  for  a  heavy  beam  12  ins.  in  depth. 
Weight  of  beam  should  be  added  to  load. 

Inversely.— If  the  load  is  required,  divide  load  in  table  by  span  of  beam  in  feet, 
and  subtract  weight  of  beam. 

To    Compute    Deflection    of  Like    Beams. 
RULE. — Divide  square  of  span  in  feet  by  70  times  depth  of  beam  in  ins. 
ILLUSTRATION. — Assume  beam  as  preceding. 


70X12.25      857.5 


22  ., 

=  -—?-=. 262  ins. 


STRENGTH    OP    MATERIALS. TRANSVERSE. 


Comparative    Strength,    and.    Deflection    of   Cast-iron. 
Flanged.    Beams. 


DESCRIPTION  OF  BEAM. 

Comp. 
Strength. 

DESCRIPTION  OF  BEAM. 

Comp. 

Strength. 

Beam  of  equal  flanges 

« 

Beam  with  flanges  as  i  to  4  5 

78 

"  with  only  bottom  flange  .  . 
"  with  flanges  as  i  to  2  
u  with  flanges  as  i  to  4.  ... 

"f* 

•63 

•73 

u  with  flanges  as  i  to  5.  5  ... 
u  with  flanges  as  i  to  6  
"  with  flanges  as  i  to  6.  73  .  . 

:£ 

i 
.92 

Rolled    "Wrought-iron    Bea'fiis— Itinglish. 
Safe  Stress  fa  a  Span  of  10  Feet.     (D.  K.  Clark.) 


Depth. 

Breadth 
of  Flanges. 

Thic 
Web. 

mess. 
Flanges. 

Weight  per 
Lineal  Foot. 

Ultimate 
Strength. 
Loaded 
in  Middle. 

Safe  Stress 

Uniformly 
Distributed. 

Ins. 

Ins. 

Inch. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

3 

2 

-1875 

.2187 

5-5 

2800 

910 

3 

3 

•25 

•3I25 

10 

5600 

i860 

3-i25 

1-625 

•1875 

.2187 

5-5 

2490 

830 

4 

2 

•25 

•3*25 

8 

549° 

1830 

4 

3 

•25 

•375 

12 

8  510 

2830 

4-75 

2 

•25 

•3125 

8 

6940 

2310 

5 

3 

•3I25 

•4375 

13 

13440 

4480 

5 

4-5 

•375 

•  5 

23 

19270 

6420 

5-5 

2 

•375 

•4375 

10 

11880 

3960 

6 

5 

•4375 

•5625 

30 

23830 

7940 

6.25 

2 

•3125 

•4375 

u 

13440 

4440 

6.25 

2.25 

•  3I25 

•375 

18 

13000 

4330 

6.25 

3-25 

•3125 

.4062 

12.5 

17470 

5820 

7 

2.25 

.281 

•375 

14 

14790 

4930 

7 

2.25 

•3125 

•4375 

14 

17020 

5670 

7 

3-625 

•3125 

•4375 

'9 

23300 

7760 

7 

3.625 

•3125 

•5 

J9 

25980 

8660 

8 

2-375 

-3125 

•4375 

15 

20830 

6940 

8 

2-5 

•375 

•375 

15 

21  280 

7090 

8 

4 

•375 

•  5 

21 

34500 

11500 

8 

5 

•375 

•5625 

29 

44800 

H930 

8 

5-125 

•4375 

•5625 

29 

47040 

15680 

9-25 
95 

3-75 
4-5 

•4375 
•375 

.'6875 

24 
30 

41560 
59360 

13850 
19750 

0 

4-5 

•4375 

•5625 

32 

56000 

18660 

o 

4-75 

•4375 

•5625 

32 

58240 

19410 

0 

4-75 

•75 

•  625 

36 

76160 

2539° 

2 

5 

•5625 

•8175 

42 

IOO8OO 

33600 

2 

6 

•5625 

•9375 

56 

136640 

45530 

4 

5-5 

•5625 

•875 

60 

150020 

50000 

4 

6 

•5625 

8175 

60 

152  260 

50750 

6 

5625 

•75 

•8175 

62 

188  160 

62720 

TVr  ought-iron  Rectangular  Grirders  or  Tu.~bes.  (Riv'd.) 

Supported  at  Both  Ends.    Loaded  in  Middle. 

— - —  =  W.  A  representing  area  of  section  in  sq.  ins.,  d  depth  in  ins.,  I  length  be- 
tween supports  in  feet,  and  W  destructive  weight  in  Ibs. 

ILLUSTRATION.— What  is  the  destructive  weight  of  a  rectangular  girder,  35.75  ins. 
in  depth  by  24  in  breadth,  metal  .75  inch  thick,  and  length  between  supports  45  feet  ? 

Assume  C  or  coefficient  =  37  oo,  as  per  case  (18)  in  preceding  table,  page  806, 
andaiea  =  87.375  ins. 

Then  8?-375  X  35-75  X  37QQ  =  '»  557  528  =  ^  ^  ^ 

By  experiment  it  was  257  080  Us.     By  Inversion  77-7  =  A,  and  — —  =  d. 

\j  a  A  O 

HODGKINSON'S  formula  would  give  a  result  of  259373  Ibs.,  and  MOLESWOBTH'S 
303907  Ibs. 


8lO         STRENGTH    OF    MATERIALS.  —  TRANSVERSE. 

TJneqvially    Loaded.    Beains,  etc. 
I3  W 

-  =  w.    I  representing  length  between  supports,  and  m  and  n  distances  from 

points  of  support,  all  in  like  denomination,  and  W  and  w  destructive  and  safe  weights, 
also  in  tike  denomination. 

To   Compute   .Destructive  Weight  and  Area  of  Bottom 
IPlate. 

—  -  —  =  W;     -^rr  =  A;    5Z?d  ••!-  =  A.     A  representing  area  of  plate  in  sq. 

ins.,  d  and  I  depth  and  length,  in  and  n  distances  of  ioad  at  other  points  than  in 
middle,  all  in  feet,  and  W  weight  in  Ibs. 

NOTE.—  Sufficient  metal  should  be  provided  in  sides  to  resist  transverse  and 
shearing  stress,  and  in  upper  flange  to  resist  crushing. 

ILLUSTRATION.—  What  area  of  wrought  iron  is  necessary  in  bottom  plate  of  a  rec- 
tangular tubular  girder,  3  feet  in  depth,  supported  at  both  ends,  and  loaded  in  middle 
with  130000  Ibs.  ? 

C,  ascertained  by  experiment  for  destructive  stress,  180000  Ibs.,  and  area  7.  i  sq.ins. 

130000X30 

—  ~  -  —  7.  22  sq.  ins. 
180000X3 

•Wrougnt-iron.    Cylindrical   Beams   or   Tubes. 

-  =  W.        ILLUSTRATION.  —  What  is  destructive  weight  of  a  cylindrical  tube, 
12.4  ins.  in  diameter,  .131  inch  in  thickness,  and  10  feet  between  its  supports? 

Area  of  metal  =  5.05  sq.  ins.,  and  C  =  2856,  as  in  the  igth  case  of  table,  page  806. 
Then  5.05X12.4X2856  =     884  2  ^ 
10 

D.  K.  CLARK.     ^-^  -  =  W.    d  representing  diameter,  t  thickness  of  metal,  and 
I  length,  all  in  ins.,  S  tensile  strength  of  metal  per  sq.  inch,  and  W  weight,  both  in  Ibs, 


MOLESWORTH'S  formula  gives  a  result  of  23  286.1  Ibs. 

"Wrouglit-iron   Elliptical  Beams   or   Tubes. 

-  -.—  W.      ILLUSTRATION.—  -Assume  diameter  of  tube  9.75  and  15  ins.,  metal 
143  inch  in  thickness,  and  distance  between  supports  10  feet. 
A  =  5.56  sq.  ins.    C  =  3147,  as  per  case  (20)  in  preceding  table,  page  806. 
Then  S.56XI5X3I47  =  ^^?  =  26  245.9  «*• 

10  10 

D.  K.  CLARK.     *'57  *  —  —  —  -  -  =  W.    6  and  d  representing  conjugate  and  trans- 

verse  diameter,  I  length  between  supports,  t  thickness  of  metal,  all  in  ins.,S  tensile 
strength  of  metal  per  sq.  inch,  and  W  destructive  weight,  buth  in  Ibs. 


NOTE.  —  B.  Baker,  in  his  work  on  Strength  of  Beams,  etc.,  London,  1870,  page  26, 
shows  that  ordinary  method  of  computing  transverse  strength  of  a  hollow  shaft  by 
difference  of  diameter  alone  is  erroneous,  in  consequence  of  Joss  of  resistance  to 
flexure  in  a  hollow  beam. 

Grirders   and  Beams  of  TJnsymmetrical   Section. 

4        =W.    S  representing  tensile  resistance  of  metal,  and  W  destructive  weight, 

both  in  Ibs.,  d  distance  between  centres  of  compression  and  extension,  or  crushing  and 
tensile  resistances,  in  ins.,  and  I  length  between  supports,  in  feet. 
NOTE.—  To  ascertain  d,  see  Rule,  page  819, 


STRENGTH    OF   MATERIALS.  -  TRANSVERSE.          8ll 

ILLUSTRATION.—  Dimensions  of  a  rolled  wrought-iron  girder,  n  feet  in  length  be- 
tween its  supports,  is  as  follows  : 
Top  flange  ................  2.5  X  i  inch,  j  Bottom  flange  ............  4  X  .38  inch. 

Web  ......................  325       **     j  Depth  ...................  7  ins. 

What  is  its  destructive  weight  ? 

d  =  5.22  ins     S  assumed  at  45000  IDS.    Then  4  X  4f^°°*  5'**  =  7118.18  Ibs. 

Strength  of  Riveted  Beams  or  Girders,  compared  with  Solid,  is  less,  and  deflec- 
tion is  greater 

"Wrought-iron.  Inclined  .Beams,  etc. 

—  —  =  w     L  and  I  representing  lengths  or  inclination,  and  horizontal  line,  in  like 

denominations,  and  W  and  w  destructive  and  safe  weights  on  horizontal  line  and  in- 
clination, also  in  like  denominations. 

Plate  Grirders. 

—  -  —  =  W.    A  representing  section  in  sq.  ins.  ,  d  depth  in  ins.t  and  I  length  be- 
tween supports  in  feet. 
ILLUSTRATION     What  load  will  destroy  a  wrought-iron  plate  girder  or  beam  of 
following  dimensions,  10  feet  in  length  between  its  supports? 
Top  flange  ..............  4.5  X  .375  inch. 

Bottom  flange  .........  4-5  X.  375    " 

Angle  pieces  ...........  2    X  -3125  u 

Area  of  Section  =  13  sq.  ins. 
Assume  coefficient  of  5180  as  per  case  (14)  in  preceding  Table,  page  806. 

Then  I3  X  *+**  X  5l 

MOLESWORTH.     —  =  S.     L  representing  load  equally  distributed,  and  S  stress  on 

8  d 
centre,  both  in  tons,  and  d  effective  depth  of  girder  in  feet. 

By  actual  experiment  L  =  48  tons  for  16.5  feet  between  supports;  hence, 
10:16  5:148  79.2  tons  —  39.  6  when  supported  in  middle,  and  14.25  ins.  =  i.iSjsfset. 

Then  Q39'6  x  10  _  39$  _.  ^^  which  x  224Q  _  ^  ^  2  ^ 
8  X  1-1075      9.5 

D.  K  CLARK.     d  '4q+^»55«)  _  w      d  representing  &>pfa  Of  gira€r  or  beam 

less  depth  of  lower  flange  in  ins.,  a  and  a'  areas  of  sections  oj  bottom  flange  and  of 
web,  at  its  reputed  depth,  both  in  sq.  ins.,  and  I  length  between  supports  in  feet. 
d  =  14.  25  —  .375  =  13.875  ins.     a  —  3,  and  a  =  5  sq.  ins. 


I 


Widthof  web 375  inch. 

Depthofweb 13.5     ins. 

Depthofbeam 14.25     •• 


Mr  Clark  assumes,  however,  that  for  girders  of  like  construction  the  destructive 
stress  should  be  taken  at  two  thirds  of  that  deduced  by  the  formula. 

I  Girders  or  Beams  mthout  Upper  and  Lower  Flanges. 

ILLUSTRATION.  —  Assume  angles  2.125  X  .28  above,  2.125  X  .3  below,  web 
.25,  depth  7  ins.,  and  length  between  supports  7  feet 

Area  of  section  =  6.35  sq.  ins.,  and  C  =  3840,  as  per  case  (15)  in  preceding  Table, 
page  Sod. 


Approximate.    -  ^  --  —  W.    a  representing  area  of  sections  of  upper 

and  lower  angles,  a'  area  of  section  of  web  for  total  depth,  both  in  sq.  ins.,  d  depth  of 
girder  in  ins.  ,  and  W  load  or  stress  in  Ibs. 


8  12          STRENGTH    OF   MATERIALS.  —  TRANSVERSE. 

a  =  4.6  sq.  ins.  ,  and  a'=  7  X  -25  =  1.75  sg.  ins. 


Then  -  -  -  -  =  21—  =  I3.687,  which  x  2240  =  30658.8  Ibs. 

IRON  AND  STEEL  RAILS. 

Symmetrical    Section. 

To   Compute   Transverse    Strength.    (D.  K.  dark.) 


-±  -  !l_-  -  -  =  W,  and  -  —  -^  -  -  =  S.  S  representing  ten 
(4a  —  +  i.i55<<»2) 

ftte  strength  in  Ibs.  or  tons  per  sq.  inch,  a  area  of  one  head  or  flange  exclusive  of  cen- 
tral portion  composing  web,  in  sq.  ins.,d'  depth  or  distance  between  centres  of  heads, 
d  depth  of  rail,  t  thickness  of  web,  I  distance  between  supports,  all  in  ins.,  and  W 
weight  in  Ibs.  or  tons,  alike  to  S. 

ILLUSTRATION  i.—  What  is  destructive  weight  of  a  wrought-iron  double-headed 
rail,  5.4  ins.  deep,  having  a  web  of  .8  ins.,  an  area  of  head  of  i.o  sq.  ins.,  distance 
between  centres  of  its  heads  4.2  ins.,  and  between  its  supports  5  feet? 
S  assumed  at  50  ooo  Ibs. 


soooo 


= 

5  X  12  60 

43  125  Ibt. 

2.  —What  is  destructive  weight  of  a  Bessemer  steel  double-headed  rail,  5.4  ins. 
deep,  having  a  web  of  .75  inch,  an  area  of  head  of  2  sq.  ins.,  and  distance  between 
heads  4.  2  ins.? 

S  assumed  at  80000  Ibs. 


W^A     *         -r  ,.+         80000X51.39 

Then  — -= 3    **  =  68  520  Z6s. 

5X12  60 

NOTE.— Transverse  strength  of  Bessemer  Rails  increases  very  generally,  in  direct  proportion  with 
the  proportion  of  Carbon  in  it. 

TJn  symmetrical    Section. 

'92     =  W.     d"  representing  vertical  distance  between  centres  of  tension 

I  h 

and  compression,  h  height  of  neutral  axis  above  base  of  section,  and  I  length  between 
supports,  all  in  ins.,  and  A  sum  of  products,  obtained  by  multiplying  areas  of  strips 
of  reduced  section  under  tensile  stress,  by  their  mean  distances,  respectively,  that  is, 
the  distances  of  their  centres  of  gravity,  from  the  neutral  axis,  in  ins. 

Bowstring   Grirder. 

To  Compute  Diameter  of  a  "Wrought-iron  Tie-rod  of  an 
Arched   or    33o\vstring    G-irder   of  Cast    Iron. 

j-  =  d.    W  representing  weight  distributed  over  beam  in  Ibs.,  I  length 

4500  X  n> 

between  piers  or  supports  in  feet,  and  h  height  between  centre  of  area  of  section  of 
girder  and  centre  of  rod  in  ins. 

ILLUSTRATION.— Required  diameter  of  tie-rod  for  an  arched  girder,  25  feet  be- 
tween its  piers,  and  30  ins.  between  centres  of  its  area  and  of  rod,  to  safely  support 
a  uniformly  distributed  load  of  25  ooo  Ibs.  ? 

725000X25          7625000        .    , 
—  =  A  / — =  -v/4-62  =  2. 15  ins. 
4500X30     V 135000 

If  two  rods  are  used     Then     /— -  =  1.52  ins.  =  diameter  of  each  rod. 


STRENGTH    OF    MATERIALS. — TRANSVERSE. 


CAST  IRON. 
Transverse   Strength,  of  GJ-irders   and.   Beams. 

(Deduced  from  Experiments  of  Barlow,  Hodgkinson,  Hughes,  Bramah,  Cubitt, 

Tredgold,  and  others.) 

Reduced  to  a  Uniform  Measure  of  One  Foot  in  Length. 
Supported  at  Both  Ends.    Stress  or  Weight  applied  in  Middle. 


SKCTIOX. 

Flanges. 

Web. 

Depth. 

Distance. 

Area. 

Destrncti 
For  Dis- 
tance. 

re  Weight, 
of  One  Foot. 

JW_ 
A<F~" 

Ins. 

Ins. 

Ins. 

Feet.  Ins. 

Sq.  Ins. 

Lbs. 

Lbs. 

/ 

— 

i 

I 

i 

I 

2240 

2240 

2240 

( 



i 
3 

i 
3 

4    6 
13    6 

I 
9 

500 
5080 

2250 
68580 

2250 

2540 

I    J 

—  • 

i 
i 

3 
4 

4    6 
4    6 

3 

4 

5100 
10300 

22950 
4635,0 

2550 
2896 

•Jo 

4       X2 

2 

4 

5 

12 

6720 

33600 

700 

Q 

1.52  X    .78 

,56 

4.07 

4    6 

2-35 

6666 

30000 

3136 

L 

i-5    X    .5 

•5 

3 

3    * 

2 

5208 

16145 

2676 

Y 

i-5   X   .5 

•5 

3 

3    l 

2 

4536 

14062 

2331 

JL 

i-5    X   -5 

•5 

4 

3    * 

X 

7104 

22420 

5475 

T 

'•5   X   .5 

•  5 

4 

3    i 

I 

33- 

10267 

2553 

x 

X-53XX 

•  5 

2.04 

4 

2.6 

4004 

16016 

3019 

H 

a       X    .51 

X 

2.02 

4 

2.59 

2569 

10276 

1963 

• 

- 

— 

2.52 

5 

4.98 

4H3 

20715 

1650 

^ 

- 

- 

2.83 

5 

4 

2988 

14940 

1320 

/ 

2.28  X    .53 

{  '425 

}5,3 

4    6 

2.28 

9503 

42763 

3656 

' 

23.9   X3-I2 

3-29 

36.1 

20 

183-5 

403312 

8  066  240 

1220 

| 

1.76  X    .4 

.29 

5-13 

4    6 

2.82 

6678 

30512 

2077 

[ 

I-74X    .26 
1.78  X    .55 

}  -3 

5-13 

4    6 

2.87 

7368 

33200 

2250 

JLJ 

1.07  X    .3 

2.1     X     -57 

P 

5-13 

4    6 

3-02 

8270 

37215 

2402 

i 

'•54  X    .32 
6.5    X    .51 

}    -34 

5-13 

4    6 

5.41 

21  OO9 

94540 

3406 

" 

2.5    Xi-5 
375X1.4 

ji-25 

8.l8 

IX 

15 

35620* 

39i  853 

3193 

*  Stirling  iron. 


Hence,  —  —  =  W.  A  representing  area  of  section,  d  depth  in  ins.,  I  length  in  feet, 
and  W  destructive  weight  in  Ibs. 

NOTE.—  When  lengths  are  less  than  those  instanced,  breaking  weight  will  be  in- 
creased, in  consequence  of  increased  stability  of  girder. 


8  14          STRENGTH    OF   MATERIALS.  —  TRANSVERSE. 

To  Compute  Transverse  Strength  or  Destructive  Stress 
of  Cast-iron.    Beams    or    Grirclers,  of  various    Figures. 

Supported  at  Both  Ends.     Weight  applied  in  Middle. 

When  Section  of  Beam  or  Girder  is  alike  to  any  of  Examples  given  in 
preceding  Table.  RULE  i.*  —  Divide  product  of  area  of  section  and  depth 
in  ins.,  and  Coefficient  for  girder,  etc.,  from  preceding  Table,  by  length  be- 
tween supports  in  feet,  and  quotient  will  give  breaking  weight  in  Ibs. 

EXAMPLE.  —  Dimensions  of  a  beam,  having  top  and  bottom  flanges  in  proportion 
of  i  to  6,  give  an  area  of  section  of  25.6  sq.  ins.,  a  depth  of  15.5  ins.,  and  a  length 
between  its  supports  of  18  feet;  what  is  its  destructive  weight? 

NOTE.—  In  consequence  of  increased  area  -of  metal  over  case  No.  21  im  Table,  Coef- 
ficient of  3402  is  reduced  to  3300. 

Dimensions.—  Top  flange,  3  X  .75  ins.  ;  bottom,  18  X  .75  a  =  13.5  sq  ins.  ;  web, 
15.  5  X  •  7  a'  =  10.  8  sq.  ins.  ,  and  a'  =  15.  5  —  .  75  =.  14.  75  ins. 

Then  25.6X15.5X3300  =  .309440  =         6  6  lbs 

lo  18 

D.  K.  CLARK.     —  —  -  —  -^—  2—i  =  W.    a  representing  area  of  bottom  flange,  a' 

of  web  at  depth  a'  of  beam,  less  depth  of  bottom  flange  in  sq.  ins.,  I  length  between 
supports  in  feet,  and  W  destructive  weight  in  tons. 


.4-75  (7  X.  3.  .     =  =  which  x  224Q  -m 

3  X  io  54 

HODGKINSON'S  formula  would  give  a  result  of  53491.2  lbs.,  and  MOLESWORTH'S 
54  248.  3  «w. 

RULE  2.  —  From  product  of  breadth  and  square  of  depth  in  ins.  of  rec- 
tangular solid,  the  dimensions  of  which  are  the  depth  and  greatest  breadth  of 
beam  in  its  centre,  subtract  product  of  breadths  and  square  of  depths  of 
that  part  of  the  beam  which  is  required  to  make  it  a  rectangular  solid,  and 
then  determine  its  resistance  by  rule  for  the  particular  case  as  to  its  being 
supported  or  fixed,  etc. 

This  rule  is  applicable  only  in  case  referred  to,  viz.,  when  area  of  section  is  great 
compared  with  area  of  extreme  dimensions. 

Mr.  Baker,  in  case  of  a  hollow  cylindrical  shaft,  where  thickness  of  metal  is  but 
one  eighth  of  extreme  diameter,  computes  result  at  but  .4  of  that  of  a  solid  beam. 
This  is  in  consequence  of  resistance  to  flexure  in  hollow  beam  being  more  than 
proportionally  greater  than  in  solid. 

EXAMPLE.  —  Take  7th  case  from  preceding  Table,  page  813,  for  length  of  one  foot. 

Coefficient  for  cold-blast  iron  =  500. 

Then  1.52  X  4.072  —  1.52  X  2.5i2  X  4  X  500  =  (25.17  —  9.58)  X  2000  =  31  180  lbs. 

Result  as  by  experiment,  30000  lbs. 

NOTE  i.—  These  rules  are  applicable  to  all  cases  where  flange  of  beam  is  as  shown 
in  Table,  and  beam  rests  upon  two  supports,  or  contrariwise,  as  to  position  of  flange, 
when  beam  is  fixed  at  one  end  only. 

2.—  When  case  under  consideration  is  alike  in  its  general  character  to  one  in 
Table,  but  differs  in  some  one  or  more  points,  an  increase  or  decrease  of  metal  is  ob- 
tained by  an  increase  or  reduction  of  the  Coefficient,  according  as  the  differences  may 
affect  resistance  of  beam. 

3.—  The  Coefficients  here  given  are  based  altogether  upon  experiments  with  Eng- 
lish iron. 

*  Utility  of  these  rules  in  preference  to  those  of  Hodgkinson,  Fairbairn,  Tredgold,  Hughes,  and 
Barlow  is  manifest,  as  in  one  case  the  Coefficient  of  the  metal  is  considered,  and  in  the  other  cases  the 
metal  is  assumed  to  be  of  a  uniform  value  or  strength. 

Only  variable  element  not  embraced  in  this  rule  is  that  consequent  upon  any  peculiarity  of  form  of 
section.;  as,  for  instance,  in  that  of  a  Hodgkinson,  or  like  beam,  where  area  of  one  flange  greatly  ex- 
ceeds the  rest  of  section,  and  this  flange  is  other  than  below,  when  beam  rests  upon  two  supports  or  is 
fixed  at  both  ends,  or  than  above,  when  beam  is  fixed  at  one  or  both  ends. 

This  deficiency  is  met  to  some  extent  by  the  three  cases  in  table,  where  proportion  of  flanges  are  i  to 
*,  i  to  3,  and  i  to  6.5. 

t  For  thick  castings  put  7,  and  put  Coefficient  same  as  tensile  strength  of  metal  in  tons  per  sq.  inch. 


STRENGTH    OF   MATERIALS. TRANSVERSE.          815 

Flanged   Hollow   or   Annular   Seams    of   Symmetrical 
Sections.    (D.  K.  Clark.) 

When  Depth  is  Great  Compared  with  Thickness  of  Flanges.— Figs,  i,  2,  and  3, 

d  X  S  (4  a  -j-  1. 155  a') 
i  — i =  W.   a  representing  area  of  one 

flange,  a'  area  of  web  or  ribs,  both  in  sq.  ins.,  d  depth  of 
beam,  less  depth  of  one  flange,  and  I  distance  between  sup- 
ports, both  in  ins.,  S  tensile  strength  of  metal,  and  W 
weight  between  supports,  both  in  Ibs. 

When  Depth  of  Flanges  is  Great  Compared  with  Depth  of  Beam.— Figs. 

4  and  5. 

4.  5.        ^         d'2      . 

— j —    =  "VT.    a  representing  area  of  one  flange  less 

Jjfr      thickness  of  web,  in  sq.  ins.,  t  thickness  of  web,  d'  reputed  depth  or 
distance  between  centres  of  flanges,  and  d  depth  of  beam,  all  in  ins. 

When  Section  of  Circular  or  Elliptic  Beam  is  Small  Compared  with  Diam- 
eter.—Figs.  6, 7,  and  8. 
8.         __ 

'~  =  W. 


b  and  d  representing  mean  breadth  and  depth. 

ILLUSTRATION  i.— Assume  Figs,  i,  2,  and  3,  20  ins.  in  depth,  width  of  flanges  on 
top  and  bottom  ribs  5  ins.,  thickness  of  flanges  and  webs  i  inch,  and  of  sides  of 
Fig.  3  .5  inch;  length  between  supports  10  feet,  and  S  20000  Ibs. ;  what  would  be 
breaking  weight  of  each? 

Then  ao-'Xaoooo(4X5  +  ».»55Xi8)  =  3^000^20  +  2079)  = 

10  X  12  120 

2.— Assume  Figs.  4  and  5, 6  ms.  in  depth,  area  of  flanges  3  ins.,  widths  of  webs  i 
Inch,  and  length  and  S  as  in  preceding  case. 


, 


10        12  120 

3.  —  Assume  Fig.  6  10  ins.  in  diameter.  Fig.  7,  7.  5  ins.  in  depth  and  12  ins.  in  width, 
and  Fig.  8,  12  ins.  in  depth  and  7.5  ins.  in  width,  and  thickness  of  all  metal  i  inch. 

Then,  Fig.  6  *HX  K.°X  .  X  ~ooo  =  6,80000^^,,  »•.,  which  is  .4  of 

IO  X  12  I2O 

that  of  solid  cylinder. 

Figs.  7  and  8  '-S7X(a'  +  7.5->X.X*»ooo  =  6*7850  ^ 

IO  X  12  I2O 

NOTE.  —  For  all  ordinary  purposes,  operation  of  computing  their  strength,  by  first 
computing  that  of  their  circumscribing  figure,  and  then  deducting  from  it  strength 
due  to  difference  between  it  and  section  of  beam  under  computation,  will  be  suf- 
ficiently accurate.  See  Illustration,  page  814. 

If  greater  accuracy  is  required,  see  page  810,  or  D.  K.  Clark's  Manual,  pp.  513-17. 

NOTE.—  To  compute  location  of  neutral  axis  of  beams  of  unsymmetrical  section, 
see  also  D.  K.  Clark,  pp.  514-15. 

*  This  result  agrees  with  deduction  of  Mr.  Baker,  aa  given  by  him  in  his  work  on  Strength  of  Beams, 
etc..  pp.  26-7,  for  hollow  or  annular  beams  of  small  area  of  section  compared  with  that  of  diameter. 
even  up  to  a  thickness  of  metal  of  one  eighth  of  diameter.  He  assigns  their  strength  to  low  M  .4  at 
that  of  solid  cylinder,  in  consequence  of  loss  of  resistance  to  flexure. 


STRENGTH    OF    MATERIALS. — TRANSVERSE. 

Q-eneral    Formulas    for   Destructive   AVeiglit  of  Solid 
Beams    of  Symmetrical    Section.. 

Supported  at  Both  Ends.     Weight  applied  in  Middle. 
Line  of  Neutral  Axis  runs  through  centre  of  gravity  of  section. 

2  a  drS  _  w  and     l  W    =  S.    In  square  beams  for  a  d  put  d3.    a  and  d  rep 

I  zadr 

resenting  area  and  depth  of  section,  r  radius  of  gyration  (half  depth  of  learn  =  i), 
I  length  of  beam  between  its  supports  in  ins.,  W  destructive  weight  in  tons  or  Ibs., 
and  S  tensile  strength  of  material  in  like  tons  or  Ibs.  per  sq.  inch. 

ILLUSTRATION.— Assume  dimensions  of  cast-iron  beams,  Figs,  i,  2,  3,  4,  and  5,  as 
follows,  viz. :  i  and  2,  5  X  5  ins. ;  3,  2.5  X  10;  4,  5.64  diameter;  and  5,  7.25  X  4- 39' 
or  equal  areas;  distance  between  supports  60  ins.,  and  tensile  strength  of  iron  = 
26  ooo  Ibs. 


3't  3>  -5775»  4, 


Areas  of  each  25  sq.  ins.     Radius  of  gyration,  No.  i,  -5775;  2>  '4 
,5l  and  5,  1.43. 

2  X  25  X  10  X  -5775  X  26000 


60 


-  =  125 125  Ibs. 


,.     «  X  25  X  7-07*  X.  4083X26  ooo  =  fc        -. 

60 


4.  For  formula  for  square  beams  substitute  —  —  =  W 


Then  4. 


'5X5.64X26oco 

DO 


=6i  .  and  • 


.7854  X  4-39  X  7-252  X  26000 
60 


=  78  532  Ibs. 


These  formulas  give  a  result  equal  to  a  transverse  strength  for  Cast  iron  of  550  foi 
a  tensile  strength  of  26000  Ibs.,  and  of  Wrought  iron  of  600  Ibs.  for  a  like  strength 
of  50000  Ibs.  (as  per  table,  page  788). 


—  =  W.    C  representing  coefficient  of  strength  of  metal  in  Ibs. ,  b  and  d 
breadth  and  depth  in  ins.,  I  length  in  feet,  and  W  destructive  weight  in  tons. 

R* —  r* 

6.  — - —  4. 7  =  6  d2.    R  and  r  representing  external  and  internal  radius. 


-  =  6  d2.    6'  and  d'  representing  interior  breadth  and  depth. 
!2.          9.  =2  W.    d  representing  depth  or  height. 


10.  6  d2  -{-  2  &'  d'2  =  W.     6  and  d  representing  breadth  and  depth  of  centre  and 
vertical  rib,  and  b'  and  d'  breadth  and  depth  of  horizontal  rib,  external  to  central  rib. 

Values  of  C  550  for  a  tensile  strength  of  Cast  Iron  of  26000  Ibs.  per  sq.  inch,  and 
of  600  for  a  like  strength  of  Wrought  Iron  of  50000  Ibs.,  and  pro  rata. 


Diagonal  of  square. 


t  In  square  beams  <J3  s  a  X  d. 


STRENGTH    OF   MATERIALS. TRANSVERSE.          817 

Flanged  Beams  of  TJnsymmetrical  Section.    (D.K.Clark.) 

„   i  L  1  ?  I. 

4— —  =  W.    S  representing  total  tensile  strength  of  section  in  Ibs.  per  sq.  inch,  d 

vertical  distance  between  centres  of  tension  and  compression  in  ins.,  I  length  in  ins., 
andW  weight  in  Ibs. 

ILLUSTRATION.— If  the  sectional  area  of  a  beam  of  cast  iron  is  5.9  sq.  ins.,  the 
depth  or  distance  between  centres  of  tension  and  compression  5.6  ins.,  distance  be- 
tween supports  5.5  feet,  and  tensile  strength  of  metal  30000  Ibs.  per  sq.  inch. 

Then  4X5.9X30000X5.6  =  ^er0  =  6°°72'7  m' 

STEEL. 
To    Compute    Transverse    Strength   of   Steel    Bars. 

Supported  at  Both  Ends.     Weight  applied  in  Middle. 

—  =  W.    S  representing  tensile  strength  in  Ibs.,  I  length  between  supports 
in  ins. ,  and  W  weight  in  Ibs. 

ILLUSTRATION.— What  is  ultimate  destructive  stress  of  a  bar  of  Crucible  steel 
2  ins.  square,  and  2  feet  between  supports?  8  =  90000  Iba 

„,         1.155  X  90000  X  2»  __  831600  _ 

2  X  12  24 

To  Compute  Section  of  Lower  Flange  of  a  G-irder  or 
Cylindrical  Shaft  of  Cast  Iron  to  Sustain  a  Safe  Load 
in  its  Middle.  (Baker.) 

=  M.    I  representing  distance  between  supports  in  feet,  d  depth  of  girder,  etc. 
G 
in  ins.,  W  weight  in  tons,  C  coefficient,  and  M  moment  of  weight  around  support. 

ILLUSTRATION.— What  should  be  section  of  a  girder,  12  ins.  deep,  to  sustain  a  safe 
load  of  10  tons  in  its  middle,  between  supports  16  feet  apart? 

Stress  assumed  2  tons  per  sq.  inch,  and  Factor  of  safety  4. — — —  =480= M. 

M 
And  1 — o  =  a"    s  representing  stress  assumed  in  tons,  and  a  area  of  section  of 

a  X  Q 
flange  in  sq.  ins.  480 

Then  — =  20  sq.  ins. 

12X2 

For  Rectangular,  IDiagonal,  or  Circular  Beam  or  Shaft. 

General  Formulas  for  Computation  of  Destructive 
\Veight  of  a  Beam  or  Grirder  of  any  form  of  Cross 
Section  and  of  any  Material.  (B.  Baker.) 

Load  applied  at  Middle. 
—      '        =  W.    S  representing  tensile  strength  of  material  per  sq.  inch  in  tons, 

M  moment  of  resistance  of  section  =  product  of  effective  depth  of  girder  or  beam,  and 
effective  area  of  flange  portion  of  section,  in  sq.  ins.,  Q  resistance  due  to  flexure,  I  dis- 
tance between  supports  in  feet,  and  Q'  =  Q  X  thickness  of  web  of  section,  both  in  ins. 

Average  Values  of  Sfor  Various  Materials. 


Cast  Iron 7 

Wrought  Iron 21 


Steel 40  to  50 

"   Plates 35 

32 


Tons. 


Oak 2.5  to  4  5 


Pine 


'3-5 


8i8 


STRENGTH    OF    MATERIALS. TRANSVERSE. 


Substituting  Values  of  S  and  Q  in  a  General  Equation. 


Wrought 
Iron  . 


=1-75 


d* 


W=.5625 


2.  625  to  4.25  — 


=  .  I  tO     l6  y 


d*b 


=  . 08  tO.  14  y 


=.o6tO.IIy 


25  y     =2  tO  3-25  y  =.o8tO.«, 

d  representing  depth  of  a  rectangular  bar,  side  of  a  square,  or  diameter  of  a  round, 
b  breadth  of  a  vertical  bar,  all  in  ins.,  and  I  distance  between  supports  in  feet. 

Moment   of  Resistance. 

Moment  of  Resistance  of  a  cross  section  is  the  static  force  resisting  an  ex- 
ternal force  of  tension  or  compression,  and  it  is  equal  to  moment  of  Inertia, 
divided  by  distance  of  centre  of  effect  of  the  area  of  fibres  which  are  respec- 
tively the  most  extended  or  compressed  from  the  neutral  axis  of  the  section. 


Xo   Comprite   Moment   of  Resistance. 


—  =  M.    I  representing  moment  of  inertia,  and  d  distance  of  centre  oj  effect  of 
area  of  fibres  of  extension  or  compression. 

\Vorlt   of  Resistance. 

Under  a  Quiescent  Load.  —  Intensity  of  Elastic  resistance  increases  uni- 
formly with  total  space  through  which  action  of  stress  operates  ;  hence,  it 
may  be  defined  by  a  triangular  section. 

Consequently,  .5  s  L  =  R.  5  representing  space  passed  through,  L  load,  and  R  re- 
sistance. 

To   Compute    Moment   of  Resistance. 

—  j—  and  —  —  =  R.     C  a  coefficient  =  one  sixth  of  destructive  weight,  I  moment 
n  ti 

of  inertia,  h  height  of  neutral  axis  from  base  of  section,  R  moment  of  resistance,  and 
M  modulus  of  rupture. 

NOTE.  —  Neutral  axis,  for  all  practical  purposes,  is  at  centre  of  gravity  of  any 
section. 

For  Radius  of  Gyration,  see  Centre  of  Gyration,  page  609. 

For  other  rule  for  computation  of  Moment  of  Resistance,  see  Strength  of  Beams, 
B.  Baker,  London,  1870. 

Moment   of  Inertia. 

Moment  of  Inertia  is  resistance  of  a  beam  to  bending,  and  moment  of  any 
transverse  section  is  equal  to  sum  of  products  of  each  particle  of  its  area  into 
square  of  their  distance  from  neutral  axis  of  section. 

B       ILLUSTRATION.  —  If  transverse  section  of  a  beam,  A  B  C  D,  Fig.  i,  is 
"5   8  X  20  ins.,  its  neutral  axis  will  be  at  middle  of  its  depth,  o  r;  divide 
A  B,  o  r,  into  any  number  of  equal  spaces,  as  shown,  then  each  space 
will  be  2X2  =  4  sq.  ins.,  and  the  distances  of  the  centre  of  each 
square  from  neutral  axis  will  be  as  follows  • 


2X2X4Xi2:=i6 
2X2X4X32  =  i44 
2  X  2  X  4  X  52  =  4°o 


4,4. 
5,5- 


2X2X4X7^784 
2  X  2  X  4  X  Q2—  "9^ 

3,  3.    2  X  2  X  4  X  52  =  4°o  2640  x  2  for  low- 

er half  =  5280  =  moment. 

NOTE.  —  If  the  area  of  the  figure  in  illustration  had  been  more  minutely  divided, 
the  result  would  have  approximated  more  nearly  to  the  above  result. 
For  Moment  of  Inertia  of  a  Revolving  Body,  see  Centre  of  Gyration,  page  609. 


STRENGTH    OF   MATERIALS. — TRANSVERSE. 


819 


To  Compute  Moment  of  Inertia  of  a  Solid  Beam.—  Fig.  3. 
ILLUSTRATION.— Take  elements  of  preceding  case. 


Then 


64000 


=  5333-  33  moment. 


Or,  .3  t3  n9  b  —  M.     t  representing  breadth  of  vertical  divisions,  n  number  of  hori- 
zontal divisions  from  plane  of  neutral  axis,  b  breadth,  and  d  depth  of  beam. 

ILLUSTRATION.— Take  elements  of  preceding  case. 

i  =  2,  n  =  s,  and  6  =  8. 
Then  .3X2*Xs*X8  =  2400  X  2  for  lower  half—  4800  =  moment. 

•   Beams  of  Various  Figures.— Figs.  3, 4,  5. 

*  ~13~Jd",  4  aid  5-  -    !=i*I£!  =  ll. 

b'  and  d'  representing  respectively  breadth  lest 
thickness  of  web,  and  depth  less  thickness  of  flanges. 


r  representing  radius,  t  transverse  and  c  conjugate  diameters,  and  s  side. 

To  Compute  Common  Centre  of  Q-ravity  and  Vertical 
Distance  "bet-ween  Centres  of  Crushing  and  Tensile 
Stress  of  a  Q-irder  or  Beam. 

RULE.  —  Multiply  surface  of  section  of  each  part  or  figure  composing 
whole,  by  distance  of  its  centre  from  centre  of  one  of  the  two  extreme  parts 
or  figures,  as  • ;  divide  sum  of  their  products  by  sum  of  surfaces  of  sec- 
tion, and  result  will  give  distance  of  common  centre  of  gravity  from  centres 
of  each  extreme  part  or  figure. 

EXAMPLE.— Take  annexed  figure. 

2.5     X 1X0  =2.5     xo      =    .o 

•325  X  (^~  +  -M  =  -325  X  3-31  =  1.076 

.38   X4X  (~*  +  5-62 +  -H  =  1.52   X  6.31  =9. 591 
4-345  10.667 

Dividing  10.667  by  4. 34$  =  2- 455  =  distance  of  common  centre  from  centre  of  upper 
part. 

1.52   Xo  =1.52   Xo      =     .o 


Above 


Below 


325  X  5-62  X  +  v       =1-826X3      =  5.478 


2-5 


(-1  +  5.62  +  '-!^  =2.5     X6.3i  =  i5.775 

5-846  21.253 

Dividing  21.225  by  5.846  =  3.631  =  distance  of  common  centre  from  centre  of  lower 

part. 
Hence,  3.631  +  —  =  3.821  =  distance  of  common  centre  from  bottom,  and  3.631  + 

1.652  =  6.283  =  distance  between  centres  of  gravity. 


82O         STRENGTH    OF   MATERIALS. TRANSVERSE. 

To  Compete  Neutral  Axis  of  a  Beam  of  TJnsymmetrical 
Section.— Figs.  3,  4,  &,  6,  7,  8,  and.    9.     (D.  K.  Clark.) 

OPERATION. — Divide  section  as  reduced  into  its  simple  elements,  and 
assume  a  datum-line  from  which  moments  of  elements  are  to  be  computed. 
Multiply  area  of  each  element  by  distance  of  its  own  centre  of  gravity  from 
datum-line,  to  ascertain  its  moment.  Divide  sum  of  these  moments  by  to- 
tal reduced  area ;  and  quotient  is  distance  of  centre  of  gravity  of  reduced 
section,  or  of  neutral  axis  of  whole  section,  from  datum-line. 
ILLUSTRATION.— Fig.  8  annexed  is  12  ins.  deep,  12  ins.  wide,  and  i  inch  thick. 
Extend  web,  c  d,  to  the  lower  surface  at  d'  and  d",  leaving  5.5  ins. 
of  web,  a  d'  and  d"  6,  on  each  side.  Reduce  this  width  in  the  ratio 
of  1.73  to  i,  or  to  (5.5-:- 1. 73  —  )  3.2  ins.,  and  set  off  d'  a'  and  d"  6' 
\l  each  equal  to  3. 2  ins.  Then  reduced  flange,  a'  6',  is  (3. 2  X  2  =  6. 4  -f- 

V.  ..g        i  — )  7.4  ins.  wide,  and  reduced  section  consists  of  two  rectangles, 
a'  b'  and  c  d.    Assume  any  datum  line,  as  «/,  at  upper  end  of  sec- 
tion, and  bisect  depths  of  rectangles,  or  take  intersections  of  their 
diagonals  at  g  and  o,  for  their  centres  of  gravity.     Distances  of  these 
__  from  datum  line  are  5.5  and  11.5  ins.  respectively,  and  areas  of  the 
a'd'<Tif   &  rectangles  are  n  X  i  =  n  sq.  ins.,  and  7.4  X  i  =  7.4  sq.  ins. 
Then,    cd  =  n    x   5-5=  6<>-5 
a'b'=.  7.4  X  11.5=  85.1 

18.4  145.6  =  7.91  ins. 

Showing  that  centre  of  gravity  of  reduced  section,  being  neutral  axis  of  whole 
section,  is  7.91  ins.  below  upper  edge,  in  line  ii.  Centre  of  gravity  of  entire  section 
at  •  ,  it  may  be  added,  is  8.65  ins.  below  upper  edge,  or  .74  inch  lower  than  that  of 
reduced  section. 

Neutral  axes  of  other  sections,  Figs.  3  to  7,  found  by  same  process,  are  marked  on 
the  figures.  Section  of  a  flange  rail,  No.  7,  which  is  very  various  in  breadth,  may  be 
treated  in  two  ways:  either  by  preparatorily  averaging  projections  of  head  and 
flange  into  rectangular  forms,  or,  by  taking  it  as  it  is,  and  dividing  it  into  a  con- 
siderable number  of  strips  parallel  to  base,  for  each  of  which  the  moment,  with  re- 
spect to  assumed  datum-line,  is  to  be  ascertained.  First  mode  of  treatment  is  ap- 
proximate; second  is  more  nearly  exact. 

To  Compute  Ultimate  Strength  of  Homogeneous  Beams 
of  TJnsymmetrical   Section. 

OPERATION. — Resuming  section,  Fig.  9,  for  which  neutral  axis  has  been 
ascertained, 

To  Compute  Tensile  Resistance, 

Divide  portion  below  neutral  axis  t  z,  Fig.  9,  with  reduced  width  of 
flange,  a!  6",  into  parallel  strips,  say  .5  inch  deep,  as  shown, 
and  multiply  area  of  each  strip  by  its  mean  distance  from 
neutral  axis  for  proportional  quantity  of  resistance  at 
strip.  Divide  sum  of  products,  amounting  in  this  case 
to  31.3,  by  extreme  depth  below  neutral  axis  =  4.09  ins., 
and  multiply  quotient  by  1.73  S  (ultimate  tensile  resist- 
ance at  lower  surface).  The  final  product  is  total  tensile 
resistance  of  section ;  or, 
31.3X1.738 


S  representing  ultimate  tensile  strength  of  material  per  sq.  inch. 
Again,  multiply  area  of  each  strip  by  square  of  its  mean  distance  from  neu- 
tral axis,  and  divide  sum  of  these  new  products,  amounting  to  104.64,  by 
sum  of  first  products.    The  quotient  is  distance  of  resultant  centre  of  tensile 
stress,  d,  from  neutral  axis.    Or,  resultant  centre  is, 

*°f'  *  =  3. 34  ins.  below  neutral  axis. 
This  process  is  that  of  ascertaining  centre  of  gravity  of  all  the  tensile  resistances 


8TBENGTH   OF   MATEBIALS. — TKANSVEESE. 


821 


By  a  similar  process  for  upper  portion  in  compression,  sum  of  first  products  is 
ascertained  to  be  same  as  for  lower  portion  =:  31.3. 

But  maximum  compress!  ve  stress  at  upper  portion  is  greater  than  maximum 
tensile  stress  at  lower  portion,  in  ratio  of  their  distances  from  neutral  axis,  or  as 

«-73SX  —  =3.34  S,  and  3I'3  X  3'34  S=  13.24  S  total  compressive  resistance, 

4.09  7.91 

which  is  same  as  total  tensile  resistance,  in  conformity  with  general  law  of  equal 
ity  of  tensile  and  corapressive  stress  in  a  section. 

Sum  of  products  of  areas  of  stress,  divided  by  squares  of  their  distances  respec- 
tively from  neutral  axis,  is  164.9,  *&&  resultant  centre  c,  Fig.  9,  is 
ins.  above  neutral  axis. 

Sum  of  distances  of  centres  of  stress  or  of  resistance  from  neutral  axis,  3.34  -4- 
5.27  =  8.61  in*.  =  distance  apart  of  these  centres  as  represented  by  central  line,  c'  d'. 

Abbreviated  Computation.  —  As  upper  part  of  section  is  a  rectangle,  its  resultant 
centre  =  f  of  height,  or  7.91  x  j=  5.27  ins.  above  neutral  axis.  Average  resist- 
ance is  half  maximum  stress,  viz.,  that  at  upper  portion,  which  is  3.34  S  per  sq. 
inch. 

Area  of  rectangle  therefore  =  7.  91  x  i  =  7.91  *<?.  in*.  ,  and  7'9*  X  3'34    =  13.  21  S 
compressive  resistance,  as  before  determined. 
Moment  of  tensile  resistance  =  13.  21X8.61  ins.  =  113.  768,  also  =  —  ,  or  i-y—  =* 

W.     S  representing  total  resistance  of  section  in  Ibs.,  d  vertical  distance  apart  of 
centres  of  tension  and  compression,  and  I  length  between  supports,  all  in  ins. 

Strength  of  Beam  Inverted.  —  When  inverted,  maximum  tensional  resistance  of 
beam  at  its  lower  surface  c,  Fig.  8,  is  1.73  S. 


Area  of  rectangle  i  i  0  =  7.91  sq.  ins.,  and 


—  *'73 


=  6.79  S  total  tensile  re- 


sistance, or  about  one  half  of  beam  in  its  normal  position. 

NOTE. — For  other  rule  for  computation  of  centre  of  gravity,  see  Strength  of  Beams,  etc.    B.  Baktr, 
London,  1870. 

Comparative   Qualities   of*  Various   Metals.    Major  Wade. 


METALS. 

Density. 

Compres- 
sion. 

Tensile. 

Torsion. 

Trans- 
verse. 

Tensile 
pression. 

Hard, 
ness. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

(  Least.  .  .  . 

6.9 

84529 

9000 

416 

to  9.4 

4-57 

Cast  Iron  <  Greatest. 

7-4 

174120 

45970 

— 

958 

"  3-8 

33-51 

(  Mean.  .  .  . 

7-225 

144916 

31829 

8614 

680 

"4.6 

22.34 

Wrought  Iron  {(j^aJest* 

7.704 
7.858 

40000 
127  720 

38027 
74592 

3643 

542 

"  I 

10.45 
12.14 

Cast  Steel.  ...  {  Greatest' 

7.729 
8-953 

198944 
391  985 

128000 

28280 

1916 

I  tO  3.  1 

— 

T>                      (  Least.  .  .  . 

7-978 

17698 

1852 

— 

— 

4-57 

lze  (Greatest. 

8-953 

— 

56786 

2656 

— 

— 

£94 

Factors   of  Safety. 

Girders,  Beams,  etc.,  of  cast  iron  should  not  be  subjected  to  a  greater  stress 
than  one  sixth  of  their  destructive  weight,  and  they  should  not  be  subjected 
to  an  impulsive  stress  greater  than  one  eighth. 

The  following  are  submitted  by  English  Board  of  Trade,  Commission- 
ers, etc. 


STRUCTTEB. 

Stress. 

Factor. 

STBUCTURB. 

Stress. 

FactoT. 

CAST  IRON. 

Dead 

q  tO  6 

WROUGHT  IRON. 
Girders  

Dead 

Columns.     •  •  •  •  •  • 

t« 

36 

Live 

it 

Bridges  

Mixed 

4..  ,....,,,', 

Machinery  

Live 

8 

STEEL. 

Shock 

10 

Bridges.  

Mixed 

3 

Z* 

822         STRENGTH   OF   MATERIALS. — TRANSVERSE. 

GJ-irders,  Beams,  Lintels,  etc. 

Transverse  or  Lateral  Strength  of  any  Girder,  Beam,  Breast-summer, 
Lintel,  etc.,  is  in  proportion  to  product  of  its  breadth  and  square  of  its 
depth,  and  area  of  its  cross-section. 

Best  form  of  section  for  Cast-iron  girders  or  beams,  etc.,  is  deduced 
from  experiments  of  Mr.  E.  Hodgkinson,  and  such  as  have  this  form  of 
section  T  are  known  as  Hodgkiuson's. 

Rule  deduced  from  his  experiments  directs,  that  area  of  bottom  flange 
should  be  6  times  that  of  top  flange — flanges  connected  by  a  thin  ver- 
tical web,  sufficiently  rigid,  however,  to  give  the  requisite  lateral  stiff- 
ness, tapering  both  upward  and  downward  from  the  neutral  axis ;  and 
in  order  to  set  aside  risk  of  an  imperfect  casting,  by  any  great  dispro- 
portion between  web  and  flanges,  it  should  be  tapered  so  as  to  connect 
with  them,  with  a  thickness  corresponding  to  that  of  flange. 

As  both  Cast  and  Wrought  iron  resist  compression  or  crushing  with  a 
greater  force  than  extension,  it  follows  that  the  flange  of  a  girder  or  beam 
of  either  of  these  metals,  which  is  subjected  to  a  crushing  strain,  according 
as  the  girder  or  beam  is  supported  at  both  ends,  or  fixed  at  one  end,  should  be 
of  less  area  than  the  other  flange,  which  is  subjected  to  extension  or  a  ten- 
sile stress. 

When  girders  are  subjected  to  impulses,  and  sustain  vibrating  loads,  as  in 
bridges,  etc.,  best  proportion  between  top  and  bottom  flange  is  as  i  to  4 ;  as 
a  general  rule,  they  should  be  as  narrow  and  deep  as  practicable,  and  should 
never  be  deflected  to  more  than  .002  of  their  length. 

In  Public  Halls,  Churches,  and  Buildings  where  weight  of  people  alone 
are  to  be  provided  for,  an  estimate  of  175  Ibs.  per  sq.  foot  of  floor  surface 
is  sufficient  to  provide  for  weight  of  flooring  and  load  upon  it.  In  comput- 
ing other  weight  to  be  provided  for  it  should  be  that  which  may  at  any  time 
bear  upon  any  portion  of  their  floors ;  usual  allowance,  however,  is  for  a 
weight  of  280  Ibs.  per  sq.  foot  of  floor  surface  for  stores  and  factories. 

In  all  uses,  such  as  in  buildings  and  bridges,  where  the  structure  is  ex- 
posed to  sudden  impulses,  the  load  or  stress  to  be  sustained  should  not  ex- 
ceed from  .2  to  .16  of  breaking  weight  of  material  employed ;  but  when  load 
is  uniform  or  stress  quiescent,  it  may  be  increased  to  .3  and  .25  of  breaking 
weight. 

An  open-web  girder  or  beam,  etc.,  is  to  be  estimated  in  its  resistance  on 
the  same  principle  as  if  it  had  a  solid  web.  In  cast  metals,  allowance  is  to 
be  made  for  loss  of  strength  due  to  unequal  contraction  in  cooling  of  web 
and  flanges. 

In  Cast  Iron,  the  mean  resistances  to  Crushing  and  Extension  are,  for 
American  as  4.55  to  i,  and  for  English  as  5.6  to  7  to  i ;  and  in  Wrought  Iron 
are,  for  American  as  1.5  to  i,  and  for  English  as  1.2  to  i ;  hence  the  mass  of 
metal  below  neutral  axis  will  be  greatest  in  these  proportions  when  stress  is 
intermediate  between  ends  or  supports  of  girders,  etc. 

Wooden  Girders  or  Beams,  when  sawed  in  two  or  more  pieces,  and  slips 
are  set  between  them,  and  whole  bolted  together,  are  made  stiffer  by  the 
operation,  and  are  rendered  less  liable  to  decay. 

Girders  cast  with  a  face  up  are  stronger  than  when  cast  on  a  side,  in  the 
proportion  of  i  to  .96,  and  they  are  strongest  also  when  cast  with  bottom 
flange  up. 

Most  economical  construction  of  a  Girder  or  Beam,  with  reference  to  at- 
taining greatest  strength  with  least  material,  is  as  follows :  The  outline  of 


STRENGTH    OF   MATERIALS. — TRANSVERSE.          823 


top,  bottom,  and  sides  should  be  a  curve  of  various  forms,  according  as 
breadth  or  depth  throughout  is  equal,  and  as  girder  or  beam  is  loaded  only 
at  one  end,  or  in  middle,  or  uniformly  throughout. 

Breaking  Weights  of  Similar  Beams  are  to  each  other  as  Squares  of  their 
like  Linear  Dimensions. 

By  Board  of  Trade  regulations  hi  England,  iron  may  be  strained  to  5  tons 
per  sq.  inch  in  tension  and  compression,  and  by  regulation  of  the  Fonts  et 
Chausse'es,  France,  3.81  tons. 

Rivets  .75  and  i  inch  in  diameter,  and  set  3  ins.  from  centre  in  top  <A 
girder,  and  4  ins.  at  bottom. 

Character  of  fracture,  as  to  whether  it  is  crystalline  or  fibrous,  depends 
upon  character  of  blows ;  thus,  sharp  blows  will  render  it  crystalline,  and 
slow  will  not  disturb  its  fibrous  structure. 

For  spans  exceeding  40  feet,  wrought  iron  is  held  to  be  preferable  to 
cast  iron. 

Riveting,  when  well  executed,  is  not  liable  to  be  affected  by  impact  or 
velocity  of  load. 

A  Coupled  Girder  or  Beam  is  one  composed  of  two,  fastened  together,  and 
set  one  over  the  other. 

Trussed.   Beams   or  GHrders. 

Wrought  and  Cast  Iron  possess  different  powers  ef  resistance  to  tension  and  com- 
pression; and  when  a  beam  is  so  constructed  that  these  two  materials  act  in  uni- 
son  with  each  other  at  stress  due  to  load  required  to  be  borne,  their  combination  wili 
effect  an  essential  economy  of  material.  In  consequence  of  the  difficulty  of  adjust- 
ing a  tension  rod  to  the  stress  required  to  be  borne,  it  is  held  to  be  impracticable  to 
construct  a  perfect  truss  beam. 

Fairbairn  declares  that  it  is  better  for  tension  of  truss-rod  to  be  low  than  high, 
which  position  is  fully  supported  by  following  elements  of  the  two  metals  • 

Wrought  Iron  has  great  tensile  strength,  and,  having  great  ductility,  it  undergoes 
much  elongation  when  acted  upon  by  a  tensile  force.  On  the  contrary,  Cast  Iron 
has  great  crushing  strength,  and,  having  but  little  ductility,  it  undergoes  but  little 
elongation  when  acted  upon  by  a  tensile  stress;  and,  when  these  metals  are  re- 
leased from  the  action  of  a  high  tensile  stress,  the  set  of  one  differs  widely  from 
that  of  the  other,  that  of  wrought  iron  being  the  greatest. 

Under  same  increase  of  temperature,  expansion  of  wrought  is  considerably  great- 
er than  that  of  cast  iron;  1.81*  tons  per  sq.  inch  is  required  to  produce  in  wrought 
iron  same  extension  as  in  cast  iron  by  i  ton. 

Fairbairn,  in  his  experiments  upon  English  metals,  deduced  that  within  limits 
of  stress  of  13440  Ibs.  per  sq.  inch  for  cast  iron,  and  30240  Ibs.  per  sq.  inch  for 
wrought  iron,  tensile  force  applied  to  wrought  iron  must  be  2.25  times  tensile  force 
applied  to  cast  iron,  to  produce  equal  elongations. 

Relative  tensile  strengths  of  cast  and  wrought  iron  being  as  i  to  1.35,  and  their 
resistance  to  extension  as  i  to  2.25,  therefore,  where  no  initial  tension  is  applied  to 
a  truss-rod,  cast  iron  must  be  ruptured  before  wrought  iron  is  sensibly  extended. 

Resistance  of  cast  iron  in  a  trussed  beam  or  girder  is  not  wholly  that  of  tensile 
strength,  but  it  is  a  combination  of  both  tensile  and  crushing  strengths,  or  a  trans- 
verse strength ;  hence,  in  estimating  resistance  of  a  trussed  beam  or  girder,  trans- 
verse strength  of  it  is  to  be  used  in  connection  with  tensile  strength  of  truss. 

Mean  transverse  strength  of  a  cast  -  iron  bar,  one  inch  square  and  one  foot  in 
length,  supported  at  both  ends,  stress  applied  in  the  middle,  without  set,  is  about 
900  Ibs. ;  and  as  mean  tensile  strength  of  wrought  iron,  also  without  set,  is  about 
20000  Ibs.  per  sq.  inch,  ratio  between  sections  of  beams  and  of  truss  should  be  in 
ratio  of  transverse  strength  per  sq.  inch  of  beam  and  of  tensile  strength  of  truss. 

Girders  under  consideration  are  those  alone  in  which  truss  is  attached  to  beam 
at  its  lower  flange,  in  which  case  it  presents  following  conditions: 

*  Elongation  of  cact  and  wrought  iron  being  5500  and  10  ooo,  hence  to  ooo  -f-  5500  =  i.8x. 


824         STRENGTH    OF   MATERIALS. — TRANSVERSE. 

i.  When  truss  runs  parallel  to  lower  flange.  2.  When  truss  runs  at  an  inclination 
to  lower  flange,  being  depressed  below  its  centre.  3.  When  beam  is  arched  upward, 
and  truss  runs  as  a  chord  to  curve. 

Consequently,  in  all  these  cases  section  of  beam  is  that  of  an  open  one  with  a 
cast-iron  upper  flange  and  web,  and  a  wrought-iron  lower  flange,  increased  in  its  re- 
sistance  over  a  wholly  cast-iron  beam  in  proportion  to  the  increased  tensile  strength 
of  wrought  iron  over  cast  iron  for  equal  sections  of  metals. 

From  various  experiments  made  upon  trussed  beams,  it  is  shown  : 

i.  That  their  rigidity  far  exceeds  that  of  simple  beams;  in  some  cases  it  was  from 
7  to  8  times  greater.  2.  That  when  truss  resists  rupture,  upper  flange  of  beam  be- 
ing broken  by  compression,  there  is  a  great  gain  in  strength.  3.  That  their  strength 
is  greatly  increased  by  upper  flange  being  made  larger  than  lower  one.  4.  That 
their  strength  is  greater  than  that  of  a  wrought-iron  tubular  beam  containing  same 
area  of  metal. 

Comparative    Value    of*  'Wrought-iron    Bars,  Hollow 
Q-irders,  or   Tutoes   of*  Various   Figures  (English). 

Circular,  uniform  thickness 1.7 

Plate  beams 1.7 

Elliptic,  uniform  thickness 1.8 

Rectangular,  uniform  thickness 2 


Circular  tubes,  riveted . 

Flanged  beams , 1.2 

Elliptic  tubes,  riveted 1.3 

Rectangular  tubes,  riveted i.  5 


General  Deductions  from  Experiments  of  Stephenson,  Fairbairn,  Cubitt, 
Hughes,  etc. 

Fairbairn  shows  in  his  experiments  that  with  a  stress  of  about  12  320  Ibs.  per  sq. 
inch  on  cast  iron,  and  28000  Ibs.  on  wrought  iron,  the  sets  and  elongations  are 
nearly  equal  to  each  other. 

A  cast-iron  beam  may  be  bent  to  .3  of  its  breaking  weight  if  load  is  laid  on  grad- 
ually ;  and  16  of  it,  if  laid  on  at  once,  will  produce  same  effect,  if  weight  of  beam 
is  small  compared  with  weight  laid  on.  Hence,  beams  of  cast  iron  should  be  made 
capable  of  bearing  more  than  6  times  greatest  weight  which  will  be  laid  upon  them. 

In  beams  of  cast  or  wrought  iron,  if  fixed  or  supported  at  both  ends,  flanges 
should  be  in  proportion  to  relative  resistances  of  material  to  crushing  or  extension. 

Breaking  weights  in  similar  beams  are  to  each  other  as  squares  of  their  like  linear 
dimensions;  that  is,  breaking  weights  of  beams  are  computed  by  multiplying  to- 
gether area  of  their  section,  depth,  and  a  Constant,  determined  from  experiments  on 
beams  of  the  particular  form  under  investigation,  and  dividing  product  by  distance 
between  supports. 

Cast  and  wrought-iron  beams,  having  similar  resistances,  have  weights  nearly  as 
2.44  to  i. 

A  box  beam  or  girder,  constructed  of  plates  of  wrpught-lron,  compared  to  a  single 
rib  and  flanged  beam  I,  of  equal  weights,  has  a  resistance  as  100  to  93. 

Resistance  of  beams  or  girders,  where  depth  is  greater  than  their  breadth,  when 
supported  at  top,  is  much  increased.  In  some  cases  the  difference  is  fully  one  third. 

When  a  beam  is  of  equal  thickness  throughout  its  length,  its  curve  of  equilibrium, 
to  enable  it  to  support  a  uniform  stress  with  equal  resistance  in  every  part, 
should  be  an  Ellipse,  and  if  beam  is  an  open  one,  its  curve  of  equilibrium,  for  a  uni- 
form load,  should  be  that  of  a  Parabola.  Hence,  when  middle  portion  is  not  wholly 
removed,  its  curve  should  be  a  compound  of  an  ellipse  and  a  parabola,  approaching 
nearer  to  the  latter  as  the  middle  part  is  decreased. 

Girders  of  cast  iron,  up  to  a  span  of  40  feet,  involve  a  less  cost  than  of  wrought 
Iron. 

Cast-Iron  beams  and  girders  should  not  be  loaded  to  exceed  .2,  or  subjected  to  a 
greater  stress  than  .  166  of  their  destructive  weight;  and  when  the  stress  is  attended 
with  concussion  and  vibration,  this  proportion  must  be  increased. 

Simple  cast-iron  girders  may  be  made  50  feet  in  length,  and  best  form  is  that  of 
Hoogkinson ;  when  subjected  to  a  fixed  load,  flanges  should  be  as  i  to  6,  and  when 
to  a  concussion,  etc.,  as  i  to  4. 

Forms  of  girders  for  spaces  exceeding  limit  of  those  of  simple  cast  iron  are  vari- 
ous; principal  ones  adopted  are  those  of  straight  or  arched  cast-iron  girders  in 
separate  pieces,  and  bolted  together  —  Trussed,  Bowstring,  and  wrought-iron  Box 
and  Tubular. 


STRENGTH    OF    MATERIALS. — TRANSVERSE. 


82S 


Straight  or  Arched  Girder,  formed  of  separate  castings,  is  entirely  dependent 
upon  bolts  of  connection  for  its  strength. 

Trussed  or  Bowstring  Girder  is  made  of  one  or  more  castings  to  a  single  piece. 
and  its  strength  depends,  other  than  upon  the  depth  or  area  of  it,  upon  the  proper 
adjustment  of  the  tension,  or  the  initial  strain,  upon  the  wrought-irou  truss. 

Box  or  Tubular  Girder  is  made  of  wrought  iron,  and  is  best  constructed  with 
cast-iron  tops,  in  order  to  resist  compression:  this  form  of  girder  is  best  adapted  to 
afford  lateral  stiffness. 

When  a  girder  has  four  or  more  supports,  its  condition  as  regards  a  stress 
upon  its  middle  is  essentially  that  of  a  beam  fixed  at  both  ends. 

The  following  results  of  the  resistances  of  materials  will  show  how  they 
should  be  distributed  in  order  to  obtain  maximum  of  strength  with  minimum 
of  dimensions : 


To  Tension. 

To  Crushing. 

To  Tension. 

To  Cnwh'g. 

Cast  iron  

(  21  OOO 

90300 

Oak,  white,  mean. 

II  OOO 

7500 

"       English.. 
Granite  

(32000 
(13000 

(23000 

578 

140500 
58000 
116000 
15000 

*'     English   "    . 
Wrought  iron  

6500 
(45000 
(59000 
{31000 

3100 
47000 
83000 
40000 

Limestone  

(   670 

4000 

53000 

65000 

The  best  iron  has  greatest  tensile  strength,  and  least  compressive  or  crushing. 

Conditions  of  Forms  and  Dimensions  of  a  Symmetrical 
Beam   or  Girder. 

When  Fixed  at  One  End,  and  Loaded  at  the  Other. 

1.  When  Depth  is  uniform  throughout  entire  Length,  section  at  every  point 
must  be  in  proportion  to  product  of  length,  breadth,  and  square  of  depth,  and 
as  square  of  depth  is  in  every  point  the  same,  breadth  must  vary  directly  as 
length  ;  consequently,  each  side  of  beam  must  be  a  vertical  plane,  tapering 
gradually  to  end. 

2.  When  Breadth  is  uniform  throughout  entire  Length,  depth  must  vary 
as  square  root  of  length  ;  hence  upper  or  lower  sides,  or  both,  must  be  deter- 
mined by  a  parabolic  curve. 

3.  When  Section  at  every  point  is  similar,  that  is,  a  Circle,  an  Ellipse,  a 
Square,  or  a  Rectangle,  Sides  ofivhich  bear  a  fixed  Proportion  to  each  other, 
the  section  at  every  point  being  a  regular  figure,  for  a  circle,  the  diameter 
at  every  point  must  be  as  cube  root  of  length  ;  and  for  an  ellipse  or  a  rec- 
tangle, breadth  and  depth  must  vary  as  cube  root  of  length. 

ILLUSTRATION.—  A  rectangular  beam  as  above,  6  ins.  wide  and  i  foot  in  depth  at 
its  extreme  end,  and  4  feet  in  length,  is  capable  of  bearing  6480  Ibs.  ;  what  should 
be  its  dimension  at  3  feet?  a/4  =  1<s87j  and  ^3  =  z  ^ 

Then  1.587  :  1.442  ::  i  :  9086,  and  6  and  12  x  .9086  =  5.452  and  10.9. 


Hence  ™ 


=  2l6,  and 


When  Fixed  at  One  End,  and  Loaded  uniformly  throughout  its  Length. 

1.  When  Depth  is  uniform  throughout  its  entire  Length,  breadth  must  in- 
crease as  the  square  of  length. 

2.  When  Breadth  is  uniform  throughout  its  entire  Length,  depth  will  vary 
directly  as  length. 

3.  When  Section  at  every  point  is  similar,  as  a  Circle,  Ellipse,  Square,  and 
Rectangle,  section  at  every  point  being  a  regular  figure,  cube  of  depth  must 
be  in  ratio  of  square  of  length. 


826         STRENGTH    OP   MATERIALS.  —  TRANSVERSE. 

ILLUSTRATION.—  Take  preceding  case. 

Then  4*  :  3*  ::  12  3  :  972,  and  V972  =  9-9  in  depth. 

When  Supported  at  Both  Ends. 

1.  When  Loaded  in  the  Middle,  Coefficient  or  Factor  of  Safety  of  the  beam, 
or  product  of  breadth  and  square  of  depth,  must  be  in  proportion  to  distance 
from  nearest  support  ;  consequently,  whether  the  lines  forming  the  beam  are 
straight  or  curved,  they  meet  in  the  centre,  and  of  course  the  two  halves  are 
alike. 

2.  When  Depth  is  Uniform  throughout,  breadth  must  be  in  ratio  of  length. 

3.  When  Breadth  is  Uniform  throughout,  depth  will  vary  as  square  root 
of  length. 

4.  When  Section  at  every  point  is  similar,  as  a  Circle,  Ellipse,  Square,  and 
Rectangle,  section  at  every  point  being  a  regular  figure,  cube  of  depth  will 
be  as  square  of  distance  from  supported  end. 

When  Supported  at  Both  Ends,  and  Loaded  uniformly  throughout  its 
Length. 

1.  When  Depth  is  Uniform,  breadth  will  be  as  product  of  length  of  beam 
and  length  of  it  on  one  side  of  given  point,  less  square  of  length  on  one  side 
of  given  point. 

2.  When  Breadth  is  Uniform,  depth  will  be  as  square  root  of  product  of 
length  of  beam  and  length  of  it  on  one  side  of  given  point,  less  square  of 
length  on  one  side  of  given  point. 

3.  When  Section  at  every  point  is  similar,  as  a  Circle,  Ellipse,  Square,  and 
Rectangle,  section  at  every  point  being  a  regular  figure,  cube  of  depth  will 
be  as  product  of  length  of  beam  and  length  of  it  on  one  side  of  given  point, 
less  square  of  length  on  one  side  of  given  point. 


Elliptical-sided   Beams. 
To  Determine  Side  or  Curve  of  an  Elliptical-sided  Beam. 

/——-  —  d.     L  representing  load  in  Ibs.,  I  length  in  feet,  C  coefficient,  and  b 

*^    2  C  O 

breadth  in  ins. 

ILLUSTRATION.— What  should  be  depth  in  centre  of  a  beam  of  white  pine,  10  feet 
in  length  between  its  supports,  and  5  ins.  in  breadth,  to  support  a  load  of  10000  Ibs.? 

/IOOOO  X  10  /1 00  000 

Assume  C  =  100.    Then  A  / ==    / =  10  ins. 

V  2  X  ioo  X  5      V     1000 

Hence,  outline  of  beam  is  that  of  a  semi-ellipse,  having  10  feet  for  its  transverse 
diameter,  and  9  ins.  for  its  semi-conjugate. 
NOTE.— Weight  of  Girder,  Beam,  etc.,  should  in  all  cases  be  added  to  stress  or  load 

Miscellaneous   Illustrations. 

j._What  should  be  side  of  a  rectangular  white  oak  beam,  2  ins.  in  width,  and  6 
feet  between  its  supports,  to  sustain  a  load  of  360  Ibs.  ? 
Assume  stress  at  .2  of  breaking  weight  of  150  Ibs.  =s  30. 


Y4X2X3o~V   240 
2.-  -What  should  be  breadth  and  depth  of  such  a  beam  if  square? 


3.—  What  should  be  diameter  of  a  cylinder? 


STRENGTH    OF   MATERIALS. — TRANSVERSE. 

STEEL. 
To    Compute   Transverse   Strength,   of*  Steel    liars. 

Supported  at  Both.  Ends.     Weight  applied  in  Middle. 
i.i55S&da  __  w     g  representing  t&nyile  strength  in  Ibs. ,  I  length  between  supportt 

in  ins. ,  and  W  weight  in  Ibs. 

ILLUSTRATION.— What  is  ultimate  destructive  stress  of  a  bar  of  Crucible  steel, 
2  ins.  square,  and  2  feet  between  supports?  S  =  90000  Ibs. 

...55X90QOOX.'  =  83.600  =  m 

2  X  12  24 

Elastic  Transverse  Strength  is  50  per  cent,  of  its  ultimate  strength. 

Hardening  in  oil  increases  its  strength  from  12  to  56  per  cent.    Thus, 

Soft  steel,  121  520  Ibs. ;  soft  steel,  cooled  in  water,  90 160  Ibs. ;  soft  steel, 
cooled  in  oil,  215  120  Ibs. 

Krupp's  is  about  .45  of  its  tensile  breaking  weight,  .24  of  its  compressive 
or  crushing  strength,  .38  of  its  transverse,  and  .39  of  its  torsional. 

Friction  of  a  steel  shaft  compared  to  one  of  wrought  iron  is  as  .625  to  i. 

Capacity  of  steel  to  resist  a  transverse  stress  is  much  less  than  to  resist 
torsion. 

Relative  diameters  of  steel  and  wrought-iron  shafts,  to  resist  equal  trans- 
verse stress,  are  as  .98  to  i,  and  weight  of  such  a  proportion  of  steel  shaft 
compared  with  one  of  wrought  iron  will  be  about  4  per  cent,  less,  and  friction 
of  bearing  will  be  6  per  cent.  less. 

CYLINDERS,  FLUES,  AND  TUBES. 
Hollcrw   Cylinders.    Cast   Iron. 

To    Compute    Elements   of  Hollo^w   Cylinders   within 
Limits   of*  Elastic    Strength.  .(D.K.Clark.) 

S  x  hyp.  log.  R  =  P.  fe^  ^  R  =  S.  —  =  hyp.  log.  R.      S  representing 

elastic  tensile  strength  of  metal  in  Ibs.  per  sq.  inch,  R  ratio  of  external  diameter  to  in- 
ternal, =  —  —  — ,  and  P  internal  pressure  in  Ibs.  per  sq.  inch,  d  and  a'  representing 
inte)-nal  and  external  diameter,  and  r  and  r'  internal  and  external  radii,  all  in  ins. 

Norn.— Hyperbolic  Logarithm  of  a  number  is  equal  to  product  of  ita  common  logarithm  and  2.3026. 

ILLUSTRATION  i.  — Diameters  of  a  hydrostatic  cylinder  5.3  by  13.125  ins. ;  what 
pressure  within  its  elastic  strength  will  it  sustain  per  sq.  inch? 

Assume  S  =  xoooo  Ibs.    Hyp.  log.  R  =  I3'125  x  2. 3026  =  log.  2. 5  X  2.3026  =  .92. 
Then  10000  X  .92  =  9200  Ibs.  per  sq.  inch. 

NOTE.— For  Bursting  Strength  take  maximum  strength  of  metal. 

2. — A  water-pipe  .75  inch  thick  has  an  internal  diameter  of  10  ins.,  what  is  its 
bursting  pressure? 

S  =  30 ooo  Ibs.     Hyp.  log.  IO  +  -75X2  =  . x 398. 

Then  30000  X  .1398  =  4194  Ibs. 

3.— If  it  were  required  of  a  hydrostatic  press  to  sustain  a  pressure  of  589050  Iba 
upon  a  ram  of  5  ins.  in  diameter,  what  would  be  pressure  on  ram,  and  what  should 
be  thickness  of  metal,  assuming  it  equal  to  an  elastic  tensile  stress  of  15000  Jbs. 
per  sq.  inch? 

Area  of  5  ins.  =  19.635.     —  —  =  30000  =  pressure  per  sq.  inch  on  ram. 

Then  3000°  sis  2,  which  =  hyp.  log.  R  —  7. 39,  and  7. 39  X  5  =  36  95  —  external  di 

15000 
•meter.    36.95  —  5  =  31.95,  which  -4-  2  =  is-975  »»*•  thickness  of  metal. 


828         STRENGTH   OP   MATERIALS. — TRANSVERSE. 
"Wrought  Iron   and   Steel. 


-S  =  P. 


B-fr^i^  —  i 

ILLUSTRATION  i.— If  diameters  of  a  wrought  iron  cylinder  are  5  and  15  ins  ,  and 
ultimate  or  destructive  strength  of  metal  is  40000  Ibs.  per  sq.  inch,  what  is  its  break- 
ing pressure?  15 

-j.  =  3.    Hyp.  log.  3  =  .477 12  X  2.3026  =  1.0986. 

Then  —  —  X  40000  =  61 972  Ibs.  per  sq.  inch  =  61 972  X  5  -5- 15  —  5  = 

30986.2  Ibs.  per  sq.  inch  of  section  of  metal. 

2.— A  steam-boiler  6  feet  in  internal  diameter,  of  wrought-iron  plates  .375  inch 
thick  and  double  riveted  longitudinally,  burst  at  a  joint  by  a  pressure  of  300  Ibs.  pel 
sq.  inch;  what  was  resistance  of  joint  per  sq.  inch  of  its  section? 


7g-f- -375X2 
72 

Then 


=  1.0104.    Hyp.  log.  1.0104  =  .010345. 
2  x  300  600 


=  29  405  Ibs.  per  sq.  inch  of  section  of  joint. 


i.  0104-}-. oio  345  — i      .020745 

SHIP  AND  BOILER  PLATES. 

(See  pages  751-757  for  Boiler  Riveting.) 

Ultimate  Tensile  Strength,  of  Riveted  and.  "Welded 

Joints    of  "Wronght-iron   3?lates.    (O.K.  Clark.) 

Entire  Plate  =  100. 


JOINTS. 

Plate. 
•375 

•4375 

Aver- 
age. 

JOINTS. 

•5 

Plate. 

•375 

•4375 

Aver- 
age. 

Scarf-  welded  



1  02 

106 

104 

Double  riv'd,  snap-  ( 

Lap-welded     .       .  . 

CO 

66 

60 

62 

headed  J 

59 

72 

70 

67 

Single  hand  riveted. 

3*J 

40 

60 

y 
50 

50 

"        "  counter-  ) 

"        "     snap-) 
headed  J 

50 

56 

52 

53 

sunk  and  snap-  J 
headed  ) 

53 

69 

72 

65 

"     "  by  machine 

4° 

52 

54 

49 

"    "with  single) 

"     "    counter-  1 
sunk  head  ...  \ 

44 

52 

50 

49 

welt,  counters'k  [ 
and  snap-headed) 

52 

65 

60 

59 

Strength  of  Riveted  Joints  per  Sq.  Inch  of  Single  Plate.    ( Wm.  Fairbaim.) 

Single  Lapped. — Machine  riveted.    Pitch  3  times,  25  ooo  Ibs. 

Hand  riveted.    Pitch  3  times,  24  ooo  Ibs. 

Rivets  "  staggered,"  and  equidistant  from  centres,  30  500  Ibs. 

Abut  Joints. — Hand  riveted.  Rivets  not  "staggered,"  and  equidistant 
from  centres,  single  cover  or  strap,  30000  Ibs. 

Rivets  "square,"  single  cover  or  strap,  42 ooo  Ibs. ;  double  covers  or 
straps,  55  ooo  Ibs. 

Comparative    Strength,   of  Riveted.   Joints. 

Entire  Plate  .375  ins.  thick  =  zoo. 

Double  riveted,  double  strap,  or  fish- )  R    I  Double  riveted,  single  strap,  or  fish-)  , 

plated  joi nt }  80        plated  joint }  °5 

Double  riveted  lap  joint 72  |  Single  riveted  lap  joint 60 

For  all  joints  of  plates  over  .5  inch,  other  than  double  welded,  these  proportions 
are  too  high. 

A  closer  pitch  of  rivets  should  be  adopted  in  single  than  in  double  riveted  abuti, 
etc. 


STRENGTH    OF   MATERIALS. — TRANSVERSE. 


829 


Dimensions    of*  Rivets,   Pitch,   .Lap,   etc. 


Plat*. 

fhickneu. 

Diarn. 
of  Rivet. 

Length 
from  Head. 

Pitch. 

Single. 

Lap. 

Double. 

Staggered. 

Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

•25 
•3125 

;|25 

1.125 

1-375 

1.625 

1.5625 

2 

2-75 

3-4375 

2-4375 
3 

•375 

•75 

1.625 

1.75 

2-4375 

4.125 

3-625 

•5 

.8125 

2.25 

2.125 

2.625 

4-4375 

3-9375 

•5625 

•9375 

2-75 

2-375 

3 

5-1875 

4-5625 

.625 

i 

3 

2.625 

3-25 

5-5 

4-8125 

1.125 

3-25 

3 

3.625 

6.1875 

5-4375 

•875 

1.25 

4 

3-375 

4 

6.875 

6.0625 

i 

1.5 

4-5 

4-375 

4-873 

8.25 

7-25 

Straps.  —Single,  .125  thicker  than  the  plate;  Double,  each  625  of  thickness  of 
)late. 

To    Compute    Diameter    of  Rivet. 

Ordinarily,  T  i.  25  -f .  1875  =  d.    T  representing  thickness  of  plate,  and  d  diameter 
if  rivet. 


Plates. 

Single.... 
Staggered. 


Pitch 


Metal  between  the 
Holes. 


of  Rivets.    (Nelson  Foley.) 
Plates. 


52  to  62  per  cent 
68  to  75   "      " 


Diarn. 
of  Rivets. 


i. 4  to  2.3 

1.4  tO  2.1 


Metal  between  the 
Holes. 


70  to  78  per  cent. 
76  to  80   "      " 


Diam. 
of  Rivets. 


.99  to  i. 7 
.77  to  i 


Proportions    of  Single    Rivet   "Wronght-iron   Joints. 

(French.) 


Thickness 
of  Plate. 

Diameter 
of  Rivets. 

Pitch  of 
Rivets. 

Width  of 
Lap. 

Thickness 
of  Plate. 

Diameter 
of  Rivets. 

Pitch  of 
Rivets. 

Width  of 
Lap. 

Mil's.  Inch.!  Mil's 

Inch. 

Mil's 

Ins. 

Mil's 

ns. 

Mil's 

Inch. 

Mil's 

Ins. 

Mil's 

Ins. 

Mil's 

Ins. 

3 

.118      8 

•315 

27 

i.  06 

30 

.18 

10 

•394 

20 

.787 

56 

2.2 

58 

2.28 

4 

.158 

IO 

•394 

32 

1.26 

34 

•34 

ii 

•433 

21 

.827 

57 

2.  24  j    60 

2.36 

5 

.197 

12 

•472 

37 

1.46 

40 

.58 

2 

.472 

22 

.866 

58 

2.28 

60 

2.36 

6 

.236 
.276 

3 

:ir 

43 
48 

1.69 
1.89 

44 
50 

•73 
•97 

3 

4 

.512 
•551 

23 
24 

.906 
•945 

60 

62 

2-36 

2-44 

62 
64 

2-44 
2.52 

8 

•3i5 

17 

.669 

5' 

2.01 

54 

5 

•591 

25 

.984 

63 

2.48 

66 

2.6 

9 

•354 

*9 

.748 

54 

2-I3I    S^ 

2.2 

6 

-63 

26 

1.024 

65 

2.56 

68 

2.68 

Double-Riveted    and    Douttole-S trapped    Plate   Joints. 

( Mr.  Brunei. ) 
Plates,  20  ins.  in  width,  .5  inch  thick,  Abut  jointed,  with  a  Strap  or  Fish-plate  on 

each  side,  10  ins.  in  width.     Holes  punched. 

20  .6875  inch  rivets,  4  ins.  pitch,  set  "square,"  tensile  strength  77  per  cent. 
18  .75        u        "  "       "       "staggered,"      "  "         78.6     " 

24  -75        "        u      5    "       "  "square,"       "  "         84        " 

To    Compute    IP    and    Economy  of  a    Steam-Boiler. 

Steam  at  70  Ibs.  m.  g.,  and  Evaporation  0/30  Ibs.  of  Water  per  Hour  from  212°. 

(2°)— FXW 


and  (T  +  32°)-FxW^E 

(^+320)  _  2I2o  x  30  (t- 212°)  X  C 

T  representing  total  heat  from  the  water  at  32°  at  pressure  of  steam,  t  total  heat  from 
the  water  at  70  Ibs.,  and  F  temperature  of  feed  water,  all  in  degrees,  W  weight  of 
water  evaporated,  C  weight  of  fuel  consumed,  and  E  evaporation,  all  in  Ibs.  per  hour. 
ILLUSTRATION. —Assume  steam  at  98  Ibs.,  water  at  135°,  evaporation  10505  Ibs., 
and  consumption  of  fuel  1105  Ibs.  per  hour. 

1184.1°+ 32°--  1 35°  X  io  505.  jp      1184.  i°+  32°— 135°  X  io  505  _  ;. 

(n77.9°+32°)-2i2°  X  30  (1177-9°- 212°)  X  1105 

4A 


830         STRENGTH    OF   MATERIALS. — TRANSVERSE. 


H/ulls   of  "Vessels. 

Diameter  of  Rivets. 


Plate. 

U.  S.  and 
British 

Lloyds. 

Liverpool 
Regry. 

Admiralty, 
Eng. 

Millwall, 
Eng. 

Pitch 
of  Rivets. 

Length  c 
Count«r- 

f  Rivets. 
Snap- 
headed, 

Inch. 

Inch. 

Ins. 

Ins. 

Inch. 

Ins. 

Ins. 

Ins. 

•3'2S 

.625 

•5 

.625 

1-75 

.125 

1-5 

•375 

.625 

5 

.625 

.625 

2 

•25 

1.625 

•4375 

.625 

.625 

•75 

.625 

2.125 

•375 

i-75 

•5 

•75 

•75 

•75 

•75 

2.25 

.5 

2 

•5625 

•75 

•75 

.875 

•75 

2-437 

.6875 

2.1875 

.625 

•75 

.8125 

•875 

•875 

2.56 

•9375 

2-375 

.6875 

•87S 

•875 

•«75 

•875 

2.8l2 

-1875 

2.625 

•75 

•875 

•875 

•875 

3-I25 

•375 

2-75 

.8125 

.875 

•9375 

.875 

3-375 

•5 

2.875 

.875 

i 

X 

.125 

X 

3-625 

.625 

3 

•9375 

z 

1.0625 

.125 

X 

3-875 

'I5 

3-125 

X 

i  • 

1.125 

.125 

X 

4-125 

-875 

3-25 

Lap  of  Joint  or  Course  should  be  .5  pitch  of  rivets  added  to  .3  diam.  of  rivet. 

NOTE. — Lloyd's  requires  a  spacing  of  4.5  diameter.  Liverpool  Registry,  4.  Ad- 
miralty, 4.5  to  5  in  edges  and  abuts  of  bottom  and  bulkhead  plates,  and  5  to  6  in 
other  water-tight  work.  Bureau  Veritas,  4  diameters  for  single  riveting,  and  4. 5 
for  double. 

STEEL  PLATES. 

Steel  Plates,  according  to  M.  Barba,  .354  inch  thick  are  equal  to  wrought 
iron  .472  inch  thick,  or  as  3  to  4 ;  consequently,  when  iron  rivets  are  used, 
their  diameter  should  be  in  proportion  to  an  iron  plate. 

It  is  ascertained  also  that  they  are  best  united  by  iron  rivets. 

A  steel  plate  .3125  inch  thick  requires  an  iron  rivet  .5625  inch  in  diam- 
eter, and  1.375  ins.  apart. 

Bridge   Elates   and    Rivets. 

Plates  .25  to  .5  inch  thick.  Rivets  .75  to  i  inch  diameter,  and  3  ins.  apart 
from  centres  in  upper  flange  or  girder,  and  4  ins.  in  lower 

Rivet  Heads. 

i./%1T*\       Ellipsoidal,  Fig.  i.  —  D  diameter,  R  radius  of  head  =  D,  r  radius  of 
flange  =  .4  D,  c  depth  at  centre  —  .5  D. 

Segmented,  Fig.  2. — D  diameter,  c  depth  at  centre  —  .625 
D,  R  radius  of  head  = .  75  D,  o  depth  below  head  = .  125  D. 

Countersunk. — Head  1.52  3,  angle  60°.    Countersink  .45  diam.  of  plate. 
Cheesehead  or  heads,  sect,  n  of  which  is  a  parallelogram.     Head  .45  D, 
diameter  1.5  D. 

Rivets. 

Shearing  strength  of  a  Lowi  oor  rivet  =  40  320  d2  or  18  ce*  in  tons. 
d  representing  diameter  of  rivet   n  ins. 

3V-    ^moranda. 

Punching  holes  for  riveting  weakens  plates,  varying  from  10  to  20  per  cent,  ac- 
cording to  their  temper,  hardest  losing  most. 

Countersunk  riveting  does  not  impair  strength  of  joint,  as  compared  with  ex- 
ternal head. 

Diagonal  abut  joints  are  stronger  than  square. 

Shearing  strength  of  rivets  should  not  exceed  that  of  plates. 

Maximum  strength  of  joint  is  attained  at  90  to  100  per  cent,  of  net  section  of  plata 

Shearing  strength  of  English  wrought  iron  is  taken  at  80  per  cent,  of  its  tensil* 
strength. 


STRENGTH    OF   MATERIALS. — TRANSVERSE.          83! 


LEAD   PIPE. 


Resistance   of  Lead.   Pipe  to   Internal   Pressure. 


I.Q.I/  /cuwty,  «/urtt*7ie,  ana  fairoaim.) 

Diam. 

Thick- 
ness. 

Weight 

IDta. 

Thick- 
ness. 

Weight 
per 
Foot. 

Bursting 
Pressure. 

Diam. 

Thick- 
ness. 

Weight 
Foot. 

Burst!  -ig 
Pressure. 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Lbs. 

.5 

.2 

2.3 

1.25 

.21 

5-3 

683 

2 

.21 

9.2 

498 

.625 

.2 

2.6 

1-5 

.24 

7-  * 

734 

2 

.2 

•75 

.22 

3-8 

i-5 

.2 

— 

528 

3 

•25 

— 

364 

i 

.2 

4.1 

i-5 

.2 

— 

626 

3 

•25 

— 

374 

Tensile  strength  of  metal  =  2240  Ibs.  per  sq.  inch. 

To  Compute  Thickness  of  a  Lead  Pipe  when  Diameter 
and    [Pressure    in    L"bs.  per    Sq..  Inch    is   given. 

RULE.—  Multiply  pressure  in  Ibs.  per  sq.  inch  by  internal  diameter  of  pipe 
in  ins.,  and  divide  product  by  twice  tensile  resistance  of  metal  in  Ibs.  per  sq. 
inch. 

ILLUSTRATION.—  Diameter  of  a  lead  pipe  is  3  ins.,  and  pressure  to  which  it  is  to 
be  submitted  is  370  Ibs.  per  sq.  inch;  what  should  be  thickness  of  metal? 
370X3      ^o          8  . 

2240  X  2         4480 

Difference  in  Weight  between  Pipes  of  "Common,"  "Middling,"  and  "Strong" 
is  12  per  cent. 

To   Compute  "Weight  of  Lead   Pipe. 

D2  —  d2  3.  86  —  W.  D  and  d  representing  external  and  internal  diameters  in  in*., 
and  W  weight  of  a  lineal  foot  in  Ibs. 

To  Compute  Miaximtim  or  Bursting  Pressure  that  may 
"be   toorne   "by   a   Lead    Pipe. 

RULE.  —  Multiply  tenoile  resistance  of  metal  in  Ibs.  per  sq.  inch  by  twice 
thickness  of  pipe,  and  divide  product  by  internal  diameter,  both  in  ins. 

ILLUSTRATION.—  What  is  bursting  pressure  of  a  lead  pipe  3  ins.  in  diameter  and 
.5  inch  thick? 


Assume  a  column  of  water  34  feet  in  height  to  weigh  15  Ibs.  per  sq.  inch;  what 
head  of  water  would  such  a  pipe  sustain  at  point  of  rupture? 
15  :  34  '•'•  746-6  :  1692.  3  feet. 

Resistance   of  Grlass    G-lobes   and    Cylinders   to   Internal 
Pressure   and    Collapse.    (Flint  Glass.) 

Bursting  Pressure. 

GLOBES.  II  CYLINDER. 


Diameter. 

Thickness. 

Per  Sq.  Inch. 

Diameter. 

Length. 

Thickness. 

Per  Sq.  Inch 

Ins. 

Inch. 

Lbs. 

Ins. 

Ins. 

Inch. 

Lbs. 

4 

.024 

84 

4 

7 

.079 

282 

5 

.022 

90 

Elliptical  (Crown  Glass). 

6 

.059 

152 

II        4-i 

7        1      -019 

109 

Collapsing  Pressure. 

5 

.014 

29* 

3 

«4 

.0X4 

85 

4 

.025 

1000* 

4 

7 

•034 

202 

6 

•059 

900* 

11         4 

«4 

.064 

29; 

»  Unbroken. 

832          STRENGTH    OF    MATERIALS,  —  TRANSVERSE. 


se   Bronze. 

Manganese  Bronze,  No.  2,  has  a  Tensile  strength  of  72  ooo  to  78  600  Ibs,. 
per  sq.  inch,  its  elastic  limit  is  from  35000  to  50000  Ibs.,  its  ultimate  elon- 
gation 12  to  22  per  cent.,  and  its  hardness  alike  to  that  of  mild  steel. 

Transverse  Strength.  —  Destructive  stress  of  a  bar  i  inch  square,  supported 
at  both  ends  at  a  distance  of  i  foot  =  4200  Ibs.,  bending  to  a  right  angle  be- 
fore breaking,  and  requiring  1700  Ibs.  to  give  it  a  permanent  set. 

MEMORANDA. 
Cast  Iron. 

Beams  cast  horizontally  are  stronger  than  when  cast  vertically. 

Relative  strength  of  columns  of  like  material  and  of  equal  weights  is  : 
Cylindricaljiooi  Square,  93;  Cruciform,  98-,  Triangular,  no.  (Hodgkinson.) 

If  strength  of  a  cylindrical  column  is  100,  one  of  a  square,  a  side  of  which 
is  equal  to  diameter  of  the  cylinder,  is  as  150. 

Repetition  of  Stress.  —  A  piece  submitted  to  transverse  stress  broke  at 
i956th  strain,  with  a  stress  .75  of  that  of  its  original  ultimate  resistance. 

Resistance  to  Bursting  of  Thick  Cylinders.  —  Mean  resistance  to  bursting, 
of  chambers  of  cast-iron  guns  is  as  follows  (Major  Rodman,  U.S.A.)  : 

Thickness  of  metal  =  i  calibre,  length  =  3  calibres,  52  217  Ibs.  per  sq.  inch. 

Thickness  of  metal  =  .5  calibre,  length  =  3  calibres,  49  100  Ibs.  per  sq.  inch. 

The  tensile  strength  of  the  iron  being  18820  Ibs. 

Diam.  of  cylinder  2  ins.,  length  12  ins.,  metal  2  ins.,  80229  Ibs.  per  sq.  inch. 

Diam.  of  cylinder  3  ins.,  length  12  ins.,  metal  3  ins.,  93  702  Ibs.  per  sq.  inch. 

Tensile  strength  of  iron  being  26  866  Ibs. 

Sudden  Applications  of  Stress.  —  Loss  of  strength  by  sudden  application 
of  load  was,  by  experiment,  18.6  per  cent,  in  excess  of  load  applied  gradually, 
and  its  elongation  20  per  cent,  greater. 

Low  Temperature.  —  Tensile  strength  at  23°  under  sudden  application  of 
load,  was  reduced  3.6  per  cent.,  and  elongation  18  per  cent. 

"Wrought   Iron. 

Increased  Hammering  gives  20  per  cent,  greater  strength  with  decreased 
elongation. 

Hardening.  —  Water  increases  strength  more  than  oil  or  tar.  A  bar  .87 
inch  in  diameter,  forged  and  hardened  in  water,  attained  a  tensile  strength 
of  73  448  Ibs.  (Mr.  Kirkaldy.) 

Case  Hardening.—  Loss  of  tensile  strength  4950  Ibs.  per  sq.  inch. 

Cold  Rolling  added  18.5  per  cent,  to  tensile  strength,  and  when  plates 
were  reduced  .33  in  thickness,  strength  was  nearly  doubled,  with  but  .1  per 
cent,  elongation.  Specific  gravity  was  reduced. 

Fibre.  —  Plates  are  about  12  per  cent,  stronger  with  fibre  than  across  it. 

Angles,  Tees,  etc.,  have  from  2200  to  4500  Ibs.  less  tensile  strength  than 
rectangular  bars. 

Galvanizing  does  not  perceptibly  affect  strength. 

Welding.  —  Strength  as  affected  by  welding  varies  by  experiment  from  2.6 
to  43.8  per  cent,  less,  average  being  19.4. 

Elastic  Strength  is  about  .45  of  its  tensile  breaking  weight,  .15  of  its  com- 
pressive  or  crushing  strength,  and  .5  of  its  transverse  strength. 

Effect  of  Screw  Threads.—  i  inch  bolts  lose  by  dies  6.11  per  cent.,  and  by 
chasing  28  per  cent. 

Steel. 

Steel  can  be  hardened  in  water  at  a  temperature  of  310°. 


STRENGTH    OF   MATERIALS. — TRANSVERSE. 


833 


WOODS. 
To   Compnte   Transverse   Strength   of  Large   Timber. 

Destructive  Stress. 
Fixed  at  One  End,  and  Loaded  at  the  Other.    '3S*d2  =  W. 

Fixed  at  Both  Ends,  and  Loaded  in  Middle.    -  —  ^  -  =  W. 
*  Supported  at  Both  Ends,  and  Loaded  in  Middle.    -  ^—L  - 


=  W. 


Fixed  at  Both  Ends,  and  Loaded  at  any  other  point  than}  -45Sfrd2_w 
the  Middle.  j  I 

Supported  at  Both  Ends,  and  Loaded  at  any  other  point  )  .3&bd2l  _  w 
than  the  Middle.  j  mn 


*  Hence, 

7   - 


ir*=s'andF55='-2S- 


tances  of  load  from  nearest  supports  in  ins. 

When  a  beam  is  uniformly  loaded,  the  stress  is  twice  that  if  applied  in  its  middle 
or  at  one  end. 

"Valnes   of  1.3    S. 

Hence,  for  other  coefficients,  as  .3, 1.8,  etc.,  the  values  will  be  proportional. 


WOODS. 

1.2  S 

WOODS. 

1.2  S 

Ash  white  

2.38 

3-7 

2.4 

Mahogany,  Honduras  

2.3 

*'     English  ..  

2.46 

Oak  Pa.  

2 

Beech  

2-  55 

'    Va.  

2.3 

Birch  

2.5 

2.5 

Cedar  

1.6 

'    English  

l.J 

1.6 

4    Dantzic  

1.35 

Chestnut  

i.  s^ 

'    French  

2.44 

.85 

Pine,  Va  

3 

Elm,  English  

1.  12 

"     pitch  

2.2 

"     Rock  Canada  

2.6l 

"     white  

2.71 

Fir  Dantzic 

"     yellow 

o  87 

Greenheart     *.       .       ........ 

•a  8l 

"         "      Canada  

3-oy 
I.o 

Gum  blue  

2 

Redwood,  Cal  

1.  1 

Hackmatack  
Iron  wood  

1.36 

3  64 

Spruce  
Teak  

1.2 

3-  17 

Larch  .  .  . 

1.77 

Walnut,  black.  .  . 

1.25 

ILLUSTRATION  i.—  What  is  destructive  stress  of  a  beam  of  English  oak,  2  ins. 
square,  and  6  feet  between  its  supports? 

1.2  from  table  =  1.7,  and  S  =  .66  of  5700  (mean  of  tensile  and  crushing  strength) 
=  3762  Ibs. 

x.7X2X2*X3762^5ii63  6 

6  X  12  72 

By  experiment  of  Mr.  Laslett  it  was  688  Ibs. 

2.  —  What  is  destructive  stress  of  a  beam  of  yellow  pine,  3  ins.  by  12,  and  14  feet 
between  its  supports? 

1.2  from  table  =  3.  87,  and  S  =  .  66  of  10200  (mean  of  tensile  and  crushing  strength) 
=6732  Ibs. 

3.87  X  3  X.  2*  X  6732       11254827 
14  X  12 


168 


7  =  66        ^ 


If  the  beam  was  fixed  at  both  ends  then  3.87  would  be  5.8. 
Or,  as  1.2  :  1.8  ::  3.87  :  5.8. 

4  A* 


834          STRENGI 

d 

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ERSE. 

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£    0           •§ 

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)o,  Spruce  550,  White  Pine  500,  Chestnut  480,  and  Hemlock  450. 

'  ,  tit 

®         1       £~    N 

fl         ^s 

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s  o/  TaWe. 
2._What  should  be  depth  of  a  like  beam,  4  ins. 
between  its  supports,  to  bear  a  statical  weight  of  4 

/IoX45oo_  /45000 

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Illustration 
i.  —  What  is  safe  statical  load  for  a  White-pine  beam,  4  ins.  by  12,  and 
15  feet  between  its  supports,  loaded  in  middle? 
A  like  beam,  i  inch  in  width,  12  ins.  in  depth,  and  x  foot  between  its 

Knrmnrts  will  hpar  JIR  nor  tahlp  -,f\  o™  Ihc 

Hence  16  200  x  4  -r-  15  =  4320  Ibs. 
Hatfield  gives  Georgia  Pine  850,  White  Oak  650,  Canadian  Oak  5c 

?j         H      ^ 

3^11  13  I 
«s^i  Bw 

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STRENGTH    OF   MATERIALS.  —  TRANSVERSE.  835 

Floor    Beams    of  "Wood. 

Condition  of  stress  borne  by  a  Floor  beam  is  that  of  a  beam  supported  at 
both  ends  and  uniformly  loaded. 

To    Compute    Capacity   of   Floor    Beairxs,   Q-irders,   etc. 
Supported  at  Both  Ends. 

RULE.  —  Divide  product  of  breadth  and  square  of  depth,  in  ins.,  and  Coeffi- 
cient for  material,  by  length  in  feet,  and  result  will  give  weight  in  Ibs. 

Or,  b^2°  =  W.     When  Fixed  at  Both  Ends.     T-56d2C  =  w 

EXAMPLE.  —  The  dimensions  of  a  white-pine  floor  timber  are  4  by  12  ins.,  and  its 
length  between  supports  15  feet;  what  weight  will  it  sustain  in  its  centre? 

.  C,  as  per  preceding  table  =  1  12.  5.     Then  4  X  iz2  X^igj  =  64800  =  ^^  ^ 
When  Uniformly  Loaded.    Multiply  the  result  by  2. 

To    Compute    Depth,    of  a    Floor    Beam    or    Grirder. 
Supported  at  Both  Ends. 

When  Length  between  Supports  and  Breadth  are  Given.  RULE.  —  Divide 
product  of  length  in  feet,  and  weight  to  be  borne  in  Ibs.,  by  product  of 
breadth  in  ins.,  and  Coefficient  for  material,  and  square  root  of  quotient  will 
give  depth  in  ins.,  for  distance  between  centres  of  one  foot. 

When  the  Computation  is  made  Independent  of  the  Preceding  Table. 

/-^  =  d.     When  Fixed  at  Both  Ends.       /  l!L  =  d. 

V  4  &C  \  6bC 

C  as  may  be  assumed  or  ascertained. 

When  Distance  between  Centres  of  Beams  is  greater  or  less  than  one  Foot. 
RULE.  —  Divide  product  of  square  of  depth  of  the  beam,  when  distance  be- 
tween centres  is  one  foot,  and  distance  given,  by  12,  and  square  root  of  quotient 
will  give  depth  of  beam. 

EXAMPLE.  —Assume  beam  in  preceding  case  to  be  set  15  ins.  from  centres  of  ad- 
joining beams;  what  should  be  its  depth? 


/I22X   15  /2l6o 

V—  ^=V"^=1 


To    Compute    Breadth,    of   a    Floor    Beam    or    GHrder. 
Supported  at  Both  Ends. 

When  Length  and  Depth  are  given.  RULE.  —  Divide  product  of  length  in 
feet,  and  weight  to  be  borne  in  Ibs.,  by  product  of  square  of  depth  in  ins., 
and  Coefficient  for  material,  and  quotient  will  give  breadth  in  ins. 

Or,  L£  =  6.     When  Fixed  at  Both  Ends.         l™  „  =  b. 
d  C  i.  5  d2  C 

When  Uniformly  Loaded,  multiply  the  result  by  2. 
EXAMPLE.  —  Take  elements  of  a  preceding  case,  page  834. 
15  X  4320  _  64  800  _ 
i22  X  112.5      16200 

When  Distance  between  Centres  of  Beams  is  greater  or  less  than  One  Foot. 
RULE.  —  Divide  product  of  breadth  for  a  beam,  when  distance  between  centres 
is  one  foot,  and  distance  given,  by  12,  and  residt  will  give  breadth. 


836  STRENGTH    OF    MATERIALS.  —  TRANSVERSE. 

EXAMPLE.  _  Assume  beam,  as  in  preceding  case,  to  be  set  15  ins.  from  centre  of 
adjoining  beams;  what  should  be  its  breadth  ? 


When  Weight  is  Suspended  or  Stress  borne  at  any  other  point  than  the  Middle, 
See  Formulas,  page  801. 

Header    and.    Trimmer    Beams. 

Conditions  of  stress  borne,  or  to  be  provided  for  by  them  are  as  follows  : 

Header  supports  .5  of  weight  of  and  upon  tail  beams  inserted  into  or  at- 
tached to  it,  and  stress  upon  it  is  due  directly  to  its  length,  weight  of  and 
that  upon  tail  beams  it  supports,  alike  to  a  girder  loaded  at  different  points. 

Trimmer  beams  support,  in  addition  to  that  borne  by  them  directly  as 
floor  beams,  each  .5  weight  on  headers. 

NOTE.  —  In  consequence  of  effect  of  mortising  (when  stirrups  or  bridles  are  not 
used),  a  reduction  of  fully  one  inch  should  be  made  in  computing  the  capacity  of 
deptti  of  headers  and  trimmers. 

To    Compute    Breadth,    of  a    Header    Beam. 

When  Uniformly  Loaded.  RULE.  —  Compute  weight  to  be  borne  in  Ibs.  by 
tail  beams,  divide  it  by  two  (one  half  only  being  supported  by  header),  mul- 
tiply result  by  length  of  beam  in  feet,  and  divide  product  by  product  of 
twice  Coefficient  of  material  and  square  of  depth,  and  result  will  give  breadth 
in  ins.  _ 

Or>  —  TT^T  =  b-     w  representing  weight  in  Ibs.  per  sq.  foot. 
2  (j  d 

EXAMPLE.—  What  should  be  breadth  of  a  Georgia  pine  header,  13  ins.  in  depth, 
10  feet  in  length,  supporting  tail  beams  12  feet  in  length,  bearing  200  Ibs.  per  sq. 
foot  of  area  ? 

C,  a?  per  preceding  table,  112.5,  and  depth  =  13  —  i  =  12  ins. 

12  X  10  X  200-7-2  X  10         120000 


2  X  II2-5  X  I22  32400 


=  3. ^  ins. 


To  Compnte  the   Capacity  of  a  ITloor  Uniformly  loaded 
•when   one   of  its   Sides  rests  npon   a  Header  Beam. 

1.  Determine  the  capacity  of  a  trimmer  and  header  beam  at  the  point 
of  their  connection.     Assume  the  less,  as  the  limit  of  their  capacity  to  sus- 
tain a  load,  and  twice  this  capacity  will  represent  that  of  one  half  of  the 
floor  at  the  points  of  connection  of  the  header  and  trimmers,  the  other  half 
resting  on  the  wall. 

2.  Compute  area  of  floor  in  square  feet,  first  by  its  length  from  wall  of 
building  to  face  of  header  beam,  and  its  width  from  the  centre  of  the  spaces 
between  the  trimmers  and  the  beams  beyond ;  then  add  that  determined  by 
the  width  of  the  trimmer  and  the  centre  of  the  space  between  it  and  the 
beams,  and  the  length  of  it  by  the  width  of  the  opening  between  the  face  of 
the  header  and  the  wall,  as  hatch  or  stairway,  and  this  combined  area  will 
be  that  which  rests  upon  the  header  and  trimmer. 

3.  Divide  the  capacity  of  the  header  and  trimmers  as  obtained,  by  the  half 
area  of  the  floor  resting  thereon,  less  the  area  required  or  allotted  for  passage 
way  (but  not  considered  by  the  Department  of  Buildings),  and  the  quotient 
will  give  the  capacity  of  the  floor  in  Ibs.  per  square  foot,  from  which  is  to 
be  deducted  the  weight  per  square  foot  of  the  beams,  flooring,  ceiling,  etc. 


STRENGTH    OF    MATERIALS.  —  TRANSVERSE. 


837 


To    Compute    Depth    of   a    Header    Beam. 
RULE.—  See  rule  for  depth  of  a  floor  beam,  page  835,  with  the  exception 
that  a  header,  alike  to  a  trimmer,  is  assumed  to  be  always  uniformly  loaded. 


V    4&C    ~ 
To    Compute    Breadth    of    a    Trimmer    Beam. 

With  One  Header  and  One  Set  of  Tail  Beams.     RULE. — Proceed  as  for 
computation  of  dimension  of  a  beam  loaded  at  any  other  point  than  middle. 

Uniformly  Loaded.    1 


e'  X  —  -\-n  c  X  W  =  L,  product  of  area  of  floor  and  load 

per  sq.  foot,  and  L  -;  --  -  —  =  breadth. 

H  representing  length  of  header,  c  distance  between  centres  of  beams,  m  and  n, 
lengths  of  tail  beams  and  width  of  hatch  or  stairway,  c'  sum  of  half  distance  ofc, 
added  to  half  of  an  assumed  width  of  trimmer,  and  I  length  of  trimmer,  all  infect, 
W  load  per  sq.  foot  on  floor,  and  C  coefficient  in  Ibs.,  b  breadth  of  one  sq.  inch  of  the 
material,  and  d  depth  of  beam  in  ins. 

EXAMPLE.—  What  should  be  breadth  of  a  trimmer  or  carriage  beam  of  Georgia 
pine.  23  feet  in  length,  15  ins.  in  depth,  sustaining  a  header  10  feet  in  length,  with 
tail  beams  19  feet,  distance  between  centres  one  foot,  and  designed  fora  load  of  270 
Ibs.  per  sq  foot  of  floor? 

Assume  C  =  275,  as  assigned  by  the  Department  of  Buildings,  N.  Y.,m  and  n  =  ig 
and  4  feet,  (7—15  —  1  —  14,  and  c'  =  .75. 

io-f  .75~xf^  X4-75  +  4X  273=:  15828.75  and  *5  —  =  6.  75  in*. 

I  X  196  X  275  -r-  23 

NOTE  i.—  Depth  of  trimmer  beams  is  usually  determined  by  depth  of  floor  beams; 
where  not,  proceed  to  determine  it  as  for  a  header. 

2.  —  When  a  trimmer  beam  is  mortised  to  receive  headers,  it  is  proper  to  deduct 
i  inch  from  its  depth,  as  in  preceding  illustrations.  When  bridle  or  stirrup  irons 
are  used  to  suspend  headers,  a  deduction  of  the  thickness  of  the  iron  only  is  neces- 
sary, usually  .5  inch. 

With  Two  Headers  and  One  Set  of  Tail  Beams.—  Fig.  i. 

OPERATION.  —  Proceed  for  each  weight  or  load  as  for  a  beam,  when  weights 
are  sustained  or  stress  borne  at  other  point  than  the  middle. 

—  =  W  and  w.    a  representing  area  of  floor  in  sq.  feet,  L  load  per  sq.foot,  and 
W  and  w  weights  or  loads  at  points  of  rest  on  trimmers. 

NOTE.—  Hatfleld  and  some  other  authors  give  complex  and  extended  formulas, 
to  deduce  the  dimensions  of  a  Girder  or  Beam,  under  a  like  stress. 

Upon  consideration,  however,  it  will  be  readily  recognized  that  a  beam  loaded  at 
more  than  one  point  is  simply  two  or  more  beams  of  proportionate  width,  as  the 
case  may  be,  loaded  at  different  points,  and  connected  together. 


Fig.  i  . 


-....-m--.fr- 


-/- 


ILLUSTRATION.  —  What  should  be  breadth  of  a 
trimmer  beam  of  Yellow  pine  25  feet  in  length,  u 
ins.  in  depth,  sustaining  two  headers  12  feet  in 
length,  set  at  15  feet  from  one  wall  and  5  feet  from 
the  other,  to  support  with  safety  300  Ibs.  per  sq. 
foot  of  floor  ? 


w    W 
12  X  10  —  5  X  3°o  _  l8  °°° 


and  d  =  n  —  i  =  iofor  loss  by  mortising. 

12  X  5  X  300  18000 

4  4 

=  4500  Ibs.  at  w. 


,  at  W,  and 


838 


STRENGTH    OF    MATERIALS. — TRANSVERSE. 


Then 


aud 


> 

25Xio2Xi25       312500 
tns.t  and  2.i6-|-  1.44  =  3.6  ins.  combined  breadth. 


_  = 

25X10^X125       312500 


Fig.  2. 


With  Two  Headers  and  Two  Sets  of  Tall  Beams.—  Fig.  2. 

OPERATION.  —  Proceed  as  directed  for  Fig.  i. 

ILLUSTRATION.  —  What  should  be  breadth  of  a 
trimmer  beam  of  Yellow  or  Georgia  pine,  25  feet  in 
length,  12  ins.  in  depth,  sustaining  two  headers  12 
feet  in  length,  one  set  at  15  feet  from  one  wall 
and  the  other  at  5  feet  from  the  other,  to  support 
with  safety  300  Ibs.  per  sq.  foot  of  floor? 


W 


112.5,  and  d  =  i2  —  1  =  11  for  loss  by  mortising. 


54000 


.  at 


Then 


'5  X  io  X  13  SOP      2  025  OOP 


=         ^      breadth  for  load  on  header  at  15  feet. 
or  load  on  header  at  5  feet,  and 


25  X  n2  X  112.5       340312 
_  20X5X4500    ._  450000  _         ins 
25  X  ii2  X  112.5      340312 
5.94  -f-  1.32  =  7.26  ins.  combined  breadth. 

With  Three  Headers  and  Two  Sets  of  Tail  JSeaws.—  Fig.  3. 


w'    w 


W 


OPERATION.  —  Proceed  as  directed  for  Fig.  i. 

ILLUSTRATION.  —  What  should  be  breadth  of  a  trim- 
mer  beam  of  Yellow  pine,  20  feet  in  length,  13  ins.  in 
depth,  sustaining  3  headers  15  feet  iu  length,  set  at  3, 
7,  and  13  feet  from  one  wall,  to  sustain  a  load  of  200 
lbs.  per  sq.  foot  of  floor  ? 


=  i2  ins.,  and  C=  125. 

15  X  7  X  200 2 

4 


=  5250  a*  at  w; 


and 

20Xl2 

bined  breadth. 


=  . 

360000 

oo  i 


ins.;     7X13X3000^000 

20Xl22Xl25          36°  000 


H 


3  =  *5a  ins.  com. 


360000 

Stirrnps    or    Bridles. 

Stirrups  are  resorted  to  in  flooring  designed  for  heavy  loads,  in  order  to 
avoid  the  weakening  of  the  trimmers  by  mortising. 

Average  wrought  iron  will  sustain  from  40000  to  50000  lbs.  per  sq.  inch. 

Hence  45  ooo  lbs.  as  a  mean,  which  -=-  5  for  a  factor  of  safety,  =  9000  lbs. 

A  stirrup  supports  one  half  weight  of  header,  and  being  doubled  (looped), 
the  stress  on  it  is  but  .5  -=-  2  =  .25  of  load  on  header. 

To    Compete    Dimensions    of   Stirrnps    or    JBridles. 

W-r-2  area 

-  —  area.    Hence  —  —  •  -  =  width. 

2  x  C  thickness 

ILLUSTRATION.—  What  should  be  area  and  width  of  .75  inch  wrought-iron  stirrup 
irons  for  a  weight  on  a  header  beam  of  240000  lbs.  ? 


oooo     24°00°-f"2  =  '-^^ 
2X9000         18000 


=  6.  66  sq.  ins.  ,  and  —  ^  8.  8  ins.  =  width. 
-75 


STRENGTH    OF   MATERIALS. — TRANSVERSE.          839 

Grirder. 

Condition  of  stress  borne  by  a  Girder  is  that  of  a  beam  fixed  or  supported 
at  both  ends,  as  the  case  may  be,  supporting  weight  borne  by  all  beams 
resting  thereon,  at  the  points  at  which  they  rest 

To    Coxnpvite    Dimensions    of  a   Girder. 

RULE. — Multiply  length  in  feet  by  weight  to  be  borne  in  Ibs.,  divide 
product  by  twice  *  the  Coefficient,  and  quotient  will  give  product  of  breadth 
and  square  of  depth  in  ins. 

j  vrr  I  I  TTT 

Or,  —  =  b  and  d2,  and  A  /-^-  =  d. 

2  C  V  2  6  C 

EXAMPLE.— It  is  required  to  determine  dimensions  of  a  Yellow-pine  girder,  15  feet 
between  its  supports,  to  sustain  ends  of  two  lengths  of  beams,  at  distances  of  5  feet, 
each  resting  upon  it  and  adjoining  wall,  15  feet  in  length,  having  a  superincumbent 
weight,  including  that  of  beams,  of  200  Ibs.  per  sq.  foot. 

Condition  of  stress  upon  such  a  girder  is  that  of  a  number  of  beams,  15  feet  in 
length,  supported  at  their  ends,  and  sustaining  a  uniform  stress  along  their  length, 
of  200  Ibs.  upon  each  superficial  foot  of  their  supporting  area. 

Coefficient  =  137. 5. 

15X15X200^-2  (for  half  support  on  the  uall)  —  22  500  Ibs. 

Then  — — —^=  1227.2  =  &  and  d2.    Assume  b  =  12  ins.,  then    /I227-2  —  I0  x 

2X137-5  V      12 

ins.  the  depth. 

To  Compute  Greatest  .Load,  upon  a  Girder,  and  Dimen- 
sions   thereof'.— Fig.  1. 

When  a  Beam  is  Loaded  at  Two  Points. 
Fi&.  i.  < 4- »  _  ee 


—  =  effect  of  weight  at  2, 

™  (W  n  -f  w  s)  —  the  two  effects 
W  w  l 

at  1,  and  j  (w  r  -f  W  m)  =  tivo  effects  at  2. 

Then,  for  weight  and  dimensions,  same  formulas  will  apply. 

ILLUSTRATION. —Assume  weight  of  8000  Ibs.  at  3  feet  from  one  end  of  a  white-pine 
beam  io  inches  in  depth  and  12  feet  in  length  between  its  bearings,  and  another 
weight  of  3000  Ibs.  at  5  feet  from  other  end.  C  =  112.5. 

8000  X  3  X  12  —  3  =  216  ooo  effect  of  weight  at  location  i,  and  3000  X  5  X  12  —  5 
•fe  105  ooo  effect  of  weight  at  location  2.  Hence  i,  being  greatest,  =  AV,  and  2  =  w. 

Then,  - — -  x  8000  =  18  ooo  at  W,  and  - — -  x  7000  =  8750  at  w ;  and 

'TO  TO  •*  '  J 


—  (8000  x  9  +  3000  x  5)  =  21  750  =  total  effect  at  W,  and  4-  (3000  X  7  +  8000  x  3) 

—  1  8  750  =  total  effect  at  w. 

Hence,  to  ascertain  total  effect  and  dimensions. 

2x750x3x9  ^.fr^afc 

12  X  io2  X  112.5 

Verification.—  Breadth  at  W.        l8oo°X3X9    =3.6  ins.     Then  21  750  —  18  ooo 
12  X  io2  x  112.5 


*  For  being  uniformly  loaded. 


840 


STRENGTH    OF    MATERIALS. TRANSVERSE. 


Beam     Loaded    Uniformly    and    at    Several    .Points. 
To  Determine  Equal  Weiyht  at  Centre  Fig.  2. 

Fig. 2.4 7. ^  ILLUSTRATION. — What  should  be 

breadth  of  a  beam  of  Georgia  pine, 
20  feet  in  length,  15  ins.  in  depth, 
'  uniformly  loaded  with  4000  Ibs., 
and  sustaining  3  headers  or  con- 
centrated loads  of  6000  Ibs.,  at  re- 

20'       ±  ^L  spective  distances  of  4  and  9  feet 

from  one  end  and  7000  Ibs.  at  6 
feet  from  the  other  end  ? 

m  =  g,   7i=  1 1,   r  =  i6,    0  =  4,    s  —  20  —  u  14,    d— 15  — 1  =  14,    L  =  4ooo,    and 
_      _  su  m  n  o  r  .  L I 

C  =  800 -r- 4  =  200.      — —^=w;       — — =W;      — —  =  w;     and  —  -r-2  =  load  in 

centre  uniformly  distributed. 

4  X  16  X  6000  9  X  1 1  X  6000  6  X  14  X  7000 

=19  200  \ —  =  29700;       • —  20400  76s. 

on  r>r\  *  r>n 


Then  -5-  (6000  X  i  1  +  7000  X  6)  +  6000 
0 


=  -^-x  108000+  13200  =  61  800  7bs. 


omitting  uniformly  distributed  load  = =  2000  Ibs.  concentrated  at  centre,  of  A  B. 

Then  to  obtain  total  effect  at  W,  10  :  9  ::  2000  :  1800  —  effect  of  load.     Hence, 
9  X. »X  1800  which  61800  =  70710  Ibs.,  and  7°7I°X9Xl1 

20  20  X  I42  X  200 

=  8.93  ins.  breadth  of  beam. 

Operation  deduced  by  Graphic  Delineation  of  Greatest  Stress  without  uni- 
form Load. 
Fig.  3.  < -j— . . }.  Moments  of  weights  = 

At  i  2 —  3  T  B     40'  or       W  mn          ^   w  s  u 

_;    -- .;  and  —  = 

19  200,  29  700,  and  29  400,  and 
let  fall  perpendiculars  i,  2,  and  3 
proportionate  thereto. 

Connect  w',  W,  and  w  with 
A  B,  and  sum  of  distances  of  in- 
tersections of  these  lines  upon  perpendiculars,  from  x,  2,  and  3,  respectively,  will 
give  stress  upon  A  B  at  these  points. 
Whence,  greatest  stress  at  greatest  load  will  be  ascertained  to  be  61 800  Ibs. 


as  in  Fig.  2.  I 

ILLUSTRATION.— Take  elements  of  above  case,  omitting  uniformly  distributed  load. 

—  (6000  X  ii  X  7000  X  6)  -f-  6000  "  X4  =  —  X  108  000+  13  200  =  61  800  Ibs. 
20  20         20 

Deflection   of  Girders   and   Beams. 


ing  length  in  feet,  b  and  d  breadth  and  depth,  and  D  deflection  in  ins. 
Values  of  C  for  Various  Woods.    (Hatfield.) 


Ash.. 


,  4000 


Chestnut 2550 

Hemlock 2800 

Hickory 3850 


Larch 2093 

Oak,  white 3100 

"    English,  mean. .  2686 
Spruce 3500 


Pine,  Georgia 5900 

"     pitch 2836 

u     white 2900 

"     red 4259 


ILLUSTRATION.— What  would  be  deflection  of  a  floor  beam  of  white  pine,  10  feet 
in  length,  4  ins.  in  breadth,  and  8  in  depth,  with  4000  Ibs.  loaded  in  its  middle? 

4000X10'        4000000  . 

C  =  2900.  ^  ^ — ^  g3  =  —      —  =  -674  tncA. 


2900  X  4  X  8  3 


5  939  20° 


*  Load  uniformly  distributed. 


STRENGTH   OF   MATERIALS.  —  TRANSVERSE.          84! 
When  Weight  is  Uniformly  Distributed. 


.625  W?» 

"D 


3  /29™*3Xr*X*  =  3/^ 

V      -625  x  6000         v 


Hence,  Deflection  in  preceding  illustration  would  be  ,674  x  .625  =  .421  ins. 
ILLUSTRATION.  —  What  should  be  length  of  a  white-pine  beam  3  by  10  ins.,  to  sup- 
port 6000  Ibs.  uniformly  distributed,  with  a  deflection  of  2  ins.  ?  C  =  2900. 

^400000 

3750 

A  fair  allowance  for  deflection  of  floor  beams,  etc.,  is  .03  inch  per  foot  of  length; 
.04  inch  may  be  safely  resorted  to. 

"Weights   of  Floors   and.   of  Loads. 

Dwellings.  —  Weight  of  ordinary  floor  plank  of  white  pine  or  spruce,  3  Iba. 
per  sq.  foot,  and  of  Georgia  pine,  4.5  Ibs. 

Plastering,  Lathing,  and  Furring  will  average  9  Ibs.  per  sq.  foot. 

Clay  Blocks  (Flat  Arch)  5.25  x  7.25  ins.  in  depth  and  i  foot  in  length, 
21  Ibs.  =  80  Ibs.  per  cube  foot  of  volume. 

Floors  of  dwellings  will  average  5  Ibs.  per  sq.  foot  for  white  pine  or  spruce, 
and  on  iron  girders  will  average  from  17  to  20  Ibs.  per  sq.  foot. 

Weight  of  men,  women,  and  children  over  5  years  of  age,  105.5  Ibs.,  and 
one  third  of  each  will  occupy  an  average  area  of  12  x  16  ins.  =  192  sq.  ins. 
=  78.5  Ibs.  per  sq.  foot. 

Of  men  alone  15  x  20  ins.  =  300  sq.  ins.  =48  in  100  sq.  feet. 

Bridges,  etc.  —  Weight  of  a  body  of  men,  as  of  infantry  closely  packed,  = 
138  Ibs.  each,  and  they  will  occupy  an  area  of  20  X  15  ins.  =  300  sq.  ins.  = 
66.24  Ibs.  per  sq.  foot  of  floor  of  bridge,  and  as  a  live  or  walking  load,  80  Ibs. 
per  sq.  foot. 

Weight  of  a  dense  and  stationary  crowd  of  men,  120  Ibs.  per  sq.  foot. 

Bridging  of  Floor  Beams  increases  their  resistance  to  deflection  in  a  very 
essential  degree,  depending  upon  the  rigidity  and  frequency  of  the  bridges. 

Weight  on  Floors,  etc.,  in  addition  to  Weight  o-f  Struct- 
ure, per   Sq..  Foot. 


Ballrooms 85  Ibs. 

Brick  or  stone  walls 115  to  150 

Churches  and  Theatres. . .  80 

Dwellings 40 

Factories 200  to  400 

Grain 100 


Roofs,  wind  and  snow 30  to  35  Ibs. 

Slate  roofs 45  " 

Snow,  per  inch .  5  Ib. 

Street  bridges 80  Ibs. 

Warehouses 250  to  50x3  " 

Wind 50  " 


Scarfs. 

Relative  resistance  of  scarfs  in  Oak  and  Pine,  2  ins.  square,  and  4  feet  in 
length,  by  experiments  of  Col.  Beaufoy. 

Scarf  12  ins.  in  Length  and  13  ins.  from  End,  or  i  inch  from  Fulcrum. 

Vertical. — no  Ibs.  gave  away  in  scarf. 

Horizontal,  large  end  uppermost  and  towards  fulcrum. — 101  Ibs.  fastenings 
drew  through  small  end  of  scarf ;  small  end  uppermost,  etc.,  87  Ibs.  gave 
away  in  thick  part  of  scarf. 

Factors   of  Safety. 

Statical  or  Dead  Load  at  .2  of  destructive  stress,  but  for  ordinary  pur- 
poses it  may  be  increased  to  .25,  and  in  some  cases  with  good  materials  to  .3. 
Live  Load  at  .1  to  .125  of  destructive  stress. 
See  also  page  802. 


842 


SUSPENSION   BRIDGE. 


SUSPENSION   BKIDGE. 
To    Compute   Elements. 


sin.  z 

=  stress  at  • .  C  representing  chord  or  span,  a  half  chord,  and  v  versed  sine  of 
chord  or  curve  of  deflection,  in  feet,  L  distributed  load  inclusive  of  suspended  struct- 
ure, Q  load  per  lineal  foot,  and  S  stress  at  centre,  all  in  tons,  x  distance  of  any  point 
from  centre  of  curve,  and  h  height  of  chain  at  x  above  centre  of  it,  both  in  feet,  s 
stress  on  chain  at  any  point,  as  x,from  centre  of  span,  s  stress  on  any  tension-rod, 
and  t  stress  at  abutments,  all  in  tons,  n  number  of  tension-rods,  o  angle  of  tangent 
of  chain  with  horizon  at  any  point,  as  x,  r  angle  of  chain  with  vertical  at  abutments, 
I  length  of  chain,  in  feet,  and  z  angle  of  direction  of  chain. 

Assume  C  =.  300 feet,  L  =  1000  tons,  v  =  25  feet,  x  =  ioofeet,  n  =  30,  r  —  71°  34', 
and  o  =  12°  32'. 


1500 


tan. 


4X  ii.n 

2X  100 


=  . 2222  =  I2U  32  —  = 


4>_  =  cot  angle  r. 


/7T 

V      \ 


s  =  ,;  and- 


2X25 


-  =  .3162=18°  26'. 


For  a  deflection  of  .125  of  span,  horizontal  stress  is  equal  to  total  load. 
To  Construct  curve,  see  Geometry,  page  230. 

To  Compute  Ratio  which  Stress  on  Chains  or  Cables  at 
either  Point  of  Suspension  Bears  to -whole  Suspended 
"Weight  of  Structxire  and  Load. 

sjp  9  =  R.    R  representing  ratio. 
ILLUSTRATION.—  Assume  elements  of  preceding  case. 

— — —  =  1.58  ratio.    By  a  preceding  formula  it  would  be  1.536. 

Stress  on  Back  Stays. — The  cables  being  led  over  rollers,  having  free  mo- 
tion, tension  upon  them  is  same,  whether  angle  i  is  same  as  that  of  r  or  not. 

Stress  on  Piers.— When  angles  r  and  i  are  alike,  stress  on  piers  will  be 
vertical,  but  when  angle  of  i  is  greater  or  less  than  r,  stress  will  be  oblique. 

To    Compute    Horizontal    Stress    and   'Vertical    Pressure 
on    Piers. 

S  cos.  z  —  S  i,  S  cos.  n  =  S  o,  S  sin.  z—  P  i,  and  S  sin.  n  —  P  o.  Si  and  S  o 
representing  stress,  and  P  i  and  P  o  pressure,  inward  and  outward. 

NOTE.— Span  of  New  York  and  Brooklyn  Bridge  1595.5  feet,  deflection  128  feet, 
angle  of  deflection  at  piers  from  horizontal  15°  10'. 


TRACTION. 


843 


TRACTION. 

Results  of  Experiments  on  Traction  of  Roads 
and.  IPavements.  (At.  Morin.) 

i  st.  Traction  is  directly  proportional  to  load,  and  inversely  proportional 
to  diameter  of  wheel. 

2d.  Upon  a  paved  or  Macadamized  road  resistance  is  independent  of 
width  of  tire,  when  it  exceeds  from  3  to  4  ins. 

3d.  At  a  walking  pace  traction  is  same,  under  same  circumstances,  for 
carriages  with  or  without  springs. 

4th.  Upon  hard  Macadamized,  and  upon  paved  roads,  traction  increases 
with  velocity :  increments  of  traction  being  directly  proportional  to  incre- 
ments of  velocity  above  velocity  of  3.28  feet  per  second,  or  about  2.25  miles 
per  hour.  The  equal  increment  of  traction  thus  due  to  each  equal  increment 
of  velocity  is  less  as  road  is  more  smooth,  and  carriage  less  rigid  or  better 
hung. 

5th.  Upon  soft  roads  of  earth,  sand,  or  turf,  or  roads  thickly  gravelled, 
traction  is  independent  of  velocity. 

6th.  Upon  a  well-made  and  compact  pavement  of  dressed  stones,  traction 
at  a  walking  pace  is  not  more  than  .75  of  that  upon  best  Macadamized 
roads  under  similar  circumstances ;  at  a  trotting  pace  it  is  equal  to  it. 

yth.  Destruction  of  a  road  is  in  all  cases  greater  as  diameters  of  wheels 
are  less,  and  it  is  greater  in  carriages  without  springs  than  with  them. 

Experiments  made  with  the  carriage  of  a  siege  train  on  a  solid  gravel 
road  and  on  a  good  sand  road  gave  following  deductions : 

1.  That  at  a  walk  traction  on  a  good  sand  road  is  less  than  that  on  a  good 
firm  gravel  road. 

2.  That  at  high  speeds  traction  on  a  good  sand  road  increases  very  rapidly 
with  velocity. 

Thus,  a  vehicle  without  springs,  on  a  good  sand  road,  gave  a  traction  2.64! 
tunes  greater  than  with  a  similar  vehicle  on  same  road  with  springs. 


Results   -with,   a   Dynamometer. 

Wagon  and  Load  2240  Ibs.  * 


ROADWAY. 


Relat'e  num- 
ber of  horses 
for  like  effect. 


On  railway,  8  Ibs 

On  best  stone  tracks,  12.5  Ibs. 
Good  plank  road,  32  to  50  Ibs. 


. 
4  to  6.25 


Telford  road,  46  Ibs 

Broken  stone  or  con'te,  46  Ibs. 

Gravel  or  earth,  140-147  Ibs.  I 
Common  earth  road,  200  Ibs. . 


Relat'e  nnm- 
ber  of  horses 
for  like  effect. 


5-75 
5-75 
17-5 
18.37 
25 


Stone  block  pavement,  32.5  "         4.06 
Macadamized  road,  65  Ibs. ...        8.12 

NOTE.— By  recent  experiments  of  M.  Dupuit,  he  deduced  that  traction  is  inversely 
proportional  to  square  root  of  diameter  of  wheel. 

Relation  of  force  or  draught  to  weight  of  vehicle  and  load  over  6  different  con^ 
Et  ructions  of  road,  gave  for  different  speeds  as  follows: 

Walk.       Trot.  Walk.      Trot. 

Stage  coach,  5  tons.  .1.3  x      |     Carriage,  seats  only,  on  springs.  .1.29       i 

Resistance   to    Traction    on    Common    Roads. 

On  Macadamized  or  Uniform  Surfaces.    (M.  Dupuit.) 

1.  Resistance  is  directly  proportional  to  pressure. 

2.  It  is  independent  of  width  of  tire. 

3.  It  is  inversely  as  square  root  of  diameter  of  wheel. 

4.  It  is  independent  of  speed. 

*  See  Treatise  on  Roads,  Streets,  and  Pavements,  by  Brev.  Maj.-Gen'l  Q.  A.  Gillmore,  U.  8.  A. 
t  Telford  estimated  it  at  3.5. 


844 


TRACTION. 


On  Paved  and  Rough  Roads. 

Resistance  increases  with  speed,  and  is  diminished  by  an  enlargement  of 
tire  up  to  a  moderate  limit. 

Traction  on  Various  Roads.— Traction  of  a  wheeled  vehicle  is  to  its  weight 
upon  various  roads  as  follows : 

Per  Ton. 
46  to    78 
46  to   90 


Per  Ton. 

Stone  track,  best  12.5  to  15 
"         "      ....  28     to  39 
"     pavement.  14     to  36 
Asphalted    22     to  28 

Per  100  Ibs. 
•  55  to    .58 
1.25  to  1.3 

.5   to  1.5 
i      to  1.25 

Plank  22     to  45 

.98  tO  2 

Block  stone       }         t 
pavement....  J32 

1.4    to  1.6 

Tel  ford  road 

Macadamized. . . 

"       loose    67  to  1 

Gravel 134  to  180 

Sandy 140  to  313 

Earth 200  to  290 


Per  ito  Ibi. 
2.1  tO     3.5 

2  to    4 

3  t(>    5 
6     to   8 
6.3  to  14 
9     to  13 


Hence,  a  horse  that  can  draw  140  Ibs.  at  a  walk,  can  draw  upon  a  gravel  road 
X  JOG  =  2000  Ibs. 


Resistance   on   Common.   Roads   or   Fields 

(Bedford  Experiments,  1874.*) 


GRAVELLED  ROAD. 
(Hard  and  dry,  rising  i  in  430.) 

Maxi- 
mum 
Draft. 

Average 
Draft. 

Average 
Speed 

dour. 

IP  de- 
veloped 
per 
Minute. 

Draft  per  Ton 
on  Level. 

Work 
per  HP 

Horse. 

Lbs. 

Lbs. 

Miles. 

IP. 

Lbs. 

IP. 

2  horse  wagon  without  springs. 

320 

159 

2-5 

i.  06 

43.  5  or.  0192 

•£3 

4     " 

400 

251 

2.6 

1.74 

44-5  "-°2 

.87 

2     "         "      with 

300 

133 

2.47 

.88 

34.7  "  .015 

•44 

i     "     cart      without      " 

180 

49-4 

2.65 

•35 

28      "  .0125 

•35 

ARABLE  FIELD. 

(Hard  and  dry,  rising  i  in  1000.) 

2  horse  wagon  without  springs. 

IOOO 

700 

2-35 

4-36 

210    or  .099 

2.18 

4     "          " 

I2OO 

997 

2.52 

6.7 

194    "  .083 

3-35 

I     "         «       with            " 

IOOO 

710 

2-35 

4-45 

210      "  .099 

1.22 

i     "     cart      without      " 

400 

212 

2.61 

1.48 

140      "  .0625 

I.48 

Fore  wheels  of  wagons  were  39  ins.,  and  hind  57  ins.  in  diam. ;  tires  varying  from 
2.25  to  4  ins. ;  and  wheels  of  cart  were  54  ins.  in  diam.,  and  tires  3.5  and  4  ins. 

Springs  reduced  resistance  on  road  20  per  cent.,  but  did  not  lessen  it  in  the  field. 

From  these  data  it  appears,  that  on  a  hard  road,  resistance  is  only  from  25  to  .  16 
of  resistance  in  field.  Lowest  resistance  is  that  of  cart  on  road  —  28  Ibs.  per  ton ; 
due,  no  doubt,  to  absence  of  small  wheels  alike  to  those  of  the  wagons. 

Assuming  average  power  without  springs  to  be  .6  IP  on  road,  as  average  for  a 
day's  work,  it  represents  .6  X  33000  =  19800  foot-lbs.  per  minute  for  power  of  a 
horse  on  such  a  road. 

Resistance  of  a  smooth  and  well-laid  granite  track  (tramway),  alike  to  those  in 
London  and  on  Commercial  Road,  is  from  12.5  to  13  Ibs.  per  ton. 

Omnibus. t    ( Weight  5758  Ibs.) 

Average  Speed  per  Hour.      Per  Ton.  Total. 

Granite  pavement  (courses  3  to  4  ins.) 2.87  miles.      17.41  Ibs.  44.75  Ibs 

Asphalt  roadway 3.56     "           27.14  "  69.75    " 

Wood  pavement 3.34     "  41.6     '  106.88    " 

Macadam  road,  gravelly 3.45     "  44-48  "  114-32    " 

"           "     granite,  new 3.51     "  101.09  "  2598     " 

NOTE.— The  resistance  noted  for  an  asphalt  roadway  is  apparently  inconsistent 
with  that  for  a  granite  pavement,  for  when  it  is  properly  constructed  it  is  least 
resistant  of  all  pavements. 


*  See  report  in  Engineering,  July  10,  1874,  pagt  23. 


t  Report  Soc.  Arts,  London,  1875. 


TRACTION.  845 

Wagon .    (Sir  John  Macneil. ) 

Weight  2342  Ibs.     Speed  2.5  Miles  per  Hour. 

Resistance. 
Per  Ton.  Total. 

Well-made  stone  pavement 31.2  Ibs  33  Ibs. 

Road  made  with  6  ins.  of  broken  hard  stone,  on  a  foundation)               4<  ,    u 

of  stoues  in  pavement,  or  upon  a  bottom  of  concrete >  44 

Old  flint  road,  or  a  road  made  with  a  thick  coating  of  broken  )  ,        u  6     tt 

stone,  on  earth f 

Road  made  with  a  thick  coating  of  gravel,  on  earth 140      "  147   " 

Stage   Coach.     (Sir  John  Macneil.) 
Weight  3192  Ibs.      Gradients  i  to  20  to  600. 
Speed.  Metalled  Road. 

At   6  miles  per  hour 62  Ibs.  per  ton. 

"     8        «  73     " 

"  10 79     ' 

NOTE. — It  was  found  that,  from  some  unexplained  cause,  the  net  frictional  resistance  at  equal  speeds 
varied  considerably,  according  to  gradient,  resistances  being  a  maximum  for  steepest  gradient,  and  a 
minimum  for  gradients  of  i  in  30  to  i  in  40 ;  for  these  they  are  less  than  i  in  600.  Mode  of  action  of 
the  horses  on  the  carriage  may  nave  been  an  influential  element.  (I).  K.  Clark.) 

To  Compxite  Resistance  to  Traction  on  "Various  Roads. 

(Sir  John  Macneil.) 
ON   A  LEVEL. 

RULE. — Divide  weight  of  vehicle  and  load  in  Ibs.  by  its  unit  in  following 
table,  and  to  quotient  add  .025  of  load ;  add  sum  to  product  of  velocity  of 
vehicle  in  feet  per  second,  and  Coefficient  in  following  table  for  the  particular 
road,  and  result  will  give  power  required  in  Ibs. 

Or,  —  .  W  -{-  w  .025 -|-  C  v  =  T.    W  and  w  representing  weights  of  vehicle  and  load, 
unit 

Coefficients  for  Traction  of  Various  Vehicles. 

Stage  coach too  I  2  horse  wagon  without  springs 54 

Heavy  wagon 93    2     "          "      with  "      42 

4  horse  wagon  without  springs 55  I  i     "     cart      without       "      36 

Coefficients  for  Roads  of  Various  Construction. 


Pavement 2 

Broken  stone,  dry  and  clean 

"      covered  with  dust. . . . 
"          "      muddy 10 


Macadamized  road 4.3 

Gravel,  clean 13 

"       muddy 32 

Stone  tramway 1.2 


Sand  and  Gravel 12.1 

ILLUSTRATION.— What  is  the  traction  or  resistance  of  a  stage  coach  weighing  2200 
Ibs.,  with  a  load  of  1600  Ibs.,  when  driven  at  a  velocity  of  9  feet  per  second  over  a 
dry  and  clean  broken  stone  road  ? 

22OO-4-l6oO    .    ,   • 

• 1-  1600  X  .025  -f  5  X  9  =  123  Ibs. 

To  Compute  JPower  necessary  to  Sustain  a  "Vehicle  upon 
an  Inclined  Road,  and  also  its  Pressure  thereon ^omit>- 
ting  Effect  of  Friction. 

AT   AN    INCLINATION. 

W  :  A  C  ::  o  :  B  C,  and  W  :  A  C  ::  p  :  A  B. 
Or,  r  e  :  e  o  : :  A  B  :  B  C;  W  :  e  o  : :  I :  h:  whence, 
W  j  =  eo. 

Assume  A  B  of  such  a  length  that  vertical  rise 
BC  =  i  foot;  then, 


x/.<TB2+i 


846 


TRACTION. 


W  W  V  W^  W  I'2 

Or,  —  =  P;     -  =  »;    or,  —  =  P.  and   —  =r>.     W  representing 

1  l  vV2+i  Vr2+i 

weights  of  vehicle  and  load  o,  and  Y  power  or  force  necessary  to  sustain  load  on  road, 
p  pressure  of  load  on  surface,  all  in  Ibs.,  h  height  of  plane,  I  inclined  length  of  road 
or  plane,  and  V  horizontal  length,  all  in  feet. 

ILLUSTRATION.—  What  is  power  required  to  sustain  a  carriage  and  its  load,  weigh- 
ing 380x3  Ibs.,  upon  a  road,  inclination  of  which  is  i  in  35,  and  what  is  its  pressure 
upon  road? 

Sin.  A  =  .028  56.     Cos.  A  =  .999  59.     I  =  35.014. 

Then  3800  X-Q28  56=  108.53  Jt)S-  —  power,  and  3800  X  .99959=3798.44  Ibs.  pressure. 

To  Compute  Resistance  of  a  Load,  on  an  Inclined.  Road. 

RULE.  —  Ascertain  the  tractive  power  required,  and  add  to  it  the  power 
necessary  to  sustain  load  upon  inclination,  if  load  is  to  ascend,  and  subtract 
it  if  to  descend. 

EXAMPLE  i.  —  In  preceding  example  tractive  power  required  is  123  Ibs.,  and  sus- 
taining power  for  that  inclination  108.53;  hence  123  +  108.53^231.53  Ibs. 

2.  —  If  this  load  was  to  be  drawn  down  a  like  elevation. 
Then  123  —  108.53  =  14.47  Ms. 

To  Compute  Power  necessary  to  Move  and  Sustain  a 
"Vehicle  either  Ascending  or  Descending  an  Elevation, 
and  at  a  given  Velocity,  omitting  Effect  of  ifriction. 

f  —    —  -f-  —  J  cos.  L  =F  (W  -f  w)  sin.  L.  -j-  Fc  =  R.     L  representing  angle  of 

elevation  for  a  stage  wagon  and  a  stage  coach,  and  t  units  as  preceding  ;  upper  sign 
taken  when  vehicle  descends  the  plane,  and  lower  when  it  ascends. 

ILLUSTRATION.—  Assume  a  stage  coach  to  weigh  2060  Ibs.,  added  to  which  is  a 
load  of  i  zoo  Ibs.,  running  at  a  speed  of  9  feet  per  second  over  a  broken  stone  road 
covered  with  dust,  and  having  an  inclination  of  i  in  30;  what  is  power  necessary 
to  move  and  sustain  it  up  the  inclination,  and  what  down  it  ? 

v  =  <),    c  =  8,    sin.  of  L.  =sin.  of  i°  54'-J-:r=.o333,    and  cos.  L.  =.9995. 

Then  (2°  ^oo"00  -T-  ^)  X  .9995  +  (2060+  i  loo  )X.  0333  +  8X9  =  59-°7  + 
105.23  -j-  72  3=  236.3  Ibs.  up  inclination. 


"  —      "°° 


ADd    "  —  100        +      r    X  *9995  +  8X9  —  (2060  +  i  ioo)  X.  0333  =  59.07  -f  72 
—  105.23  =  25.84  Ibs.  down  inclination. 

Tractive  and  Statical  Resistance  of  Elevations.     (Gillmore.) 

T 

:  =  g'.    T  representing  traction  in  Ibs.  per  ton,  W  weight  of  load  in  Ibs., 


VW2  —  T2 
and  g'  grade  of  road. 

ILLUSTRATION.— Assume  traction  as  per  preceding  table,  page  844,  200,  and  weight 
of  vehicle  2  tons;  what  should  be  least  grade  of  road? 

200  X  2  I     . 

, =^  =  .0897  =  -+. 

V  448o2  —  200  X  2 

Showing  that,  for  a  road  upon  which  traction  is  200  Ibs.  per  ton,  the  grade  should 
not  exceed  one  in  height  to  one  eleventh  fall  of  base;  hence,  generally,  the  proper 
grade  of  any  description  of  road  will  be  equal  to  force  necessary  to  draw  load  upon 
like  road  when  level. 

Practically,  greatest  grade  of  a  Telford  or  Macadamized  road  in  good  condition 
=  .05,  and  a  horse  can  attain  at  a  walk  a  required  height  upon  this  grade,  without 
more  fatigue  and  in  nearly  same  time  that  he  would  require  to  attain  a  like  height 
over  a  longer  road  with  a  grade  of  .033,  that  he  could  ascend  at  a  trot. 

For  passenger  traffic,  grades  should  not  exceed  .033. 


TRACTION. 


847 


Resistance    of*  Gravity    at    Different    Inclinations    of 
Grade.     For  a  Load  of  ioo  Lbs. 


Grade. 

R 

Grade. 

R 

Grade. 

R 

Grade. 

R 

i  in    5 

i  in  10 
i  in  15 
i  in  20 

Lbs. 
19.61 
9-95 
6.65 
4.99 

i  in  25 
i  in  30 
i  in  35 
i  in  40 

Lbs. 

4 
3-33 
2.85 

2-5 

i  in  45 

i  in  50 

i  in  55 
i  in  60 

Lbs. 
2.22 
2 
1.82 
1.67 

i  in   70 
i  in    80 
i  in   90   . 
i  in  ioo 

Lbs. 
1-43 
1-25 
i.  ii 

I 

Inclination  of  Roads.—  Power  of  draught  at  different  inclinations  and  velocities 
Is  as  follows  (Sir  John  Macneil) : 


Inclination. 

Angle. 

Feet 
per  Mile. 

Tractioi 
6  Miles. 

i  at  Speed 
Hour  of 
8  Miles. 

a  of  per 

10  Miles. 

Frictior 
Ton  at  S 
6  Miles. 

al  Resists 
jeedsofpt 
8  Miles. 

nee  per 
r  Hour  of 
10  Miles. 

in  20 
in  26 
in  30 
in  40 
in  60 

2o  5< 

2°  12 

i°55' 

I°26' 

57-5' 

264 
203.4 
176 
132 
88 

268 
2I3 

«*s 

1  60 
III 

296 
219 
196 
166 

120 

318 
225 
200 
I72 
128 

% 

456 

72 

P 

61 

78 

112 

& 

ii 

Grade. 

Grade  of  a  road  should  be  reduced  to  least  of  practicable  attainment,  and 
ae  a  general  rule  should  be  as  low  as  i  in  33,  and  steepest  grade  that  is  ad- 
missible on  a  broken  stone  road  is  i  in  20. 


The  condition  of  traction  is  /-f  sin.  a  L,  which  should  not  exceed  P,  and  sin.  a 

P 
should  not  exceed  -= /,  or/    f  representing  coefficient  of  friction,  a  angle  ofin- 

Lt 

clination,  L  load,  and  P  power  in  Ibs. 

ILLUSTRATION.— In  case,  page  846,  weight  or  load  =  2060 4-1100=3160  Ibs.,  Co- 
efficient of  friction  for  such  a  road  =  .042  per  ioo  Ibs. ,  and  sin.  i°  54'  =  .033 16. 

Then  .042  +  .033 16  X  3160  =  237. 5  Ibs. 

Traction  of  a  Vehicle  compared  to  its  Weight  on  Different  Roads. 
(F.  Robertson,  F.  R.  A.  S.) 

Stone  pavement i  in  68  I  Flint  foundation i  in  34 

Macadamized  road i  "  49  |  Gravel  road i  "  15 

Sandy  road i  in  7. 

Assuming  a  horse  to  have  a  tractive  force  of  140  Ibs.  continuously  and  steadily  at 
a  walk,  he  can  draw  at  a  walk  on  a  gravel  road  15  x  140  =  2100  Ibs. 

Friction   of  Roads. 

Friction  of  Roads. — According  to  Babbage  and  others,  a  wagon  and  load 
weighing  1000  Ibs.  requires  a  traction  as  follows : 


Of  Load. 

Fresh  earth  125 

(  -035 
1  .067 
jw,  iron 

Friction 

Per  Ton. 
1  86 

157 
141 
117 

63 

34 

101 

74 

Sled,  hard  sn 
Coefficients  of  , 

Per  ioo  Ibs. 

Gravel  road  new  083 

Common  road,  bad  order.  .  .07 
Sand  road  063 

Broken  stone,  rutted  052 
"          "    fair  order.  .  .  .028 
"          "    perfect  order  .015 
Macadamized  new  045 

"           gravelly  02 
Warth,  good  ovder.  0.15 

Macadamized 

Dry  high  road. .. 
Well  paved  road. 

Railroad 


shod 033  of  load. 

in  Proportion  to  Load. 

Per  ioo  Ibs. 

Wood  pavement 019 

Asphalt  roadway 012 

Stone  pavement 015 

Granite      "        008 

Stone         "    very  smooth  .006 
Plank  road 01 


Of  Load. 

033 
025 
014 

f  -0035 
I  -0059 


Stone  track 05 


Per  Ton 
42 

27 
34 


13 
us 


848 


TRACTION. 


To    Compute    Friction al    Resistance    to    Traction    of  a 
Stage  Coach,  on  a  Metalled  Road,  in  Grood  Condition. 

30 -|-  4  v  +  \/10  v  =  &    v  representing  speed  in  miles  per  hour,  and  R  frictional 
resistance  to  traction  per  ton. 
NOTE.— Formula  is  applicable  to  wagons  at  low  speeds. 

Canal,  Slackwater,  and.   River. 

On  a  canal  and  water,  resistance  to  traction  varies  as  square  of  velocity, 
from  that  of  2  feet  per  second  to  that  of  11.5  feet. 

When  velocity  is  less  than  .33  miles  per  hour,  resistance  varies  in  a  less 
degree. 

In  towing,  velocity  is  ordinarily  i  to  2.5  miles  per  hour. 

Resistance  of  a  boat  in  a  canal  depends  very  much  upon  the  comparative 
areas  of  transverse  sections  of  it  and  boat,  it  being  reduced  as  difference 
increases. 

In  a  mixed  navigation  of  canal  and  slack-water,  3  horses  or  strong  mules 
will  tow  a  full-built,  rough-bottomed  canal  boat,  with  an  immersed  sectional 
area  of  94.5  sq.  feet,  and  a  displacement  of  240  tons,  1.75  to  2  miles  per  hour 
for  periods  of  12  hours. 

With  a  section  of  but  24.5  sq.  feet,  or  a  displacement  of  65  tons,  an  aver- 
age speed  of  2.5  miles  is  attained  for  a  like  period. 

By  the  observations  of  Mr.  J.  F.  Smith,  Engineer  of  the  Schuylkill  Navigation 
Co.,  a  canal  boat,  with  an  immersed  section  alike  to  that  above  given,  can  be  towed 
for  10  hours  per  day  as  follows: 

Per  Hour. 


By  i  horse  or 
mule. 

By  2  horses  or 

mules. 

By  3  horses  or 
mules. 

By  4  horses  or 
mules. 

By  8  horses  or 
mules. 

i  mile. 

i.  5.  miles. 

1.75  miles. 

1.875  miles. 

2.5  miles. 

Assuming  then,  the  tractive  power  of  a  horse  as  given  in  table,  page  437,  the  above 
elements  determine  results  as  follows: 


Horses. 

Miles. 

Tractive  Power 
divided  by  Load. 

in  Lbs.  per  Ton. 

fraction 
in  Lbs.  per  Sq.  Foot  of 
immersed  Section. 

250  —  240 

1.04 

2.65 

165  X  2  —  240 

1.38 

3-49 

3  

I  7^ 

140  X  3  —  240 

4.  44 

1.871; 

132  X  3  —  240 

1.65 

4.19 

125  X  3  —  240 

i  56 

•^.08 

?  (fight) 

2.  S 

loo  X  ^  —   6q 

4.61 

12.24 

Upon  a  canal  of  less  section  and  depth,  a  displacement  of  105  tons,  with  a:i  im- 
mersed section  of  43  sq.  feet,  a  speed  of  2  miles  with  2  borses  was  readily  obtained, 
which  would  give  a  traction  of  2.38  Ibs.  per  ton,  and  of  5.71  Ibs.  per  sq.  foot  of  im- 
mersed section. 

Maximum   Power  of  a  Horse   on   a  Canal.    (Molesworth.) 
Miles  per  hour 2.5        3        3.5       4         5         678          9       10 

D".™.:n}     »-5        *        M        4-5      *.9      -       «.S     —5      .9        -73 

Load  drawn  in  tons ..  520       243    153       102       52       30     19     13          9        6.5 

Street  Railroads   or  Tram-ways.    (Gen'l  Gillmore.*) 

Upon  a  level  road,  and  at  a  speed  of  5  miles  per  hour,  the  power  required  to  draw 
a  car  and  its  load  is  from  -g^-Q  to  ^^  of  total  weight,  varying  with  condition  of 
rails  and  dryness  or  moisture  of  their  surface. 

*  Treatise  on  Roads,  Streets,  and  Pavements.     D.  Van  No«trand,  1876,  N.  Y. 


TJRACTION.  —  WATEK.  849 

To    Compute    Resistance   of*  a   Car. 

TX6=/;    -   —  =  c;    -     —  =  r;  and  /+  c  +  r  =  R.    T  representing  weight 

in  tons,  f  friction  in  Ibs.,  v  speed  in  miles  per  hour,  a  area  of  front  or  section  of  car 
in  sq.feet,  c  concussion,  r  resistance  of  atmosphere,  and  R  total  resistance,  all  in  Ib*. 
ILLUSTRATION.—  Assume  a  car  and  load  of  8960  Ibs.,  with  an  area  of  section  of  56 
sq.  feet,  and  a  speed  of  5  miles  per  hour. 


Then  =  4  tons  ;     4  x  6  =  24  Ibs.  friction  ;     1-   =  6.66  Ibs.  concwrion  ; 

5  -  5_  =  3.  5  Ibs.  resistance  of  air  ;    and  24  +  6.  66  +  3.  5  =  34.  16  Ibs. 


400 

In  average  condition  of  a  road,  the  resistance  of  a  car  may  be  taken  at  T^,  which, 
in  preceding  case,  would  be  74.66  Ibs.  On  a  descending  grade,  therefore,  of  i  in 
74  66,  the  application  of  a  brake  would  mot  be  required. 


WATER. 

FRESH  WATER.    Constitution  of  it  by  weight  and  measure  is 

By  Weight.        By  Measure.    I  By  Weight.          By  Mewur*. 

Oxygen...  88.9  i         |    Hydrogen.,  n.i  2 

Cube  inch  of  distilled  water  at  its  maximum  density  of  39.  i°,  barom- 
eter at  30  ins.,  weighs  252.879  grains,  and  it  is  772.708  times  heavier 
than  atmospheric  air. 

Cube  foot  (at  39.1°)  weighs  998.8  ounces,  or  62.425  Ibs. 

NOTE. — For  facility  of  computation,  weight  of  a  cube  foot  of  water  is 
usually  taken  at  1000  ounces  and  62.5  Ibs. 

At  a  temperature  of  32°  it  weighs  62.418  Ibs.,  at  62°  (standard  tem- 
perature) 62.355  Ibs.,  and  at  212°  59.64  Ibs.  Below  39.1°  its  density 
decreases,  at  first  very  slow,  but  progressing  rapidly  to  point  of  conge- 
lation, weight  of  a  cube  foot  of  ice  being  but  57.5  Ibs. 

Its  weight  as  compared  with  sea-water  is  nearly  as  39  to  40. 

It  expands  .085  53  its  volume  in  freezing.  From  40°  to  12°  it  ex- 
pands .00236  its  volume,  and  from  40°  to  212°  it  expands  .0467  — 
times  =  .ooo  271  5  for  each  degree,  giving  an  increase  in  volume  of  i 
cube  foot  in  21.41  feet. 

Volumes,    Height,    and.    Pressure    of    Pure     Water. 

Cube  Ins.        Feet. 


At   32° 

27.684 

— 

2.307 

)           i  Lb. 

At  62° 

i  Ton 

—  35-923   cube  feet. 

"    39-i° 

27.68 

— 

2.3067 

_  Pressure 

"    " 

i  Lb. 

= 

27.71 

'     ins. 

"    62° 

27.712 

s±s 

2-3°93 

per 

\\   39ul0 

i  Tonneai 

I  == 

35.3156 

"     feet. 

"  212° 

28.978 

= 

2.4148 

sq.  inch. 

i  Kilogr. 

= 

0353 

((             U 

Height  of  a  Column  of  Water  at  62°  or  62.355  Ibs. 

i  Ib.  per  sq.  inch  =  2. 3093  feet,  and  at  pressure  of  atmosphere  =  33. 947  feet  = 
10.347  meters. 

Ice   and    Snow. 

Cube  foot  of  Ice  at  32°  weighs  57.5  Ibs.,  and  i  Ib.  has  a  volume  of 
30.067  cube  ins. 

Volume  of  water  at  32°,  compared  with  ice  at  32°,  is  as  i  to  1.085  53?  ex- 
pansion being  8.553  per  cent. 

Cube  foot  of  new  fallen  snow  weighs  5.2  Ibs.,  and  it  has  12  times  bulk  of 
water. 


850 


WATER. 


Rainfall. 
Annual  Fall  at  different  Places. 


LOCATIOW. 

Ins. 

LOCATION. 

Ins. 

LOCATION. 

Ins. 

Alabama 

Ft  Crawford  Wis.. 

29  54 

Michigan  

•3-1  e 

Albany       

41.  3^ 

Ft  Gibson,  Ark  

30.  64 

45 

7.  ye 

Ft.  Snelling,  Iowa.. 

30.  32 

Mobile,  1842  

S4.  Q4 

Allegheny    ....... 

,666 

Fortr  Monroe,  Va.. 

52.  53 

Naples  

41  8 

Antigua 

Florence      

O.C    Q 

Newburg 

Archangel     .... 

14   ^2 

Frankfort  Oder... 

21.3 

New  York  

4°-5 
36 

"         Main 

»6  A 

Ohio  

•3.6 

Bahamas 

Geneva  

iu.4 

02  6 

Palermo  

22  8 

Baltimore    

OQ    Q 

Gibraltar  

47-  20 

Paris  

23.  1 

55.87 

21.3 

Philadelphia  

49 

Bath  Me      

1,4.  «;8 

31 

Plymouth  (Eugl  )  . 

Belfast 

* 

Gordon  Castle  Sc'd 

Port  Philip 

20  16 

Biskra  

.2 

Granada  | 

105 

Poughkeepsie  

32.06 

Bombay  
Bordeaux 

1  10 

Great  Britain  ..... 

126 

Providence  
Rochester  

36.74 

Boston     

OQ    27 

Greenock  

61.8 

Rome  

on 

Brussels       • 

Halifax 

Santa  Fe 

£Le 

Buffalo       

Hanover  

22.4. 

Savannah  

55 

52 

Schenectadv  

47-  77 

Calcutta 

RT 

Hong-kong    

8l   3S 

Siberia              .... 

Cape  St.  Francois.  . 
Cape  Town  

150 
23-31 

Hudson  
India                     { 

39-32 
60 

Sierra  Leoiie  
Sitka  

84 
85.79 

Charleston  
Cherbourg  

54 

OQ.  7 

Jamaica  

I30 

•3  A     -31 

St.  Bernard  
St.  Domingo  

48 
1  20 

Cologne  
Copenhagen    

24 

23 

Jerusalem  
Key  West  

J4-31 

65 

31   3Q 

St.  Petersburg  
State  of  N  Y  

17.6 

OO    7Q 

Cracow  

10.  oo 

Khassaya,  India.  .  . 

610 

Sydney  

4Q 

Demerara      ... 

Lewiston  

Tasmania  

"       1849  

10,2.  21 

Liverpool  

34  *2 

Trieste  

46.4 

Dover  (Engl.  )  
Dublin  

37-52 
30.87 

London  

25-2 
51  85 

Ultra  Mullay,  India 
Utica  

263.21 
on  -3 

Dumfries  

36.92 

22 

Venice  

34-  I 

East  Hampton 

•38  S2 

4Q 

Vera  Cruz  

62 

/ 

49 
ofi  IA. 

Vienna 

10  6 

Edinburgh  J 

Manchester  j 

Washington      .   . 

19.0 

Fairfield... 

32.03 

West  Point  .  .  . 

48.7 

Average  rainfall  in  England  for  a  number  of  years  was,  South  and  East,  34  ins.; 
West  and  hilly,  43.02  to  50  ins.,  and  percolation  of  it  was  estimated  at  30  per  cent. 
Mean  volume  of  water  in  a  cube  foot  of  air  in  England  is  3.789  grains. 

Globe,  mean  depth 36     ins. 

Cape  of  Good  Hope  in  1846 in  3  hours,  6.2    " 

At  Khassaya,  in  6  rainy  months 550  ins. ;  in  i  day,    25. 5    " 

Evaporation. — Mean  daily  evaporation,  in  India  .22  inch;  greatest  .56;  in  Eng- 
land .08.  At  Dijon,  when  mean  depth  of  rainfall  was  26.9  ins.  in  7  years,  evapora- 
tion was  for  a  like  period  26.1  ins.,  and  in  Lancashire,  Eng.,  when  fall  was  45.96 
ins.,  evaporation  was  25.65. 

"Volume   of  Rainfall. 

Rainfall,  depth  in  ins. ,  X  2  323  200  =  cube  feet  per  sq.  mile. 

X  17.378  74  =  millions  of  gallons  per  sq.  raila 
X  3630  =  cube  feet  per  acre. 
'•  "        "       X  27  154.3  =  gallons  per  acre. 

Mineral  Waters  are  divided  into  5  groups,  viz. : 

1.  Carbonated,  containing  pure  Carbonic  acid  — as,  Seltzer,  Germany;  Spa,  Bel- 
gium; Pyrmont,  Westphalia;  Seidlitz,  Bohemia;  and  Sweet  Springs,  Virginia. 

2.  Sulphurous,  containing  Sulphuretted  hydrogen— as,  Harrowgate  and  Chelten- 
ham, England;  Aix-la-Chapelle,  Prussia;  Blue  Lick,  Ky. ;  Sulphur  Springs,  Va.,  etc 

3.  Alkaline,  containing  Carbonate  of  soda— these  are  rare,  as,  Vichy,  Ems. 


WATER. 


4.  Chalybeate,  containing  Carbonate  of  iron — as,  Hampstead,  Tunbridge,  Chelten- 
ham, and  Brighton,  England;   Spa,  Belgium;   Ballston  and  Saratoga,  N.  Y. ;   and 
Bedford,  Penn. 

5.  Saline,  containing  salts— as,  Epsom,  Cheltenham,  and  Bath,  England;  Baden- 
Baden  and  Seltzer,  Germany;   Kissingen,  Bavaria;   Plombieres,  France;   Seidlitz, 
Bohemia  ;   Lucca,  Italy  ;   Yellow  Springs,  Ohio  ;   Warm  Springs,  N.  C. ;  Congress 
Springs,  N.  Y. ;  and  Grenville,  Ky. 

Brief  Rules  for  Qualitative  Analysis  of  Mineral  Waters. 
First  point  to  be  determined,  in  examination  of  a  mineral  water,  is  to  which  of 
above  classes  does  water  in  question  belong. 

1.  If  water  reddens  blue  litmus  paper  before  boiling,  but  not  afterwards,  and  blue 
color  of  reddened  paper  is  restored  upon  warming,  it  is  Carbonated. 

2.  If  it  possesses  a  nauseous  odor,  and  gives  a  black  precipitate,  with  acetate  of 
lead,  it  is  Sulphurous. 

3.  If,  after  addition  of  a  few  drops  of  hydrochloric  acid,  it  gives  a  blue  precipitate, 
with  yellow  or  red  prussiate  of  potash,  water  is  a  Chalybeate. 

4.  If  it  restores  blue  color  to  litmus  paper  after  boiling,  it  is  Alkaline. 

5.  If  it  possesses  neither  of  above  properties  in  a  marked  degree,  and  leaves  a 
large  residue  upon  evaporation,  it  is  a  Saline  water. 


Re-agents. 

When  water  is  pure  it  will  not  become  turbid,  or  produce  a  precipitate 
with  any  of  following  Re-agents : 

Baryta  Water,  if  a  precipitate  or  opaqueness  appear,  Carbonic  Acid  is  present. 

Chloride  of  Barium,  indicates  Sulphates,  Nitrate  of  Silver,  Chlorides,  and  Oxalate 
of  Ammonia,  Lime  salts.  Sulphide  of  Hydrogen,  slightly  acid,  Antimony,  Arsenic, 
Tin,  Copper,  Gold,  Platinum,  Mercury,  Silver,  Lead,  Bismuth,  and  Cadmium;  Sul- 
phide of  Ammonium,  solution  alkaloid  by  ammonia,  Nickel,  Cobalt,  Manganese, 
Iron,  Zinc,  Alumina,  and  Chromium.  Chloride  of  Mercury  or  Gold  and  Sulphate  of 
Zinc,  organic  matter. 

Filter   Beds. 

Fine  sand,  2  feet  6  ins. ;  Coarse  sand,  6  ins. ;  Clean  shells,  6  ins. ,  and  Clean  gravel 
2  feet,  will  filter  700  gallons  water  in  24  hours  per  square  foot,  by  gravitation. 

SEA  WATER.     Composition  of  it  per  volume  : 
Chloride  of  Sodium  (common  salt). .  2.51 


Sulphuret  of  Magnesium 53 

Chloride  of  "        33 


Carbonate  of  Lime        ) 

"        of  Magnesia  J °2 

Sulphate  of  Lime 01 

Water 96.6 

By  analysis  of  Dr.  Murray,  at  specific  gravity  of  1.029,  it  contains 

Muriate  of  Soda 220.01  I  Muriate  of  Magnesia 42.08 

Sulphate  of  Soda 33. 16  |  Muriate  of  Lime 7.84 

303.09 

Or,  i  part  contains  .030309  parts  of  salt  =  ^  part  of  its  weight. 
Mean  volume  of  solid  matter  in  solution  is  3.4  per  cent.,  .75  of  which  is 
common  salt. 

Boiling    Points    at   Different   Degrees    of  Saturation. 


Salt,  by  Weight, 
in  100  Parts. 

Boiling 
Point. 

Salt,  by  Weight, 
in  100  Parts. 

Boiling 
Point. 

Salt,  by  Weight, 
in  loo  Parts. 

Boiling 
Point. 

3-03  =  A 

213.2° 

«5-«5  =  & 

217.9° 

27.28  =  inf 

222.5° 

6.06  =  ^ 

214.4° 

18.18  =  -^ 

2I9° 

30-31  =  if 

223.7° 

9-°9  =  inr 

215.5° 

»»•"=* 

22O.2° 

33-34  =  -H- 

224.9° 

"•"-A 

2l6.7° 

24.25  =  ^ 

221.4° 

*3°-37  =  if 

236° 

852  WATER. — WAVES    OF   THE    SEA. 

J3ep»c»sits   at  Different  Degrees    of  Saturation   and.   Tern 

peratnre. 
When  1000  Parts  are  reduced  by  Evaporation. 


Volume  of  Water. 

Boiling  Point. 

Salt  in  100  Parts. 

Nature  of  Deposit. 

1000 

299 

102 

214° 
217° 
228° 

10 

29-5 

None. 
Sulphate  of  Lime. 
Common  Salt 

It  contains  from  4  to  5.3  ounces  of  salt  in  a  gallon  of  water. 
Saline  Contents  of  Water  from  several  Localities. 


Baltic 6.6 

Black  Sea 21.6 

Arctic 28.3 


British  Channel 35.5 


Mediterranean 


39-4 


Equator 39.42 


South  Atlantic 41,2 

North  Atlantic  ....    42.6 
Dead  Sea 385 


There  are  62  volumes  of  carbonic  acid  in  1000  of  sea- water. 

Cube  foot  at  62°  weighs  64  Ibs.    Its  weight  compared  with  fresh  water 
being  very  nearly  as  40  to  39. 

Height  of  a  Column  of  Water  at  60°  or  64.3125  Ibs. 

At  62°,  i  Ton  =  35  cube  feet,     i  Lb.  per  sq.  inch  =  2.239  ^ee^  au(^  at  pressure  of 
atmosphere  =  32.966  feet  =  10.048  meters. 

Weights. 
A  ton  of  fresh  water  is  taken  at  36,  and  one  of  salt  at  35  cube  feet 


WAVES    OF   THE   SEA. 

Arnott  estimated  extreme  height  of  the  waves  of  an  ocean,  at  a  distance 
from  land  sufficiently  great  to  be  freed  from  any  influence  of  it  upon  their 
culmination,  to  be  20  feet. 

French  Exploring  Expedition  computed  waves  of  the  Pacific  to  be  22jeet 
in  height. 

By  observations  of  Mr.  Douglass  in  1853,  ne  deduced  that  when  waves  had 
heights  of 

8  feet,  there  were     35     in  number  in  one  mile,  and  8  per  minute. 
15    "  "         5  and  6  "  5          " 

20    "  3  4          « 

J.  Scott  Russell  divides  waves  into  2  classes — viz.  : 
Waves  of  Translation,  or  of  ist  order;  of  Oscillation,  or  of  2d  order. 
"Waves    of  tlxe   First   Order. 

1.  Velocity  not  affected  by  intensity  of  the  generating  impulse. 

2.  Motion  of  the  particles  always  forward  in  same  direction  as  wave,  and 
same  at  bottom  as  at  surface. 

3.  Character  of  wave,  a  prolate  cycloid,  in  long  waves,  approaching  a  true 
cycloid.    When  height  is  more  than  one  third  of  length,  the  wave  breaks. 

"Waves    of  the    Second    Order. 

1.  Ordinary  sea  waves  are  waves  of  second  order,  but  become  waves  of  the 
first  order  as  they  enter  shallow  water. 

2.  Power  of  destruction  directly  proportional  to  height  of  wave,  and  great- 
est when  crest  breaks. 

3.  A  wave  of  10  feet  in  height  and  32  feet  in  length  would  only  agitate 
the  water  6  ins.  at  10  feet  below  surface ;  a  wave  of  like  height  and  100  feet 
in  length  would  only  disturb  the  water  18  ins.  at  same  depth. 

Average  force  of  waves  of  Atlantic  Ocean  during  summer  months,  as  de- 
termined by  Thomas  Stevenson,  was  611  Ibs.  per  sq.  foot;  and  for  winter 
months  2086  Ibs.  During  a  heavy  gale  a  force  of  6983  Ibs.  was  observed, 


WAVES    OF   THE    SEA. 


853 


J.  Scott  Riissell  deduced  that  a  wave  30  feet  in  height  exerts  a  force  of  i 
ton  per  sq.  foot,  and  that,  in  an  exposed  position  in  deep  water,  1.75  tons 
may  be  exerted  upon  a  vertical  surface. 

At  Cassis,  France,  when  the  water  is  deep  outside,  blocks  of  15  cube  me- 
ters were  found  insufficient  to  resist  the  action  of  waves. 

Breakwaters  with  vertical  walls,  or  faces  of  an  angle  less  than  i  to  i,  will 
reflect  waves  without  breaking  them.  Waves  of  oscillation  have  no  effect 
on  small  stones  at  22  feet  below  the  surface,  or  on  stones  from  1.5  to  2  feet, 
12  feet  below  surface. 

A  roller  20  feet  high  will  exert  a  force  of  about  i  ton  per  sq.  foot. 

Greatest  force  observed  at  Skerryvore,  3  tons  per  sq.  foot ;  at  Bell  Rock, 
1.5  tons  per  sq.  foot. 

Waves  of  the  second  order,  when  reflected,  will  produce  no  effect  at  a  depth 
of  12  feet  below  surface. 

Action  of  waves  is  most  destructive  at  low-water  line. 

Waves  of  first  order  are  nearly  as  powerful  at  a  great  depth  as  at  surface. 

To   Compnte  "Velocity. 

When  I  is  less  than  d.     . 55  ^/l  or  i. 818  ^Jl  =  V. 


When  I  exceeds  1000  d.    -^32.17  d  =  V,  and  When  Height  of  Wave  becomes  a  sen- 


sible Proportion  to  Depth,  */32. 17(1  +  3  —\=V. 


To   Compute   Height  of*  "Waves   in   Reservoirs,  etc. 


(2.  5  —  Vf1)  —  height  in  feet,    L  representing  length  of  Reservoir,  Pond, 
etc.,  exposed  to  direction  of  wind,  in  miles. 

Tidal   Waves. 

Wave  produced  by  action  of  sun  and  moon  is  termed  Free  Tide  Wave. 
Semi-diurnal  tide  wave  is  this,  and  has  a  period  of  12  hours  24+  minutes. 
'    Professor  A  iry  declared  that  when  length  of  a  wave  was  not  greater  than 
depth  of  the  water,  its  velocity  depended  only  upon  its  length,  and  was  pro- 
portionate to  square  root  of  its  length. 

When  length  of  a  wave  is  not  less  than  1000  times  depth  of  water,  velocity  of  it 
depends  only  upon  depth,  and  is  proportionate  to  square  root  of  it;  velocity  being 
same  that  a  body  falling  free  would  acquire  by  falling  through  a  height  equal  to  half 
depth  of  water. 

For  intermediate  proportions,  velocity  can  be  obtained  by  a  general  equation. 

Under  no  circumstances  does  an  unbroken  wave  exceed  30  or  40  feet  in  height. 

A  wave  breaks  when  its  height  above  general  level  of  water  is  equal  to  general 
depth  of  it. 

Diurnal  and  other  tidal  waves,  so  far  as  they  are  free,  may  be  all  considered  as 
running  with  the  same  velocity,  but  the  column  of  the  length  of  wave  must  be 
doubled  for  diurnal  wave. 

Length  of  Wave. 


Depth  of  Water. 

Feet. 

i 

Feet. 
10 

Feet. 
ZOO 

Feet. 

1000 

Feet. 
IOOOO 

Feet. 

IOO  000 

Velocity  per  Second. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

i 

2.26 

5-34 

5-67 

10 

2.26 

7-iS 

16.88 

17.92 

'7-93 

— 

IOO 

— 

7-»5 

22.62 

53-19 

56.67 

56.71 

1000 

— 

22.62 

71-54 

168.83 

179.21 

KOOOO 

— 

— 

— 

71-54 

326.714 

533-9 

854  WHEEL   GEARING. 

WHEEL  GEARING. 

Pitch  Line  of  a  wheel  is  circle  upon  which  pitch  is  measured,  and  it 
is  circumference  by  which  diameter,  or  velocity  of  wheel,  is  measured. 

Pitch  is  arc  of  circle  of  pitch  line,  is  determined  by  number  of  teeth 
in  wheel,  and  necessarily  an  aliquot  part  of  pitch  line. 

True  or  Chordial  Pitch,  or  that  by  which  dimensions  of  tooth  of  a 
wheel  are  alone  determined,  is  a  straight  line  drawn  from  centres  of 
two  contiguous  teeth  upon  pitch  line. 

Line  of  Centres  is  line  between  centres  of  two  wheels. 

Radius  of  a  wheel  is  semi  -  diameter  bounded  by  periphery  of  the 
teeth.  Pitch  Radius  is  semi-diameter  bounded  by  pitch  line. 

Length  of  a  Tooth  is  distance  from  its  base  to  its  extremity. 

Breadth  of  a  Tooth  is  length  of  face  of  wheel. 

Depth  of  a  Tooth  is  thickness  from  face  to  faoe  at  pitch  line. 

Face  •/  a  Tooth,  or  Addendum,  is  that  part  of  its  side  which  extends 
from  its  pitch  line  to  its  top  or  Addendum  line. 

Flank  of  a  Tooth  is  that  part  of  its  side  which  extends  from  pitch 
line  to  line  of  space  at  base  of  and  between  adjacent  teeth ;  its  length, 
as  well  as  that  of  face  of  tooth,  is  measured  in  direction  of  radius  of 
wheel,  and  is  a  little  greater  than  face,  of  tooth,  to  admit  of  clearance 
between  end  of  tooth  and  periphery  of  rim  of  wheel  or  rack. 

Cog  Wheel  is  general  term  for  a  wheel  having  a  number  of  cogs  or  teeth  set  in  or 
upon,  or  radiating  from,  its  circumference. 

Mortice  Wheel  is  a  wheel  constructed  for  reception  of  teeth  or  cogs,  which  are 
fitted  into  recesses  or  sockets  upon  face  of  the  wheel. 

Plate  Wheels  are  wheels  without  arms. 

Rack  is  a  series  of  teeth  set  in  a  plane. 

Sector  is  a  wheel  which  reciprocates  without  forming  a  full  revolution. 

Spur  Wheel  is  a  wheel  having  its  teeth  perpendicular  to  its  axis. 

Bevel  Wheel  is  a  wheel  having  its  teeth  at  an  angle  with  its  axis. 

Crown  Wheel  is  a  wheel  having  its  teeth  at  a  right  angle  with  its  axis. 

Mitre  Wheel  is  a  wheel  having  its  teeth  at  an  angle  of  45°  with  its  axis. 

Face  Wheel  is  a  wheel  having  its  teeth  set  upon  one  of  Hs  sides. 

Annular  or  Internal  Wheel  is  a  wheel  having  its  teeth  convergent  to  its  centre. 

Spur  Gear. — Wheels  which  act  upon  each  other  in  same  plane. 

Bevel  Gear. — Wheels  which  act  upon  each  other  at  an  angle. 

Inside  Gear  or  Pin  Gearing. — Form  of  acting  surfaces  of  teeth  for  a  pitch-circle 
in  inside  gearing  is  exactly  same  with  those  suited  for  same  pitch-circle  in  outside 
gearing,  but  relative  position  of  teeth,  spaces,  and  flanks  are  reversed,  and-  adden- 
dum-circle is  of  less  radius  than  pitch-circle. 

A  Train  is  a  series  of  wheels  in  connection  with  each  other,  and  consists  of  a 
series  of  axles,  each  having  on  it  two  wheels,  one  is  driven  by  a  wheel  on  a  preced- 
ing axis  and  other  drives  a  wheel  on  following  axis. 

Idle  Wheel.— A  wheel  revolving  upon  an  axis,  which  receives  motion  from  a  pre- 
ceding wheel  and  gives  motion  to  a  following  wheel,  used  only  to  affect  direction  of 
motion. 

Trundle,  Lantern,  or  Wallower  is  when  teeth  of  a  pinion  are  constructed  of  round 
bars  or  solid  cylinders  set  into  two  disks.  Trundle  with  less  than  eight  staves  can- 
not be  operated  uniformly  by  a  wheel  with  any  number  of  teeth. 

Spur,  Driver,  or  Leader  is  term  for  a  wheel  that  impels  another;  one  impelled  is 
Pinion,  Driven,  or  Follower. 


WHEEL    GEARING.  855 

Teeth  of  wheels  should  be  as  small  and  numerous  as  is  consistent  with 
Strength. 

When  a  Pinion  is  driven  by  a  wheel,  number  of  teeth  in  pinion  should  not 
be  less  than  8. 

When  a  Wheel  is  driven  by  a  pinion,  number  of  teeth  in  pinion  should  not 
be  less  than  10. 

When  2  wheels  act  upon  one  another,  greater  is  termed  Wheel  and  lesser  Pinion. 

When  the  tooth  of  a  wheel  is  made  of  a  material  different  from  that  of  wheel  it  is 
termed  a  Cog  ;  in  a  pinion  it  is  termed  a  Leaf,  in  a  trundle  a  Stave,  and  on  a  disk 
a  Pin. 

Material  of  which  cogs  are  made  is  about  one  fourth  strength  of  cast  iron. 
Hence,  product  of  their  b  d2  should  be  4  times  that  of  iron  teeth. 

Number  of  teeth  in  a  wheel  should  always  be  prime  to  number  of  pinion ; 
that  is,  number  of  teeth  in  wheel  should  not  be  divisible  by  number  of  teeth 
in  pinion  without  a  remainder.  This  is  in  order  to  prevent  the  same  teeth 
coming  together  so  often  and  uniformly  as  to  cause  an  irregular  wear  of  their 
faces.  An  odd  tooth  introduced  into  a  wheel  is  termed  a  Hunting  tooth  or  Cog. 

The  least  number  of  teeth  thai;  it  is  practicable  to  give  to  a  wheel  is  regu- 
lated by  necessity  of  having  at  least  one  pair  always  in  action,  in  order  to 
provide  for  the  contingency  of  a  tooth  breaking ;  and  least  number  that  can 
be  employed  in  pinions  having  teeth  of  following  classes  is :  Involute,  25 ; 
Epicycloidal,  12;  Staves  or  Pins,  6. 

Velocity  Ratio  in  a  Train  of  Wheels.— To  attain  it  with  least  number  of 
teeth,  it  should,  in  each  elementary  combination,  approximate  as  near  as 
practicable  to  3.59.  A  convenient  practical  rule  is  a  range  from  3  to  6. 

ILLUSTRATION.        i        6        36        216        1296  velocity  ratio. 

123  4  elementary  combination. 

To  increase  or  diminish  velocity  in  a  given  proportion,  and  with  least 
quantity  of  wheel-work,  number  of  teeth  in  each  pinion  should  be  to  number 
of  teeth  in  its  wheel  as  i :  3.59.  Even  to  save  space  and  expense,  ratio 
should  never  exceed  i :  6.  (Buchanan.) 

To   Compute   Fitch. 

RULE. — Divide  circumference  at  pitch-line  by  number  of  teeth. 
EXAMPLE.— A  wheel  40  ins.  in  diameter  requires  75  teeth;  what  is  its  pitch? 

3.1416X40-^-75  =  1-6755  »«*• 

To   Compute   True   or   Ch.ordial    Pitcli. 
RULE. — Divide  180°  by  number  of  teeth,  ascertain  sine  of  quotient,  and 
multiply  it  by  diameter  of  wheel. 

EXAMPLE.— Number  of  teeth  is  75,  and  diameter  40  ins. ;  what  is  true  pitch? 
i8o-f-  75  =  2°  24',  and  sin.  of  2°  24'  =  .041 88,  which  x  40  =  1.6752  ins. 

To    Compute    Diameter. 

RULE. — Multiply  number  of  teeth  by  pitch,  and  divide  product  by  3.1416. 
EXAMPLE.— Number  of  teeth  in  a  wheel  is  75,  and  pitch  1.6755  ins. ;  what  is  di- 
ameter of  it?  75  x  ,.6755-:- 3. 1416  =  4o  ins. 

When  the  True  Pitch  is  given.    RULE. — Multiply  number  of  teeth  hi  wheel 
by  true  pitch,  and  again  by  .3184. 
EXAMPLE.— Take  elements  of  preceding  case. 

75  X  1.6752  X  .3184  =  40  ins. 

Or,  Divide  180°  by  number  of  teeth,  and  multiply  cosecant  of  quotient  by 
pitch. 

180-:-  75  =  2°  24',  and  cos.  2°  24'  =  23.88,  which  x  1.6752  =  40  int. 


WHEEL   GEARING. 

To   Compute   Number  of  Teeth. 
RULE. — Divide  circumference  by  pitch. 

To  Compute   Number  of  Teeth  in.  a  Pinion  or  Follo-wer 

to  have  a  given  "Velocity. 

RULE. — Multiply  velocity  of  driver  by  its  number  of  teeth,  and  divide 
product  by  velocity  of  driven. 

EXAMPLE  i.— Velocity  of  a  driver  is  16  revolutions,  number  of  its  teeth  54,  arid 
velocity  of  pinion  is  48;  what  is  number  of  its  teeth? 
16X54-:-  48=18  teeth. 

2.  —A  wheel  having  75  teeth  is  making  16  revolutions  per  minute;  what  is  num- 
ber of  teeth  required  in  pinion  to  make  24  revolutions  in  same  time? 
16X75-^24  =  50  teeth. 

To  Compute  Proportional  Radius  of  a  Wheel  or  Pinion. 

RULE. — Multiply  length  of  line  of  centres  by  number  of  teeth  in  wheel, 
for  wheel,  and  in  pinion,  for  pinion,  and  divide  by  number  of  teeth  in  both 
wheel  and  pinion. 

EXAMPLE.— Line  of  centres  of  a  wheel  and  pinion  is  36  ins.,  and  number  of  teeth 
in  wheel  is  60,  and  in  pinion  18 ;  what  are  their  radii  ? 

_£    vx    _O 
=  27.691715. 


To    Compxite   Diameter   of*  a   Pinion. 

When  Diameter  of  Wheel  and  Number  of  Teeth  in  Wheel  and  Pinion  are 
given.  RULE. — Multiply  diameter  of  wheel  by  number  of  teeth  in  pinion, 
and  divide  product  by  number  of  teeth  in  wheel. 

EXAMPLE.— Diameter  of  a  wheel  is  25  ins.,  number  of  its  teeth  210,  and  number 
of  teeth  in  pinion  30;  what  is  diameter  of  pinion? 

25X30-^-210  =  3.57  ins. 

To  Compute    Number   of  Teeth    required  in    a  Train  of 
"Wheels    to    produce    a   given    Velocity. 

RULE. — Multiply  number  of  teeth  in  driver  by  its  number  of  revolutions, 
and  divide  product  by  number  of  revolutions  of  each  pinion,  for  each  driver 
and  pinion. 

EXAMPLE. — If  a  driver  in  a  train  of  three  wheels  has  90  teeth,  and  makes  2  revo- 
lutions, and  velocities  required  are  2,  10,  and  18,  what  are  number  of  teeth  in  each 
of  other  two? 

10  :  90  ::  2  :  1 8  =  teeth  in  2d  wheel         18  :  90  ::  2  :  10  =  teeth  in  $d  wheel 
To    Compute   "Velocity   of  a   Pinion. 

RULE.— Divide  diameter,  circumference,  or  number  of  teeth  in  driver,  as 
case  may  be,  by  diameter,  etc.,  of  pinion. 

When  there  are  a  Series  01*  Train  of  Wheels  and  Pinions.  RULE. — Divide 
continued  product  of  diameter,  circumference,  or  number  of  teeth  in  wheel* 
by  continued  product  of  diameter,  etc.,  of  pinions. 

EXAMPLE  i  — Tf  a  wheel  of  32  teeth  drives  a  pinion  of  10,  upon  axis  of  which  there 
is  one  of  30  teeth,  driving  a  pinion  of  8,  what  are  revolutions  of  last? 

^X^  =  9^=12  revolutions. 

IO  O  OO 

2. — Diameters  of  a  train  of  wheels  are  6,  9, 9,  10,  and  12  ins. ;  of  pinions,  6,  6,  6, 6, 
and  6  ins. ;  and  number  of  revolutions  of  driving  shaft  or  prime  mover  is  10;  what 
are  revolutions  of  last  pinion? 

6  X  9  X  9  X  10  X  12  X  io  _  583  200  _ 

"~"     ""~   ~  -          =  7S  revolutlons' 


WHEEL    GEARING. 


857 


To  Compute  Proportion,  that  Velocities  of  Wheels  in 
a  Train,  should,  bear  to  one  another. 

RULE. — Subtract  less  velocity  from  greater,  and  divide  remainder  by  one 
less  than  number  of  wheels  in  train  ;  quotient  is  number,  rising  in  arithmet- 
ical progression  from  less  to  greater  velocity. 

EXAMPLE. — What  should  be  velocities  of  3  wheels  to  produce  18  revolutions,  the 
driver  making  3? 

—  =  7. 5  =  number  to  be  added  to  velocity  of  driver  =  7. 5  -f-  3  =  10. 5,  and 
jo.  5  -j-  7.5  =  18  revolutions.    Hence  3,  10. 5,  and  18  are  velocities  of  three  wheels. 

Pitch,   of  "Wheels. 

To    Compute    Diameter   of   a    "Wheel    for    a   given    Pitch,, 
or    Pitch    for   a  given    Diameter. 

From  8  to  192  Teeth. 


No.  of 
Teoth. 

Diame- 
ter. 

No.  of  i 
Teeth. 

Diame- 
ter. 

No.  of 
Teeth. 

Diame- 
ter. 

No.  of 
Teeth. 

Diame- 
ter. 

No.  of 
Teeth. 

Diame- 
ter. 

8 

2.61 

45 

14-33 

82 

26.11 

119 

37-88 

156 

49-66 

9 

2.93 

46 

14.65 

83 

26.43 

120    38.2 

157   49.98 

10 

3-24 

47 

14.97 

84 

26.74 

121 

38-52 

158    50.3 

ii 

3-55 

48 

15.29 

85 

27.06 

122 

38.84 

159   50.61 

12 

3.86 

49 

I5.6l 

86 

27.38 

123 

39.16 

160   50.93 

13 

4.18 

50 

15-93 

87 

27.7 

124 

39-47 

161  i  51.25 

14 

4-49 

5i 

16.24 

88 

28.02 

125 

39-79 

162 

5L57 

15 

4.81 

52 

16.56 

89 

28.33 

126 

40.11 

163 

51.89 

16 

5-12 

53 

16.88 

90 

28.65 

I27 

40-43 

164 

52.21 

17 

5-44 

54 

17.2 

9i 

28.97 

128 

40-75 

165 

52.52 

18 

5.76 

55 

17.52 

92 

29.29 

I29 

41.07 

166 

52.84 

19 

6.07 

56 

I7.8 

93 

29.61 

130 

41.38 

167 

53-i6 

20 

6-39 

57 

I8.I5 

94 

29-93 

131 

41.7 

168 

53.48 

21 

6.71 

58 

18.47 

95 

30.24 

132 

42.02 

169  |  53-8 

22 

7-03 

59 

18.79 

96 

30-56 

133 

42.34 

170 

54-12 

23 

7-34 

60 

19.11 

97 

30.88 

134 

42.66 

171 

54-43 

24 

7.66 

61 

19.42 

98 

31.2 

135 

42.98 

172 

54-75 

25 

7.98 

62 

19.74 

99 

31-52 

136 

43-29 

J73 

55-07 

26 

8-3 

63 

20.06 

100 

31.84 

137 

43.61 

\  174 

55-39 

27 

8.61 

64 

20.38 

IOI 

32.15 

138 

43-93 

|  175   55-71 

28 

8-93 

65 

20.7 

102 

3247 

139 

44-25 

176 

56.02 

29 

9-25 

66 

2I.O2 

103 

32.79 

I4O 

44-57 

177 

56.34 

30 

9-57 

67 

21-33 

104 

33-ii 

141 

44.88 

178 

56.66 

31 

9.88 

68 

21.65 

105 

33-43 

142 

45-2 

179 

56.98 

32 

10.2 

69 

21.97 

106 

33-74 

*43 

45-52 

180 

57-23 

33 

10.52 

70 

22.29 

107 

34.06 

144 

45.84 

181 

5762 

34 

10.84 

71 

22.6l 

108 

34.38 

145 

46.16 

182 

5793 

35 

ii.  16 

72 

22.92 

109 

34-7 

146 

46.48 

183 

58.25 

36 

11.47 

73 

23.24 

no 

35-02 

147 

46.79 

184 

58.57 

37 

11.79 

74 

23.56 

III 

35-34 

148 

47.11 

185 

58.89 

38 

12.  II 

75 

23.88 

112 

35.65 

149 

47-43 

186 

59.21 

39 

12.43 

76 

24.2 

"3 

35-97 

150   47-75 

187 

59-53 

40 

12.74 

77 

24.52 

114 

36.29 

151   48.07 

188 

59.84 

4i 

13.06 

78 

24.83 

"5 

36.61 

152 

48.39 

189 

60.  16 

42 

13.38 

79 

25.15 

116 

36.93 

153 

48.7 

190 

60.48 

43 

13-7 

80 

25-47 

117   37.25 

154 

49.02 

191 

60.81 

44 

14.02 

81 

25.79 

"8   37.56 

155   49-34 

192 

61.13 

Pitch  in  this  table  is  true  pitch,  as  before  described. 

To    Compute    Circumference    of  a   "Wheel. 
RULE. — Multiply  number  of  teeth  by  their  pitch. 
4.  C* 


858  WHEEL   GEARING. 

Xo    Compute   Revolxitions   of  a   Wheel   or   Pinion. 
RULE. —  Multiply  diameter  or  circumference  of  wheel  or  number  of  its 
teeth  in  ins.,  as  case  may  be,  by  number  of  its  revolutions,  and  divide  prod- 
uct by  diameter,  circumference,  or  number  of  teeth  in  pinion. 

EXAMPLE.— A  pinion  10  ins.  in  diameter  is  driven  by  a  wheel  2  feet  in  diameter, 
making  46  revolutions  per  minute;  what  is  number  of  revolutions  of  pinion? 
2Xi2X46-=-io=:iio.4  revolutions. 

To  Compute  Numtoer  of  Teeth  of  a  Wheel    for   a  given 

Diameter   and.    Pitch. 

RULE.— Divide  diameter  by  pitch,  and  opposite  to  quotient  in  preceding 
table  is  given  number  of  teeth. 

EXAMPLE.— Diam.  of  wheel  is  40  ins.,  and  pitch  1.675;  what  is  number  of  its  teeth? 
40-7-1.675  =  23.88,  and  opposite  thereto  in  table  is  75  =  number  of  teeth. 

To  Compute  Diameter  of  a  Wheel  for  a  given  Pitch  and 

Number  of  Teeth. 

RULE. — Multiply  diameter  in  preceding  table  for  number  of  teeth  by 
pitch,  and  product  will  give  diameter  at  pitch  circle. 
EXAMPLE.— What  is  diameter  of  a  wheel  to  contain  48  teeth  of  2.5  ins.  pitch? 
15. 29  X  2. 5  =  38. 225  ins. 

To  Compute  Pitch  of  a  Wheel  for  a  given  Diameter  and. 
Number  of  Teeth. 

RULE. — Divide  diameter  of  wheel  by  diameter  in  table  for  number  of 
teeth,  and  quotient  will  give  pitch. 

EXAMPLE.— What  is  pitch  of  a  wheel  when  diameter  of  it  is  50.94  ins.,  and  num- 
ber of  its  teeth  80?  5a94  _._  25>47  _.  2  intm 

G-eiieral    Illustrations. 

i. — A  wheel  96  ins.  in  diameter,  making  42  revolutions  per  minute,  is  to  drive  a 
shaft  75  revolutions  per  minute ;  what  should  be  diameter  of  pinion  ? 

96  X  42  -J-75  =  53-  76  in*. 

2.— If  a  pinion  is  to  make  20  revolutions  per  minute,  required  diameter  of  an- 
other to  make  58  revolutions  in  same  time. 

58  -r-  20  =  2.9  =  ratio  of  their  diameters.  Hence,  if  one  to  make  20  revolutions  is 
given  a  diameter  of  30  ins.,  other  will  be  30-^2.9  =  10.345  ins. 

3.— Required  diameter  of  a  pinion  to  make  12.5  revolutions  in  same  time  as  one 
of  32  ins.  diameter  making  26. 

32  X  26  -t- 12. 5  =  66. 56  ins. 

4.— A  shaft,  having  22  revolutions  per  minute,  is  to  drive  another  shaft  at  rate 
of  15,  distance  between  two  shafts  upon  line  of  centres  is  45  ins. ;  what  should  be 
diameter  of  wheels? 

Then,  ist.  22  + 15  :  22  ::  45  :  26.75  =  ins.  in  radius  of  pinion. 

2d.    22  -}-  15  :  15  ::  45  :  18.24  =  *w*.  in  radius  of  spur. 

5.— A  driving  shaft,  having  16  revolutions  per  minute,  is  to  drive  a  shaft  81  revo- 
lutions per  minute,  motion  to  be  communicated  by  two  geared  wheels  and  two  pul- 
leys, with  an  intermediate  shaft;  driving  wheel  is  to  contain  54  teeth,  and  driving 
pulley  upon  driven  shaft  is  to  be  25  ins.  in  diameter;  required  number  of  teeth  in 
driven  wheel,  and  diameter  of  driven  pulley. 

Let  driven  wheel  have  a  velocity  of  Vi6x  81  =  36,  a  mean  proportional  between 
extreme  velocities  16  and  81. 

Then,  ist.  36  :  16  ::  54  :  24      =  teeth  in  driven  wheel. 

2d.    81  :  36  ::  25  :  u.u  =ins.  diameter  of  driven  pulley. 

6.— If,  as  in  preceding  case,  whole  number  of  revolutions  of  driving  shaft,  num- 
ber of  teeth  in  its  wheel,  and  diameters  of  pulleys  are  given,  what  are  revolutions 
of  shafts  ? 

Then,  ist.  18  :  16  ::  54  :  48  =  revolutions  of  intermediate  shaft. 
2d.    15  :  48  ::  25  :  80  =  revolutions  of  driven  shaft. 


WHEEL    GEARING. TEETH    OF    WHEELS. 


859 


Teeth,   of  TVTieels. 

Epicycloidal. — In  order  that  teeth  of  wheels  and  pinions  should  work 
evenly  and  without  unnecessary  rubbing  friction,  the  face  (from  pitch  line 
to  top)  of  the  outline  should  be  determined  by  an  epicycloidal  curve  (see 
page  228),  and  that  of  the  flank  (from pitch  line  to  base)  by  an  hypocycloidal 
(see  also  page  228). 

When  generating  circle  is  equal  to  half  diameter  of  pitch  circle,  hypocy- 
cloidal described  by  it  is  a  straight  diametrical  line,  and  consequently  out- 
line of  a  flank  is  a  right  line,  and  radial  to  centre  of  wheel. 

If  a  like  generating  circle  is  used  to  describe  face  of  a  tooth  of  other  wheel 
or  pinion  respectively,  the  wheel  and  pinion  will  operate  evenly. 

ILLUSTRATION.— Determine  all  elements  of  wheel 
—viz. ,  Pitch  circle,  Number  of  teeth,  Pitch,  Length, 
Face,  and  Flank. 

Cut  a  template  A  to  pitch  circle  c  c  of  wheel,  and 
secure  it  temporarily  to  a  board. 

Having  determined  depth  of  tooth,  set  it  off  on 
pitch  line,  as  a  o,  Fig.  i,  and  above  it  apply  a  sec- 
ond template,  a;  radius  of  wheel  is  equal  to  half 
radius  of  pinion;  insert  into,  or  attach  exactly  at  its  edge,  a  tracer  .,  roll  template 
a  along  A,  and  tracer  will  describe  an  epicycloidal  curve,  a  r,  and  by  inverting  a 
describe  o  ?•,  and  faces  of  a  tooth  are  delineated. 

To  describe  flanks,  define  pitch  line  c  c,  Fig.  2,  and  arc  n  n, 
drawn  at  base  of  teeth  or  board  A  (as  in  Fig.  i),  secure  a  strip 
of  wood,  w,  equal  in  length  to  radius  of  wheel,  and  locate 
centre  of  it,  a;,  draw  radii  x  a  and  *  o,  and  they  will  define 
flanks,  which  should  be  filleted,  as  shown  at  ss.  Define  arc 
zz,  and  length  of  tooth  is  determined. 

Proceed  in  like  manner  conversely  for  teeth  of  pinion,  and 
wheel  and  pinion  thus  constructed  will  operate  truly. 

In  construction  of  the  teeth  of  a  wheel  or  pinion  in 
the  pattern-shop,  it  is  customary  to  construct  the  wheel 
or  pinion  complete,  out  to  face  of  wheel  at  base  of  teeth, 
and  then  to  insert  the  teeth  in  rough,  approximately 
shaped  blocks,  by  a  dovetail  at  their  base,  fitting  into  face  of  wheel,  and  then 
the  outline  of  a  tooth  is  described  thereon ;  the  block  is  then  removed,  fin- 
ished as  a  tooth,  replaced,  fastened,  and  filleted. 

Involute. 

Teeth  of  two  wheels  will  work  truly  together  when  their  face  is  that  of  an 
involute  (see  page  229),  and  that  two  such  wheels  should  work  truly,  the 
circles  from  which  the  involute  lines  for  each  wheel  are  generated  must  be 
concentric  with  the  wheels,  with  diameters  in  same  ratio  as  those  of  the  wheels. 
Assume  Ac,  Be,  Fig.  3.  pitch  radii  of  two  wheels  designed 
t&  work  together,  through  c,  draw  a  right  line,  e  i,  and  with 
perpendiculars  e  c,  i  c,  describe  arcs  n  o,  r  s,  and  involutes 
n  c  o  and  res  define  a  face  of  each  of  the  teeth. 

To  describe  teeth  of  a  pair  of 

wheels  of  which  Ac,  Be,  Fig.  4, 

are  pitch  radii,  draw  c  t,  c  e,  per- 
pendicular to  radials  B  i  and  A  e, 

and  they  are  to  be  taken  as  the 

radials  of  the  involute  arcs  from 

which  the  faces  of  the  teeth  are 

to  be  defined ;  then  fillet  flanks  at 

base,  as  before  described,  Fig.  2. 

Involute  teeth  will  work  with  truth,  even  at  varying 
distances  apart  of  the  centres  of  the  wheels,  and  any  wheels  of  a  like  pitch  will  work 
in  union,  however  varied  their  diameters. 


86o 


WHEEL   GEARING. — TEETH   OF   WHEELS. 


Circular  teeth  are  defined  as  follows : 

«  Assume  A  A,  Fig.  5,  pitch-line,  6  b  line  of  base 

of  teeth,  and  t  t  face  line.  Set  off  on  pitch-line 
divisions  both  of  pitch  and  depth  of  teeth,  then 
with  a  radius  of  1.25  pitch  describe  arcs  as  o  s 
upon  pitch  line  for  faces  of  teeth,  then  draw  ra- 
dial  lines  o  v,  r  «,  to  centre  of  wheel  for  flanks, 
°  strike  arc  1 1  to  define  length  of  tooth,  and  fillet 
flanks  at  base  as  before  described. 

Proportions  of  Teeth. 
In  computing  dimensions  of  a  tooth,  it  is  to 
be  considered  as  a  beam  fixed  at  one  end, 
weight  suspended  from  other,  or  face  of  beam  \ 
and  it  is  essential  to  consider  the  element  of  velocity,  as  its  stress  in  opera- 
tion, at  high  velocity  with  irregular  action,  is  increased  thereby. 

Dimensions  of  a  tooth  should  be  much  greater  than  is  necessary  to  resist 
direct  stress  upon  it,  as  but  one  tooth  is  proportioned  to  bear  whole  stress 
upon  wheel,  although  two  or  more  are  actually  in  contact  at  all  times ;  but 
this  requirement  is  in  consequence  of  the  great  wear  to  which  a  tooth  is  sub- 
jected, shocks  it  is  liable  to  from  lost  motion,  when  so  worn  as  to  reduce  its 
depth  and  uniformity  of  bearing,  and  risk  of  the  loss  of  a  tooth  from  a  defect. 
A  tooth  running  at  a  low  velocity  may  be  materially  reduced  in  its  dimen- 
sions, compared  with  one  running  at  a  high  velocity  and  with  a  like  stress. 

Result  of  operations  with  toothed  wheels,  for  a  long  period  of  time,  has 
determined  that  a  cast-iron  (Eng.)  tooth  with  a  pitch  of  3  ins.  and  a  breadth 
of  7.5  ins.  will  transmit,  at  a  velocity  of  6.66  feet  per  second,  power  of  59.16 
horses. 

To  Compute    Dimensions    of  a  Tooth  to  Resist   a   given 

Stress. 

RULE. — Multiply  extreme  pressure  at  pitch-line  of  wheel  by  length  of 
tooth  in  decimal  of  a  foot,  divide  product  by  Coefficient  of  material  of  tooth, 
and  quotient  will  give  product  of  breadth  and  square  of  depth. 

S  I 
Or,  —  =  6  d2.     S  representing  stress  in  Ibs.,  and  I  length  in  feet. 

The  Coefficient  of  cast  iron  for  this  or  like  purposes  may  be  taken  at  from  50  to  70. 


Pitch  A  B  =  i. 
Length  c  o  = .  75. 
Working  length  c  c  =  .7. 
Clearance  e  to  o  —  .05. 


Depth  r  s  =  .+^ 
Space  s  v  =  .55. 
Play  s  v  — r  s  =  .i 
Face  B  c  =  .3s. 


NOTE.  —  It  is  necessary  first  to  determine  i 
order  to  obtain  either  length  or  depth  of  a  tootb 


itch,  io 


EXAMPLE.  —  Pressure  at  pitch  line  of  a  cast- 
iron  wheel  (at  a  velocity  of  6.66  feet  per  sec- 
ond) is  4886  Ibs.  ;  what  should  be  dimensions 
of  teeth,  pitch  being  3  ins.  ? 

3  X  75  —  2.  25  length  of  tooth,  which  -f-  12  =  .  1875  =  length  in  decimal  of  a  foot. 
Coefficient  of  material  is  taken  at  60. 

—    ''  7   =15.  27.     If  length  =  2.  25,  pitch  =  3,  and  depth  =  i.  35  ins. 
Pitches  of  Equivalent  Strength  for  Cast  Iron  and  Wood.—  Iron  i.    Hard  wood  1.26. 
Then 


=  8.39  ins.  breadth. 


When  Product  ofbd2  is  obtained,  and  it  is  required  to  ascertain  eithet 
dimension.  ~  =  depth,  and  b--  =  breadth. 


WHEEL   GEAKING. — TEETH    OF   WHEELS.  86 1 

To   Compute   Depth   of  a   Tooth. 

1.  When  Stress  is  given.    RULE. — Extract  square  root  of  stress,  and  mul* 
tiply  it  by  .02  for  cast  iron,  and  .027  for  hard  wood. 

2.  When  H?  is  given.     RULE. — Extract  square  root  of  quotient  of  EP  di- 
vided by  velocity  in  feet  per  second,  and  multiply  it  by  .466  for  cast  iron, 
and  .637  for  hard  wood. 

EXAMPLE.— H*  to  be  transmitted  by  a  tooth  of  cast  iron  is  60,  and  velocity  of  it 
at  its  pitch-line  is  6.66  feet  per  second;  what  should  be  depth  of  tooth? 

/  60 
\/6~66  *  -466  =  Zl398  *** 

To   Compute   KP   of  a   Tooth. 

RULE.— Multiply  pressure  at  pitch-line  by  its  velocity  in  feet  per  minute, 
and  divide  product  by  33  ooo. 

EXAMPLE.— What  is  H?  of  a  tooth  of  dimensions  and  at  velocity  given  in  preced- 
ing example. 

4886  X  6. 66  X  60"  -f-  33  ooo  —  59. 16  horses. 

To  Compute  Stress  that  may  be  borne  "by  a  Tooth. 

RULE. — Multiply  Coefficient  of  material  of  tooth  to  resist  a  transverse 
strain,  as  estimated  for  this  character  of  stress,  by  breadth  and  square  of  its 
depth,  and  divide  product  by  extreme  length  of  it  in  decimal  of  a  foot. 

EXAMPLE. — Dimensions  of  a  cast-iron  tooth  in  a  wheel  are  1.38  ins.  in  depth,  2.1 
ins.  in  length,  and  7.5  ins.  in  breadth;  what  is  the  stress  it  will  bear? 


Coefficient  assumed  at  60.  '  -  4^97 

deductions  by  the  rules  of  different  authors  for  like 
cast-iron  tooth: 
Pitch  .......  3  ins.  \  Depth.  ...  1.38  ins.  \  Breadth.  .  .  7.5  ins.  \  Length.  ...  2.1  ins. 


Following  deductions  by  the  rules  of  different  authors  for  like  elements  are  sub- 
mitted for  a  cast-iron  tooth: 


ACTUAL  POWER  IN  STRESS  EXERTED 
at  a  velocity  of  400  feet  per  wan.,  4886  Ibs. 

Depth  of 
Tooth. 

ACTUAL  POWER  IN  STRESS  EXERTED 
at  a  velocity  of  400  feet  per  min.  ,  4886  Ibs. 

Depth  of 
Tooth. 

/H 
By  Above  Rule     1  —  X  •  446  

Ins. 
1.398* 

i-75 
1.76 

By  Rankine    /—  .  .  . 

In*. 
1.8 

2.25 
2.24 

V  1500 
"  Tredgold  —    /—... 

"  Imperial  Journal     /  —  -;  .. 

4  V  « 
"Buchanan    /556H.. 

V     v 

H  representing  horse-power  (60),  W  stress  in  Ibs.,  and  v  velocity  in  feet  per  second. 

Depth,  IPitoh,  and   Breadth.    (M.  Aforin.) 

Cast  iron 028  <^W  =  d.  .057  Vw  =  p- 

Hard  wood 038  v\Y  =  d.  .079  vw  =  p- 

W  representing  weight  or  stress  upon  tooth  in  Ibs.,  d  depth  of  tooth,  and  P  pitch 

i  ins 


in  ins 


When  velocity  of  pitch-circle  does  not  exceed  5  feet  per  second  b  =  4  d, 
when  it  exceeds  5  feet  b  =  5  d,  and  if  wheels  are  exposed  to  wet  6  =  6  d. 

b  representing  breadth. 

ILLUSTRATION.— Assume  pressure  at  pitch-line  of  a  cast-iron  wheel  upon  a  tooth 
equal  6000  Ibs.,  and  velocity  5  feet  per  second.  • 

Then  .028  ^6000=  2. 17  ins.  Depth,  and  .057  v/6ooo  =  4.41  ins.  Pitch. 

NOTE.  —  For  farther  Illustrations  of  Formation  of  Teeth,  Bevel  Gearing,  Willis's  OdontoprapW, 
Staves,  Trundles,  etc.,  see  Mosely's  Engineering,  Shelton's  Mechanic's  Guide,  Fairbairn's  Mechanism 
and  Machinery  of  Construction,  etc. 

*  This  depth,  with  a  breadth  of  7.5  int.,  is  .1  of  ultimate  strength  of  average  strength  of  America* 
Cast  Iron. 


862          TEETH    OF   WHEELS. — WINDING   ENGINES. 

PROPORTIONS  OP  WHEELS. 

With  six  flat  A  rms  and  Ribs  upon  one  side  of  them,  as  cmmfy ;  or  a  Web 
in  centre,  as  e&^aa. 

Rim. — Depth,  measured  from  base  of  teeth,  .45  to  .5  of  pitch  of  teeth,  hav- 
ing a  web  upon  its  inner  surface  .4  of  pitch  in  depth  and  .25  to  .3  of  it  in 
width. 

NOTE.— When  face  of  wheel  is  mortised,  depth  of  rim  should  be  1.5  times  pitch, 
and  breadth  of  it  1.5  times  breadth  of  tooth  or  cog. 

Hub. — When  eye  is  proportionate  to  stress  upon  wheel,  hub  should  be 
twice  diameter  of  eye.  In  other  cases  depth  around  eye  should  be  .75  to  .8 
of  pitch. 

Arm. — Depth  .4  to  .45  of  pitch.  Breadth  at  rim  1.5  times  pitch,  increas- 
ing .5  inch  per  foot  of  length  toward  hub. 

Rib  upon  one  edge  of  arm,  or  Web  in  its  centre,  should  be  from  .25  to  .3 
pitch  in  width,  and  .4  to  .45  of  it  in  depth. 

When  section  of  an  arm  differs  from  those  above  given,  as  with  one  with 
a  plane  section,  as  <mmm ,  or  with  a  double  rib,  as  jL&zJ ,  its  dimensions 
should  be  proportioned  to  form  of  section. 

In  a  wheel  of  greater  relative  diameter,  length  of  hub  and  breadth  of  arms, 
or  of  the  rib  or  web,  according  as  plane  of  arm  is  in  that  of  wheel,  or  con- 
trariwise, should  be  made  to  exceed  breadth  of  face  of  wheel  (at  the  hub) 
in  order  to  give  it  resistance  to  lateral  strain. 

Number  of  arms  in  wheels  should  be  as  follows*. 

1.5  to  3.25  feet  in  diameter. 4  I  5  to  8.5  feet  in  diameter 6 

3-25  "  5        "  "       5  I  8.5  "  16      "  "        8 

16  to  24  feet  in  diameter 10 

With  light  wheels,  number  of  arms  should  be  increased,  in  order  better  to 
sustain  rigidity  of  rim. 

Mortise  Wheels. — Their  rim  or  face  should  be  .9  pitch  of  tooth,  and  twice 
depth  of  rim  of  a  solid  wheel. 


WINDING  ENGINES. 

With  Winding  Engines,  for  drawing  coals,  etc.,  out  of  a  Pit,  where  it 
is  required  to  give  a  certain  number  of  revolutions,  it  is  necessary  to 
have  given  diameter  of  Drum  and  thickness  of  rope,  which  is  flat  made, 
and  contrariwise. 

To    Compnte    Diameter   of  a   Drnm. 

Where  Flat  Ropes  are  used,  and  are  wound  one  part  over  the  other.  RULE. 
—Divide  depth  of  pit  in  ins.  by  product  of  number  of  revolutions  and  3.1416, 
and  from  quotient  subtract  product  of  thickness  of  rope  and  number  of  rev- 
olutions ;  remainder  is  diameter  in  ins. 

EXAMPLE.— If  an  engine  makes  20  revolutions,  depth  of  pit  being  600  feet  and 
rope  i  inch,  what  should  be  diameter  of  drum  ? 

600  X  12        7200 

— 2  —  i  X  20  =  - — 20  =  04. 50  ins. 

20X3-1416  62.832 

To    Compxite    Diameter    of  Roll. 

RULE.— To  area  of  drum  add  area  or  edge  surface  of  rope ;  then  ascertain 
by  inspection  in  table  of  areas,  or  by  calculation,  diameter  that  gives  this 
area,  and  it  is  the  diameter  of  Roll. 


WINDING   ENGINES. — WINDMILLS.  863 

EXAMPLE.— What  is  diameter  of  roll  in  preceding  example? 

Area  of  94. 59  =  7027. 2  -f-  (area  of  7200  X  i)  =  7200  =14 887. 2,  and  -^14227.2-:- 
.7854  —  151.85  ins. 
Or,  Radius  of  drum  is  increased  number  of  revolutions  multiplied  by  thickness 

of  rope ;  as  —  ^  -f  20  x  i  =  67. 295  ins. 

To    Compute   Num"ber   of*  Revolutions. 

RULE. — To  area  of  drum  add  area  of  edge  surface  of  rope ;  from  diameter 
of  the  circle  having  that  area  subtract  diameter  of  drum,  and  divide  re- 
mainder by  twice  thickness  of  rope ;  quotient  will  give  number  of  revolutions. 

EXAMPLE.— Length  of  a  rope  is  2600  ins.,  its  thickness  i  inch,  sind  diameter  of 
drum  20  ins. ;  what  is  number  of  revolutions  ? 

Area  of  20 -{-area  of  rope  =  314. 16-1-2600  =  2914.16,  diameter  of  which  is  60.91, 

60.  QI  — 2O 

and  — =  20.45  revolutions. 

Or,  subtract  diameter  of  drum  from  diameter  of  roll,  and  divide  remainder  by 
twice  thickness  of  rope ;  as  60. 91  —  20  =  40. 91,  and  40. 91  -r- 1  x  2  =  20. 45  revolutions. 

To    Compute    Point   of   Meeting    of  Ascending    and    De- 
scending  Buckets   when    two   or   more   are   used. 

To  Compute  Point  of  Meeting  of  Buckets.  RULE. — Divide  sum  of  length 
of  turns  of  rope  by  2,  and  to  quotient  add  length  of  last  turn ;  divide  sum 
by  2,  multiply  quotient  by  half  number  of  revolutions,  and  product  will 
give  distance  from  centre  of  drum  at  which  buckets  will  meet. 

NOTE  i. — Meetings  will  always  be  below  half  depth  of  pit. 

2. — At  half  number  of  revolutions  buckets  will  meet. 

EXAMPLE. — Diameter  of  a  drum  is  9  feet,  thickness  of  rope  i  inch,  and  revolu- 
tions 20;  what  is  depth  of  pit,  and  at  what  distance  from  too  will  buckets  meet? 

28.  54-4-  ^8.48   .    -— — 20       71.00X10 

54  7       ~  +  38.48-^2  XT  =  7    "2A       =  35-995  X  10  =  359-95/e<*. 

To  Compute  this  Depth.  RULE. — To  diameter  of  drum  add  thickness  of 
rope  in  feet,  and  ascertain  its  circumference ;  to  diameter  of  drum  add  quo- 
tient of  product  of  twice  thickness  of  rope  and  number  of  revolutions  less  i, 
divided  by  12  for  a  diameter,  and  circumference  of  this  diameter  is  length 
of  last  turn,  also  in  feet ;  add  these  two  lengths  together,  multiply  their  sum 
by  half  number  of  revolutions,  and  product  will  give  depth  of  pit. 

9  -f  thickness  of  rope  =  9  -f  ^  of  i  =  9.083,  which  x  3. 1416  =  28. 54  feet  =  length 

of  first  turn.     9. 0833  -|-  - — -  —  X  3. 1416  =  38. 48  feet  =  length  of  last  turn. 

Then  28.54  -f  38.48  X  —  =  67.02  X  10  =  670.2  feet,  depth  of  pit. 


WINDMILLS. 

Driving  Shaft  of  a  vertical  windmill  should  be  set  at  an  elevating  angle 
with  horizon  when  set  upon  low  ground,  and  at  a  depressing  angle  when  set 
upon  elevated  ground.  Range  of  these  angles  is  from  3°  to  15°.  A  velocity 
of  wind  of  10  feet  per  second  is  not  generally  sufficient  to  drive  a  loaded 
mill,  and  if  velocity  exceeds  35  feet  per  second  the  force  is  generally  too 
great  for  ordinary  structures. 

Angle  of  Sails  should  be  from  18°  to  30°  at  their  least  radius,  and  from 
7°  to  17°  at  their  greatest  radius,  mean  angle  being  from  15°  to  17°  to  plane 
of  motion  of  sails.  Length  of  a  whip  (arm)  is  divided  into  7  parts,  sails  ex- 
tending over  6  parts. 


864 


WIND-MILLS. 


Whip  in  parts  of  its  length  :  Breadth  .033,  at  top  .016  ;  Depth  .025,  at  top 
.0125  ;  Width  of  sail  .33,  at  axis  .2.  Distance  of  sail  from  axis  .014  of 
length  of  whip,  and  cross-bars  16  to  18  ins.  from  centres. 

To    Compute    Angles    of  Sails. 

23°  —  -  —  —  =  angle  of  sail  with  plane  of  its  motion  at  any  part  of  it  d  repre- 
senting distance  of  part  of  sail  from  its  axis,  and  r  extreme  radius  of  sail,  both  in  feet, 

ILLUSTRATION.—  Assume  r  =  14,  and  length  of  sail  12  feet,  d  =  .  5  of  12  or  three 
sixths  of  sail  =  .5  X  i2-|~(i4  —  12)  =  2  =  8  feet. 


Then  23°  —  =  23  —  5.  88°  =  17.  12°. 

Hence,  angle  of  sail  with  axis  =  90°  —  17.  12°  =  72.88°. 

If  radius  of  sails  is  divided  into  6  equal  parts,  angles  at  each  of  these  parts  will 
be  as  follows: 

Distance  from  A^cis. 
123456 

Angle  of  sail  with  a«is  .......     ...........  67.5°    69°    71.5°    75®    79.5°    85° 

"       "       with  plane  of  motion  .........  22.5°    21°    18.5°    15*"    10.5°      5° 

To    Compute    Elements   of  \Vindmills- 

7.16  v  11.5  v  Ar3 

~-.  -  =  n;  —  £—  =rc;  .1047  n  =  av;  —  -  -  =  IP; 

r'sm.  x  r  1080000 

IPX  1  080000  /R24-r2 

-  -  -  =  A  ;        A  /  -  '-  -  =  r'.    v  representing  velocity  of  wind  per  sec- 

v        2 

ond,  r'  radius  of  centre  of  percussion  of  sails,  and  R  and  r  outer  and  inner  radii  of 
sails,  all  in  feet,  x  mean  angle  of  sail  to  plane  of  motion,  n  number  of  revolutions  oj 
arms  per  minute,  a  v  angular  velocity,  A  area  of  sails  in  sq.feet,  and  IP  horse-power. 

ILLUSTRATION.  —  If  a  windmill  has  4  arms  of  28  feet,  with  a  mean  angle  (a;)  of  16°, 
with  an  area  of  sail  of  150  sq.  feet  each,  having  an  inner  radius  of  4  feet,  and  is  op- 
erated by  wind  at  a  velocity  of  40  feet  per  second;  what  are  its  elements? 


Deductions  from  "Velocities  varying  from  4  to  9  Feet  per 
Second.     (Mr.  Smeaton.) 

1.  Velocity  of  windmill  sails,  so  as  to  produce  a  maximum  effect,  is  near- 
ly as  velocity  of  wind,  their  shape  and  position  being  same. 

2.  Load  at  maximum  is  nearly,  but  somewhat  less  than,  as  square  of  ve- 
locity of  wind,  shape  and  position  of  sails  being  same. 

3.  Effects  of  same  sails,  at  a  maximum,  are  nearly,  but  somewhat  less 
than,  as  cubes  of  velocity  of  wind. 

4.  Load  of  same  sails,  at  maximum,  is  nearly  as  squares,  and  their  effect 
as  cubes  of  their  number  of  turns  in  a  given  time. 

5.  In  sails  where  figure  and  position  are  similar,  and  velocity  of  wind  the 
same,  number  of  revolutions  in  a  given  time  will  be  reciprocally  as  radius  or 
length  of  sail. 

6.  Load,  at  a  maximum,  which  sails  of  a  similar  figure  and  position  will 
overcome  at  a  given  distance  from  centre  of  motion,  will  be  as  cube  of  radius. 

7.  Effects  of  sails  of  similar  figure  and  position  are  as  square  of  radius. 

8.  Velocity  of  extremities  of  Dutch  sails,  as  well  as  of  enlarged  sails,  in 
all  their  usual  positions  when  unloaded,  or  even  loaded  to  a  maximum,  is 
considerably  greater  than  that  of  wind. 


WINDMILLS. — WOOD  AND   TIMBER. 


865 


Results   of  Experiments   on   Effect   of*  Windmill    Sails. 
When  a  vertical  windmill  is  employed  to  grind  corn,  the  millstone  usu- 
ally makes  5  revolutions  to  i  of  the  sail. 

1.  When  velocity  of  wind  is  19  feet  per  second,  sails  make  from  n  to  12 
revolutions  in  a  minute,  and  a  mill  will  grind  from  880  to  990  Ibs.  in  an 
hour,  or  about  22  440  Ibs.  in  24  hours. 

2.  When  velocity  of  wind  is  30  feet  per  second,  a  mill  will  carry  all  sail, 
and  make  22  revolutions  in  a  minute,  grinding  19184  Ibs.  of  flour  in  an  hour, 
or  47  616  Ibs.  in  24  hours. 

Results   of  Operation   of  Windmills.     (A.  R.  Woolf,  M.  E.) 

Velocity  of  Wind  15  to  20  Miles  per  Hour. 
Revolutions  of  Wheel  and  Gallons  of  Water  raised  per  Minute. 


Desig- 

nation 
of  Mill. 

Revolutions 
of 
Wheel. 

Wat. 
25  Feet. 

8r  raised  to 
50  Feet. 

an  E  leva  tic 
loo  Feet. 

n  of 
200  Feet. 

Power 
developed. 

Coetpc 
Actual.* 

r  Hoar. 
PerH>. 

Feet. 
8-5 

10 

14 

18 

20 
25 

No. 
70  to  75 
60  1065 

So  to  55 
40  to  45 

35  to  4«> 
30  to  35 

Gallons. 
6.16 
19.18 

45-14 

97.68 
124.95 
212.38 

Gallons. 
3.02 
9-56 
22.57 
52.16 

63t*7f 
106.96 

Gallons. 

4-75 
11.25 
24.42 
3i-25 
49-73 

Gallons. 

5 

12.21 

15-94 
26.74 

IP 

.04 

.12 

.28 
.6l 
.78 
1.34 

Cents. 
.60 
•70 
1-63 
2.83 
3.56 
4.26 

Cents. 
15 
5-8 
5-8 
4.6 
4-5 

3-3 

*  Including  interest  at  5  per  cent,  per  annum. 


WOOD   AND  TIMBER. 

Selection  of  Standing  Trees. —  Wood  grown  in  a  moist  soil  is  lighter, 
and  decays  sooner,  than  that  grown  in  dry,  sandy  soil. 

Best  Timber  is  that  grown  in  a  dark  soil,  intermixed  with  gravel. 
Poplar,  Cypress,  Willow,  and  all  others  which  grow  best  in  a  wet  soil, 
are  exceptions. 

Hardest  and  densest  woods,  and  least  subject  to  decay,  grow  in  warm 
climates ;  but  they  are  more  liable  to  split  and  warp  in  seasoning. 

Trees  grown  upon  plains  or  in  centre  of  forests  are  less  dense  than 
those  from  edge  of  a  forest,  from  side  of  a  hill,  or  from  open  ground. 

Trees  (in  U.  S.)  should  be  selected  in  latter  part  of  July  or  first  part 
of  August;  for  at  this  season  leaves  of  sound,  healthy  trees  are  fresh 
and  green,  while  those  of  unsound  are  beginning  to  turn  yellow.  A 
sound,  healthy  tree  is  recognized  by  its  top  branches  being  well  leaved, 
bark  even  and  of  a  uniform  color.  A  rounded  top,  few  leaves,  some  of 
them  turned  yellow,  a  rougher  bark  than  common,  covered  with  parasitic 
plants,  and  with  streaks  or  spots  upon  it,  indicate  a  tree  upon  the  de- 
cline. Decay  of  branches,  and  separation  of  bark  from  the  wood,  are 
infallible  indications  that  the  wood  is  impaired. 

Green  timber  contains  37  to  48  per  cent,  of  liquids.  By  exposure  to 
air  in  seasoning  one  year,  it  loses  from  17  to  25  per  cent.,  and  when 
seasoned  it  retains  from  10  to  15  per  cent. 

According  to  M.  Leplay,  green  wood  contains  about  45  per  cent,  of  its 
weight  of  moisture.  In  Central  Europe,  wood  cut  in  winter  holds,  at  end  of 
following  summer,  fully  40  per  cent,  of  water,  and  when  kept  dry  for  sev' 
eral  years  retains  from  15  to  20  per  cent,  of  water. 

Felling  Timber. — Most  suitable  time  for  felling  timber  is  in  midwinter  and 
in  midsummer.  Recent  experiments  indicate  latter  season  and  month  of  July. 

4D 


866  WOOD    AND   TIMBER. 

A  tree  should  be  allowed  to  attain  full  maturity  before  being  felled.  Oak 
matures  at  75  to  100  years  and  upwards,  according  to  circumstances ;  Ash, 
Larch,  and  Elm  at  75 ;  and  Spruce  and  Fir  at  80.  Age  and  rate  of  growth 
of  a  tree  are  indicated  by  number  and  width  of  the  rings  of  annual  increase 
which  are  exhibited  in  a  cross-section  of  its  body. 

A  tree  should  be  cut  as  near  to  the  ground  as  practicable,  as  the  lower 
part  furnishes  best  timber. 

Dressing  Timber. — As  soon  as  a  tree  is  felled,  it  should  be  stripped  of  its 
bark,  raised  from  the  ground,  reduced  to  its  required  dimensions,  and  its 
sap-wood  removed. 

Inspection  of  Timber. — Quality  of  wood  is  in  some  degree  indicated  by  its 
color,  which  should  be  nearly  uniform,  and  a  little  deeper  towards  its  cen- 
tre, and  free  from  sudden  transitions  of  color.  White  spots  indicate  decay. 
Sap-wood  is  known  by  its  white  color ;  it  is  next  to  the  bark,  and  soon  rots. 

Defects    of  Timber. 

Wind-shakes  are  serious  defects,  being  circular  cracks  separating  the  con- 
centric layers  of  wood  from  each  other. 

Splits,  Checks,  and  Cracks,  extending  toward  centre,  if  deep  and  strongly 
marked,  render  timber  unfit  for  use,  unless  purpose  for  which  it  is  intended 
will  admit  of  its  being  split  through  them. 

Brash  is  when  wood  is  porous,  of  a  reddish  color,  and  breaks  short,  with- 
out splinters.  It  is  generally  consequent  upon  decline  of  tree  from  age. 

Belted  is  that  which  has  been  killed  before  being  felled,  or  which  has  died 
from  other  causes.  It  is  objectionable. 

Knotty  is  that  containing  many  knots,  though  sound ;  usually  of  stinted 
growth. 

Twisted  is  when  grain  of  it  winds  spirally ;  it  is  unfit  for  long  pieces. 

Dry-rot  is  indicated  by  yellow  stains.  Elm  and  Beech  are  soon  affected, 
if  left  with  the  bark  on. 

Large  or  decayed  knots  injuriously  affect  strength  of  timber. 

Heart-shake. — Split  or  cleft  in  centre  of  tree,  dividing  it  into  segments. 

Star-shake. — Several  splits  radiating  from  centre  of  timber. 

Cup-shake. — Curved  splits  separating  the  rings  wholly  or  in  part. 

Rind-gall. — Curved  swelling,  usually  caused  by  growth  of  layers  over  spot 
where  a  branch  has  been  removed. 

Upset. — Fibres  injured  by  crushing. 

Foxiness. — Yellow  or  red  tinge,  indicating  incipient  decay. 

Doatiness. — A  speckled  stain. 

Seasoning   and.    Preserving   Timber. 

Seasoning  is  extraction  or  dissipation  of  the  vegetable  juices  and  moisture 
or  solidification  of  the  albumen.  When  wood  is  exposed  to  currents  of  air 
at  a  high  temperature,  the  moisture  evaporates  too  rapidly,  and  it  cracks ; 
and  when  temperature  is  high  and  sap  remains,  it  ferments,  and  dry-rot 
ensues. 

Wood  requires  time  in  which  to  season,  very  much  in  proportion  to  density 
of  its  fibres. 

Water  Seasoning  is  total  immersion  of  timber  in  water,  for  purpose  of 
dissolving  the  sap,  and  when  thus  seasoned  it  is  less  liable  to  warp  and  crack, 
but  is  rendered  more  brittle. 


WOOD    AND    TIMBER.  867 

For  purpose  of  seasoning,  it  should  be  piled  under  shelter  and  kept  dry; 
should  have  a  free  circulation  of  air,  without  being  exposed  to  strong  cur- 
rents. Bottom  pieces  should  be  placed  upon  skids,  which  should  be  free 
from  decay,  raised  not  less  than  2  feet  from  ground ;  a  space  of  an  inch 
should  intervene  between  pieces  of  same  horizontal  layers,  and  slats  or  piling- 
strips  placed  between  each  layer,  one  near  each  end  of  pile,  and  others  at 
short  distances,  in  order  to  keep  the  timber  from  winding.  These  strips 
should  be  one  over  the  other,  and  in  large  piles  should  not  be  less  than  i  inch 
thick.  Light  timber  may  be  piled  in  upper  portion  of  shelter,  heavy  timber 
upon  ground  floor.  Each  pile  should  contain  but  one  description  of  timber, 
and  they  should  be  at  least  2.5  feet  apart. 

It  should  be  replied  at  intervals,  and  all  pieces  indicating  decay  should  be 
removed,  to  prevent  their  affecting  those  which  are  still  sound. 

It  requires  from  2  to  8  years  to  be  seasoned  thoroughly,  according  to  its 
dimensions,  and  it  should  be  worked  as  soon  as  it  is  thoroughly  dry,  for  it 
deteriorates  after  that  time. 

Gradual  seasoning  is  most  favorable  to  strength  and  durability  of  timber. 
Various  methods  have  been  proposed  for  hastening  the  process,  as  Steaininy, 
which  has  been  applied  with  success;  and  results  of  experiments  of  va  ions 
processes  of  saturating  it  with  a  solution  of  Corrosive  sublimate  and  Anti- 
septic fluids  are  very  satisfactory.  Such  process  hardens  and  seasons  wood, 
at  the  same  time  that  it  secures  it  from  dry-rot  and  from  attacks  of  worms. 

Woods  are  densest  and  strongest  at  the  roots  and  at  their  centres.  Their 
strength  decreasing  with  the  decrease  of  their  density. 

Oak  timber  loses  one  fifth  of  its  weight  in  seasoning,  and  about  one  third 
in  becoming  perfectly  dry. 

Pitch  pine,  from  the  presence  of  pitch,  requires  time  in  excess  of  that  due 
to  the  density  of  its  fibre. 

Mahogany  should  be  seasoned  slowly,  Pine  quickly.  Whitewood  should 
not  be  dried  artificially,  as  the  effect  of  heat  is  to  twist  it. 

Salt  water  renders  wood  harder,  heavier,  and  more  durable  than  fresh. 

Condition  of  timber,  as  to  its  soundness  or  decay,  is  readily  recognized 
when  struck  with  a  quick  blow. 

Timber  that  has  been  for  a  long  time  immersed  in  water,  when  brought 
into  the  air  and  dried,  becomes  brashy  and  useless. 

When  trees  are  barked  in  the  spring,  they  should  not  be  felled  until  the 
foliage  is  dead. 

Timber  cannot  be  seasoned  by  either  smoking  or  charring ;  but  when  it 
is  exposed  to  worms  or  to  the  production  of  fungi,  it  is  proper  to  smoke  or 
char  it,  and  it  may  be  partially  seasoned  by  being  boiled  or  steamed. 

Timber  houses  are  best  provided  with  blinds  which  keep  out  rain  and 
snow,  but  which  can  be  turned  to  admit  air  in  fine  weather,  and  the  houses 
should  be  kept  entirely  free  from  any  pieces  of  decayed  wood. 

Kiln-drying  is  suited  only  for  boards  and  pieces  of  small  dimensions,  as  it 
is  apt  to  cause  cracks  and  to  impair  the  strength,  unless  performed  very 
slowly. 

Charring,  Painting,  or  covering  the  surface  is  highly  injurious  to  any  but 
seasoned  wood,  as  it  effectually  prevents  drying  of  the  inner  part  of  the 
wood,  in  consequence  of  which  fermentation  and  decay  soon  take  place. 

Timber  is  subject  to  Common  or  Dry-rot,  former  occasioned  by  alternate 
exposure  to  moisture  and  dry  ness,  and  as  progress  of  it  is  from  the  exterior, 
covering  of  the  surface,  if  seasoned,  with  paint,  tar,  etc.,  is  a  preservative. 


868  WOOD    AND   TIMBER. 

Common-rot  is  the  consequence  of  its  being  piled  in  badly-ventilated  sheds. 
Outward  indications  are  yellow  spots  upon  ends  of  pieces,  and  a  yellowish 
dust  in  the  checks  and  cracks,  particularly  where  the  pieces  rest  upon  pil- 
ing-strips. 

Dry  or  Sap-rot  is  inherent  in  timber,  and  it  is  the  putrefaction  of  the  veg- 
etable albumen.  Sap  wood  contains  a  large  proportion  of  fermentable  ele- 
ments. 

Insects  attack  wood  for  the  sugar  or  gum  contained  in  it,  and  fungi  subsist 
upon  the  albumen  of  wood ;  hence,  to  arrest  dry-rot,  the  albumen  must  be 
either  extracted  or  solidified. 

Most  effective  method  of  preserving  timber  is  that  of  expelling  or  ex- 
hausting its  fluids,  solidifying  its  albumen,  and  introducing  an  antiseptic 
liquid. 

Strength  of  impregnated  timber  is  not  reduced,  and  its  resilience  is  improved. 

In  desiccating  timber  by  expelling  its  fluids  by  heat  and  air,  its  strength 
is  increased  fully  15  per  cent. 

The  saturation  of  wood  with  creosote,  tar,  antiseptics,  etc.,  preserves  it 
from  the  attack  of  worms.  Jarrow  wood,  from  Australia,  is  not  subjected 
to  their  attack. 

In  a  perfectly  dry  atmosphere  durability  of  woods  is  almost  unlimited. 
Rafters  of  roofs  are  known  to  have  existed  1000  years,  and  piles  submerged 
in  fresh  water  have  been  found  perfectly  sound  800  years  from  period  of 
their  being  driven. 

Resistance  of  woods  to  extension  is  greater  than  that  of  compression. 

Impregnation    of  "Wood.. 

Several  of  the  successful  processes  are  as  follows : 

Kyan,  1832. — Saturated  with  corrosive  sublimate.  Solution  i  Ib.  of  chlo- 
ride of  mercury  to  4  gallons  of  water. 

Burnett  (Sir  Wm.),  1838.  —  Impregnation  with  chloride  of  zinc  by  sub- 
mitting the  wood  endwise  to  a  pressure  of  150  Ibs.  per  sq.  inch.  Solution, 
i  Ib.  of  the  chloride  to  4  gallons  of  water. 

Boucheri. — Impregnation  by  submitting  the  wood  endwise  to  a  pressure 
of  about  15  Ibs.  per  sq.  inch.  Solution,  i  Ib.  of  sulphate  of  copper  to  12.5 
gallons  of  water. 

Bethel. — Impregnation  by  submitting  the  wood  endwise  to  a  pressure  of 
150  to  200  Ibs.  per  sq.  inch,  with  oil  of  creosote  mixed  with  bituminous 
matter. 

Robbins,  1865. — Aqueous  vapor  dissipated  by  the  wood  being  heated  in  a 
chamber,  the  albumen  solidified,  then  submitted  to  vapor  of  coal  tar,  resin, 
or  bituminous  oils,  which,  being  at  a  temperature  not  less  than  325°,  readily 
takes  the  place  of  the  vapor  expelled  by  a  temperature  of  212°. 

Hayford,  187-. — Aqueous  vapor  dissipated  by  the  wood  being  heated  in  a 
chamber  to  a  temperature  of  from  250°  to  270°,  the  albumen  solidified,  then 
air  introduced  to  assist  the  splitting  of  the  outer  surfaces.  When  vapor  is 
dissipated,  dead  oils  are  introduced  under  a  pressure  of  75  Ibs.  per  sq.  inch. 

Planks,  Deals,  and  Battens. — When  cut  from  Northern  pine  (Plnus  Sylve- 
stris)  are  termed  yellow  or  red  deal,  and  when  cut  from  spruce  (Abies,  alba, 
etc.)  they  are  termed  white  deal. 

Desiccated  wood,  when  exposed  to  air  under  ordinary  circumstances,  ab- 
sorbs 5  per  cent,  of  water  in  the  first  three  days ;  and  will  continue  to  absorb 
it  until  it  reaches  from  14  to  16  per  cent.,  the  amount  varying  according 
to  condition  of  the  atmosphere. 


WOOD   AND    TIMBER.  869 

Durability   of  Various   Woods, 

Pieces  2  feet  in  Length,  1.5  ins.  Square,  driven  28.5  ins.  into  the  Earth. 


WOOD. 

C 

After  2.5  Years. 

ondition 
After  5  Years. 

Good  

(Externally  decayed,  rest  per- 
(     fectly  sound. 
Decayed. 
Sound  as  when  driven. 
Tolerable. 
Entirely  decayed. 
Decayed. 
Much  decayed. 
(  Attacked  in  part  only,  rest  fair 
\     condition. 
Very  rotten. 

(Some  moderately,  most  very 
\     much,  decayed. 
(  Attacked  in  part  only,  rest  fair 
\     condition. 
Much  decayed. 
Very  rotten. 
Somewhat  soft,  but  good. 

Ash  Amer  

Much  decayed  

Cedar,  Va  

'  *       Lebanon 

Very  good  
Good                    

Elm    Eng  

':      Ca.ii 

Fir  

"      attacked  

Surface  only  attacked  
Very  much  decayed        • 

Oak  Can 

"  '  Memel 

"     Dautzic  

"     Chestnut  

Pine  pitch  

Surface  only  attacked  
Attacked 

4  '     yellow 

"     white  .       .   . 

Very  much  decayed  

Teak  .  . 

Very  good  

Effect   of  Creosoting. 

Results  of  Experiments  with  Various  Woods  (E.  R.  Andrews). 


WOOD. 

Water 
absorbed. 

WOOD. 

Water 
absorbed. 

Spruce  

Onlr 

f  dried  

Per  cent. 
•2543 
.0261 

.2 
.0 
.714 

•347 

Hard  pine.  .  .  . 
Gum,  black  .  . 
Birch,  white  . 

dried  

creosoted. 
dried  .... 
creosoted. 
dried  

creosoted. 

Per  cent. 
16 
o 
i 
«5 
43 
124 

\  creosoted  .  .  . 
(dried  

Cotton-wood 

\  creosoted  .  .  . 
|  dried  

{  creosoted  .  .  . 

Sesquoia  Gigantea  of  California,  dried,  .4722;  creosoted,  .o. 
Fluids  will  pass  with  the  grain  of  wood  with  great  facility,  but  will  not 
enter  it  except  to  a  very  limited  extent  when  applied  externally. 

.Absorption,  of  Preserving   Solntioii  "by  different  "Woods 
for   a   Period   of  7"   Days.     Average  Lbs.  per  Cube  Foot. 

Black  Oak 3.6  I  Hemlock 2.6  I  Rock  Oak 3.9 

Chestnut 3     |  Red  Oak 3.9  |  White  Oak 3.1 

Proportion    of  "Water   in    various   "Woods. 


Alder  (Betula  alnus) 41.6 

Ash  (Fraxinus  excelsior) 28.7 

Beech  (Fagus  sylvatica) 33 

Birch  (Betula  alba) 30.8 

Elm  ( Ulmus  campestris) 44. 5 


Horse-chestnut  (^Esculus  hippocast. )  38. 2 

Larch  (Pinus  larix). 48.6 

Mountain  Ash  (Sorbus  aucuparia). .  28.3 
Oak  (Quercus  robur) 34.7 


Pine  (Pinus  Sylvestris  L.) 39.7 

Red  Beech  (Fagus  sylvatica) 39.7 

Red  Pine  (Pinus  picea  dur) 45.2 

Spruce  (Abies,  alba,  nigra,  rubra,  \ 
excelsa) )    35 


Sycamore  (Acer  pseudo-platanus) . .  27 

White  Oak  (Quercus  alba) 36.2 

White  Pine  (Pinus  abies  dur) 37.  i 


White  Poplar  (Populus  alba) 50.6 

Willow  (Salis  caprea) 26 

Decrease   in    Dimensions   of  Timber  t>y   Seasoning. 
Ins.  WOODS.  Ins.  Int. 

to    13.25          Pitch  Pine,  South 18.375  to  18.25 

to    10.75          Spruce 8.5     to    8.375 

to    11.625        White  Pine,  American..  12        1011.875 


WOODS. 

Cedar,  Canada. 

Elm 

Oak,  English 12 


Pitch  Pine,  North...  ioXioto  9.75X9-75 


Yellow  Pine,  North 18        to  17.875 


Weight  of  a  beam  of  English  oak,  when  wet,  was  reduced  by  seasoning 
from  972.25  to  630.5  Ibs. 

4  D* 


8/o 


WOOD   AND   TIMBER. 


Weight   of  a  Cube    Foot   of  Oak   and  Yellow  !Pine. 


AOB. 

White  < 
Round. 

3ak,  Va. 
Square. 

Yellow  1 
Round. 

Pine,  Va. 
Square. 

Live  Oak. 

78.7 
66.  T 

Green  

64.7 
53-6 
4.6 

67.7 
53-5 

AQ.Q 

47-8 
39-8 
n.  ^ 

39-2 
34-2 

7Q.  C 

i  Year  

2  Years.  .  . 

In  England,  Timber  sawed  into  boards  is  classed  as  follows : 
6.5  to  7  ins.  in  width,  Battens;  8.5  to  10  ins.,  Deals;  and  n  to  12  ins., 
Planks.     (See  also  page  62.) 

Distillation.— 'From  a  single  cord  of  pitch  pine  distilled  by  chemical  ap- 
paratus, following  substances  and  in  quantities  stated  have  been  obtained : 


Charcoal 50  bushels. 

Illuminating  Gas about  1000  cu.  feet. 

Illuminating  Oil  and  Tar. . .  50  gallons. 
Pitch  or  Resin 1.5  barrels. 


Pyrol igneous  Acid 100  gallons. 

Spirits  of  Turpentine 20      " 

Tar i  barrel. 

Wood  Spirit 5  gallons. 


Strength,   of  Timber. 

Results  of  experiments  have  satisfactorily  proved:  That  deflection  was 
sensibly  proportional  to  load ;  That  extension  and  compression  were  nearly 
the  same,  though  former  being  the  greater ;  That,  to  produce  equal  deflection, 
load,  when  placed  in  the  centre,  was  to  a  load  uniformly  distributed,  as  .638 
to  i ;  That  deflection  under  equal  loads  is  inversely  as  breadths  and  cubes 
of  the  depths,  and  directly  as  cubes  of  the  spans.  ( M.  Morin.) 

It  has  also  been  shown,  that  density  of  wood  varies  very  little  with  its  age. 
That  coefficient  of  elasticity  diminishes  after  a  certain  age,  and  that  it  de- 
pends also  on  the  dryness  and  the  exposure  of  the  ground  where  the  wood 
is  grown.  Woods  from  a  northerly  exposure,  on  dry  ground,  have  a  high 
coefficient,  while  those  from  swamps  or  low  moist  ground  have  a  low  one. 
That  tensile  strength  is  influenced  by  age  and  exposure.  The  coefficient 
of  elasticity  of  a  tree  cut  down  in  full  vigor,  or  before  it  arrives  at  this 
condition,  does  not  present  any  sensible  difference.  That  there  is  no  limit 
of  elasticity  in  wood,  there  being  a  permanent  set  for  every  extension. 

Average  Result  of  Experiments  on  Tensile  Strength  of  Wood  in  Various 
Positions  per  Sq.  Inch.     (MM.  Chevandier  and  Wertheim.) 

With  the  fibre,  6900  Ibs.    Radially,  683  Ibs.,  and  Tangentially,  723  Ibs. 


To   Compute   Volume   of  an    Irregular   Body. 

By  "  Simpson's  Rule." 

OPERATION. — Take  a  right  line  in  the  figure  for  a  base  line,  as  A  B,  divide  the  fig- 
ure into  any  number  of  equal  parts,  and  compute  the  areas  of  their  plane  sections 
as  i,  2,  3,  etc. ,  at  the  points  of  division,  by  rules  applicable  to  area  of  a  plane.  Then, 
operate  these  areas  as  if  they  were  the  ordinates  of  a  plane  curve  or  figure  of  same 
length  as  the  figure,  and  result  will  give  volume  required. 

ILLUSTRATION.—  Assume  a  figure  having  areas  as  follows,  and  A  B  —  24  feet. 

Sections,  i     Areas,  3  feet     Multiplier,  i     Products,   3 


14 
36 
ii 

84" 


and  84  X>4-J- 4-7-3  =  168  cube  feet. 


MISCELLANEOUS   MIXTURES.  8/1 

MISCELLANEOUS   MIXTURES. 
Cements. 

Much  depends  upon  manner  in  which  a  cement  is  applied  as  upon  the 
cement  itself,  as  best  cement  will  prove  worthless  if  improperly  applied. 
Following  rules  must  be  rigorously  adhered  to  to  attain  success : 

1.  Bring  cement  into  intimate  contact  with  surfaces  to  be  united.    This  is  best 
done  by  heating  pieces  to  be  joined  in  cases  where  cement  is  melted  by  heat,  as 
with  resin,  shellac,  marine  glue,  etc.    Where  solutions  are  used,  cement  must  be 
well  rubbed  into  surfaces,  either  with  a  brush  (as  in  case  of  porcelain  or  glass), 
or  by  rubbing  the  two  surfaces  together  (as  in  making  a  glue  joint  between  pieces 
of  wood). 

2.  As  little  cement  as  practicable  should  be  allowed  to  remain  between  the  united 
surfaces     To  secure  this,  cement  should  be  as  liquid  as  practicable  (thoroughly 
melted  if  used  with  heat),  and  surfaces  should  be  pressed  closely  into  contact  until 
cement  has  hardened. 

-  3.  Time  should  be  allowed  for  cement  to  dry  or  harden,  and  this  is  particularly 
the  case  in  oil  cements,  such  as  copal  varnish,  boiled  oil,  white  lead,  etc.  When 
two  surfaces,  each  .5  inch  across,  are  joined  by  means  of  a  layer  of  white  lead 
placed  between  them,  6  months  may  elapse  before  cement  in  middle  of  joint  be- 
comes hard.  At  the  end  of  a  month  the  joint  will  be  weak  and  easily  separated;  at 
end  of  2  or  3  years  it  may  be  so  firm  that  the  material  will  part  anywhere  else  than 
at  joint.  Hence,  when  article  is  to  be  used  immediately,  the  only  safe  cements 
are  those  which  are  liquefied  by  heat  and  which  become  hard  when  cold.  A  joint 
made  with  marine  glue  is  firm  an  hour  after  it  has  been  made.  Next  to  cements 
that  are  liquefied  by  heat  are  those  which  consist  of  substances  dissolved  in  water 
or  alcohol.  A  glue  joint  sets  firmly  in  24  hours;  a  joint  made  with  shellac  varnish 
becomes  dry  in  2  or  3  days.  Oil  cements,  which  do  not  dry  by  evaporation,  but 
harden  by  oxidation  (boiled  oil,  white  lead,  red  lead,  etc. )  are  slowest  of  all. 

Stone. — Resin,  Yellow  Wax,  and  Venetian  Red,  each  i  oz. ;  melt  and  mix. 
Aquarium. 

Litharge,  fine  white  dry  Sand,  and  Plaster  of  Paris,  each  i  gill ;  finely  pulverized 
Resin,  .33  gill. 

Mix  thoroughly  and  make  into  a  paste  with  boiled  linseed  oil  to  which  drier  has  been  added.  Beat 
well,  and  let  stand  4  or  5  hours  before  using  it.  After  it  has  stood  for  15  hours,  however,  it  loses  its 
strength.  Glass  cemented  into  a  frame  with  this  cement  will  resist  percolation  for  either  salt  or  fresh 
water. 

Adhesive    for   Fractures    of  all    Kinds. 

White  Lead  ground  with  Linseed-oil  Varnish,  and  kept  from  contact  with  the  air. 

Requires  a  few  weeks  to  harden. 

t  Stone  or  Iron. 

Compound  equal  parts  of  Sulphur  and  Pitch. 

Brass   to   G-lass. 

Electrical. — Resin,  5  ozs. ;  Beeswax,  i  oz. ;  Red  Ochre  or  Venetian  Red,  in  pow- 
der, i  oz.  Dry  earth  thoroughly  on  a  stove  at  above  212°.  Melt  Wax  and  Resin 
together  and  stir  in  powder  by  degrees.  Stir  until  cold,  lest  earthy  matter  settle 
to  bottom. 

Used  for  fastening  brass-work  to  glass  tubes,  flasks,  etc. 

Chinese   "Waterproof. 

Schio-liao.—To  3  parts  of  Fresh  Beaten  BJood  add  4  parts  of  Slaked  Lime  and  a 
little  Alum;  a  thin,  pasty  mass  is  produced,  which  can  be  used  immediately. 

Materials  which  are  to  be  made  specially  waterproof  are  painted  twice,  or  at  most  three  times. 
Wooden  public  buildings  of  China  are  painted  with  gchio-liao,  which  gives  them  an  unpleasant  red- 
dish appearance,  but  adds  to  their  durability.  Pasteboard  treated  with  it  receives  appearance  and 
strength  of  wood. 

China. 

Curd  of  milk,  dried  and  powdered,  10  ozs. ;  Quicklime,  i  oz. ;  Camphor,  2  drachma 

Mix,  and  keep  air-tight.    When  used,  a  portion  is  to  be  mixed  with  a  little  water  into  a  paste. 

Cisterns   and    "Water-casks. 
Melted  Glue,  8  parts;  Linseed  oil,  boiled  into  a  varnish  with  Litharge,  4  parts. 

This  cement  hardens  in  about  48  hours,  and  renders  the  joints  of  wooden  cisterns  and  casks  air  and 
water  tight. 


8/2 


MISCELLANEOUS   MIXTURES. 


Cloth   or   Leather. 

Shellac,  i  part;  Pitch,  2  parts;  India  Rubber,  4  parts;  and  Gutta  Percha,  10 
parts;  cut  small;  Linseed  oil,  2  parts;  melted  together  and  mixed 

Earthen    and    Glass   Ware. 

Heat  article  to  be  mended  a  little  above  212°,  then  apply  a  thin  coating  of  gum 
Shellac  upon  both  surfaces  of  broken  vessel. 

Or,  dissolve  gum  Shellac  in  alcohol,  apply  solution,  and  bind  the  parts  firmly  to- 
gether until  cement  is  dry. 

Or,  dilute  white  of  egg  with  its  bulk  of  water  and  beat  up  thoroughly.  Mix  to 
consistence  of  thin  paste  with  powdered  Quicklime. 

Use  immediately. 

Entomologists'. 

Thick  Mastic  Varnish  and  Isinglass  size,  equal  parts. 

Gutta   Percha. 

Melt  together,  in  an  iron  pan,  2  parts  Common  Pitch  and  i  part  Gutta  Percha. 

Stir  well  together  until  thoroughly  incorporated,  and  then  pour  liquid  into  cold  water.  When  cold 
it  is  black,  solid,  and  elastic ;  but  it  softens  with  heat,  and  at  100°  is  a  thin  fluid.  It  may  be  used  as  a 
soft  paste,  or  in  liquid  state,  and  answers  an  excellent  purpose  in  cementing  metal,  glass,  porcelain, 
ivory,  etc.  It  may  be  used  instead  of  putty  for  glazing. 

GUass. 

SoreVs.—  Mix  commercial  Zinc  White  with  half  its  bulk  of  fine  Sand,  add  a  solu- 
tion of  Chloride  of  Zinc  of  1.26  spec,  grav.,  and  mix  thoroughly  in  a  mortar. 

Apply  immediately,  as  it  hardens  very  quickly. 

Holes    in    Castings. 

Sulphur  in  powder,  i  part;  Sal-ammoniac,  2  parts;  powdered  Iron  turnings,  80 
parts.     Make  into  a  thick  paste. 
Make  only  as  required  for  immediate  use. 

Hydraulic   Paint. 

Hydraulic  cement  mixed  with  oil  forms  an  incombustible  and  waterproof  paint 
for  roofs  of  buildings,  outhouses,  walls,  etc. 

Iron   "Ware. 

Sulphur,  2  parts;  fine  Black-lead,  i  part.  Heat  sulphur  in  an  iron  pan  until 
it  melts,  then  add  the  lead ;  stir  well,  and  remove.  When  cool,  break  into  pieces 
as  required.  Place  upon  opening  of  the  ware  to  be  mended,  and  solder  with  an 
iron. 

[Kerosene    Lamps,  etc. 

Resin,  3  parts;  Caustic  Soda,  i;  Water,  5,  mixed  with  half  its  weight  of  Plaster 
of  Paris. 

It  seta  firmly  in  about  three  quarters  of  an  honr.  Is  of  great  adhesive  power,  not  permeable  to  kero« 
•ene,  a  low  conductor  of  heat,  and  but  superficially  attacked  by  hot  water. 

Leather  to    Iron,  Steel,  or   G-lass. 

i. — Glue,  i  quart,  dissolved  in  Cider  Vinegar;  Venice  Turpentine,  i  oz. ;  boil  very 
gently  or  simmer  for  12  hours. 

Or,  Glue  and  Isinglass  equal  parts,  soak  in  water  10  hours,  boil  and  add  tannin 
until  mixture  becomes  "ropy;"  apply  warm. 

Remove  surface  of  leather  where  it  is  to  be  applied. 

2. — Steep  leather  in  an  infusion  of  Nutgall,  spread  a  layer  of  hot  Glue  on  sur- 
face of  metal,  and  apply  flesh  side  of  leather  under  pressure. 

Leather   Belting. 

Common  Glue  and  Isinglass,  equal  parts,  soaked  for  10  hours  in  enough  water  to 
cover  them.  Bring  gradually  to  a  boiling  heat  and  add  pure  Tannin  until  whole  be- 
comes ropy  or  appears  alike  to  white  of  eggs. 

Clean  and  rub  surfaces  to  be  joined,  apply  warm,  and  clamp  firmly. 

^Molding   and    Temporary-    Adhesion. 

Soft.—  Melt  Yellow  Beeswax  with  its  weight  of  Turpentine,  and  color  with  finely 
powdered  Venetian  red. 
When  cold  it  has  the  hardness  of  soap,  but  is  easily  softened  and  molded  with  the  fingers. 


MISCELLANEOUS   MIXTURES.  8/3 

Maltha,  or   Grreek    Mastic. 

Lime  and  Sand  mixed  in  manner  of  mortar,  and  made  into  a  proper  consistency 
with  milk  or  size  without  water. 

Marble. 

Plaster  of  Paris,  in  a  saturated  solution  of  Alum,  baked  in  an  oven,  and  reduced 
to  powder.  Mixed  with  water,  and  color  if  required. 

JMetal   to    Glass. 

Copal  Varnish,  15  parts;  Drying  Oil,  5;  Turpentine,  3.  Melt  in  a  water  bath  and 
add  10  of  Slaked  Lime. 

Mending   Shells,  etc. 
Gum  Arabic,  5  parts;  Rock  Candy,  2;  and  White  Lead,  enough  to  color. 

Large    Objects. 

Wollastori's  White.—  Beeswax,  i  oz. ;  Resin,  4  ozs. ;  powdered  Plaster  of  Paris,  5 
oz.  Melt  together. 

Warm  the  edges  of  the  object  and  apply  warm. 

By  means  of  this  cement  a  piece  of  wood  may  be  fastened  to  a  chuck,  which  will  hold  when  cool ;  and 
when  work  is  finished  it  may  be  removed  by  a  smart  stroke  with  tool.  Any  traces  of  cement  may  be 
removed  by  Benzine. 

Marble    Workers    and    Coppersmiths. 

White  of  egg,  mixed  with  finely-sifted  Quicklime,  will  unite  objects  which  are 
not  submitted  to  moisture. 

Porcelain. 

Add  Plaster  of  Paris  to  a  strong  solution  of  Alum  till  mixture  is  of  consistency 
of  cream. 

It  sets  readily,  and  is  suited  for  cases  in  which  large  rather  than  small  surfaces  are  to  be  united. 

Rust   Joint. 

(Quick  Setting.)—  Sal-ammoniac  in  powder,  i  Ib. ;  Flour  of  Sulphur,  2  Ibs. ;  Iron 
borings,  80  Ibs.     Made  to  a  paste  with  water. 
(Slow  Setting.)— Sal-ammoniac,  2  Ibs. ;  Sulphur,  i  Ib. ;  Iron  borings,  200  Ibs. 

The  latter  cement  is  best  if  joint  is  not  required  for  immediate  use. 

Steam    Boilers,  Steam-pipes,  etc. 

Finely  powdered  Litharge,  2  parts;  very  fine  Sand,  i;  and  Quicklime  slaked  by 
exposure  to  air,  i. 

This  mixture  may  be  kept  for  any  length  of  time  without  -'njuring.  In  using  it,  a  portion  is  mixed 
into  paste  with  linseed  oil,  boiled  or  crude.  Apply  quickly,  :-.s  it  soon  becomes  hard. 

Soft.—  Red  or  White  Lead  in  oil,  4  parts;  Iron  borings,  2  to  3  parts. 
Hard.—  Iron  borings  and  salt  water,  and  a  small  quantity  of  Sal-ammoniac  with 
fresh  water. 

Transparent— GHass. 

India-rubber,  i  part  in  64  of  chloroform ;  gum  Mastic  in  powder,  16  to  24  parts. 
Digest  for  two  days,  with  frequent  shaking. 

Or,  pulverized  Glass,  10  parts;  powdered  Fluor-spar,  20;  soluble  Silicate  of  Soda, 
60.  Both  glass  and  fluor-spar  must  be  in  finest  practicable  condition,  which  is  best 
done  by  shaking  each  in  fine  powder,  with  water,  allowing  coarser  particles  to  de- 
posit, and  then  by  pouring  off  remainder,  which  holds  finest  particles  in  suspension. 

The  mixture  must  be  made  very  rapidly,  by  quick  stirring,  and  applied  immediately. 

TJniting    Leather   and.    Mietal. 

Wash  metal  with  hot  Gelatine;  steep  leather  in  an  infusion  of  Nutgalls,  hot, 
and  bring  the  two  together. 

"Waterproof  Mastic. 

Red  Lead,  i  part;  ground  Lime,  4  parts;  sharp  Sand  and  boiled  Oil,  5  parts. 
Or,  Red  Lead,  i  part;  Whiting,  5;  and  sharp  Sand  and  boiled  Oil,  10. 

"Wood    to    Iron. 

Litharge  and  Glycerine.—  Finely  powdered  Oxide  of  Lead  (litharge)  and  Concen- 
trated Glycerine. 

The  composition  is  insoluble  in  most  acids,  is  unaffected  by  action  of  moderate  heat,  sets  rapidly, 
and  acquires  an  extraordinary  hardness. 

Turner's.—  Melt  i  Ib.  of  Resin,  and  add  .25  Ib.  of  Pitch. 

While  boiling  add  Brick  dust  to  give  required  consistency.  In  winter  it  may  be 
necessary  to  add  a  little  Tallow. 


874  MISCELLANEOUS   MIXTURES. 

GLUES. 
[Marine. 

Dissolve  India  Rubber,  4  parts,  in  34  parts  of  Coal-tar  Naphtha;  add  powdered 
Shellac,  64  parts. 

While  mixture  is  hot  pour  it  upon  metal  plates  in  sheets.    When  required  for 
use,  heat  it,  and  apply  with  a  brush. 

Or,  India  Rubber,  i  part;  Coal  Tar,  12  parts;  heat  gently,  mix,  and  add  powdered 
Shellac,  20  parts.     Cool.     When  used,  heat  to  about  250° 

Or,  Glue,  12  parts;  Water,  sufficient  to  dissolve;  add  Yellow  Resin,  3  parts;  and, 
when  melted,  add  Turpentine,  4  parts. 

Strong  Glue.  —  Add  Powdered  Chalk  to  common  Glue. 

Mix  thoroughly. 

3VEvicilage. 

Curd  of  Skim  Milk  (carefully  freed  from  Cream  or  Oil),  washed  thoroughly,  and 
dissolved  to  saturation  in  a  cold  concentrated  solution  of  Borax. 

This  mucilage  keeps  well,  and,  as  regards  adhesive  power,  far  surpasses  gum  Arabic. 

Or,  Oxide  of  Lead,  4  Ibs.  ;  Lamp-black,  2  Ibs.  ;  Sulphur,  5  ozs.  ;  and  India  Rubber 
dissolved  in  Turpentine,  10  Ibs. 

Boil  together  until  they  are  thoroughly  combined. 

Preservation  of  Mudlage.  —  A  small  quantity  of  Oil  of  Cloves  poured  into  a  bottle 
containing  Gum  Mucilage  prevents  it  from  becoming  sour. 

To   Resist   Moistixre. 

Glue,  5  parts;  Resin,  4  parts;  Red  Ochre,  2  parts;  mixed  with  least  practicable 
quantity  of  water. 

Or,  Glue,  4  parts;  Boiled  Oil,  i  part,  by  weight,  Oxide  of  Iron,  i  part. 
Or,  Glue,  i  lb.,  melted  in  2  quarts  of  skimmed  Milk. 

JParchxn  exit  . 

Parchment  Shavings,  i  lb.  ;  Water,  6  quarts. 

Boil  until  dissolved,  then  strain  and  evaporate  slowly  to  proper  consistence. 

Rice,  or   Japanese. 

Rice  Flour;  Water,  sufficient  quantity. 

Mix  together  cold,  then  boil,  stirring  it  during  the  time. 


Glue,  Water,  and  Vinegar,  each  2  parts.  Dissolve  in  a  water-bath,  then  add  Al- 
cohol, i  part. 

Or,  Cologne  or  strong  Glue,  2.2  Ibs.  ;  Water,  i  quart;  dissolve  over  a  gentle  heat; 
add  Nitric  Acid  36°,  7  ozs.  ,  in  small  quantities. 

Remove  from  over  fire,  and  cool. 

Or,  White  Glue,  16  ozs.  ;  White  Lead,  dry,  4  ozs.  ;  Rain  Water,  2  pints.  Add  Al- 
cohol, 4  ozs.,  and  continue  heat  for  a  few  minutes. 

Elastic    and    Sweet.  —Stamps    or    Rolls. 

Elastic.—  Dissolve  good  Glue  in  water  by  a  water-bath.  Evaporate  to  a  thick  con- 
sistence, and  add  equal  weight  of  Glycerine  to  Glue;  submit  to  heat  until  all  water 
is  evaporated,  and  pour  into  molds  or  on  plates. 

Sweet.—  Substitute  Sugar  for  the  Glycerine. 

To   Adhere    Engravings   or   Lithographs   upon   Wood. 

Sandarach,  250  parts;  Mastic  in  tears,  64  parts;  Resin,  125  parts;  Venice  Tur- 
pentine, 250  parts;  and  Alcohol,  1000  parts  by  measure. 

BROWNING,  OR  BRONZING,  LIQUID. 
Sulphate  of  Copper,  i  oz.  ;  Sweet  Spirit  of  Nitre,  i  oz.  ;  Water,  i  pint 

Mix.    Let  stand  a  few  days  before  use. 


MISCELLANEOUS    MIXTURES.  8/5 

Q-tm.   Barrels. 

Tincture  of  Muriate  of  Iron,  i  oz. ;  Nitric  Ether,  i  oz. ;  Sulphate  of  Copper,  4 
scruples ;  rain  water,  i  pint.  If  the  process  is  to  be  hurried,  add  2  or  3  grains  of 
Oxy muriate  of  Mercury. 

When  barrel  is  finished,  let  it  remain  a  short  time  in  lime-water,  to  neutralize  any  acid  which  may 
have  penetrated ;  then  rub  it  well  with  an  iron  wire  scratch-brush. 

After  Browning.  —Shellac,  i  oz. ;  Dragon's-blood,  .25  oz. ,  rectified  Spirit,  i  qt. 
Dissolve  and  filter. 

Or,  Nitric  Acid,  spec.  grav.  1.2;  Nitric  Ether,  Alcohol,  and  Muriate  of  Iron,  each  i 
part.  Mix,  then  add  Sulphate  of  Copper  2  parts,  dissolved  in  Water  10  parts. 

["  LACQUERS. 

Small    Arms,  or    Waterproof*  IPaper. 

Beeswax,  13  Ibs. ;  Spirits  Turpentine,  13  gallons;  Boiled  Linseed  Oil,  i  gallon. 

All  ingredients  should  be  pure  and  of  best  quality.  Heat  them  together  in  a  copper  or  earthen  ves- 
sel over  a  gentle  fire,  in  a  water-bath,  until  they  are  well  mixed. 

Bright    Iron    Work. 

Linseed  Oil,  boiled,  80.5  parts;  Litharge,  5.5  parts;  White  Lead,  in  oil,  11.25  parts; 
Resin,  pulverized,  2.75  parts. 

Add  litharge  to  oil ;  simmer  over  a  slow  fire  3  hoars ;  strain,  and  add  resin  and  white  lead ,  keep  it 
gently  warmed,  and  stir  until  resin  is  dissolved. 

Or,  Amber,  6  parts;  Turpentine,  6  parts;  Resin,  i  part;  Asphaltum,  i  -part;  and 
Drying  Oil,  3  parts;  heat  and  mix  well. 
Or,  Shellac,  i  Ib. ;  Asphaltum,  6  Ibs. ;  and  Turpentine,  i  gallon. 

Iron   and.    Steel. 

Clear  Mastic,  10  parts;  Camphor,  5  parts;  Sandarac,  15  parts;  and  Elimi  Gum, 
5  parts.  Dissolve  in  Alcohol,  filter,  and  apply  cold. 

Brass. 

Shellac,  8  ozs. ;  Sandarac,  2  ozs. ;  Annatto,  2  ozs. ;  and  Dragon's-blood  Resin,  .25 
oz. ;  and  Alcohol,  i  gallon. 

Or,  Shellac,  8  ozs. ;  and  Alcohol,  i  gallon.  Heat  article  slightly,  and  apply  lacquer 
with  a  soft  brush. 

Wood,  Iron,  or  "Walls,  and.  rendering  Cloth,  Paper,  etc., 
'Waterproof. 

Heat  120  Ibs.  Oil  Varnish  in  one  vessel,  33  Ibs.  Quicklime  in  22  Ibs.  water  in  an- 
other. Soon  as  lime  effervesces,  add  55  Ibs.  melted  India  Rubber.  Stir  mixture, 
and  pour  into  vessel  of  hot  Varnish.  Stir,  strain,  and  cool 

When  used,  thin  with  Varnish  and  apply,  preferably  hot. 

To   Clean    Soiled    Engravings. 
Ozone  Bleach,  i  part;  Water,  10;  well  mixed. 

INKS. 

Indelible,  for  Marking   Linen,  etc. 
i.— Juice  of  Sloes,  i  pint;  Gum,  .5  oz. 

This  requires  no  "  preparation  "  or  mordant,  and  is  very  dnrable. 

2  —Nitrate  of  Silver,  i  part;  Water,  6  parts;  Gum,  i  part;    Dissolve. 

3.— Lunar  Caustic,  2  parts;  Sap  Green  and  Gum  Arabic,  each  i  part;  dissolve  with 
distilled  water. 

"Preparation."— Soda,  i  oz. ;  Water,  i  pint;  Sap  Green,  .5  drachm.  Dissolve, 
and  wet  article  to  be  marked,  then  dry  and  apply  the  ink. 

Perpetual,  for  Tomb-stones,  Marble,  etc.— Pitch,  n  parts;  Lamp-black,  i  part; 
Turpentine  sufficient.  Warm  and  mix. 

Copying  Ink.— Add  i  oz.  Sugar  to  a  pint  of  ordinary  Ink. 

SOLDERING. 
Base  for   Soldering. 

Strips  of  Zinc  in  diluted  Muriatic,  Nitric,  or  Sulphuric  Acid,  until  as  much  is  de- 
composed as  acid  will  effect.  Add  Mercury,  let  it  stand  for  a  day;  pour  off  the 
Water,  and  bottle  the  Mercury. 

When  required,  rub  surface  to  be  soldered  with  a  cloth  dipped  in  the  Mercury. 


8/6  MISCELLANEOUS   MIXTURES. 

VARNISHES. 
'Waterproof. 

Flour  of  Sulphur,  i  Ib. ;  Linseed  Oil,  i  gall. ;  boil  them  until  they  are  thoroughly 
combined. 

Good  for  waterproof  textile  fabrics. 

Harness. 

India  Rubber,  .5  Ib. ;  Spirits  of  Turpentine,  i  gall. ;  dissolve  into  a  jelly;  then  mix 
hot  Linseed  Oil,  equal  parts  with  the  mass,  and  incorporate  them  well  over  a  slow  fire. 

Fastening    I-jeather   on    Top    Rollers. 

Gum  Arabic,  2.75  ozs.,  and  a  like  volume  of  Isinglass,  dissolved  in  Water. 

To    IPreserve    GHass    from    the    Su.n. 

Reduce  a  quantity  of  Gum  Tragacanth  to  fine  powder,  and  dissolve  it  for  24  hours 
in  white  of  egg  well  beat  up. 

"Water-color   Drawings. 

Canada  Balsam,  i  part;  Oil  of  Turpentine,  2  parts. 
Mix  and  size  drawing  before  applying. 

Objects   of  Natural   History,  Shells,  Fish,  etc. 

Mucilage  of  Gum  Tragacanth  and  of  Gum  Arabic,  each  i  oz. 

Mix,  and  add  spirit  with  Corrosive  Sublimate,  to  precipitate  the  more  stringy  por- 
tion of  the  Gum. 

Iron    and.    Steel. 

Mercury,  120  parts;  Tin,  10  parts;  Green  Vitriol,  20  parts;  Hydrochloric  Acid  of 
1.2  sp.  gr.,  15  parts,  and  pure  Water,  120  parts. 

Blackboards. 

Shellac  Varnish,  5  gallons;  Lamp-black,  5  ozs.;  fine  Emery,  3  ozs.;  thin  with 
Alcohol,  and  lay  in  3  coats. 

Black. 

Heat,  to  boiling,  Linseed  Oil  Varnish,  10  parts,  with  Burnt  Umber,  2  parts,  and 
powdered  Asphaltum,  i  part. 

When  cooled,  dilute  with  Spirits  of  Turpentine  as  may  be  required. 

Balloon. 

Melt  India  Rubber  in  small  pieces  with  its  weight  of  boiled  Linseed  Oil. 
Thin  with  Oil  of  Turpentine. 

Transfer. 
Alcohol,  5  ozs. ;  pure  Venice  Turpentine,  4  ozs. ;  Mastic,  i  oz. 

To   render   Canvas   "Waterproof  and   Pliable, 
Yellow  Soap,  i  Ib  ,  boiled  in  6  pints  of  Water,  add,  while  hot,  to  112  Ibs.  of  oil  Paint 

'Waterproof  Bags. 
Pitch,  8  parts,  Wax  and  Tallow,  each  i  part. 

To    Clean   Varnish. 

Mix  a  lye  of  Potash  or  Soda,  with  a  little  powdered  Chalk. 

STAINING. 
Wood   and    Ivory. 

Yellow.  —Dilute  Nitric  Acid  will  produce  it  on  wood. 

Red.—  An  infusion  of  Brazil  Wood  in  Stale  Urine,  in  the  proportion  of  i  Ib.  to  a 
gallon,  for  wood,  to  be  laid  on  when  boiling  hot,  also  Alum  water  before  it  dries. 
Or,  a  solution  of  Dragon's-blood  in  Spirits  of  Wine. 

Black.—  Strong  solution  of  Nitric  Acid. 

Blue.—  For  Ivory:  soak  it  in  a  solution  of  Verdigris  in  Nitric  Acid,  which  will  turn 
it  green;  then  dip  it  into  a  solution  of  Pearlash  boiling  hot. 

Purple. — Soak  Ivory  in  a  solution  of  Sal-ammoniac  into  four  times  its  weight  of 
Nitrous  Acid. 

Jf dhogany.—  Brazil,  Madder,  and  Logwood,  dissolved  in  water  and  put  on  hot. 


MISCELLANEOUS   MIXTURES.  8/7 

MISCELLANEOUS. 
Blacking    for    Harness. 

Beeswax,  .5  Ib. ;  Ivory  Black,  2  ozs. ;  Spirits  of  Turpentine,  i  oz. ;  Prussian  Blue 
ground  in  oil,  i  oz. ;  Copal  Varnish,  .25  oz. 

Melt  wax  and  stir  it  into  other  ingredients  before  mixture  is  quite  cold;  make  it 
into  balls.  Rub  a  little  upon  a  brush,  and  apply  it  upon  harness,  then  polish  lightly 
with  silk. 

To    Clean    Brass    Ornaments. 

Brass  ornaments  that  have  not  been  gilt  or  lackered  may  be  cleaned,  and  a  very 
brilliant  color  given  to  them,  by  washing  them  in  Alum  boiled  in  strong  Lye,  in  the 
proportion  of  an  ounce  to  a  pint,  and  afterwards  rubbing  them  with  strong  Tripoli. 

To   Harden   Drills,  Chisels,  etc. 

Temper  them  in  Mercury. 

To    Clean    Coral. 

Brush  with  equal  parts  Spirits  of  Salts  and  cold  water. 

Or,  dip  in  a  hot  solution  of  Potash  or  Chloride  of  Lime.  If  much  discolored,  let 
it  remain  in  solution  for  a  few  hours. 

Blacking,  ^vithout    Polishing. 

Molasses,  4  ozs. ;  Lamp-black,  .5  oz. ;  Yeast,  a  table-spoonful;  Eggs,  2;  Olive  Oil, 
a  teaspoonful;  Turpentine,  a  teaspoonful.     Mix  well. 
To  be  applied  with  a  sponge,  without  brushing. 

Dubbing. 
Resin,  2  Ibs. ;  Tallow,  i  Ib. ;  Train-oil,  i  gallon. 

Anti-friction    Q-rease. 

Tallow,  ioo  Ibs. ;  Palm-oil,  70  Ibs.   Boiled  together,  and  when  cooled  to  80°,  strain 
through  a  sieve,  and  mix  with  28  Ibs.  of  Soda,  and  1.5  gallons  of  Water. 
For  Winter,  take  25  Ibs.  more  oil  in  place  of  the  Tallow. 
Or,  Black  Lead,  i  part;  Lard,  4  parts. 

To   Attach,   Hair   Felt   to    Boilers. 

Red  Lead,  i  Ib. ;  White  Lead,  3  Ibs. ;  and  Whiting,  8  Ibs.  Mixed  with  boiled  Lin^ 
seed  Oil  to  consistency  of  paint. 

IPastils   for   Fumigating. 

Gum  Arabic,  2  ozs. ;  Charcoal  Powder,  5  ozs. ;  Cascarilla  Bark,  powdered,  .75  oz. ; 
Saltpetre,  .25  drachm.  Mix  together  with  water,  and  make  into  shape. 

For   "Writing    -upon    Zinc    Labels.  —Horticultural. 

Dissolve  ioo  grains  of  Chloride  of  Platinum  in  a  pint  of  water;  add  a  little  Mu- 
cilage and  Lamp-black. 

Or,  Sal-ammoniac,  i  dr. ;  Verdigris,  i  dr. ;  Lamp-black,  .5  dr. ;  Water,  10  drs.    Mix. 

To    Remove   old.    Ironmold. 

Rempisten  part  stained  with  ink,  remove  this  by  use  of  Muriatic  Acid  diluted  by 
5  or  6  times  its  weight  of  water,  when  old  and  new  stain  will  be  removed. 

To    Cut    India   Rubber. 

Keep  blade  of  knife  wet  with  water  or  a  strong  solution  of  Potash. 

Adhesive    for    Rubber    Belts. 

Coat  driving  surface  with  Boiled  Oil  or  Cold  Tallow,  and  then  apply  powdered 
Chalk. 

Liard. 

50  parts  of  finest  Rape-oil,  and  i  part  of  Caoutchouc,  cut  small.  Apply  heat  until 
it  is  nearly  all  dissolved. 

To   Preserve   Leather   Belting   or   Hose. 

Apply  warm  Castor  Oil.    For  hose,  force  it  through  it. 

To    Oil    Leather    Belting. 
Apply  a  solution  of  India  Rubber  and  Linseed  Oil. 

*£ 


8;8 


MISCELLANEOUS    MIXTURES. 


Dressing   for    Leather    Belts. 

i.— Beef  Tallow,  i  part,  and  Castor  Oil,  2  parts.     Apply  warm. 

2.— Beef  Tallow,  3  Ibs.  •,  Beeswax,  i  Ib.     Healed  and  applied  warm  to  botL  sides 

Files. 
Lay  dull  files  in  diluted  Sulphuric  Acid  until  they  are  bitten  deep  enough. 

To    Remove   Oil   from    Leather. 

Apply  Aqua-ammonia. 

To    Clean    Paint. 

Wash  with  a  solution  of  Pearlash  in  water.     If  greasy,  use  Quicklime. 
Or,  Extract  of  Litherium  diluted  with  from  200  to  300  parts  of  water. 

To   Remove    Paint. 

Mix  Soft  Soap,  2  ozs.,  and  Potash,  4  ozs.,  in  boiling  Water,  with  Quicklime,  .5  Ib. 
Apply  hot,  and  let  remain  for  i  day. 
Or,  Extract  of  Litherium,  thinly  brushed  over  the  surface  2  or  3  times. 

To    Clean    Marble. 

Chalk,  powdered,  and  Pumice-stone,  each  i  part;  Soda,  2  parts.     Mix  with  water. 
Wash  the  spots,  then  clean  and  wash  off  with  Soap  and  Water. 

Paste    for    Cleaning    Metals. 

Oxalic  Acid,  i  part;  Rottenstone,  6  parts.  Mix  with  equal  parts  of  Train  Oil  and 
Spirits  of  Turpentine. 

"Watchmaker's  Oil,  which  never  Corrodes  or  Thickens. 

Place  coils  of  thin  Sheet  Lead  in  a  bottle  with  Olive  Oil.  Expose  it  to  the  sun  for 
a  few  weeks,  and  pour  off  the  clear  oil. 

Durable   Paste. 

Make  common  Flour  paste  rather  thick  (by  mixing  some  Flour  with  a  little  cold 
water  until  it  is  of  uniform  consistency,  and  then  stir  it  well  while  boiling  water  is 
being  added  to  it);  add  a  little  Brown  Sugar  and  Corrosive  Sublimate,  which  will 
prevent  fermentation,  and  a  few  drops  of  Oil  of  Lavender,  which  will  prevent  it  be- 
coming moldy.  When  dried,  dissolve  in  water. 

It  will  keep  for  two  or  three  years  in  a  covered  vessel. 

To    Extract    Orease    from    Stone    or    Marble. 

Soft  Soap,  i  part ;  Fuller's  Earth,  2  parts ;  Potash,  i  part.    Mix  with  boiling  water. 
Lay  it  upon  the  spots,  and  let  it  stand  for  a  few  hours. 

Stains. 

To  Remove.—  Stains  of  Iodine  are  removed  by  rectified  Spirit;  Ink  stains  by  Ox- 
alic or  Superoxalate  of  Potash;  Ironmolds  by  same;  but  if  obstinate,  moisten  them 
with  Ink,  then  remove  them  in  the  usual  way. 

Red  spots  upon  black  cloth,  from  acids,  are  removed  by  Spirits  of  Hartshorn,  or 
other  solutions  of  Ammonia. 

Stains  of  Marking -ink,  or  Nitrate  of  Silver.— Wet  stain  with  fresh  solution  ol 
Chloride  of  Lime,  and,  after  10  or  15  minutes,  if  marks  have  become  white,  dip  the 
part  in  solution  of  Ammonia  or  of  Hyposulphite  of  Soda.  In  a  few  minutes  wash 
with  clean  water. 

Or,  stretch  the  stained  linen  over  a  basin  of  hot  water,  and  wet  mark  with  Tinc- 
ture of  Iodine. 

Preservative    Paste   for   Objects    of  Natural    History. 
White  Arsenic,  i  Ib. ;  Powdered  Hellebore,  2  Ibs. 

To   Preserve    Bottoms    of  Iron    Steam-boilers. 

Red  Lead,  75  parts;  Venetian  Red,  17  parts;  Whiting,  6.5  parts;  and  Litharge. 
1.5  parts  by  weight. 

To   Preserve    Sails. 

Slacked  Lime,  2  bushels.  Draw  off  the  lime-water,  and  mix  it  with  120  gallons 
water,  and  with  Blue  Vitriol,  .25  Ib. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.      8/9 

"Whitewash. 

For  outside  exposure,  slack  Lime,  .5  bushel,  iu  a  barrel;  add  common  Salt,  i  Ib. , 
Sulphate  of  Zinc,  .5  Ib. ;  and  Sweet  Milk,  i  gallon. 

To    ^Preserve    "Wood. -work. 

Boiled  Oil  and  finely  powdered  Charcoal,  each  i  part;  mix  to  the  consistence  of 
paint.  Apply  2  or  3  coats. 

This  composition  is  well  adapted  for  casks,  water-spouts,  etc. 

To   JPolish.   "Wood. 

Rub  surface  with  Pumice  Stone  and  water  until  the  rising  of  the  grain  is  removed 
Then,  with  powdered  Tripoli  and  boiled  Linseed  Oil,  polish  to  a  bright  surface. 

P*aint    for    "Window    Grlass. 

Chrome  Green,  .25  oz. ;  Sugar  of  Lead,  i  Ib. ;  ground  fine,  in  sufficient  Linseed  Oil 
to  moisten  it.  Mix  to  the  consistency  of  cream,  and  apply  with  a  soft  brush. 

The  glass  should  be  well  cleansed  before  the  paint  is  applied.  The  above  quantity  is  sufficient  for 
about  200  feet  of  glass. 

To    Alake   13 rain    Tiles    IPorous. 

Mix  sawdust  with  the  clay  before  burning. 


MISCELLANEOUS  OPERATIONS  AND   ILLUSTRATIONS. 

i. — It  is  required  to  lay  out  a  tract  of  land  in  form  of  a  square,  to  be  en- 
closed with  a  post  and  rail  fence,  5  rails  high,  and  each  rod  of  fence  to  con- 
tain 10  rails.  What  must  be  side  of  this  square  to  contain  just  as  many 
acres  as  there  are  rails  in  fence? 

OPERATION,  i  mile  =:  320  rods.  Then  320  X  320  -r- 160,  sq.  rods  in  an  acre  =  640 
acres;  and  320  X  4  sides  and  X  10  rails  =  12  800  rails  per  mile. 

Then,  as  640  acres :  12800  rails  ::  12800  acres  :  256000  rails,  which  will  enclose 
256000  acres,  and  V256  °ooX  69.5701  =  number  of  yards  in  side  of  a  sq.  acre,  and 
-r-  1760,  yards  in  a  mile  =  20  milts. 

2. — How  many  fifteens  can  be  counted  with  four  fives? 

OPERATION.     4  X  3  X  2  X  f  =  *-*  =  <. 
1X2X3          6 

3. — What  are  the  chances  in  favor  of  throwing  one  point  with  three  dice? 

OPERATION.— Assume  a  bet  to  be  upon  the  ace.  Then  there  will  be  6  X  6  x  6  =  216 
different  ways  which  the  dice  may  present  themselves,  that  is,  with  and  without  an  ace. 

Then,  if  the  ace  side  of  the  die  is  excluded,  there  will  be  5  sides  left,  and  5X5X5 
=  125  ways  without  the  ace. 

Therefore,  there  will  remain  only  216  — 125  =  91  ways  in  which  there  could  be  an 
ace.  The  chance,  then,  in  favor  of  the  ace  is  as  91  to  125 ;  that  is,  out  of  216  throws, 
the  probability  is  that  it  will  come  up  91  times,  and  lose  125  times. 

4.— The  hour  and  minute  hand  of  a  clock  are  exactly  together  at  12; 
when  are  they  next  together  ? 

OPERATION.— As  the  minute  hand  runs  n  times  faster  than  the  hour  hand,  then, 
as  ii  :  60  ::  i  :  5  win.  27^-  sec.  =  time  past  i  o'clock. 

5. — Assume  a  cube  inch  of  glass  to  weigh  1.49  ounces  troy,  the  same  of 
sea-water  .59,  and  of  brandy  .53.  A  gallon  of  this  liquor  in  a  glass  bottle, 
which  weighs  3.84  Ibs.,  is  thrown  into  sea-water.  It  is  proposed  to  deter- 
mine if  it  will  sink,  and,  if  so,  how  much  force  will  just  buoy  it  up? 

OPERATION.     3.84  X  12  -r- 1.49  =  30.92  cube  ins.  of  glass  in  bottle. 

231  cube  ins.  in  a  gallon  X  .53  =  122.43  ounces  of  brandy. 

Then,  bottle  and  brandy  weigh  3.84  X  12  +  122.43  =  168.51  ounces,  and  contair 
961.92  cube  ins.,  which  X  .59  =  154-  53  ounces,  weight  of  an  equal  bulk  of  sea-water 

And,  168.51  —154.53  =  13.98  ounces,  weight  necessary  to  support  it  in  the  water 


88O     MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 

6.— A  fountain  has  4  supply  cocks,  A,  B,  C,  and  D,  and  under  it  is  a  cis- 
tern, which  can  be  tilled  by  the  cock  A  in  6  hours,  by  B  in  8  hours,  by  C  in 
10  hours,  and  by  D  in  12  hours ;  now,  the  cistern  has  4  holes,  designated  E, 
F,  G,  and  H,  ami  it  can  be  emptied  through  E  in  6  hours,  F  in  5  hours,  G  in 
4  hours,  and  H  in  3  hours.  Suppose  the  cistern  to  be  full  of  water,  and  that 
all  the  cocks  and  holes  were  opened  together,  in  what  time  would  the  cistern 
be  emptied? 

OPERATION. — Assume  the  cistern  to  hold  120  gallons. 


bra.     gall.     hrs.    gall. 


If  6  :  120 

8  :  120 

xo  :  120 

12  :  120 


20  at  A. 
15  at  B. 
12  at  G. 
10  at  D. 


Run  in  in  i  hour,  57  gallons. 


hrs.     gall.     hrs.    gall. 


If  6 

5 
4 
3 


20  at  E. 
24  at  F. 
30  at  G. 
40  at  H. 


Run  out  in  i  hour,  114  gallons. 


57 

Run  out  in  i  hour  more  than  run  in,  57  gallons. 
Then,  as  57  gallons  :  i  hour  ::  120  gallons  :  2.105-!-  hours. 

7.— A  cistern,  containing  60  gallons  of  water,  has  3  cocks  for  discharging 
it ;  one  will  empty  it  in  i  hour,  a  second  in  2  hours,  and  a  third  in  3  hours ; 
in  what  time  wilHt  be  emptied  if  they  are  all  opened  together? 

OPERATION.— ist,  .5  would  run  out  in  i  hour  by  the  2d  cock,  and  .333  by  the  3d-, 
consequently,  by  the  3  would  the  reservoir  be  emptied  in  i  hour.  .5  -|-  .333  + 1  = 
^  -f- 1^  -f-  ^,  being  reduced  to  a  common  denominator,  the  sum  of  these  3  =  ^- ;  whence 
the  proportion,  1 1  :  60  : :  6  :  32T8T  minutes. 

8.— A  reservoir  has  2  cocks,  through  which  it  is  supplied ;  by  one  of  them 
it  will  fill  in  40  minutes,  and  by  the  other  in  50  minutes ;  it  has  also  a  dis- 
charging cock,  by  which,  when  full,  it  may  be  emptied  in  25  minutes.  If 
the  3  cocks  are  left  open,  in  what  time  would  the  cistern  be  filled,  assuming 
the  velocity  of  the  water  to  be  uniform  ? 

OPERATION. — The  least  common  multiple  of  40,  50,  and  25,  is  200. 

Then,  the  ist  cock  will  fill  it  5  times  in  200  minutes,  and  the  2d,  4  times  in  200 
minutes,  or  both,  9  times  in  200  minutes  ;  and,  as  the  discharge  cock  will  empty  it 
8  times  in  200  minutes,  hence  9  —  8  =  i,  or  once  in  200  minutes  =  3. 2  hours. 

9. — The  time  of  the  day  is  between  4  and  5,  and  the  hour  and  minute 
hands  are  exactly  together ;  what  is  the  time? 
OPERATION.—  Difference  of  speed  of  the  hands  is  as  i  to  12  =  11. 
4  hours  X  60  =  240,  which  -4-11  =  21  min.  49.09  sec.,  which  is  to  be  added  to  4  hours. 

10. — Out  of  a  pipe  of  wine  containing  84  gallons,  10  were  drawn  off,  and 
the  vessel  refilled  with  water,  after  which  10  gallons  of  the  mixture  were 
drawn  off,  and  then  10  more  of  water  were  poured  in,  and  so  on  for  a  third 
and  fourth  time.  It  is  required  to  compute  how  much  pure  wine  remained 
in  the  vessel,  supposing  the  two  fluids  to  have  been  thoroughly  mixed. 

OPERATION.     84  — 10  •=  74,  quantity  after  the  ist  draught. 

Then,  84  :  10  ::  74  :  8.8095,  and  74  —  8.8095  =  65.1905,  quantity  after  zd  draught. 

84 : 10 : :  65. 1905 : 7. 7608,  and  65. 1905  —  7. 7608  =  57. 4297,  quantity  after  yd  draught. 

84: 10 ::  57. 4297 -.6.8367,  and  57.4297—6.8367  =  50.593,  quantity  after  $th  draught, 
=.  result  required. 

ii. — A  reservoir  having  a  capacity  of  10000  cube  feet,  has  an  influx  of 
750  and  a  discharge  of  1000  cube  feet  per  day.  In  what  time  will  it  be 

emptied?  10000 

OPERATION.     -          =  4o  days. 

1000  —  750 

Contrariwise :  The  discharge  being  1000  and  the  influx  1250  cube  feet  per  houc 
In  what  time  will  it  be  filled? 

OPERATION.     =  40  hours  =  z  day  16  hours. 

1250  — 1000 


MISCELLANEOUS  OPEBATIONS  AND  ILLUSTRATIONS.      88 1 

12. — A  son  asked  his  father  how  old  he  was.  His  father  answered  him 
thus :  If  you  take  away  5  from  my  years,  and  divide  the  remainder  by  8, 
the  quotient  will  be  one  third  of  your  age ;  but  if  you  add  2  to  your  age,  and 
multiply  the  whole  by  3,  and  then  subtract  7  from  the  product,  you  will  have 
)the  number  of  years  of  my  age.  What  were  the  ages  of  father  and  son? 

OPERATION. — Assume  father's  age  37. 

Then  37  —  5  =  32,  and  32 -f-  8  =  4,  and  4  X  3  =  12,  son's  age.  Again:  12  -f  2  =  14, 
and  14  X  3  =  42,  and  42  —  7  =  35.  Therefore  37  —  35  =  2,  error  too  little. 

Again :  Assume  father's  age  45 ;  then  45  —  5  =  40,  and  40  -5-  8  =  5.  Therefore 
5  x  3  =  15,  son's  age.  Again:  15  +  2  =  17,  and  17  X  3  =  51,  and  51 — 7  =  44.  There- 
fore 45  —  44  =  i,  error  too  little. 

Hence  (45  sup.  X  2  error)  —  (37  sup.  x  i  error)  =  90  —  37  =  53,  and  2  —  i  =  i. 

Consequently,  53  is  father's  age.  Then  53  —  5  =  48,  and  48  -j-  8  =  6  = .  333  of  son's 
age,  and  6  x  3  =  18  years,  son's  age. 

13. — Two  companions  have  a  parcel  of  guineas.  Said  A  to  B,  if  you  will 
give  me  one  of  your  guineas  I  shall  have  as  many  as  you  have  left.  B  re- 
plied, if  you  will  give  me  one  of  your  guineas  I  shall  liave  twice  as  many  as 
you  will 'have  left.  How  many  guineas  had  each  of  them  ? 

OPERATION.— Assume  B  had  6. 

Then  A  would  have  had  4,  for  6  —  i  =  4  -f- 1  =  5-  Again :  4  ( A's  parcel)  —  1=3, 
and  6  -f- 1  =  7,  and  9X2  =  6.  Therefore  7  —  6  =  1,  error  too  little. 

Again:  Assume  B  had  8. 

Then  A  would  have  6,  for  8  —  i  =  6  +  i  =  7.  Again :  6  (A's  parcel)  —  1  =  5,  and 
8  -f- 1  =  9>  aQd  5  X  2  =  10.  Therefore  10  —  9=1,  error  too  great. 

Hence  8  X  i  =  8,  and  6  X  i  =  6.  Then  8  -f  6  =  14,  and  i  +  i  =  2.  Whence,  di- 
viding products  "by  sum  of  errors,  14 -f- 2  =  7  =  B's  parcel,  and  7  —  i  =  5  -J- 1  =  6 
for  A  when  he  had  received  i  ofB ;  also  5  —  1  X2  =  7  +  i=8  =  B's  parcel  when  he 
had  received  i  of  A. 

14. — If  a  traveller  leaves  New  York  at  8  o'clock  in  the  morning,  and  walks 
towards  New  London  at  the  rate  of  3  miles  per  hour,  without  intermission; 
and  another  traveller  starts  from  New  London  at  4  o'clock  in  the  'vening, 
and  walks  towards  New  York  at  the  rate  of  4  miles  per  hour  continuously; 
assuming  distance  between  the  two  cities  to  be  130  miles,  whereabouts  upon 
the  road  will  they  meet  ? 

OPERATION.  —From  8  to  4  o'clock  is  8  hours;  therefore,  8  X  3  =  24  miles,  per- 
formed by  A  before  B  set  out  from  New  London;  and,  consequently,  130  —  24  =  106 
are  the  miles  to  be  travelled  between  them  after  that. 

Hence,  as  (3  -f  4)  7  :  3  : :  106  •.  ^fi  =  45^  more  miles  travelled  by  A.  at  the  meeting; 
consequently,  Z4  -f-  45y  =  69^  miles  from  New  York  is  place  of  their  meeting. 

15. — If  from  a  cask  of  wine  a  tenth  part  is  drawn  out  and  then  it  is  filled 
with  water ;  after  which  a  tenth  part  of  the  mixture  is  drawn  out ;  again 
is  filled,  and  again  a  tenth  part  of  the  mixture  is  drawn  out :  now,  assume 
the  fluids  to  mix  uniformly  at  each  time  the  cask  is  replenished,  what  frac- 
tional part  of  wine  will  remain  after  the  process  of  drawing  out  and  replen- 
ishing has  been  repeated  four  times  ? 

OPERATION. — Since  .1  of  the  wine  is  drawn  out  at  first  drawing,  there  must  remain 
g.  After  cask  is  filled  with  water,  .1  of  whole  being  drawn  out,  there  will  remain 
9  of  mixture;  but  .9  ofthi*  mixture  is  wine;  therefore,  after  second  drawing,  there 

o2 
will  remain  .gof.g  of  wine,  or  -fra  ;  and  after  third  drawing,  there  will  remain  .9 

Q* 

°f -9  °f -9  of  wine,  or  -2-j. 

Hence,  the  part  of  wine  remaining  is  expressed  by  the  ratio  .9,  raised  to  a  powef 
txponent  of  which  is  number  of  times  cask  has  been  drawn  from. 

Therefore,  fractional  part  of  wine  is  ^  =  .6561. 
1°E* 


882     MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 

16. — Tnere  is  a  fish,  the  head  of  which  is  9  ins.  long,  the  tail  as  long  as 
the  head  and  half  the  body,  and  the  body  as  long  as  both  the  head  and  tail. 
Required  the  length  of  the  fish. 

OPERATION.— Assume  body  to  be  24  ins.  in  length.  Then  24-4-24-9  =  21,  Length 
of  tail. 

Hence  21  +  9  =  30,  length  of  body,  which  is  6  ins.  too  great. 

Again :  assume  the  body  to  be  26  ins.  in  length.  Then  26  -5-  2  -f-  9  =  22,  length  of 
tail.  Hence  22  +  9  =  31,  length  of  body,  which  is  5  ins.  too  great. 

Therefore,  by  Double  Position,  divide  difference  of  products  (see  rule,  page  99) 
by  difference  of  errors  (the  errors  being  alike),  26  X  6  —  24  X  5  =  36  =  difference  of 
products,  and  6  —  5  =  1=  difference  of  errors. 

Consequently,  36  -r- 1  =  36,  length  of  body,  and  36  -r-  2  +  9  =  27,  length  of  tail,  and 
36  +  27-1-9  =  72  ins. ,  length  required. 

17. — A  hare,  50  leaps  before  a  greyhound,  takes  4  leaps  to  the  greyhound's 
3,  but  2  leaps  of  the  hound  are  equal  to  3  of  the  hare's.  How  many  leaps 
must  the  greyhound  take  before  he  can  catch  the  hare  ? 

OPERATION. — As  2  leaps  of  the  greyhound  equal  3  of  the  hare,  it  follows  that  6  of 
the  greyhound  equal  9  of  the  hare. 

While  the  greyhound  takes  6  leaps,  the  hare  takes  8;  therefore,  while  the  hare 
takes  8,  the  greyhound  gains  upon  her  i. 

Hence,  to  gain  50  leaps,  she  must  take  50  X  8  =  400  leaps ;  but,  while  hare  takes 
400  leaps,  greyhound  takes  300,  since  number  of  leaps  taken  by  them  are  as  4  to  3. 

1 8. — If  a  basket  and  1000  eggs  were  laid  in  a  right  line  6  feet  apart,  and 
10  men  (designated  from  A  to  J)  were  to  start  from  basket  and  to  run  alter- 
nately, collect  the  eggs  singly,  and  place  them  in  basket  as  collected,  and 
each  man  to  collect  but  10  eggs  in  his  turn,  how  many  yards  would  each 
man  run  over,  and  what  would  be  entire  distance  run  over  ? 

OPERATION.  —  A's  course  would  be  6  X  2  feet  (first  term)  -f- 10  X  6  X  2  feet  (last 
term)  =  132  =  sum,  of  first  and  last  terms  of  progression. 

Then  132-7-2  X  10  =  660  feet  =  number  of  times  x  half  sum  of  extremes  =  sum  of 
all  the  terms,  or  the  distance  run  by  A  in  his  first  turn. 

B's  course  would  be  u  X6x  2  =  132/6^  (first  term)  +  20  X  6  X  2  =  240  feet  (last 
term)  =  372  =  sum  of  first  and  last  terms. 

Then  372  -=-  2  X  10  =  1860  =  sum  of  all  the  times,  or  B's  first  turn. 

A's  last  course  would  be  901  X  6  X  2  =  10812  feet  for  the  first  term,  and  910X6X2 
=  10920  feet  for  the  last  term  of  his  last  turn. 

Then  10  812  -f- 10  920  -f-  2  X  10  =  108  660  —  sum  of  the  terms,  or  distance  run. 


B's  last  course  would  be  911  x  6  X  2  =  10932  feet  for  the  first  term,  and  920X6X2 
=  11 040  feet  for  the  last  term  of  his  last  turn. 

Then  10  032  -f-  n  040-7-2  X  10  =  109  860  =  sum  of  the  terms  or  distance  run. 

Therefore,  if  A'S  first  and  last  runs  =  660  and  108  660  feet,  and  the  number  of 
terms  10,  then,  by  Progression,  the  sum  of  all  the  terms  =  546600/6^. 

And  if  B's  first  and  last  runs=:  1860  and  109860  feet,  and  the  number  of  terms  10, 
then  the  sum  of  all  the  terms  =  5586oo/ee<. 

Consequently,  558  600  —  546  600  =  12  ooo  =  common  difference  of  runs,  which,  be- 
ing added  to  each  man's  run  =  sum  of  all  runs,  or  entire  distance  run  over. 


A's  run,  546  600  =  182  200  yds. 
B's    "    558600=186200    " 
C's    "    570600  =  190200    " 
D's    "    582600  =  194200    u 
E's    "    594600=198200    " 


F's  run,  606  600  =  202  200  yds. 
G's    "    618600  =  206200    " 
H's    "    630600  =  210200    " 
I's     "    642600  =  214200    " 
J's     "    654600  =  218200    " 


6oo6ooo./ee£,  which -7-5280  =  1137.5  miles. 

19. — If,  in  a  pair  of  scales,  a  body  weighs  90  Ibs.  in  one  scale,  and  but  40 
Ibs.  in  the  other,  what  is  the  true  weight  ? 

-/(4Q  X  90)  =  60  IbS. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.       883 

20.  —  If  a  steamboat,  running  uniformly  at  the  rate  of  15  miles  per  hour 
through  the  water,  were  to  run  for  i  hour  with  a  current  of  5  miles  per  hour, 
then  to  return  against  that  current,  what  length  of  time  would  she  require 
to  reach  the  place  from  whence  she  started  ? 

OPERATION.     15  -f-  5  =  20  miles,  the  distance  run  during  the  hour. 

Then  15  —  5  =  10  miles  is  her  effective  velocity  per  hour  when  returning,  and 
20-r-  10  =  2  hours,  the  time  of  returning,  and  2  +  i  =  3  hours,  or  the  whole  time  oc- 
cupied. 

Or,  Let  d  represent  distance  in  one  direction,  t  and  tf  greater  and  less  times  of  run- 
ning in  hours,  and  c  current  or  tide. 


Then,  -^——velocity  of  boat  through  the  water,  and  V 

t  X  £  t 

2i.  —  Flood-tide  wave  in  a  given  river  runs  20  miles  per  hour,  current  of 
it  is  3  miles  per  hour.  Assume  the  air  to  be  quiescent,  and  a  floating  body 
set  free  at  commencement  of  flow  of  the  tide  ;  how  long  will  it  drift  in  one 
direction,  the  tide  flowing  for  6  hours  from  each  point  of  river  ? 

OPERATION.  —  Let  x  be  the  time  required;  202;  =  distance  the  tide  has  run  up,  to- 
gether with  the  distance  which  the  floating  body  has  moved;  3*  =  whole  distance 
which  the  body  has  floated, 

Then  20  x  —  3  x  =  6  X  20,  or  the  length  in  miles  of  a  tide. 


-  X  6  =  7  hours,  3  minutes,  31.765  seconds. 


20  —  3 

22. — A  steamboat,  running  at  the  rate  of  10  miles  per  hour  through  the 
water,  descends  a  river,  the  velocity  of  which  is  4  miles  per  hour,  and  re- 
turns in  10  hours  ;  how  far  did  she  proceed  ? 

OPERATION.— Let  x  =  distance  required,  - — —  =  time  of  going, =  time  of 

104-4  10  —  4 

returning.    Then,  —  -f-  —  =  10 ;  6x  -f  1 4*  =  840 ;  20*  =  840 ;  840  -r-  20  =  42  miles. 
14        6 

23. — From  CaldwelTs  to  Newburgh  (Hudson  River)  is  18  miles ;  the  cur- 
rent of  the  river  is  such  as  to  accelerate  a  boat  descending,  or  retard  one 
ascending,  1.5  miles  per  hour.  Suppose  two  boats,  running  uniformly  at  the 
rate  of  15  miles  per  hour  through  the  water,  were  to  start  one  from  each 
place  at  the  same  time,  where  will  they  meet  ? 

OPERATION.— Let  2;  =  the  distance  from  N.  to  the  place  of  meeting;  its  distance 
from  C.,  then,  will  be  18  —  x. 

Speed  of  descending  boat,  15  -f- 1. 5  =  16. 5  miles  per  hour  ;  of  ascending  boat,  15  — 

1.5  =  13.5  miles  per  hour.   — —  =  time,  of  boat  descending  to  point  of  meeting.    

10.5  13.5 

=  time  of  boat  ascending  to  point  of  meeting. 

These  times  are  of  course  equal;  therefore,  — — = .    Then,  13. 53  =  297  — 

io-5        *3-5 
16. 53,  and  13. 53  -f- 16. 53  =  297,  or  30*  =  297. 

Hence  *  =  —  =  9.9  miles,  the  distance  from  Newburgh. 
3° 

24. — There  is  an  island  73  miles  in  circumference ;  3  men  start  together 
to  walk  around  it  and  in  the  same  direction :  A  walks  5  miles  per  hour,  B  8, 
and  C  10 ;  when  will  they  all  come  aside  of  each  other  again  ? 

OPERATION.— It  is  evident  that  A  and  C  will  be  together  every  round  gone  by  A ; 
hence  it  remains  to  ascertain  when  A  and  B  will  be  in  conjunction  at  an  even  round, 
as  3  miles  are  gained  every  day  by  B.  Therefore,  as  3  :  i  ::  73  :  24.33-!-;  ^Ut5  as 
the  conjunction  is  a  fractional  number,  it  is  necessary  to  ascertain  what  number  of 
*  multiplier  will  make  the  division  a  whole  number. 

73  -f-  24. 33-J-  =  3,  the  number  of  days  required  in  which  A  will  go  round  5  time^ 
B  8,  and  C  10  times. 


884       MISCELLANEOUS  OPEKATIONS  AND  ILLUSTRATIONS. 


25. — Assume  a  cow,  at  age  of  2  years,  to  bring  forth  a  cow-calf,  and  then 
to  continue  yearly  to  do  the  samej  and  every  one  of  her  produce  to  bring 
forth  a  cow-calf  at  age  of  2  years,  and  yearly  afterward  in  like  manner ; 
how  many  would  spring  from  the  cow  and  her  produce  in  40  years  ? 

OPERATION.— The  increase  in  ist  year  would  be  o,  in  2d  year  i,  in  3d  i,  in  4th  2, 
in  $th  3,  in  6th  5,  and  so  on  to  40  years  or  terms,  each  term  being  =  sum  of  the  two 
preceding  ones.  The  last  term,  then,  will  be  165580141,  from  which  is  to  be  sub- 
tracted i  for  the  parent  cow,  and  the  remainder,  165  580140,  will  represent  increase 
required. 

26. — The  interior  dimensions  of  a  box  are  required  to  be  in  the  propor- 
tions of  2,  3,  and  5,  and  to  contain  a  volume  of  1000  cube  ins. ;  what  should 
be  the  dimensions  V 


And  what  for  a  box  of  one  half  the  volume,  or  500  cube  ins.,  and  retaining 
same  proportionate  dimensions  ? 

OPERATION. — 2  X  3  X  5  =  30,  and  —  =  15. 


Then, 


7. 
V 


5X6.43° 


=       6      and 


3  A5*2fi!  = 

V        3° 


3o  V        30 

27. — The  chances  of  events  or  games  being  equal,  what  are  the  odds  for 
or  against  the  following  results  ? 


Ive   Events. 


rcmr   Events. 


Odds. 

Against. 

In  favor. 

Odds. 

Against. 

In  favor. 

31       to  i 
4-33  to  i 
5  to  3  in  fa 
ing  3  and  2. 

Tl 

Odds. 

All  the  5 
4  out  of  5 
ivor  of  the  5  e 

tree   Ever 

Against. 

i  out  of  5 
2  out  of  5 
vents  result- 
its. 

In  favor. 

15     to  i 

2.2  tO  I 

5  to  3  ag{ 
the  4  events 

T 

Odds. 

All  the  4 
3  out  of  4 
linst  2  events 
do  not  result 

•wo    Even 

Against. 

i  out  of  4 
2  out  of  4 
only,  or  that 
2  and  2. 

ts. 
In  favor. 

7  to  i 
Even 

3  to  i  in  ft 
ing  2  and  i. 

All  the  3 
(2  or  all  out 
\   of  3 
ivor  of  the  3  e 

i  out  of  3 
(  2  or  all  out 
{   of  3 
vents  result- 

3  to  i 
Even 
Even  that 

Both  events 
{i  only  out 
Of  2 

the  events  re 

I  OUt  Of  2 

{i  only  out 
of  2 
suit  i  and  i. 

28. — Required  the  chances  or  probabilities  in  events  or  games,  when  the 
chances  or  probabilities  of  the  results,  or  the  players,  are  equal. 


Events 
or 
Games. 

That  a 
named  event 
occurs  a 
majority  or 
more  of 
times. 

Against  a 
named  event 
occurring 
an  exact 
majority  of 
times. 

Against  each 
event  occur- 
ring an  equal 
number 
of  times. 

Events 
or 
Games. 

That  a 
named  event 
occurs  a 
majority  or 
more  of 
times. 

Against  a 
named  event 
occurring 
an  exact 
majority  of 
times. 

Against  each 
event  occur- 
ring an  equal 
number  of 
times. 

21 

Even 

5  to  i 



II 

Even 

3-4  to  i 



90 
19 

1.33  to  i 
Even 

4-5  to  i 

4.66  tO  i 

10 

9 

1.7  to  i 
Even 

3  to  i 

3.06  to  i 

ii 

1.55  to  i 

4.4  tO  I 

8 

1-75  tO  I 

— 

2.66  to  i 

17 

Even 

4.4  to  i 

— 

7 

Even 

2.7  to  i 

— 

16 

1.5  to  i 

4.  1  tO  I 

6 

2  tO  I 

— 

2.2  tO  I 

15 

Even 

4  to  i 

— 

5 

Even 

2.2  tO  I 

__ 

14 

1.5  to  i 

— 

3.8  to  i 

4 

2.2  tO  I 



1.66  to  i 

»3 

Even 

3-7  to  i 

— 

3 

Even 

1.66  to  i 

__ 

12 

1.6  to  i 

3-44  to  i 

2 

3  to  i 

— 

Even. 

29. — The  chances  of  consecutive  events  or  results  are  as  follows  : 
&x. — 2047 to  i.  I  10. — 1023  to  i.  I  9. — 511  to  i.  I  8. — 255  to  i.  |  7. — 127  to  i.  |  6. — 63  to  i. 

Hence  it  will  be  observed  that  the  chances  increase  with  the  number  of  events 
very  nearly  in  a  duplicate  ratio. 

ILLUSTRATION.— The  chances  of  n  consecutive  events  compared  with  10,  are  as 
2047  to  1023,  or  2  to  i. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.    885 

30. — Required  the  chances  or  probabilities  of  events  or  results  in  a  given 
number  of  times. 

The  numerator  of  a  fraction  expresses  the  chance  or  probability  either  for  the  re- 
sult or  event  to  occur  or  fail,  and  the  denominator  all  the  chances  or  probabilities 
both  for  it  to  occur  or  fail. 

Thus,  in  a  given  number  of  events  or  games,  if  the  chances  are  even,  the  proba- 
bility of  any  particular  result  is  as  — ^—  =  —  ;  — —  ;  — —  ,  etc.,  being  i  out  of 

i  +  i      2      2-1-2'   3  +  3' 
2,  2  out  of  4,  etc. ,  or  even. 

If  the  number  of  events  or  games  are  3,  then  the  probability  of  any  par- 
ticular result,  as  2  and  i,  or  i  and  2,  is  determined  as  follows : 

Number  of  permutations  of  3  events  are  i  x  2  X  3  =  6,  which  represents  number 
of  times  that  number  of  events  can  occur,  2  and  i,  or  i  and  2,  to  which  is  to  be 
added  the  2  times  or  chances  they  can  occur  all  in  one  way  or  the  reverse  thereto. 

Hence,  — —  =  —  =  — —  =  —  ,  or  3  to  i  in  favor  of  result;  and  probability  of 

'2+6       44—3        i 

one  party  naming  or  winning  two  precise  events  or  results,  as  winning  2  out  of  3, 
is  determined  as  follows:  Number  of  permutations  and  chances,  as  before  shown, 

are  8.    Hence,  number  of  his  chances  being  3,  — f—  =  -|  =  —2—  =  — ,  or  3  to  5  in 

3  +  5       8        8  —  3       5 

favor  of  result;  and  probability  of  one  party  naming  or  winning  all,  or  3  events 
or  results,  is  determined  as  follows:  Number  of  permutations  and  chances  being 
also,  as  before  shown,  8.  Hence,  as  there  is  but  one  chance  of  such  a  result, 

— L_  =  —  —  _±_  =  — ,  or  i  to  7  in  favor  of  result 
i  +  7       8      8-1       7' 

If  number  of  events,  etc.,  are  4,  then  probability  of  any  particular  result, 
as  2  and  2,  or  of  winning  2  or  more  of  them,  is  determined  as  follows : 
Number  of  permutations  and  chances  of  4  events  are  16.    Hence,  as  number  of 

chances  of  such  a  result  are  n,     "     =  ^  =  -^ —  =  — ,  or  as  n  to  5  in  favor 

'  5  +  n      16      16  — ii        5 
of  the  result,  and  that  the  results  do  not  occur  precisely  2  and  2.    The  number  of 

chances  of  such  a  result  being  10,  .  '°    =  -|-  =  -^—  =  — ,  or  5  to  3  against  it. 
o-j-io        o        o  —  5        3 

If  number  of  events,  etc.,  are  5,  then  probability  of  any  particular  result, 
as  3  and  2,  is  determined  as  follows : 
Number  of  permutations  and  chances  being  32,  and  number  of  chances  of  such 

a  result  being  20,  — ^—  =  ^  =  -^—  =  ~  =  —  ,  or  as  5  to  3  in  favor  of  the 

'12  +  20      16      16  — 10      6        3 
result;  and  that  it  may  occur  precisely  3  out  of  5,  the  number  of  chances  are 


-  —  — 5—;  =  — ,  or  ii  to  5  against  it 


ic +  22      32      16       16  —  5" 

31. — What  is  the  dilatation  of  the  iron  in  a  railway  track  per  mile,  be- 
tween the  temperatures  of  —20°  and  +130°? 

OPERATION. 20°  + 130°  =  150°.  The  dilatation  of  wrought  iron  (as  per  table, 

page  519)  is,  from  32°  to  212°  =  180°  =  .001 257  5  times  its  length. 

Hence,  as  180  :  150  : :  .001 257  5  :  .001 047  9  =  —  °479  of  5280  (feet  in  a  mile)  = 
5-53  feet  Per  mile- 

32. — A  steamer  having  an  immersed  amidship  section  of  125  sq.  feet,  has 
a  speed  of  15  miles  per  hour  with  300  H?.  What  power  would  be  required 
for  one  of  like  model,  having  a  section  of  150  sq.  feet  for  a  speed  of  20  miles? 

As  power  required  for  like  models  is  as  cube  of  speeds. 

Then  —  =  1.2  relative  sections,  and        —  ooo  _         relative  powers. 

125  i5  3  =  3375 

Hence,  i  :  1.2  ::  2.37  :  2.844  times  IP. 


886  MARINE    STEAMEKS    AND    ENGINES. 

*     ife 


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MARINE    STEAMERS    AND    ENGINES. 


887 


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£  :  :  :  B  §  -g  : 

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. 


888 


MARINE,  RIVER, 


^Passenger 
Compound    ami 

Lengths  and  Hull,  in  feet  and  tenths;  Draught,  Propeller,  and  Side  Wheels 
Surfaces,  in  sq.feet;  Weights  and  Displacements,  in  Tons  0/2240  Ibs.; 
Speed,    in.    Knots    per    Hour. 


DIMENSIONS 

AND 

CAPACITIES. 

City  of 
Paris 
and 
New 
York. 

Columbia. 

Nairn- 
Bhire. 

Brerner- 
haven. 

Tyne- 
sicler. 

Simon 
Durnois 
and  Ma- 
nagua. 

Eleciric 
end 
Frolic. 

El  Sol. 

Service  

1. 

Steel. 
P  ami  F 

2. 

Steel. 
P  and  F 

3. 

Steel. 
Refrig'tor 

4. 

Steel. 

5. 

Iron. 
Fand  P 

6. 

Steel. 
Fruit 

7. 

Iron. 

8. 

Iron. 
F  and  P 

Length  on  deck  .  .  . 
"      bet.  perp'rs 
"      tonnage  .  .  . 
Beam,  do  
Hold,  do  

527.6 
525 
527.6 
63-2 
22 

474 
463 
463-5 
55-6 
35.8 

350.5 
350 
350.6 
47-7 

24.  2 

350 
340 
339-  6 
42.6 

27.3 

260.5 
260 
260 
33-7 

15.  2 

184 
174.8 
175-2 
27.8 
19 

111.9 
106.9 
107.4 
20.5 
!!•  5 

388.75 
377-2 
375 
48 
24 

Decks  

2 

2 

2 

2 

Tons  .  .      ..I''*' 

558l 

3737 

2428 

2179 

692 

5M 

7o9 

3021 

\.... 
Draught,  load  
Displacement  do.  .  . 
Imm'd  Sec'n  at  do. 

10499 
25 

7363 
24 

10000 
14..  7 

3720 
24-3 

7880 
1058 

3393 
21.3 

6600 
870 

e 

1290 
I74 

2560 
530 

7I7 

16.2 

1462 
404 
A   e 

183 

"•5 

230 
135 

21 

6760 

934 

TO 

Cylinders,  IP  
"         Int  
"         L.P..... 
Stroke  of  piston... 
Steam  pressure.  .  .  . 
Revolutions      .... 

2  Of  45 

2"    7I 

i"xx3 

60 
150 

86  < 

20f4I 

2"  66 

2  "  IOI 

66 
150 

27 

44 

I 

1  60 

25 
40 

66 

£ 

160 

•62 

28.5 
46 

75 
42 
160 

UJ 

26 
40 

£ 

12.75 
20 
32 

22 
ISO 

32 

52 
84 

& 

nf\ 

Q 

3 

Grate  surface  

1293 
5O  625 

I22O 

209 

606  o 

i54 
4800 

2?6 

6618 

63 

29 
800 

I44r706 

Condensing  do  
Propeller,  diam.  .  .  . 
Pitch  

33000 

2  Of  l8 
32 

3032 

16.5 

t8  5 

2383 
17.6 

17  6 

345o 
16 

10 

39° 
7-9 

8  6 

6400 

18 

Side  wheels,  diam.. 
Breadth  

- 

- 

Coal,  weight  
Consumption  
Combustion  
Cargo             .  .   .  . 

3300 

Blast 

Natural 

1266 
3000 
Natural 

821 

Natural 

130 
2400 

Natural 
1080 

H3 
IOO5 
Natural 

Natural 

IOOO 

5000 

NandB 

Passengers  

I372 

1006 

* 

1180 

4000 

Crew   

48 

IIP  

IQ  17^ 

13  ooo 

- 

388 

20 

20 

^ 

• 

Bte  

Bark  'ne 

7-m  Sch'r 

Brier 

•j-m  Rrh'r 

d-tn  Sr.h 

firh'r 

r»nTifii7 

A  -m  SMi 

Remarks. No.  1.  J.  &  G.  Thomson,  Glasgow,  Scotland;  Area  of  Immersed 

Horizontal  Section  at  Load-line,  16500  nfeet  =  coefficient  .5. No.  2.  Laird 

Bros.,  Birkenhead,  Eng. No.  3.    R.  &  W.  Hawthorn,  Leslie  &  Co.,  New- 


castle, Eng. ;   Hull  2375  tons,  Engines  180,  and  Boilers  156.- 


-No.  4.  Russell 


&  Co.  and  G.  Stewart  &  Co.,  Greenock,  Eng. ;   Hull  1950  tons.  Engines  and  Boilers 

33o. No.  5.  Tyne  Steam  Shipping  Co.,  Newcastle,  England  ;   Engines  135 

tons,  Boilers  220,  and  Water  60,  "Well  deck." No.  6.  Grangemouth  Dock- 
yard Co.  &  Hudson  &  Corbett,  Glasgow,  Scotland;  Hull  391  tons,  Engines  52,  Boilers 

40,  and  Water  27;  Area  of  Load-line  3850  Gfeet,  and  of  Sails  2310. No.  7. 

Earle's  Co.,  Hull,Eng. No.  8.  The  Wm.  Cramp  &  Sons  S.  and  E.  B.  Co.,  Phil- 
adelphia, Penn. No.  9  and  10.  Delaware  River  I.  S.  B.  and  E.  Co.,  Chester, 


AND   INLAND    STBAMEES. 


889 


and. 

Triple    Expansion. 

in  feet  and  ins.;  Engines,  in  ins.;  Pressure,  in  Ibs.  ;  Revolutions,  per  minute; 

Fuel,  in  Ibs.  per  Hour  ;  P  Passengers  and  F  Freight. 

Speed,    in    Allies   per    Hour. 


Santa 
Rosa. 

Puritan. 

Tuscaro- 
ra. 

City 

Racine. 

John  F. 
Smith. 

New 
York. 

Ata- 

lanta. 

Susque- 
hanna. 

Robert 
E.Lee. 

Mary- 
lani 

L  9. 

10. 

11. 

12. 

13. 

14. 

16. 

16. 

17. 

lisT 

Iron. 

Steel. 

Steel. 

Iron. 

Iron. 

Iron. 

Iron. 

Iron. 

Wood. 

Steel. 

PandF 

PandF 

F 

FandP 

PandF 

P 

Yacht 

Yacht 

PandF 

FandP 

342-5 

— 

306.7 

220 

130 

3*5 

240 

166.5 

315 

326 

402 

296.7 

— 

122 

— 

228.5 

150 

306 



332.5 
40.6 

403-5 

289.3 
40 

203.5 
35 

122 
42 

301 
40.2 

222.9 
26.33 

164 

22 

315.8 
48.5 

316.4 
42 

23 

1  8.  i 

23 

— 

9 

ii 

15-2 

13 

9.2 

20.4 

2 

i 

2 

i 

I 

i 

2 

I 

i 

2 

'335 

3075 

1937 

802 

I42-39 

1092 

284 

117 

l892 

2416 

4593 

2669 

1041 

I35-60 

'553 

568 

233 

J479 

2419 

13-7 

'3 

16 

— 

5-3 

6-33 

12 

9-3 

16 

3"5 

5" 

4775 
643 

357<> 
624 

218 

'o55 

83 

1000 

235 

1042 
246 

310 

128 

— 

4690 
650 

8-5 

7 

9-7 

— 

3 

6 

4-75 

4-5 

— 

8 

45 

75 

2f 

28 

14 

— 

30 

2  Of  40.5 

22 

86 

no 

38 
61 

50 

26 

75 

60 

28 
42 



35 
56 

54 

9  and  14 

42 

36 

18 

12 

30 

22 

IO 

44 

90 

no 

160 

no 

125 

50 

160 

I48 

160 

80 

2o4 

90 

100 

30 

128 

164 

21 

85 

4 

8 

3 

2 

i 

3 

2 

i 

9 

2 

480 

850 

162 

90 

— 

230 

I46 

65 

118 

152 

12  OOO 

26000 

5574 

4000 

1258 

5360 

4534 

2180 

3360 

4656 



15000 

Jet 

— 

624 

5700 

2226 

1470 

'5 

— 

14 

10.5 

6.4 

— 

— 

8 

— 

13.2 

24 

— 

17-5 

16.5 

9 

— 

— 

— 

— 

16 



35 

30.16 

— 

— 

39 

— 



14 

— 

— 

— 

12.5 

— 

—  • 

17 

— 

200 

200 

230 

IIO 

15 

50 

170 

5° 

— 

200 

X75p.rP 

1.9  p.  H> 

2340 

1400 

4 

2-5P-  H» 

— 

— 

Natural 

Natural 

Natural 

Natural 

Natural 

Blast 

Blast 

Blast 

Natural 

Natural 

600 

900 

2140 

310 

— 

— 

— 

— 

H25 

3100 

165 

1200 

— 

— 

150 

2100 

18 

— 

300 

8 

— 

2OO 

29 

— 

8 

50 

45 

— 

150 

*9 

3000 

7500 

1800 

750 

300 

3700 

1950 

925 

1  200 

19 

21 

16 

14.5 

13 

17-45 

18 

21 

12.5 

Sch'r 



Sch'r 

Sch'r 

— 

— 

3-m  Sch'r 

Sch'r 



Sch'r 

Perm.,  and  W.  and  A.  Fletcher  Co.,  Hoboken,  N.  J.  - 


-No.  11.  Globe  Iron 


Works,  Cleveland,  Ohio;  Hull  1240  tons,  Engines  200,  and  Boilers  70  tons. 

No.  12.   Chas.  F.  Elmes,  Chicago,  111.,  and  Burger  &  Burger,  Wis. ;    Freight  and 

Cabin  on  deck  ;   Hull  350  tons,  Engines  40,  Boilers  36,  and  Water  23. No. 

1?.  Pusey  &  Jones  Co.,  Wilmington,  Del. No.  14.   W.  &  A.  Fletcher  Co., 

Hoboken,  N.  J. ;  Water-wheel  blades,  13  of  45  ins. —     — No.  15.  Same  builders 

as  No.  8. No.  16.  The  Harlan  &  Hollingworth  Co.,  Wilmington,  Del. ;   Hull 

136  tons,  Engines  20,  and  Boilers  25.  - 


-No.  17.   Jas.   Howard  &  Co.,  Jef- 
fersonville  and  American  Foundry,  New  Albany,  Ind. ;  Water  -  wheel  blades,  22 

of  35  ins. No.  18.  Detroit  Dry  Dock  Co.,  Detroit,  Mich. ;    Hull  1250  tons, 

Engines  140,  and  Boilers  70. 

4F 


890 


STEAM-VESSELS. FERRY    AND    TOWING. 


IPassenger   and.   Team,  and.   Tow-tooats. 
Single,   Compound,   and.    Triple    Expansion. 

Length  and  Hull,  in  feet  and  tenths  ;  Draught,  Propeller,  and  Side  Wheels,  in  feet 
and  ins.;  Engines,  in  ins.;  Pressure,  in  Ibs.;  Revolutions, per  minute ;  Surfaces, 
in  sq.  feet ;  Weights  and  Displacements,  in  Tons  of  2240  Ibs.  ;  Fuel,  in  Ibs.  per 
Hour  ;  P  Passengers  and  T  Teams. 

Speed   in    Miles    per   Hour. 


DIMENSIONS 

AND 

CAPACITIES. 

Montauk 
and 
Whitehall. 

John  G. 
McCul- 
lough. 

Bergen. 

In- 
trepid. 

Maine. 
Ferry. 

Inter- 
nation- 
al. 

Meteor. 

"~7~ 

Iron. 

Towiiitf. 

879566 

£! 

8.6 

Pat- 

erson* 
and 
Mate. 

Sleel. 
PandT 

222 

217 

40 
62 

16.6 

Service  

1. 

Iron. 
PandT 
209 

I9I 
196 

37-4 
65 
14.1 

2. 

Steel. 
TandP 
215 
198.5 
198.5 

§ 

14-5 

3. 

Steel. 
PandT 
203 
200 
220.4 

11 

16.6 

4. 

Iron. 
Towing. 
118 
no 
114 
23-5 

ii.  6 

6. 

Iron. 
TandP 
189.2 
175 
174 
S^.S 
62.5 
13-3 

6. 

Iron. 

Towing. 
140 
129.6 
130 
26 

16.2 

Length  on  deck  
"      bet.  perp'rs.  . 
"      tonnage  
Beam  do              .... 

u      over  guards.  .  . 
Hold  tonnage 

Decks 

Tons  I  

839 
1088 

880 
215 
7-5 

50 
10 
50 
32 

168 
1380 
Jet 

20.5 
8.66 
5 

Natural 
IOIO 

35oo 
4130 
47° 
104 

5i 
29-5 

12 

1008 
1310 
ii 
1340 
450 
7-75 

22 

50 

36 
100 
1  2O 
2 
140 

2  Of 

734 
1117 
9-5 
560 
225 
6.9 
18.5 
27 
42 

2*4 
1  60 

162 

2 

81 
3462 

2  Of 

8 
8.91 

108.9 
217.8 
9-5 
3°3 
164 

4 

22 

42 
36 

100 

9° 

i 

71-5 
2503 
1105 
9-5 

i4&i6 

545-7 
850.3 
7.2 
678.5 
206 
5&6.5 
46 

120 
22 

24 
I 
76 
2259 
Jet 

400 

2OO 
12 

llo° 
V65 

24 
41 

3° 
160 

2 
80 
2400 
IIOO 

9-5 

55-6 
95-6 
8 
150 

16 

32 
28 

IOO 
100 

I 
45-5 
1318 
553 

14 

10.6 
75° 

6T9 

2  Of  20 

2  Of  36 
28 
125 

120 

2 

91 
3332 
2224 

2  Of 

8.6 
ii 

Draught  load 

Displacement  do  
ImmersedSec'n  at  do. 
Freeboard  .... 

Cylinders.  IP  

Int  
"         L.P  
Stroke  of  Piston  
Steam  Pressure  
Revolutions  

Boilers    

Grate  surface  

Heating  do 

Condensing  do  
Propeller,  ilium.  .  .  J 

Pitch  

Blades 

Side-wheel  diam  
"        width.... 
Coal  weight 

Natural 

453° 
5200 

12 

1580 
Natural 
1007 
3448 
4330 
321 

& 

,'!?« 

60 

Natural 

450 

*3 
20.5 
17.6 

n.s 

20.5 
8.6 
40 

Natural 
650 
3420 
2896 

12 

39-75 
29.6 

12 

270 

Natural 
800 

IS 

16 
420 
Natural 
280 

50 
26 
21 

12 

12 

Natural 
1250 
3760 
4750 
38o 
no 
80 
40 
14.  5 

Combustion  
IIP  

Team  space 

Passenger  do  
Weight  Hull 

Engine  

Boilers  

Water  

Speed  .  . 

Remarks. No.  1.  Side-wheel,  T.  S.  Marvel  &  Co.,  Newburgh,  and  Quintard 

Iron.  Works,  N.  Y. ;   Double  ends. No.  2.   Neafle  and  Levy,  Penn  Works, 

Phila.,  Pa. ;  Propeller  at  each  end. No.  8.  Hull  same  as  1,  and  Delamater 

Iron  Works,  N.Y.;  Propeller  at  each  end ;  Weights:  of  Hull  as  launched;  Engines,  not 
including  steering  and  ventilating;  donkey  pumps,  piping  and  chimney ;  augmented 
surface,  7524  ^feet. Nos.  4.  and  5.  The  Harlan  and  Hollingsworth  Co.,  Wil- 
mington, Del.,  Propeller  and  Side-wheel. No.  6.  Neafie  and  Levy,  Phila.; 

one  Wrecking  pump,  16  and  20X18  ins.,  three  8-inch  suctions  on  each  side,  capacity 
700  tons  water  per  hour;  one  fire-pump,  eight  2.5-inch  streams  ;  Electric  search- 
lights, 6000  candle  power,  several  of  2000  candle  arc -lights. No.  7.  The 

Pusey  &  Jones  Co.,  Wilmington,  Del. No.  8.  Hull  same  as  Nos.  1  and  3,  and 

engines  W.  &  A.  Fletcher  Co.,  Hoboken,  N.  J. ;  Propeller  at  each  end;  No.  8  and 
these  *  designed  bv  Col  v.  A  Stevens,  Hoboken,  N.  J. 


MARINE    STEAM    VESSELS    AND    ENGINES.  89! 

"Wood   IPropellers. 

HERRHSHOFF,  R.  N.,  VERTICAL  DIRECT  ENGINE  (Compound). — Length  on  deck,  46 
feet;  over  all,  48  feet ;  beam,  9  feet;  hold,  5  feet. 

Displacement  at  load-line,  7.44  tons.  Area  of  section  at  load-line,  217.8  sq.feet. 
Area  of  wetted  surface,  365.5  sq.feet.  Coefficient  of  fineness,  .396. 

Cylinder.— 8  and  14  ins.  in  diam.  by  9  ins.  stroke  of  piston. 

Condenser,  External.  — Surface. 

Propeller.— 4  blades,  3  feet  in  diam.  by  4  feet  i  inch  pitch. 

Blower,  42  ins.  in  diam. 

Boiler  (vertical  coil).    Heating  surface,  174  sq.  feet.     Grates,  12.5  sq:  feet. 

Pressure  of  Steam,  53  Ibs.  per  sq.  inch.  Revolutions,  333  per  minute.  IIP,  68.4. 
Speed.  10. 18  knots  per  hour.  With  129  Ibs.  and  466  revolutions,  14.26  knots.  IIP, 
169.5.  Weight  of  Engines,  Boiler,  and  Water,  5300  Ibs. 

HERRKSHOPP,  VERTICAL  DIRECT  ENGINE  (Compound).  —  Length  over  all,  86  feet; 
beam,  nfeet.  Displacement,  27  tons. 

Cylinder. — 13  and  22  ins.  in  diam.  by  12  ins.  stroke  of  piston. 
Surface  Condensing. 
Pressure,  130  Ibs.  per  sq.  inch. 

Revolutions,  460  per  minute.    Speed,  20  knots  per  hour.     IH*,  425. 
Propeller,  3  blades.     Pitch,  5  feet. 

HERRESHOFF,  R.  I.  N. — VERTICAL  DIRECT  ENGINE  (Compound). — Length  over  aW, 
60  feet;  beam,  7  feet;  Jwld,  s.^feet.  Displacement  at  load-draught  of  32  ins.,  7  toni 
(2240  Ibs.). 

Cylinders.— 8  and  14  ins.  in  diam.  by  9  ins.  stroke  of  piston.  Surface  condenser. 

Pressure  of  Steam. — 140  Ibs.  per  sq.  inch,  cut  off  at  .5. 

Revolutions,  6ob  per  minute.    Speed,  19.875  knots  per  hour. 

Cable   or    Rope    Towing. 

"NYITRA."— HORIZONTAL  DIRECT  ENGINES  (Condensing).— Length  of  boat,  138  feet; 
beam,  24. 5  feet ;  hold,  7. 5  feet. 

Immersed  section,  74.4  sq.feet.  Displacement,  200  tons  at  load-line  of  3. 7$  feet. 
Immersed  section,  263.7  S2-  feet-  Displacement,  949  tons.  Tow.—  3  barges. 

Cylinders. — 2  of  14.18  ins.  in  diam.  by  23.625  ins.  stroke  of  piston. 

IIP,  net  effective,  100.     Speed,  7.73  miles  per  hour. 

Propellers.—  Twin,  4  feet  2  ins.  in  diam. 

Stress.— Cable,  7485  Ibs.  Per  ton  of  displacement,  6.5  Ibs. ;  per  sq.  foot  of  im- 
mersed section,  22  Ibs. 

Fuel. — Per  mile  and  ton  of  displacement  (1149),  -°7^  IDS- 

Towing.    "Wood.    Side   "Wheels. 

"WM.  H.  WEBB."— HARBOR  AND  COAST.— VERTICAL  BEAM  ENGINES  (Condensing). 
^Length  upon  deck,  185.  $  feet ;  beam,  30.25  feet ;  hold,  10.8  feet. 

Immersed  Section  at  load-line,  194  sq.feet.  Displacement  498.25  tons,  at  load- 
draught  of  7. 25  feet. 

Cylinders. — 2,  of  44  ins.  in  diam.  by  10  feet  stroke  of  piston ;  volume,  211  cube  feet. 
Condensers. — Jet,  2,  volume  105  cube  feet.  Air-pumps. — 2,  volume  45  cube  feet. 

Water-wheels. —  Diam.,  30  feet.  Blades  (divided),  21;  breadth  of  do.,  4.6  feet; 
depth  of  do.,  2.33  feet.  Dip  at  load  line,  3.75  feet. 

Boilers.  — 2  (return  flue).     Heating  surface,  3280  sq.  feet.     Grates,  147.5  sq.  feet 

Smoke-pipe. — Area,  n.6  sq.  feet,  and  35  feet  in  height  above  the  grate  level 

Pressure  of  Steam.— 35  Ibs.  per  sq.  inch,  cut  off  at  .5  stroke.  Revolutions^  22  pel 
minute.  IIP,  1500. 

Fuel.—  Anthracite  or  Bituminous.     Consumption,  1680  Ibs.  per  hour. 

Speed.— 20  miles  per  hour. 

Weights. —Engines,  Wheels,  Frame,  and  Boilers,  310  579  Ibs. 


RIVER  STEAMBOATS,  SIDE  AND    STERN  WHEEL. 

Wood   Side  "Wheels. 
Passenger. 

"  MARY  POWELL,"  HUDSON  RIVER.— VERTICAL  BEAM  ENGINE  (Condensing).— Length 
on  water-line,  286  feet;  over  all,  294  feet;  beam,  34  feet  3  ins.  ;  over  all,  64  feet;  hold, 
gfeet.  Deck  to  promenade  deck,  lofeet. 

Immersed  section  at  load  •  line  of  6  feet,  200  sq.  feet  Displacement,  800  tons  at 
mean  load-araught  of  6  feet. 

Area  of  transverse  head  surface  of  hull  above  water,  2000  sq.  feet. 

Cylinder.— 72  ins.  in  diam.  by  12  feet  stroke  of  piston;  volume,  338  cube  feet. 
Clearance  at  each  end,  12.5  cube  feet. 

Steam  and  Exhaust  Valves,  14.75  ins.  in  diam.  Air-pump,  40  ins.  in  diam.  by  5 
feet  2  ins.  stroke  of  piston.  Condenser.— Jet,  128  cube  feet.  Crank-pin,  8.75  ins.  in 
diam.  x  10.75  ins. 

Beam,  22.5  feet  in  length;  centre,  9.75  in  diam. 

Water-wheels Diam.  31  feet;  blades  (divided),  26;  breadth  of  do.,  10  feet  6  ins. ; 

width,  i  foot 6  ins. ;  immersion,  3  feet  6  ins.    Shafts.—  Journal,  15.625  ins.  by  17  ins. 

Boilers.— 2  (flue  and  return  tubular),  of  steel,  u  feet  front  by  26  feet  in  length; 
shell,  10  feet  in  diam.  and  16  feet  i  inch  in  length.  Furnaces,  2  in  each,  of  4  feet 
10  ins.  by  8  feet  in  length.  Heating  Surface,  2660  sq.  feet;  and  Superheating,  340 
sq.  feet  in  each.  Grates,  152  sq.  feet.  Flues,  10  in  each,  transverse  area,  n  feet 
7  ins.  Tubes,  80  in  each,  4.5  ins.  in  diam.,  6  feet  6  ins.  in  length,  and  8  feet  7  ins. 
in  transverse  area. 

Steam  Chimneys,  8  feet  in  diam.  x  12  feet  in  height.  Smoke-pipe,  4  feet  6  ins.  in 
diam.  and  68  feet  in  height  from  grates. 

Combustion,  Blast.  Blowers,  4  feet  in  diam.  and  3  feet  in  width.  Revolutions,  78 
per  minute.  Fuel  (anthracite),  6280  Ibs.  per  hour,  or  40  Ibs.  per  sq.  foot  of  grate 
per  hour.  Per  sq.  foot  of  heating  surface,  2. 25  Ibs. 

Speed,  23.65  miles  per  hour. 

Pressure  of  Steam,  28  Ibs.  per  sq.  inch,  cut  off  at  .47  stroke;  terminal  pressure, 
16.4  Ibs. ;  throttle,  .625  open.  Vacuum,  25  ins.  Revolutions,  22.75  per  minute. 

Temperatures.— Reservoir,  120°.  Feed  water,  120°.  Chimney,  740°.  IP.— Total, 
1900.  IIP,  1560.  Net,  1450. 

Evaporation.— Water  per  Ib.  of  coal,  from  120°,  7  Ibs.;  per  Ib.  of  combustible, 
from  120°,  8.2  Ibs.  Steam  per  total  IP  per  hour,  21.1  Ibs.  Coal  per  do.  do.,  3. 14  Ibs. 

Weights.  Engine. — Frame,  keelson,  out -board  wheel -frames  donkey  engine, 
and  boiler,  blower  engines  and  blowers,  all  complete,  360000  Ibs.  Boilers.—  Iron 
return  flue,  120000  Ibs.  Steel  return  tubular,  116000  Ibs.  Water,  128000  Ibs. 

Capacity.— 2000  passengers  and  their  baggage. 

Memoranda.— This  vessel  was  originally  but  266  feet  in  length,  and  when  length- 
ened the  cylinder  of  62  ins.  in  diam.  wag  removed  and  replaced  with  one  of  72  ins. 
Engine  designed  throughout  for  original  cylinder  and  a  pressure  of  from  50  to  55 
Ibs.,  cutting  off  at  .625  of  stroke,  with  throttle  wide  open. 

Engines  and  Boilers  built  by  Fletcher,  Harrison,  &  Co.,  New  York,  1861  and  1875. 
Iron    Stern   "Wheels. 
Passenger   and.    Freight. 

HORIZONTAL  ENGINES  (Non-condensing).— Length  upon  deck,  no  feet;  beam,  14 
feet  (deck  projecting  over,  4  feet) ;  hold,  3. 5  feet. 

Immersed  section  at  load-line,  10. 25  sq.  feet.  Displacement  at  load-draught  of  1. 1 
feet,  33  tons. 

Cylinders.— Two,  of  10  ins.  in  diam.  by  3  feet  stroke  of  piston;  volume  of  piston 
space,  1.6  cube  feet. 

Wheel—  Diam.  13  feet.    Blades,  13;  breadth  of  do.,  8.5  feet;  depth  of  do.,  8  ins. 

Revolutions,  33  per  minute.  Boiler.—  One  (horizontal  tubular).  Tubes,  100  of  2 
ins.  in  diam. 

Fuel. — Bituminous  coal.     Consumption,  4480  Ibs.  in  24  hours. 

Hull— Plates,  keel,  No.  3;  bilges,  No.  4;  bottom,  No.  5;  sides,  Nos.  6  and  7. 
Frames,  2.5  x  .5  ins,,  and  20  ins.  apart  from  centres. 


RIVER  STEAMBOATS,  STERN  WHEELS. — OIL  LAUNCH.   893 


"Wood   Stern  "Wheels. 
Passenger   and   Deck    Freight. 

"  MONTANA.  "—HORIZONTAL  ENGINES  (Non-condensing).— Length  upon  deck  (over 
att),  248  feet;  at  water-line,  245  feet;  beam,  48  feet  8  ins.  (over  all,  50  feet  4  ing.); 
hold,  6  feet;  draught  of  water  at  load-line,  5. 5  feet. 

Immersed  section  at  load-line,  244  sq.  feet.  Displacement  at  mean  light  draught 
of  22  ins.,  594  tons  (2000  Ibs.) 

Cylinders.— Two,  18  ins.  in  diam.  by  7  feet  stroke  of  piston. 

Valves,  4.5  and  5  ins.  in  diam.  Piston-rod,  4  ins.  Steam-pipe,  4.5  ins.  Connect- 
ing-rod, 30  feet  in  length. 

Water-wheel,  19  feet  in  diam.  by  35  feet  face;  blades,  3  feet  in  depth.  Shaft, 
10.25  ins.  in  diam. 

Boilers.—  Four  (horizontal  tubular),  42  ins.  in  diam.  by  26  feet  in  length.  Two 
flues  in  each,  15  ins.  in  diam.  Heating  surface,  effective,  1023,  total  1431  sq.  feet 
Furnace,  6.5  X  17  feet.  Grates,  4.16  X  17  feet;  surface,  70.8  sq.  feet.  Smoke-pipes. 
•—Two,  3  feet  in  diam.  by  55  feet  3  ins.  in  height.  Exhaust  or  Blower  draught. 

Calorimeter.—  Of  Bridge,  15.27;  of  Flues,  9.82;  and  of  Chimneys,  14.14  sq.  feet. 
Areas  of  grate,  compared  to  calorimeter  of  flues,  7.2;  to  ditto,  of  chimneys,  5;  and 
of  bridge,  4.6  sq.  feet. 

Steam-room,  562 ;  and  water  space,  294  cube  feet. 

Hull— Frames,  4X6  ins.  and  15  ins.  apart  at  centres.  Intermediate  do.,  4  x  6 
ins.,  and  running  for  7. 5  feet  each  side  of  keelson.  Planking.— Bottom,  oak,  4  ins. ; 
side  do.,  2.5  to  4  ins.  Deck  beams,  pine,  3  X  6  ins.  Deck  plank,  2. 5  ins.  Keelson, 
oak;  side  do.,  eight  each  side,  one  each  7,  8.75,  and  9  ins.,  and  five  6.75  ins.  Wales, 
one  each  side,  9  and  7  ins.  by  3,  and  one  10  X  2.5  ins.  Deck  posts,  3.5  X  3  ins.  and  4 
feet  apart.  Deck  beams,  5.5  X  3  ins.  Knuckles,  oak,  6  X  12  ins.  Bulkheads,  one 
longitudinal  and  one  athwartship  at  shear  of  stern.  Sheathing  of  wrought  iron, 
.0625  to  .125  inch  from  just  below  light  to  load-line. 

Hog  Posts.—  White  pine,  8.5  and  n  ins.  square.     Chains,  1.5  ins.  in  diam. 

Weights.—  Boilers,  29  264;  water,  18  351 ;  and  boilers,  chimneys,  grates,  and  water, 
55672  Ibs.  Hull,  oak,  520560;  Pine,  91 437;  Bolts,  spikes,  etc.,  8000,  and  Deck  and 
guards,  76000  Ibs. ;  Hull  alone,  310  tons. 

Weight  of  hull  compared  to  one  of  iron  as  8  to  5,  effecting  a  difference  of  about 
loo  tons. 

"PITTSBURGH."— HORIZONTAL  ENGINES  (Non-condensing).— Length  on  deck,  252 
feet;  beam,  39  feet;  hold,  6  feet;  draught  of  water  at  load-line,  2  feet. 

Immersed  section  at  load-line,  75  sq.feet.  Displacement  at  load  draught  of  2  feet, 
380  tons  (2000  Ibs.). 

Cylinders.— Two,  21  ins.  in  diam.  by  7  feet  stroke  of  piston. 
Water-wheel—  21  feet  in  diam.  by  28  feet  face. 

Boilers.—  a  (horizontal  tubular),  47  ins.  in  diam.  by  28  feet  in  length.  Two  fires 
in  each. 

Oil   Engine   Hianncn. 

Bilemeiits   of  Engine   and   Dimensions   of  Hiannoh. 
Consumption  .9  pint  ordinary  Mineral  Oil  per  IB?  per  Hour. 


Typ«. 

H>* 

Lau 
Length. 

nch. 
Breadth. 

Weight^ 

Type. 

H?* 

Lau 
Length. 

Breadth. 

Weight^ 

No. 
6 
5 

4 

No. 

I 
2 

3 

Feet. 
16 

SI 

27 

Feet. 
4 

1 

Lb8. 

896 
1332 
1568 

No. 
3 

2 

I 

No. 
5 

10 

IS 

Feet. 
30 
40 
45 

Feet. 
7 
7 
7-5 

Lbs. 

1848 
2688 
3136 

•  Developed  by  Brake. 


4F* 


t  Of  engine  without  oil. 


894  RIVER   STEAMBOATS. — SAILING   VESSELS. 

Passenger    and.    Deck    Freight. 

"PITTSBURGH."— HORIZONTAL  ENGINES  (Non-condensing).  —  Length  on  deck,  252 
feet;  beam,  39  feet;  hold,  6  feet;  draught  of  water  at  load-line,  2  feet. 

Immersed  section  at  load-line,  75  sq.feet.  Displacement  at  load-draught  of  2  feet, 
380  tons  (2000  Ibs.). 

Cylinders.—  Two,  21  ins.  in  diam.  by  7  feet  stroke  of  piston. 

Watcr-vjliccl.—  21  feet  in  diam.  by  28  feet  face. 

Boilers.—  2  (horizontal  tubular),  47  ins.  in  diam.  by  28  feet  in  length.  Two  fires 
in  each. 

Iron.    Stern.    Wheels. 

HORIZONTAL  ENGINES  (Non-condensing).  —  Length  upon  deck,  no  feet;  beam,  14 
feet  (deck  projecting  over,  4  feet) ;  hold,  3. 5  feet. 

Immersed  section  at  load-line,  10.25  sq.feet.  Displacement  at  load-draught  o/i.i 
feet,  33  tons. 

Cylinders.— Two,  of  10  ins.  in  diam.  by  3  feet  stroke  of  piston;  volume  of  piston 
space,  1.6  cube  feet 

Wheel—  Diam.  13  feet.    Blades,  13;  breadth  of  do.,  8.5  feet;  depth  of  do.,  8  ins. 

Revolutions,  33  per  minute.  Boiler.—  One  (horizontal  tubular).  Tubes,  100  of  2 
ins.  in  diam. 

Fuel  —Bituminous  coal.     Consumption,  4480  Ibs.  in  24  hours. 

Hull— Plates,  keel,  No.  3;  bilges,  No.  4;  bottom,  No.  5;  sides,  Nos.  6  and  7. 
Frames,  2.5  X  .5  ins.,  and  20  ins.  apart  from  centres. 

Steel. 

"  CHATTAHOOCHEE.  "—INCLINED  ENGINES  (Non- condensing).— Length  on  deck,  157 
feet;  beam,  31. 5  feet;  hold,  5  feet. 

Immersed  section  at  load-line,  153  sq.feet.    Freight  capacity,  400  tons  (2000  Ibs.). 

Cylinders.— Two,  15  ins.  in  diam.  by  5  feet  stroke;  volume  of  piston  space,  12.26 
cube  feet. 

Wheel—  One,  18  feet  in  diam. ;  blades,  2  feet  in  depth. 

Boilers.—  Three  (cylindrical  flued).  Diam.  42  ins. ;  length,  22  feet;  2  flues  of  10 
ins.  in  each.  Heating  surface,  690  sq.  feet.  Grates,  48  sq.  feet. 

Pressure  of  Steam,  160  Ibs.  per  sq.  inch,  cut  off  at .  375.     Revolutions,  22  per  min. 

Consumption  of  Fuel,  12  tons  (2000  Ibs.)  in  24  hours.  Plating  of  Hull,  .1875  to 
.25  inch.  Light  draught,  21  ins. 

Iron.    Propellers. 

VERTICAL  DIRECT  ENGINES  (Non-condensing). — Length  on  deck,  jofeet;  beam,  10.5 
feet;  draught,  12  ins. 

Propellers,  2.— 2  blades,  16.  ins.  in  diam.,  set  u  ins.  below  water-Hne. 
Boiler  (tubular  coil).    Revolutions,  480  per  minute. 
Speed,  10. 49  miles  per  hour. 
Water  led  to  propellers  through  tunnels  in  bottom  at  sides. 

" LOUISE. "—VERTICAL  TANDEM  ENGINES  (Compound).— Length,  60  feet;  beam,  12 
feet;  hold,  4.25  feet. 

Displacement  at  load-draught  of  2.5  feet,  8  tons. 
Cylinders,  5  and  10  ins.  in  diam.  by  8  ins.  stroke  of  piston. 
Surface  Condenser.— Boiler  (vertical  tubular),  4  feet  in  diam.  by  8.5  in  length. 

Iron    Sailing    Vessels. 
Passenger   and    Freight. 

ENGLISH. — SHIP. — Length  upon  deck,  ijSfeet ;  do.  at  mean  load-line  ofig.  i6feet,  177 
feet;  keel,  171  feet;  beam,  ^2.8Bfeet;  depth  of  hold,  21.75/66*;  keel  (mean),  2-75/eefc 

Immersed  section  at  load-line,  387  sq.feet.  Displacement  at  load-draught  of  19.16 
feet,  1385  tons;  at  deep  load-draught  of  20  feet,  1495  tons;  and,  in  proportion  to  its 
circumscribing  parallelopipedon, .  524. 

Load-line.— Area  at  load-draught,  4557  sq.  feet.  Angle  of  entrance,  57°;  of  clear- 
ance, 64°.  Area  in  proportion  to  its  circumscribing  parallelogram,  .784. 


YACHTS. — CUTTERS. — PILOT   BOAT.  895 

Centre  of  Gravity,  6.416  feet  below  mean  load-line.  Centre  of  Displacement  (grav- 
ity of),  6.25  feet  below  load-line;  and  4.33  feet  before  middle  of  length  of  load  line. 

Immersed  Surface.— Bottom,  7370  sq.  feet.  Keel,  1130  sq.  feet.  Sails,  13  282  sq.  feet. 

Mcta-centre,  6.66  feet  above  centre  of  gravity  of  displacement.  Centre  of  Effort 
before  centre  of  displacement,  3.5  feet;  height  of  do.  above  mean  load-line,  55.5  feet. 

Laimcli.     "Wood.. 

STEAM  LAUNCH  "  HERRESHOFF. " — VERTICAL  ENGINE  (Compound).— Length,  33  feet 
i  inch ;  beam,  %.j$feet. 

Displacement  at  mean  load-draught  o/(to  rabbet  of  keel)  19  ins.,  8929  Ibs. 
Weights.—  Hull  and  Machinery,  6555  Ibs.    Coal,  1120  Ibs. 

Yachts.     Wood. 

"AMERICA,"  SCHOONER.—  Length  over  all,  oB  feet;  upon  deck,  94  feet;  at  load-line, 
90. 5  feet ;  beam,  22. 5  feet ;  at  load-line,  22  feet ;  depth  of  hold,  9. 25  feet.  Height  ait 
tide  from  under  side  ofgarboard  stroke,  1 1  feet.  Sheer,  forward,  3  feet ;  aft,  i.  5  feet, 

Immersed  section  at  load-line,  121.8  sq.feet.  Displacement  at  load-draught  0/8.5 
feet,  from  under  side  ofgarboard  stroke  and  of  u  feet  aft,  191  tons;  and,  in  pro- 
portion  to  Volume  of  circumscribing  parallelopipedon,  .375. 

Displacement  at  4  feet  (from  garboard  stroke),  43  tons  ;  at  5  feet,  66  tons  ;  at  6 
ftet,  93  tons ;  at  7  feet,  127  tons ;  and  at  8  feet,  167  tons. 

Centre  of  Gravity.  —Longitudinally,  1.75  feet  aft  of  centre  of  length  upon  load- 
line.  Sectional,  2.58  feet  below  load  line.  Of  Fore  body,  14.25  feet  forward;  and 
of  After  body,  19  feet  aft.  Meta-centre,  6. 72  feet  above  centre  of  gravity. 

Centre  of  Effort,  31  17  feet  from  load-line.  Centre  of  Lateral  Resistance,  6.33  feet 
abaft  of  centre  of  gravity.  Area  of  Load-line,  1280  sq.  feet.  Mean  girths  of  im- 
mersed section  to  load-line,  25  feet. 

Load-draught.  —Forward,  4.91  feet;  aft,  u  5  feet.    Rake  of  Stem,  17  feet 

Spars.— Mainmast,  81  feet  in  length  by  22  ins.  in  diam.  Foremast,  70.5  feet  in 
length  by  24  ins.  in  diam.  Main  boom,  58  feet  in  length.  Main  gaff,  28  feet.  Fore 
gaff,  24  feet.  Rake,  2.7  ins.  per  foot.  Drag  of  Keel,  3  feet  Tons,  170.56. 

"  JULIA,"  SLOOP.—  Length  for  tonnage,  72.25  feet;  on  water-line,  70  feet  ^  inf.; 
beam,  19  feet  8  ins.;  hold,  6  feet  8  ins.  Tons,  O.  M.  83.4;  N.  M.  43.98. 

Load-draught,  6.25  feet. 

Sails.— Mainsail,  hoist,  49.75  feet,  foot  54.25,  and  gaff  27.66;  Jib,  hoist,  49.75  feet, 
foot  39. 5,  and  stay  63. 5.  Gaff  topsail,  hoist.  24. 5  feet. 

Areas.— Mainsail,  2322  sq.  feet.    Jib,  986,  and  Topsail,  454.  - 

Cntters. 

"TAR A"  (English)  SLOOP.— Length  on  load-line,  66 feet;  beam,  11.5  feet. 

Immersed  section  at  load-line,  11.5  sq.feet.     Displacement,  75  tons. 

Spars.— Mast,  deck  to  hounds,  42  feet.  Boom,  58  feet.  Gaff,  39  feet.  Bowsprit 
outside  of  stem,  30  feet.  Mast  to  stem,  26  feet.  Topmast,  foot  to  hounds,  25  feet 
Balloon  topsail  yard,  46  feet.  Canvas,  area,  3450  sq.  feet.  Tons,  C.  H.,  9®. 

Ballast— At  Keel,  38.5  tons.    Hull,  1.5  tons. 

"MISCHIEF"  (English),  SLOOP.— Length  on  load-line,  6ifeet;  beam,  ig.gfeet 
Immersed  section  at  load-line,  60  sq.feet.    Displacement,  55  tons. 

Pilot   Boat. 

"WM.  H.  ASPINWALL,"  SCHOONER.— Length  of  keel,  74  feet;  upon 
beam,  19  feet;  hold,  7. 6  feet.    Draught  of  water,  6  feet  forward ;  aft,  9.5  feet. 

Keel,  22  ins.  in  depth.    False  keel,  12  ins.  in  depth  at  centre. 

Spars.— Mainmast,  77  feet  in  length.  Foremast,  76  feet  Main  doom,  46  feet 
Main  gaff,  21  feet.  Fore  gaff,  20  feet 

Tons.—  N.  M.,  46.32. 


896  PASSAGES    OF   STEAMBOATS. — ICE-BOATS. 

PASSAGES  OF  STEAMBOATS. 

Distances  in  Statute  Miles. 

1807,  Clermont,  of  N.  Y.,  New  York  to  Albany,  145  miles,  in  32  hours  =  4. 53  miles 
per  hour,  neglecting  effect  of  the  tide. 

1811,  New  Orleans,  of  Pittsburgh,  Penn.  (non-condensing  and  stern- wheel),  Pitts- 
burgh to  Louisville,  Ky.,  650  miles,  in  2  days  22  hours. 

1849,  Alida,  of  N.  Y.,  Caldwell's,  N.  Y.,  to  Pier  i,  North  River,  43.25  miles,  in  i 
hour  42  min. ,  ebb  tide  =  2. 75  miles  per  hour.    Speed  =  22. 19  miles  per  hour.     1860, 
3oth  Street,  N.  Y.,  to  Cozzens's  Pier,  West  Point,  50.5  miles,  in  2  hours  4  min.,  and 
to  Poughkeepsie,  74.25  miles,  in  3  hours  27  mm.,  5  landings,  flood  tide.    And  1853, 
Robinson  Street  to  Kingston  Light,  90.375  miles,  in  4  hours,  making  6  landings, 
flood  tide. 

1850,  Buckeye  State,  of  Pittsburgh,  Penn.  (non  -  condensing),  Cincinnati  to  Pitts- 
burgh, 500  miles  (200  passengers),  53  landings,  in  i  day  19  hours;  4  miles  per  hour 
adverse"  current.    Speed  =  15.63  miles  and  1.23  landings  per  hour.    Average  depth 
of  water  in  channel  7  feet. 

1852,  Reindeer,  of  N.  Y.,  New  York  to  Hudson,  116.5  miles,  in  4  hours  57  min., 
making  5  landings.     Flood  tide. 

1853,  Shotwell,  of  Louisville,  Ky.  (non-condensing),  New  Orleans  to  Louisville, 
1450  miles,  8  landings,  in  4  days  9  hours;  4.5  to  5.5  miles  per  hour  adverse  cur- 
rent.   Speed  =  18.81  miles  per  hour. 

NOTE.— In  1817-18  the  average  duration  of  a  passage  from  New  Orleans  to  Louisvill-  was  27  days, 
12  hours;  the  shortest,  25  days. 

1855,  New  Princess,  of  New  Orleans  (non-condensing),  New  Orleans,  La.,  to  Natchez, 
Miss.,  310  miles,  in  17  hours  30  min.;  3.5  to  4  miles  per  hour  adverse  current. 
Speed  =  20.98  miles  per  hour. 

1864,  Daniel  Drew,  of  N.  Y,  Jay  Street,  N.  Y.,  to  Albany,  148  miles,  in  6  hours  51 
min. ,  9  landings.  Flood  tide.  Speed  of  boat  =  22.6  miles  per  hour. 

1867,  Mary  Powell,  of  N.  Y.,  Desbrosses  Street,  N.  Y.,  to  Newburgh,  60.5  miles,  in 
2  hours  50  min.,  3  landings;  from  Poughkeepsie  to  Rondout  Light,  15.375  miles,  in 
39  mm.,  flood  tide.  1873,  Milton  to  Poughkeepsie,  light  draught  and  flood  tide,  4 
miles,  in  9  min.;  and  1874,  Desbrosses  Street  to  Piermont,  24  miles,  in  i  hour  ;  to 
Caldwell's,  43.25  miles,  in  i  hour  50  min.  Speeds  22.77  to  23  miles  per  hour. 

Runs  from  New  York  to  Albany,  146  miles,  by  different  Boats. 

1826,  Sun 12  hours  16  min. 

1826,  North  America*.  10     u     20    " 

1841,  Troy  t 8     u     10    " 

1841,  South  America  J.    7     •*     28    " 


1852,  Fr.  Skiddy  § 6  hours  24  min. 

1860,  Armenia  II 7     "     22    " 

1864,  Daniel  Drew$....  6     "     51     " 
1864,  CWncey  Vibbardl.  6     "     42    ' 


*  7  landings.  f  4  landings.  $  9  landings.  §  6  landings.  ||  u  landings. 

Timing  Distance.— From  i4th  St.,  Hudson  River,  N.  Y.,  to  College  at  Mount  St.  Vincent,  13  mile*. 

NOTE.— Where  landings  have  been  made,  and  the  river  crossed,  the  distance  between  the  pointa 
given  is  correspondingly  increased. 

1870,  R.  E.  Lee,  of  St.  Louis,  (non-condensing),  New  Orleans  to  St.  Louis,  Mo.,  1180 
miles  (without  passengers  or  freight),  4  to  5  miles  per  hour  adverse  current;  Vicks- 
burg,  i  day  38  mm.;  Memphis,  2  days  6  hours  g  min.;  Cairo,  3  days  i  hour.;  and  to 
St.  Louis,  3  days  18  hours  14  mm.,  inclusive  of  all  stoppages. 

1870,  Natchez,  of  Cincinnati,  Ohio,  from  New  Orleans  to  Baton  Rouge,  120  miles, 
in  7  hours  40  min.  42  sec. 

Runs  from  New  Orleans  to  Natchez,  295  miles,  by  different  Boats. 

1814,  Orleans,  6  days  6  hours  40  mm.         I  1856,  New  Princess,  17  hours  30  min. 
1840,  Edward  Shippen,  idayS  hours.       \  1870,  R.  E.  Lee,  16  hours  36  min.  47  sec. 

Ice-"boats. 

Distances  in  Statute  Miles. 

1872,  Haze,  of  Poughkeepsie,  N.  Y.,  to  buoy  off  Milton,  4  miles,  in  4  min. 
1872,  Whiz,  of  Poughkeepsie,  N.  Y.,  to  New  Hamburg,  8.375  miles,  in  8  min. 


PASSAGES    OF    STEAMERS   AND    SAILING   VESSELS.       897 

PASSAGES  OF  STEAMERS  AND  SAILING  VESSELS. 

Distances  in  Geographical  Miles  or  Knots. 
Steamers.    Side-wheels. 

1807,  Phoenix,  of  Hoboken,  N.  J.  (John  Stevens),  New  York,  N.Y.,  to  Philadelphia, 
Penu.  First  passage  of  a  steam  vessel  at  sea. 

1814,  Morning  Star,  of  Eng.,  River  Clyde  to  London,  Eng.  First  passage  of  an 
English  steamer  at  sea. 

1817,  Caledonia,  of  Eng.,  Margate,  Eng.,  to  Cassel,  Germ.,  180  miles,  in  24  hours. 

1819,  Savannah,  of  N.Y.,  about  340  tons  0.  M.,  Tybee  Light,  Savannah  River,  Ga.. 
to  Rock  Light,  Liverpool,  Eng.,  3640  miles,  in  25  days  14  hours;  6  days  21  hours  or 
which  were  under  steam. 

1825,  Enterprise,  of  Eng.,  500  tons,  Falmouth,  Eng.,  to  Table  Bay,  Africa,  in  57 
days;  and  to  Calcutta,  India,  in  113  days.  First  passage  of  a  steamer  to  India. 

1830,  Hugh  Lindsay,  411  tons,  80  BP,  Bombay,  India,  to  Suez,  Egypt,  3103  miles, 
in  31  days  running  time. 

1837,  Atlanta,  of  Eng.,  650  tons,  Falmouth,  Eng.,  to  Calcutta,  in  91  days. 

1839,  Great  Western,  of  Eng.,  Liverpool  to  New  York,  N.  Y.,  3017  miles,  in  12 
days  1 8  hours. 

1870,  Scotia,  of  Eng.,  Queenstown,  Ireland,  to  Sandy  Hook,  N.  J.,  2780  miles,  in 
8  days  7  hours  31  min.  1866,  New  York  to  Queenstown,  2798  miles,  in  8  days  2 
hours  48  min.;  thence  to  Liverpool,  Eng.,  270  miles,  in  14  hours  59  min.;  total,  8 
days  17  hours  47  min. 

Screw. 

1874,  India  Government  Boat,  Steel,  length  87  feet,  beam  12  feet,  draught  of  water 
3.75  feet,  mean  speed  for  one  mile  20.77  miles  per  hour,  and  maintained  a  speed,  of 
18.92  miles  in  i  hour. 

1877,  Lusitania,  of  Eng.,  London  to  Melbourne,  Australia,  via  Cape,  u  445  miles, 
in  38  days  23  hours  40  min. 

Sailing  "Vessels. 

1851,  Chrysolite  (clipper  ship),  of  Eng.,  Liverpool,  Eng.,  to  Anjer,  Java,  13000 
miles,  in  88  days.  The  Oriental,  of  N.  Y.,  ran  the  same  course  in  89  days. 

1853,  Trade  Wind  (clipper  ship),  of  N.  Y.,  San  Francisco,  CaL,  to  New  York,  N.  Y., 
13610  miles,  in  75  days. 

1854,  Lightning  (clipper  ship),  of  Boston,  Mass.,  Melbourne,  Australia,  to  Liver- 
pool, Eng.,  12 190  miles,  in  64  days. 

1854,  Comet  (clipper  ship),  of  N.  Y.,  Liverpool,  Eng.,  to  Hong  Kong,  China,  13040 
miles,  in  84  days. 

1854,  Sierra  Nevada  (schooner),  of  N.  H.,  Hong  Kong,  China,  to  San  Francisco, 
Cal.,  6000  miles,  in  34  days. 

1854,  Red  Jacket  (clipper  ship),  of  N.  Y.,  Sandy  Hook,  N.  J.,  to  Melbourne,  Aus- 
tralia, 12  720  miles,  in  69  days  n  hours  i  min. 

1855,  Euterpe  (half-clipper  ship)  of  Rockland,  Me.,  New  York  to  Calcutta,  India, 
12  500  miles,  in  78  days. 

1860,  Andrew  Jackson  (clipper  ship),  of  Boston,  New  York,  N.  Y.,  to  San  Fran- 
cisco, Cal.,  13610  miles,  in  80  days  4  hours. 

1865,  Dreadnought  (clipper  ship),  of  Boston,  Honolulu,  Sandwich  Islands,  to  New 
Bedford,  Mass.,  13470  miles,  in  82  days;  and  1859,  Sandy  Hook,  N.  J.,  to  Rock 
Light,  Liverpool,  Eng.,  3000  miles,  in  13  days  8  hours. 

1865,  Sovereign  of  the  Seas  (medium  ship),  of  Boston,  Mass.,  in  22  days  sailed 
5391  miles  =  245  miles  per  day.    For  4  days  sailed  341.78  miles  per  day,  and  for  i 
day  375  miles. 

1866,  Henrietta  (schooner  yacht),  of  N.  Y.,  Sandy  Hook,  N.  J.,  to  the  Needles, 
Eng.,  3053  miles,  in  13  days  21  hours  55  min.  16  sec. 

1866,  Ariel  and  Serica  (clipper  ships),  of  England,  Foo-chou  foo  Bar,  China,  to 
the  Downs,  Eng.,  13  500  miles,  in  98  days. 

1869,  Sappho  (schooner  yacht),  of  N.  Y.,  Light-ship  off  Sandy  Hook,  N.  J.,  to 
Queenstown,  Ireland,  2857  miles,  in  12  days  9  hours  34  min. 


ELEMENTS    OF   MACHINES   AND    ENGINES. 


ELEMENTS  OF  MACHINES  AND  ENGINES. 

BLOWING    ENGINES. 

Furnaces. — Two.    Fineries. — Two.    {England.) 

240  Tons  Forge  Pig  Iron  per  Week. 

Engine  (non-condensing). — Cylinder,  20  ins.  in  diam.  by  8  feet  stroke  of  piston. 
Boilers. — Six  (plain  cylindrical),  36  ins.  in  diam.  and  28  feet*  in  length.     Grates, 
100  sq.  feet. 

Blowing  Cylinders.— Two,  62  ins.  in  diam.  by  8  feet  stroke  of  piston.  Pressure, 
2.17  Ibs.  per  sq.  inch.  Revolutions,  22  per  minute. 

Pipes,  3  feet  in  diam.=  168  area  of  cylinder. 

Tuyere*.— Each  Furnace,  2  of  3  ins.  in  diam. ;  i  of  3.25  ins. ;  and  i,  3  of  3  ins. 
Each  Finery,  6  of  1.33  ins. ;  and  i,  4  of  1.125  ins. 

Temperature  of  Blast,  600°.     Ore,  40  to  45  per  cent,  of  iron. 

Furnaces. — Eight,  diam.  16  to  iSfeet.    Dowlais  Iron  Works  (England). 

1300  Tons  Forge  Iron  per  Week;  discharging  44000  Cube  Feet  of  Air  per 

Minute. 

Engine  (non-condensing).—  Cylinder,  55  ins.  in  diam.  by  13  feet  stroke  of  piston. 

Pressure  of  Steam.— 60  Ibs.  per  sq.  inch,  cut  off  at .  33  the  stroke  of  piston.  Valves, 
120  ins.  in  area. 

Boilers.—  Eight  (cylindrical  flued,  internal  furnace),  7  feet  in  diam.  and  42  feet  in 
length ;  one  flue  4  feet  in  diam.  Grates,  288  sq.  feet. 

Fly  Wheel.—  Diam.,  22  feet;  weight,  25  tons. 

Blowing  Cylinder,  144  ins.  in  diam.  by  12  feet  stroke  of  piston. 

Revolutions,  20  per  minute.  Blast,  3.25  Ibs.  per  sq.  inch.  Discharge  pipe,  diam. 
5  feet,  and  420  feet  in  length.  Valves.—  Exhaust,  56  sq.  feet;  Delivery,  16  sq.  feet. 

Furnaces. —  LacJceriby  (England). 

800  Tons  Iron  per  Week. 

Engine  (horizontal,  compound  condensing). — 32  and  60  ins.  in  diam.  by  4.5 
feet  stroke  of  piston. 

Blowing  Cylinders.— Two,  80  ins.  in  diam.  by  4.5  feet  stroke  of  piston.    Pressure, 
4.5  Ibs.  per  sq.  inch.     Revolutions,  24  per  minute. 
Pipe,  30  ins.  in  diam. ;  volume,  12.25  times  that  of  blowing  cylinders. 
IP.— Engine,  290  Ibs. ;  Blowing  cylinders,  258;  efficiency,  89  per  cent. 
Valves.—  Area  of  admission, .  16  of  area  of  piston ;  of  exit, .  125. 
Volume.— 190000  cube  feet  of  air  are  supplied  per  ton  of  iron. 
Blcrwer   and.    Exhausting   ITan. 
The  Huyett  &  Smith  Manufacturing  Co.,  Detroit,  Mich. 


Blower. 

Grate 
Surface. 

Outlet. 

Diam. 
Pulleyi. 

Face 
Pulleys. 

Revolu- 
tions 
at  3  oz. 

Air  at 
302. 

H?at 

3oz. 

Revolu- 
tions 
at  6  oz. 

Air  at 

6  oz. 

IP  at 

6  oz. 

No. 

Sq.  Ft. 

Sq.  Ins. 

Ina. 

Ins. 

Per  Min. 

Cube  Ft. 

No. 

Per  Min. 

Cube  Ft. 

No. 

s 

4 
10 

15 
3° 

2 

3 

I 
i-5 

5500 
3500 

930 
1870 

.38 
.76 

7500 
5000 

1330 
2670 

I.I 

2.16 

} 

14 

50 

4 

2-5 

2700 

3  120 

1.27 

4000 

4440 

3.63 

4 

18 

75 

5 

3-25 

2OOO 

4680 

I.QI 

3000 

6670 

5-5 

5 

26 

"5 

6 

4-25 

1500 

7830 

3-2 

2300 

II  100 

9.1 

6 

36 

'75 

7 

5-25 

1300 

10900 

4-47 

1800 

15600 

12.8 

7 

56 

280 

9 

6.25 

1000 

1750° 

7-iS 

1400 

24900 

20.4 

*  40  feet  would  have  afforded  economy  in  fael. 


ELEMENTS   OF   MACHINES   AND   ENGINES. 


899 


COTTON   FACTORIES.       (English.) 

For  driving  22060  Hand-mule  Spindles,  with  Preparation,  and  260  Looms, 
with  common  Sizing. 

Engine  (condensing).—  Cinder,  37  ins.  in  diam.  by  7  feet  stroke  of  piston; 
volume  of  piston  space,  53.6  cube  feet. 

Pressure  of  Steam.—  (Indicated  average)  16.73  Ibs.  per  sq.  inch.  Revolutions,  17 
per  minute. 

Friction  of  Engine  and  Shafting.—  (Indicated)  4.75  Ibs.  per  sq.  inch  of  piston. 

IIP,  125.    Total  power  =  i.     Available,  deducting  friction  =  .  717. 

{305  hand  mule  spindles,  vrith  preparation, 
S*£3«SSf?    « 
or    10.5  looms,  with  common  sizing. 
Including  preparation  : 

i  throstle  spindle  =  3  hand-mule,  or  2.25  self-acting  spindlea 
x  self-acting  spindle  =  1.2  hand-mule  spindles. 


DREDGING   MACHINES. 

Dredging  20  Feet  from  Water-line,  or  180  Tons  of  Mud  or  Silt  per  Hour 
ii  Feet  from  Water-line. 

Length  upon  deck,  123  feet  ;  beam,  26  feet.    Breadth  over  all,  41  feet. 

Immersed  section  at  load-line,  60  sq.  feet.  Displacement,  141  tons,  at  load-draught 
of  '2-83  feet. 

Engine  (non  -condensing).—  Cylinders,  two,  12.125  ms-  in  diam.  by  4  feet  stroke 
of  piston. 

Boilers.  —  Two  (cylindrical  flue),  diam.  40.5  ins.,  and  length,  20  feet  3  ins.;  two 
flues,  14.625  ins.  in  diam.  Heating  surface,  617  sq.  feet.  Grates,  37  sq.  feet. 

Pressure  of  Steam,  25  Ibs.  per  sq.  inch;  throttle  .25  open,  cut  off  at  .5  the  stroke 
of  piston.  Revolutions,  42  per  minute. 

Buckets.—  Two  sets  of  12,  2.5  feet  in  length  by  15  ins.  at  top  and  2  feet  deep;  vol- 
ume, 6.25  cube  feet.  Chain  Links,  8  ins.  in  length  by  .5  inch  diam. 

Scows  or  Camels.—  Four,  of  40  tons  capacity  each. 

STEAM   HOPPER   DREDGER.       (Wm.  Simons  $  Co.) 


"NEPTUNE"  (English).— Length,  1 50  feet;  breadth,  32  feet. 

Dredge  'from  6  Ins.  to  25  Feet.    Capacity  of  Hopper,  500  to  600  Tons. 

Engines. — Two  (compound),  375  IP,  for  dredging  and  propulsion,  and  one  for 
raising  bucket-frame  and  anchor-posts. 

A  like  designed  dredger  ot  1000  tons'  capacity  has  dredged  25  ooo  tons  silt  per 
week  and  transported  it  4  miles. 

Dredging  1000  Tons  of  Mud  or  Silt  per  Hour,  5  to  35  Feet  in  Depth. 

Capacity  of  Hopper,  1000  Ton*. 
Engines. — Two  (compound),  IP  1000.    Speed. — 9  knots  per  hoar. 

Steam    IDredging    Crane,     (English.) 
Lift,  30  Feet  per  Hour. 


21  280 
24640 


Volu 
of 
Buck 


LbB. 
II2O 

1680 


Tons. 

25 

37-5 


and 
d. 


32 


III 

w    o 


C.YdB. 


Weight 
f  Crane 


Lbs. 
18000 
33480 


Tons. 
5 
7 


lume 
of 
cket. 


Lbs. 
2240 
3360 


Tons. 
50 
60 


an 
d. 


30 
40 


9OO  ELEMENTS    OF   MACHINES    AND    ENGINES. 

Electric   Lanncli.     Steel. 

"HILDA,"  "MARY,"  "FLO,"  and  "THEO."—  Length,  40  feet;  Beam,  6.5;  Hold, 
3.  i.— Load-draught,  40  passengers,  1.66  feet.  Motor,  B?  3.5.  Revolutions,  700  psi 
minute.  Speed,  6  miles  per  hour. 

Accumulators,  under  the  seats,  and  when  fully  charged,  capacity  for  8  hours  at 
full  speed.  Charging  is  effected  at  landings  at  termination  of  route. 

Builders.— J.  B.  Seath  &  Co.,  Glasgow,  Scotland. 

HOPPER  DREDGER  "BELFAST  No.  3."  IRON  AND  STEEL. — Length  over  all,  190  feet,, 
on  deck,  189;  between  perpendiculars  and  for  tonnage,  185;  Beam,  -$S.$feet;  Hold,, 
14.1  feet;  Tonnage,  Gross,  760  tons;  Net,  372;  Mean  draught,  g. 5  feet,  loaded,  12.5. 

Displacement,  1860  tons.     Immersed  Section,  490  dfeet.     Freeboard,  2.75  feet. 

Dredging  Capacity,  1000  tons  per  hour. 

Cylinders.    Two  of  20  ins.  in  diam.  and  two  of  38. 5  ins.     Stroke  of  piston  30  ins. 

Pressure  of  Steam,  go  Ibs.  per  ninch.     Revolutions  per  minute,  80.     IP  850. 

Boilers,  two.  Grate  surface,  81  nfeet.  Heating  surface  2120,  and  Condensing  1150. 
Propeller,  9  feet  in  diameter.  Fuel,  capacity  50  tons.  Crew,  13. 

Weight,  Hull,  500  tons.     Speed,  8. 5  knots  per  hour. 

Builders.—  Wm.  Simons  &  Co.,  Renfrew,  Scotland. 

"HERCULES,"  Panama  Canal  — Length  on  deck,  100  feet ;  beams,  40,  60,  and  45 
feet;  depth  of  hold,  12  feet.     Slot,  36  feet  in  length  by  6  feet  7  ins.  in  width. 
Ways. — Two,  one  40  feet  and  one  60  feet,  by  5  feet  in  width. 
Buckets. — 38;  volume,  1.33  cube  yards.     Spuds,  2  feet  in  diam.  and  60  in  length. 
Engines. — Two  of  100  BE*  each,  and  two  of  40  B?  each. 
Boilers.— Three  (horizontal  tubular),  16  feet  in  length. 
Elevator  and  Discharge.  —Maximum,  24  cube  yards  per  minute. 

Crane.     CWood.) 

Hull— Length  on  deck,  loofeet;  beam,  wfeet;  load-draught,  4. 5  feet. 
Radius  of  crane,  46  feet;  height,  70  feet;  counter-balance,  70  tons. 
Boiler.— Heating  surface,  500  sq.  feet.    Pressure  of  Steam,  80  Ibs.  per  sq.  inch, 
IB?,  150. 

Propellers.— Two,  4.25  feet  in  diam.    Speed,  5  miles  per  hour. 
Engine  to  operate  crane.    Cylinder.— 10  ins.  in  diam.  by  12  ins.  stroke  of  piston. 

FLOUR   MILLS. 

30  Barrels  of  Flour  per  Hour. 

Water- wheels,  Overshot.—  5,  diam.  18  feet  by  14.5  feet  face.  Buckets,  15 
ins.  in  depth.  Water.—  Head,  2.5  feet.  Opening,  2.5  ins.  by  14  feet  in  length  ovei 
each  wheel. 

5  Barrels  of  Flour  per  Hour,  and  Elevating  400  Bushels  of  Grain  36  Feet. 

Water- wheel,  Overshot.  —  Diam.  22  feet  by  8  feet  face.  Buckets,  52  of  i 
foot  in  depth.  Water.—  Head,  from  centre  of  opening,  25  ins.  Opening,  1.75  ins. 
by  80  ins.  in  length. 

Revolutions,  3.5  per  minute.    Stones,  three  of  4.5  feet;  revolutions,  130. 

Three  Run  of  Stones,  Diameter  4  Feet. 

Water-wheel,  Overshot.— Diam.  19  feet  by  8  feet  face.  Buckets,  14  ins.  ifi 
depth. 

Or, 

Steam-engine  (non-condensing).—  Cylinder,  13  ins.  in  diam. by  4  feet  stroke, 
Boiler  (cylindrical  flued).— Diam.  5  feet  by  30  in  length;  two  flues  20  ins.  in  diain. 


ELEMENTS   OP   MACHINES   AND   ENGINES. 


901 


HOISTING   ENGINES. 

ITor   3Pile   Driving,  Hoisting,  Mining,  etc. 
Uidgerwood   ]Vtanuf  g   Co.,  Ne\v   York. 


SINGLE  CYLINDERS. 

DOUBLE  CYLINDERS. 

H» 

Cylinder. 

Capacity. 

Cost,  with 
Boiler.* 

H> 

Cylinder. 

Capacity. 

Cost,  with 
Boiler.* 

H7T 

Ins. 

Lbs. 

$ 

No. 

Ins. 

Lbs. 

$ 

4 

5X5 

IOOO 

600 

8 

5X8 

2OOO 

950 

6 

6X8 

I25C 

>  *   • 

67i 

> 

12 

6X8 

2500 

I 

050 

10 

7X  10 

1800 

825 

2O 

7  X  10 

3500 

1350 

15 

8  X  10 

2800 

1050 

30 

8  X  10 

6000 

550 

20 

9  X  12 

4000 

1275 

40 

9X12 

8000 

2000 

25 

10  X  12 

5000 

'375 

50 

10  X  12 

9000 

2350 

*  Complete. 

Details   and.   Operation. 

Boiler. 

Leaders. 

Lift. 

Blows 

Piles 

Fuel 

Engine. 

Drum. 

Dimen- 
sions. 

Tubes. 

Ram. 

Hoist. 

Ram. 

per 
Minute. 

per  10 
Hours. 

H?ir. 

H> 

Ins. 

Ins. 

No. 

Lbs. 

Feet. 

Feet. 

No. 

No. 

Lbs. 

10* 
20 

12  X  24 
14  X  26 

32X75 
40X84 

48  of  2  in. 
80  of  2  in. 

1953 
2700 

40 
75 

8  to  12 

8  to  12 

25 
29 

50 
100 

£ 

*  Weight  complete,  8500  Ibs. 

Mining   Engines    and    Boilers.     ( Various  Capacities. ) 
Engine,  Boiler,  etc.,  as  given  for  Pile  Driving,  page  902. 

Operation.  —  250  to  300  tons  of  coal  in  10  hours.     Fuel,  40  IDS.  coal  per  hour. 
Water,  20  gallons  per  hour. 
Weight  of  Engine  and  Boiler,  4500  Ibs. 

The    Hancock    Inspirator.     For  a  Lift  of  Water  oj  25  Feet 


No. 

Diair 

Steam-pipe. 

eter. 
Suction. 

Discharge 
at  Pressure 
of  60  Lbs. 

No. 

Diam 
Steam-pipe. 

eter. 
Suction. 

Discharge 
at  Pressure 
of  60  Lbs. 

Ins. 

Ins. 

G'lls.perh'r. 

Ins. 

Ins. 

G'lls.perh'r. 

10 

•375 

•5 

120 

30 

1.25 

i-5 

1260 

12.5 

•  5 

•75 

220 

35 

1.25 

'•5 

1740 

15 

•5 

•75 

300 

40 

1-5 

2 

2230 

20 

•75 

i 

540 

45 

i'5 

2 

2820 

25 

i 

1.25 

900 

50 

2 

2-5 

3480 

Temperature  of  feed  water  at  20  feet  lift,  100°  ;  and  on  3  feet  lift,  145°. 

HYDROSTATIC   PRESS.       (Cotton.) 

30  Bales  of  Cotton  per  Hour. 
Engine  (non-condensing).— Cylinder,  10  ins.  in  diam.  by  3  feet  stroke  of  piston. 

Pressure  of  Steam,  50  Ibs.  per  sq.  inch,  full  stroke.  Revolutions,  45  to  60  per 
minute. 

Presses.— Two,  with  J2-inch  rams;  stroke,  4.5  feet. 
Pumps.— Two,  diam.  2  ins. ;  stroke,  6  ins. 

For  83  Bales  per  Hour. 

Engine  (non-condensing).— Cylinder,  14  ins.  in  diam.  by  4  feet  stroke  of  piston. 

Boilers.  —Three  (plain  cylindrical).  30  ins.  in  diam.  and  26  feet  in  length.  Gratet, 
32  sq.  feet.  Pressure  of  Steam,  40  Ibs.  per  sq.  inch.  Revolutions,  60  per  minute. 

Presses.—  Four,  geared  6  to  i,  with  two  screws,  each  of  7.5  ins.  in  diam.  by  1.625 
to  pitch. 

Shaft  (wrought  iron).- Journal,  8.5  ins.  fly  Wheel,  16  feet  in  diam. :  weight, 
8960  Ibs. 


QO2 


ELEMENTS    OF    MACHINES    AND    ENGINES. 


LOCOMOTIVE. 

"EXPERIMENT"  (Compound)*— Cylinders,  one  each,  12  and  26  ing.  in  diam.,  and 
•lie  26  ins.  by  2  feet  stroke  of  piston. 

Boiler. — Heating  surface,  1083.5  sq.  feet.  Grate,  17.1  sq.  feet.  Pressure  of  Steam, 
150  Ibs.  per  sq.  inch,  cut  off  at  .35.  Speed,  50  miles  per  hour.  Weight.—  Empty, 
34.75  tons. 

Street   Railroad   or   Tramway   Engine. 

Cylinder,  7  ins.  in  diam.  by  n  ins.  stroke  of  piston. 

Boiler,  78  tubes  1.75  ins.  in  diam.  by  4  feet  in  length.  Heating  surface,  160  sq. 
feet.  Grate,  4.25  sq.  feet.  Wheels,  2.33  feet  in  diam.  Base,  4.5  feet.  Gauge,  4 
feet  8.5  ins. 

Cost.— Average  per  mile  in  England,  2.52  pence  sterling  =  4. 48  cents. 

PILE-DRIVING. 

Driving  One  Pile. 

Engine  (non-condensing) — Cylinder,  6  ins.  in  diam.  by  i  foot  stroke  of  piston. 

Boiler  (vertical  tubular).— 32  ins.  in  diam.,  and  6.166  feet  in  height.  Grates,  3.7 
sq.  feet.  Furnace,  20  ins.  in  height  Tubes,  35,  2  ins.  in  diam.,  4.5  feet  in  length. 

Revolutions,  150  per  minute.  Drum,  12  ins.  in  diam.,  geared  4  to  i.  Leader,  40 
feet  in  height.  Ram.—  2000  Ibs.,  2  blows  per  minute.  Fuel,  30  Ibs.  coal  per  hour. 

Driving  Two  Piles. 

Engine  (non-condensing).—  Cylinders,  two,  6  ins.  in  diam.  by  18  ins.  stroke  of 
piston. 

Boiler  (horizontal  tubular).—  Shell,  diam.  3  feet,  and  6  feet  in  length.  Furnace 
end  3.75  feet  in  width,  3.5  feet  in  length,  and  6  feet  in  height. 

Pressure  of  Steam,  60  Ibs.  per  sq.  inch.    Revolutions,  60  to  80  per  minute. 

Frame,  8.5  feet  in  width  by  26  feet  in  length.  Leaders,  3  feet  in  width  by  24  feet 
in  height.  Rams.—  Two,  1000  Ibs.  each,  5  blows  per  minute. 

PUMPING   ENGINES. 

CORLISS  STEAM-ENGINE  Co.,  Providence,  R.  7. —VERTICAL -BEAM  ENGINE  (Com- 
pound).—Cylinders.—  1 8  and  36  ins.  in  diam.  by  6  feet  stroke  of  piston. 

Pumps.—  Four  plunger,  19  ins.  in  diam.  hy  3  feet  stroke  of  piston.  Displacement 
per  revolution  of  engine,  84.96  cube  feet. 

Boilers.— Three,  vertical  fire  tubular.  Grate. — 93  sq.  feet.  Heating  surface,  1680 
sq.  feet.  Pressure  of  Steam,  125  Ibs.  per  sq.  inch,  cut  off  at  .22  feet.  Revolutions, 
36  per  minute.  IIP  313.  Fly-wheel. — 25  feet  in  diam.,  weight  62000  Ibs. 

Fuel. — Cumberland  coal,  486  Ibs.  per  hour,  inclusive  of  kindling  and  raising  steam. 
Ash  and  Clinkers,  9.4  per  cent.  Duty  for  one  week,  113  271  ooofoot-lbs. 

Water  delivered,  17  621  gallons  per  minute,  against  head  of  180  feet. 

Duty,  average  for  1883,  per  100  Ibs.  anthracite  coal,  106  048  ooofoot-lbs. 

For  Elevating  200000  Gallons  of  Water  per  Hour. 

LYNN,  Mass.—  ENGINE  (Compound).— Cylinders,  17.5  and  36  ins.  in  diam.  by  7  feet 
stroke  of  piston;  volume  of  piston  space,  61.2  cube  feet.  Air  Pump  (double  act- 
ing), 11.25  ins-  in  diam.  by  49.5  ins.  stroke  of  piston. 

Pump  Plunger,  18.5  ins.  in  diam.  by  7  feet  stroke. 

Boilers. — Two  (return  flued),  horizontal  tubular;  diam.  of  shell,  5  feet;  drum,  3 
feet ;  tubes,  3  ins.  Length  of  shell,  16  feet.  Grates,  27.5  sq.  feet. 

Pressure  of  Steam,  90.5  Ibs. ;  average  in  high-pressure  cylinder,  86  Ibs.,  cut  off  at 
i  foot,  or  to  an  average  of  44.5  Ibs. ;  average  in  low-pressure  cylinder,  27  Ibs.,  cut 
off  at  6  ins. ,  or  to  an  average  of  10. 8  Ibs. 

Revolutions,  18.3  per  minute.     Fly  Wheel. — Weight,  24000  Ibs. 

Evaporation  of  Water,  4644  Ibs.  per  hour.     Loss  of  action  by  Pump,  4  per  cent. 

Consumption  of  Coal. — Lackawanna,  291  Ibs.  per  hour. 

Duty,  205772  gallons  of  water  per  hour,  under  a  load  and  frictional  resistance  of 
73.41  Ibs.  per  square  inch,  equal  to  103923217  foot-lbs.  for  each  100  Ibs.  of  coal. 


ELEMENTS   OF   MACHINES,  MILLS,  ETC. 


903 


uGa8lcitt"at  Saratoga,  N.T. 

Engine  (Horizontal  Compound).  Cylinders.—  High  pressure,  2  of  21  ins.  diam. 
Low  pressure,  2  of  42  ins.  uiani.,  all  3  feet  stroke  of  piston  Pumps. — Two  of  20  ins. 
diam.  by  3  feet  stroke  of  piston. 

Fly  Wheel,  12.33  f66*1  'n  diam. ;  weight,  1-2000  Ibs. 

Boilers  (horizontal  tubular). — Two  of  5.5  feet  in  diam.  by  18  feet  in  length.  Heat- 
ing surface,  2957  sq.  feet.  Grates,  51  sq.  feet  of  grate;  to  heating  surface,  i  to  58, 
and  to  transverse  section  of  tubes,  i  to  7.  Chimneys,  75  feet. 

Pressure  of  Steam.—  Mean  of  20  hours,  74.25  Ibs.  per  sq.  inch.  Revolutions,  17.87 
per  minute.  IIP. — High:pressure  cylinders,  109.2;  low-pressure,  76  65.  Total,  185.8. 

Fuel—  Anthracite,  6.9  Ibs.  per  sq.  foot  of  grate  per  hour.  Evaporation,  per  sq. 
foot  of  heating  surface  per  hour,  1.175  Ibs. ;  per  Ib.  of  coal,  9.25  Ibs. ;  per  cent,  of 
non-combustible,  3.2. 

Duty,  112  899993  foot-lbs.  per  100  Ibs.  coal.    Heating  surface  per  IIP,  14.9. 

Steam  per  sq.  foot  of  surface  per  hour,  1. 19  Ibs. ;  per  sq.  foot  of  surface  per  Ib.  of 
coal  per  hour  from  212°,  11.28  Ibs. 

Ericsson's  Caloric.     For  an  Elevation  of  50  Feet. 


Dimen- 

Space 
occupied. 

Volume 

Pipes, 
Suction 
and 

Fuel 
per  Hour. 

Furnace. 

C  U  S  T 

Deep  Well  Pump. 
Extra. 

sions. 

Floor.  (Height. 

«Sur. 

Dis- 
charge. 

Nut 
Anthr. 

Gas. 

Gas. 

Coal. 

Pump. 

Pipes  P€ 
Plain. 

r  Foot. 
Galvan. 

Ins. 

Ins. 

Ins. 

Gall. 

Ins. 

Lbs. 

Cub.  ft. 

$ 

* 

$     . 

$ 

T~ 

5 

34Xi8 

48 

ISO 

•75 

— 

15 

150 

6 

39X20 

200 

•75 

2-5 

18 

200 

210 

— 

— 

— 

8 

48X21 

63 

350 

i 

25 

235 

250 

10 

.64 

.86 

12 

54X27 

63 

800 

1.5 

6 

320 

J5 

.80 

IeI5 

12* 

42X52 

65 

1600 

2 

12 

— 

— 

450 

.92 

1.25 

*  Over  90  feet,  92  cents.                                            t  Duplex. 

Including  engine  and  pump,  oil-can  and  wrench,  complete  in  all  but  suction  and 
discharge-pipe. 

SUGAR   MILLS. 

Expressing  40  ooo  Ibs.  Cane-juice  per  day,  or  for  a  Crop  of  5000  Boxes  of 
450  Ibs.  each  in  four  Months1  Grinding. 

Engine  (non-condensing).—  Cylinder,  18  ins.  in  diam.  by  4  feet  stroke  of  piston. 

Boiler  (cylindrical  flued).  —  64  ins.  in  diam.  and  36  feet  in  length  ;  two  return  flues, 
20  ins.  in  diam.  Heating  surface,  660  sq.  feet.  Grates,  30  sq.  feet. 

Pressure  of  Steam,  60  Ibs.  per  sq.  inch,  cut  off  at  .  5  the  stroke  of  piston.  Revolu 
tions,  40  per  minute. 

Rolls.  —  One  set  of  3,  28  ins.  in  diam.  by  6  feet  in  length;  geared  i  to  14.  Shafts, 
ii  and  12  ins.  in  diam.  Spur  Wheel,  20  feet  in  diam.  by  i  foot  in  width.  Fly 
Wheel,  1  8  feet  in  diam.  ;  weight,  17  400  Ibs. 

Weights.  —  Engine,  61  460  Ibs.  ;  Sugar  Mill,  65730  Ibs.  ;  Spur  Wheel  and  Connect 
ing  Machinery  to  Mill,  28  680  Ibs.  ;  Boiler,  18  520  Ibs.  ;  Appendages,  6730  Ibs.  Total, 
181  120  IbS. 


STONE    AND    ORE    BREAKERS.       (See  p  957-) 


No. 

Re- 
ceiver. 

Pul 
D'm. 

ey- 

Face. 

>    § 

&>      cr 

Weight. 

No. 

Re- 
ceiver. 

Ins. 

Feet. 

Ins. 

Feet. 

I-P. 

Lbs. 

Ins. 

A 

•I 

4X10 
5X10 

1.66 
2-75 

6 
6 

250 

iSo 

4 

5 

4000 
6700 

1 

9X15 
IlXlS 

3 

7Xio 

2 

7-5 

250 

6 

8000 

7 

13X15 

3 

5Xi5 

2-33 

8 

180 

9 

9  loo 

8 

15X20 

4 

7X15 

2-33 

9 

1  80 

9 

10490 

9 

18X24 

Face. 

In7 


> s 

Feet. 


8 


1 80 
150 
125 


Feet. 
'•5 

'•33 
'  33 

M 

NOTK.— Amount  of  product  depends  on  distance  jaws  are  set  apart,  and  speed. 
Product  given  in  Table  is  due  when  jaws  are  set  1.5  ins.  open  at  bottom,  and  ma- 
chine is  run  at  its  proper  speed  and  diligently  fed.  It  will  also  vary  somewhat  with 
character  of  stone.  Hard  stone  or  ore  will  crush  faster  than  sandstone. 

A  cube  yard  of  stone  is  about  one  and  one  third  tons. 


Weight. 


904 


ELEMENTS   OF   MACHINES. — CHIMNEYS. 


STEAM   FIRE-ENGINE. 
Airioslieag,  N".  H.     1st    Class* 

Steam  Cylinder.— Two  of  7.625  ins.  in  diara.  by  8  ins.  stroke  of  piston. 
Water  Cylinder. — Two  of  4. 5  ins.  in  diam. 

Boiler  (vertical  tubular). — Heating  surface,  175  sq.  feet.     Grates,  4.75  sq.  feet; 
Pressure  of  Steam. — 100  Ibs.  per  sq.  inch.     Revolutions,  200  per  minute. 
Discharges.— Two  gates  of  2.5  ins.,  through  hose,  one  of  1.25  ins.  and  two  of  i  inclk 
Projection.—  Horizontal,  1.25  ins.  stream,  311  feet;  two  i  inch  streams,  256  feet 
Vertical,  1.25  ins.  stream,  200  feet.   Water  Pressure. — With  1.125  ins.  nozzle,  200  Iba 
Time  of  Raising  Steam.— From  cold  water,  25  Ibs.,  4  rain.  45  sec. 
Weights.—  Engine  complete,  6000  Ibs. ;  water,  300  Ibs. 


SAW-MILL. 

Two  Vertical  Saws,  34  Ins.  Stroke,  Lathes,  etc. 

Engine  (non-condensing).  Cylinder.— 10  ins.  in  diam.  by  4  feet  stroke  of  piston. 
Boilers.— Three  (plain  cylindrical),  30  ins.  in  diam.  by  20  feet  in  length. 
Pressure  of  Steam.— go  Ibs.  per  sq.  inch.    Revolutions,  35  per  minute. 
NOTE. — This  engine  has  cut,  of  yellow-pine  timber,  30  feet  by  18  ins.  in  i  minute. 

STONE   SAWING. 

Emerson  Stone  Saw  Co.  (Diamond  Stone  Saw,  Pittsburgh,  Penn.).— 
to  IP,  150  sq.  feet  of  Berea  sandstone,  inclusive  of  both  sides  of  cut,  in  i  hour. 

CHIMNEYS. 

LAWRENCE,  Mass.  Octagonal,  222  Feet  above  Ground,  and  19  Feet  below. 
Foundation^  35  Feet  square  and  of  Concrete  7  Feet  deep.  (Hiram  F.  Mills.) 

Shaft.— 234  feet  in  height,  20  feet  at  base,  and  11.5  at  top;  28  ins.  thick  at  base 
and  8  at  top.  Core. — 2  feet  thick  for  27  feet,  and  i  foot  for  154. 

Horizontal  Flues. — 7.5  feet  square,  and  Vertical  flue  or  cylinder  of  8.5  feet,  234 
high,  with  walls  20  ins.  thick  for  20  feet,  16  for  17  feet,  12  for  52  feet,  and  8  for  145  feet. 

Purpose.— For  700  sq.  feet  grate  surface.     Weight.— 2250  tons.    Bricks,  550000. 

NEW  YORK  STEAM  HEATING  Co.  Quadrilateral,  220  Feet  above  Ground 
and  i  Foot  below.  (Chas.  E.  Emery,  Ph.D.) 

Shaft.— 220  feet  in  height,  and  27  feet  10  ins.  by  8  feet  4  ins.  in  the  clear  inside. 
Foundation.— i  foot  below  high  water.     Capacity.—  Boilers  of  16000  H>. 

Cost  of*  Steam-Engines  and.  Boilers  complete,  and  of 
Operation  per  Day  of  1O  Hours,  inclusive  of  .Labor, 
Fuel,  and  Repairs.  (Chas.  E.  Emery,  Ph.D.) 


IFF. 

Engine. 

Water 
orate< 
IHPper 
Hour. 

Evap- 
per 
Lb.of 
Coal. 

Coa 
IIP. 

Iper 
Day. 

Labor. 

Sup- 
plies 
and  Re- 
pairs. 

Cost 
of 
Coal.* 

Total 
Cost  of 
Operafn, 
including 
Coal. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

$ 

* 

$ 

$ 

6.25 

Portable  Vertical  (  &* 

42 

7-5 

56 

394 

•33 

•73 

2.86 

12.5 

"                   "          )  °1: 

38 

717 

1.75 

.41 

3.56 

29 

Horizontal  (  ^  8 

32 

8 

40 

1308 

2.25 

.60 

2-43 

5-45 

112 

Single  Condensing.  .  . 

23 

8.8 

26.1 

3-75 

1.17 

6.14 

11.66 

276 

552 

ci                ei 

22.2 
22.2 

8.8 
8.8 

25.2 
25.2 

15663 

r5 

2.12 
4.02 

14.58 
29.16 

22.27 
4'-  5* 

$  4.42  per  ton  (2240  Ibs.),  including  cartage. 


GEAPHIC    OPERATION. 


90S 


GRAPHIC  OPERATION. 
Solutions  of*  Questions   "by   a  Grraphic  Operation. 

1.  If  a  man  walks  5  miles  in  i  hour,  how  far  win  he  walk  in  4  hours? 

Operation.—  Draw  horizontal  line,  divide  it  into  equal  parts, 
as  i,  2,  3,  and  4,  representing  hours.  From  each  of  these 
points  let  fall  vertical  lines  A  C,  i  i,  etc.,  and  divide  A  C  into 
miles,  as  5,  10,  15,  and  20,  and  from  these  points  draw  equi- 
distant lines  parallel  to  the  horizontal. 

Hence,  the  horizontal  lines  represent  time  or  hours,  and 
the  vertical,  distance  or  miles. 
Therefore,  as  any  inclined  line  in  diagram  represents  both 

time  and  distance,  course  of  man  walking  5  miles  in  an  hour 

2       3      4  is  represented  by  diagonal  Ae;  and  if  he  walks  for  4  hours, 
continue  the  time  to  4,  and  read  off  from  vertical  line  A  C  the  distance  =  20  miles. 

2.  How  far  will  a  man  walk  in  2  hours  at  rate  of  10  miles  in  i  hour  ? 
His  course  is  shown  by  the  line  A  o,  representing  20  miles. 

3.  If  two  men  start  from  a  point  at  the  same  time,  one  walking  at  the 
rate  of  5  miles  in  an  hour  and  the  other  at  ro  miles,  how  far  apart  will  they 
be  at  the  end  of  2  hours?  .  ^  ,«| 

Their  courses  being  shown  by  the  lines  A  r  and  A  o,  the  distance  r  o  represents 
the  difference  of  their  distances,  10  *>  20=  10  miles. 

4.  How  long  have  they  been  walking? 

Their  courses  are  now  shown  by  the  lines  A  o  and  A  4,  the  distance  2  4  represents 
the  difference  of  their  times,  or  2  *\»  4  =  2  hours. 

5.  When  they  are  10  miles  apart,  how  long  have  they  been  walking  ? 
Their  courses  are  again  shown  by  the  lines  A  r  and  A  o,  the  distance  r  o  repre- 
sents the  difference  of  their  distances  of  10  miles,  and  A '2,  2  hours. 

6.  If  a  man  walks  a  given  distance  at  rate  of  3.5  miles  per  hour,  and  then 
runs  part  of  distance  back  at  rate  of  7  miles,  and  walks  remainder  of  dis- 
tance in  5  minutes,  occupying  25  minutes  of  time  in  all,  how  far  did  he  run  ? 

Operation.— Draw  horizontal  line,  as  A  C, 
representing  whole  time  of  25  minutes;  set 
off  point  e  representing  a  convenient  fraction 
of  an  hour  (as  10  minutes),  and  a  i  equal  to 
corresponding  fraction  of  3.5  miles  (or  .5833); 
draw  diagonal  A  n,  produced  indefinitely  to  0, 
and  it  will  represent  the  rate  of  3. 5  miles  per 
hour. 

Set  off  C  r  equal  to  5  minutes,  upon  same 
scale  as  that  of  A  C;  let  fall  vertical  r  s,  and 
draw  diagonal  C  u  at  same  angle  of  inclination 
as  that  of  A  n;  then  from  point  u  draw  diagonal  u  O,  inclined  at  such  a  rate  as  to 
represent  7  miles  per  hour;  thus,  if  i  n  represents  rate  of  3.5  miles,  s  0,  being  one 
half  of  the  distance,  will  represent  7  miles. 

The  whole  distance  between  the  two  points  fe  thus  determined  by  C  x,  and  dis- 
tance ran  by  u  s,  measured  by  scale  of  miles  employed. 

Verification. — The  distances  A  e  and  A  i  are  respectively  10  minutes  rr.i66  of  an 
hour,  and  .5833  mile  =  .i66  of  3.5  miles.  Hence,  C  x  =  . 875  mile,  and  us  =  .5833 
mile.  Consequently,  the  man  walked  A  0=1.875  mile  =  15  minutes,  ran  Qu= 
.5833  mile  =  5  minutes,  and  walked  u  C  =  .2916  mile. 

7.  If  a  second  man  were  to  set  out  from  C  at  same  time  the  man  referred 
to  in  preceding  question  started  from  A,  and  to  walk  to  A  and  return  to  C, 
at  a  uniform  rate  of  speed  and  occupying  same  time  of  25  minutes,  at  which 
points  and  times  will  he  meet  the  first  man  ? 

Operation. — As  A  C  represents  whole  time,  and  Cx  distance  between  the  two 
points,  v  z  and  t  x  will  represent  course  of  second  man  walking  at  a  uniform  rate, 
and  he  will  meet  the  first  man,  on  his  outward  course,  at  a  distance  from  his  start- 
ing-point of  A,  represented  by  A  o,  and  at  the  time  A  a;  and  on  his  return  course 
at  distance  A  v.  x  m,  and  at  the  time  A  c. 


906 


MISCELLANEOUS. 


MISCELLANEOUS. 

No.,  Diameter,  and   Number  of  Shot.    (American  Standard.} 
Compressed    Buck    Shot. 


No. 

Diam. 

Shot 
per  Lb. 

No. 

Diam. 

Shot 
per  Lb. 

No. 

Diam. 

Shot 
per  Lb. 

3 

2 

Inch. 
•25 
.27 

No. 
284 
232 

o 

Inch. 
•3 
•32 

No. 
173 
140 

00 

ooo 

Inch. 

% 

No. 
US 
98 

Balls,  .38  Inch,  85  No.  per  Ib. ;  .44  Inch,  50  No.  per  Ib. 


Chilled   Shot. 


Diam. 

Shot 
perOz. 

No. 

Diam. 

Shot 
perOz. 

No. 

Diam. 

Shot 
perOz. 

No. 

Diam. 

Shot 
per  Oz. 

Inch. 

No. 

Inch. 

No 

Inch. 

No. 

Inch. 

No. 

•05 

2385 

9 

.08 

585 

6 

.11 

223 

i 

.16 

73 

.06 

1380 

8 

Trap 

•  495 

5 

.12 

172 

B 

•17 

61 

Trap 
.07 

1130 
868 

8 
7 

"Trap 

409 
345 

4 
3 

•13 
.14 

I36 
109 

BB 
BBB 

.18 
.19 

52 
43 

Trap 

7,6 

7 

.i 

299 

2 

•15 

88 

Drop   Shot. 


No. 

Diam. 

Pellets 
per  Oz. 

No. 

Diam. 

Pellets 
per  Oz. 

No. 

Diam. 

Pellets 
perOz. 

No. 

Diam. 

Pellets 
perOz. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Extra  Fine  Dust 

.015 

84021 

9 

Trap 

688 

5 

.12 

168 

BBB 

.19 

42 

Fine  Dust 
Dust 

•03 
.04 

10784 
4565 

.08 
Trap 

568 
472 

4 

3 

•13 
.14 

132 
106 

T 

.2 

36 

12 

•05 

2  326 

8 

.09 

399 

2 

•15 

86 

TT 

.21 

31 

ii 

.06 

1346 

7 

Trap 

338 

I 

.16 

7i 

p 

10 

Trap 

1056 

7 

.1 

291 

B 

•17 

59 

.22 

27 

10 

.07 

848 

6 

.11 

218 

BB 

.18 

50 

FF 

•23 

24 

The  scale  of  the  Le  Roy  standard  (adopted  by  the  Sportsman's  Convention)  com- 
mences  with  .21  inch  for  TT  shot,  and  reduces  .01  inch  for  each  size  to  .05  inch  for 
No.  12.  The  number  of  pellets  per  oz.  being  the  actual  number  in  perfect  shot. 

The  number  of  pellets  by  this  standard  is  nearly  identical  with  that  of  the  Amer- 
ican Standard. 

Tatham's  scale  is  same  as  Le  Roy's,  but  number  of  pellets  is  deduced  mathemat- 
ically, by  computing  them  from  the  specific  gravity  of  the  lead. 

Drains,  Diameter  and   Q-rade    of,  to  Discharge   Rainfall. 

Diam. 


Diam. 

Grade 
one  in. 

Acres. 

Diam. 

Grade 
one  in. 

Acres. 

Ins. 

Ins. 

4 

30 

•5 

40 

.2 

20 

.6 

20 

•5 

5 

80 

•  5 

7 

20 

.2 

60 

.6 

60 

•5 

'JO 

i 

8 

120 

•5 

6 

60 

i 

80 

.8 

Grade 
one  in. 

Acres. 

Diam. 

Grade 
one  in. 

Acres. 

Ins. 

60 

2.1 

80 

5-8 

I2O 

2.1 

15 

240 

7-8 

80 

2-5 

1  20 

7.8 

60 

2-75 

80 

9 

120 

4-5 

60 

10 

80 

5-3 

18 

240 

10 

British  and  Metric  Measures,  Commercial  Equivalents 
of.    (£•  Johnstont  Stones,  F.  R.  S.) 


Length.  Millimeters. 

Yard 914-4 

Foot 3°*-8 

Inch 25.4 


Weight.  Grammes. 

Pound 453-6 

Ounce 28.35 


Grain. . 


Volume.  Cube  Centimeter. 

Gallon 4554 

Quart 1136 


.0648  |  Ounce 28.4 


MEMORANDA.  907 


MEMORANDA. 

Physical  and  Mechanical  Elements,  Constructions, 
and.   Results. 

Belting.  Double.  —  600  IP  (to  be  transmitted)  -=-  velocity  of  belt  in  feet  per 
minute,  or  191  IP-:-  number  of  revolutions  per  minute-i-  diameter  of  pulley  in  feet 
=  width  in  ins.  Machine  Belts.— 1500  to  2000  HP  -4-  velocity  of  belt  in  feet  per 
minute  =  width  in  ins.  (Edward  Sawyer.) 

Blast  Pipe  of*  a  Locomotive.  Best  height  is  from  6  to  8  diameters 
of  pipe,  and  best  effect  when  expanded  to  full  diam.  of  pipe  at  2  diameters  from  base. 

Boiler  Riveting.  A  riveting  gang  (2  riveters  and  i  boy)  will  drive  in  shell, 
furnace,  etc.,  a  mean  of  12.5  rivets  per  hour. 

Brick  or  Compressed  Fuel  is  composed  of  coal  dust  agglomerated 
by  pitchy  matter,  compressed  in  molds,  and  subjected  to  a  high  temperature  in  an 
oven,  in  order  to  expel  the  moisture  or  volatile  portion  of  the  pitch  and  any  fire- 
damp that  may  exist  in  the  cells  of  the  coal. 

Bridge,  Highest.  At  Garabil,  France,  413  feet  from  floor  to  surface  of  water, 
and  1800  feet  in  length. 

Bronze,  Malleable.  P.  Dronier,  in  Paris,  makes  alloys  of  copper  and 
tin  malleable  by  adding  from  .5  per  cent,  to  2  per  cent,  quicksilver. 

Building   Department,  Requirements   of.     (New  York.) 

Furnace  Flues  of  Dwelling  Homes  hereafter  constructed  at  least  8-inch  walls  on 
each  side.  The  inner  4  ins.  of  which,  from  bottom  of  flue  to  a  point  two  feet  above 
2d  story  floor,  built  of  fire-brick  laid  with  fire-clay  mortar;  and  least  dimensions  of 
furnace  flue  8  ins.  square,  or  4  ins.  wide  and  16  ins.  long,  inside  measure;  and  when 
furnace  flues  are  located  in  the  usual  stacks,  side  of  flue  inside  of  house  to  which  it 
belongs  may  be  4  ins.  thick.  If  preferred,  furnace  flues  may  be  made  of  fire  clay 
pipe  of  proper  size,  built  in  the  walls,  with  an  air  space  of  i  inch  between  them, 
and  4  ins.  of  brick  wall  on  outside. 

Boiler  Flues  to  be  lined  with  fire-brick  at  least  25  feet  in  height  from  bottom, 
and  in  no  case  walls  of  said  flues  to  be  less  than  8  ins.  thick. 

All  flues  not  built  for  furnaces  or  boilers  must  be  altered  to  conform  to  the  above 
requirements  before  they  are  used  as  such. 

Steam  Pipes  not  to  be  laid  within  two  inches  of  any  timber  or  woodwork,  unless 
it  is  protected  by  a  metal  shield,  and  then  the  distance  not  to  exceed  one  inch.  All 
floors,  ceilings,  and  partitions  to  be  protected  from  heat  by  a  metal  tube  one  inch 
in  diameter  in  excess  of  the  pipe,  and  the  intervening  space  filled  with  mineral 
wool,  asbestos,  or  other  incombustible  material 

Horizontal  and  Hot-Air  Pipes  in  stud  partitions  to  be  double,  with  an  interven- 
ing space  between  them  of  at  least  half  an  inch,  and  a  space  of  three  inches  around 
-a  pipe:  the  inner  face  of  the  partition  to  be  lined  with  tin  plate  and  the  outer  faces 
with  iron  lath  or  slate.  Hot-air  pipes  not  to  be  permitted  in  any  stud  partition  un- 
less it  shall  be  at  least  eight  feet  distant  in  a  horizontal  line  from  the  furnace  To 
shield  the  effect  of  their  heat  in  wood  or  stud  partitions,  to  have  a  double  metal 
collar,  with  two  inches  of  air  space  between  them  and  holes  for  ventilation,  or  to  be 
enclosed  in  brick  masonry  at  least  four  inches  in  thickness. 

Cement.  Iron  to  Stone. — Fine  iron  filings,  20  parts,  Plaster  of  Paris,  60,  and 
Sal  Ammoniac,  i ;  mixed  fluid  with  vinegar,  and  applied  forthwith. 

Chimney  Draught.  W  —  w  h  =  D.  W  and  w  representing  weights  of  a 
cube  foot  of  air  at  external,  and  internal  temperatures,  h  height  of  chimney  or  pipe  in 
feet,  and  D  value  of  draught.  See  Weight  of  Air,  page  521. 

Chinese  or  India  Ink  improves  with  age,  should  be  kept  in  dry  air, 
and  in  rubbing  it  down  the  movement  should  be  in  a  right  line  and  with  very  little 
pressure. 


MEMORANDA. 


Coal,  Effective  "Value  of.  Theoretical  quantity  of  heat  per  IP  is 
2564  units  per  hour,  and  average  quantity  of  heat  in  a  Ib.  of  coal  that  is  utilized 
in  the  generation  of  steam  in  a  boiler  is  8500  units;  hence,  theoretical  quantity 

of  coal  required  per  IP  per  hour  =  ~-^  =  .3  Ibs.,  after  the  water  has  been  heated 

8500 

into  atmospheric  steam,  being  theoretically  nearly  7.5  per  cent,  of  total  heat  re- 
quired to  change  30  Ibs.  water  at  60°  into  steam  of  60  Ibs.  effective  pressure. 

The  total  heat  developed  by  the  combustion  of  coal,  when  utilized  evaporatively, 
ranges  from  .55  to  .8,  but  in  practice  it  does  not  exceed  65  per  cent. 

Coast  and  Bay  Service.  A  velocity  of  current  of  2.5  feet  per  second 
will  scour  and  transport  silt,  and  5  to  6.5  feet  sand.  For  river  scour  the  velocities 
are  very  much  less. 


Cold,  Greatest. 

liquid  Nitrous  Acid. 


-220°,  produced  by  a  bath  of  Carbon,  Bisulphide,  and 


Corrosion  of*  Iron  and  Steel.  The  corrosion  of  steel  over  iron  is, 
as  a  mean,  fully  one  third  greater. 

Cost  of  Family  of  Mechanics  in  TTrance  ranges  from  $220 
to  $600  per  annum,  of  which  clothing  costs  16  parts,  food  61,  rent  15,  and  mis- 
cellaneous 8. 

Crushing  Resistance  of  Brick.  A  pressed  brick  of  Philadelphia 
clay  withstood  a  pressure  of  500000  Ibs.  for  a  period  of  5  minutes. 

Earth  -work.  Shovelling.  —  Horizontal,  12  feet.  Vertical,  6  feet.  When 
thrown  horizontal,  12  to  20  feet,  i  stage  is  required,  and  from  20  to  30,  2  stages. 
When  vertical,  6  to  10  feet,  i  stage  is  required. 

Wheelbarrow.  —Proper  distance  up  to  200  feet. 

Number  of  Loads   and   "Volume   of  Earth   per   Day. 

One  Laborer.    (C.  Herschell,  C.  E.) 


Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume.  ||  Distance. 

Trips. 

Vo  ume. 

Feet. 

20 

5° 
70 

100 

No. 
120 
110 
IOO 
98 

Cub.  Yds. 
23-5 
16.9 
14.4 
13-8 

Feet. 
ISO 
200 
250 
300 

No. 
96 

94 
92 
90 

Cub.  Yds.        Feet. 
13-3            350 
12.8           400 

12.4        450 

12           II        500 

No. 
88 
86 
84 
82 

Cu  .Y.ds. 
1.6 

1.2 
0.9 
0.5 

Volume  of  a  barrow  load,  2.5  cube  feet. 

Portable  Railroad  and  Hand  Cars.— For  a  distance  of  550  feet,  60  cube  yards  can 
be  transported  per  day. 

Horse  Cart. — Volume   of  Earth   transported   per    Day. 

One  Laborer. 

Distance.       Trips.       Volume.      Distance.        Trips.        Volume.     Distance.       Trips.        Volui 


Feet, 
joo 
500 


No. 
86 
67 


Cub.  Yds. 


13-6 


Feet. 
1000 

1500 


43 


Cub.  Yds. 
8.6 
6.4 


Feet. 
2000 
2500 


No. 
25 


Cub.  Yds. 
5 
4-3 


Volume  of  each  load,  8  cube  feet. 

Ox  Cart  is  less  in  cost  at  expense  of  time. 

Electric  Light,  Candle  IPower  of.  Maxim  Incandescent  Lamp.-* 
Current  with  30  Faure  cells,  74  volts,  1.81  Amperes,  16  standard  candles.  With  50 
like  cells,  124  volts,  and  3.2  Amperes,  333  candles.  (Paget  Hills,  LL.  D.) 

The  elavated  electric  lights  at  Los  Angeles,  Cal.,  are  distinctly  visible  at  sea  for  a 
distance  of  80  miles. 

Engine  and  Sugar  IVlill,  "Weights  of.  ENGINE  (now- condensing). 
— Cylinder. — 30  ins.  in  diam.  by  5  feet  stroke  of  piston.  Boilers  (cylindrical  flue).— 
70  ins.  in  diam.  by  40  feet  in  length.  Weights. — Engine,  105000  Ibs. ;  Boilers,  com< 
plete,  75000  Ibs. ;  Sugar-mill,  40  ins.  by  8  feet,  220050  Ibs. ;  Connecting  Machinery, 
137  179  Ibs.  Cane  carriers,  etc. ,  46  787  Ibs. 


MEMORANDA.  909 

Filtering  Stone.  Artificial—  Clay,  15  parts;  Levigated  Chalk,  1.5;  and 
Glass  Sand,  coarse,  83.5.  Mixed  in  water,  molded,  and  hard  burned. 

Fire-engine,  Steam.  Relative  effect  for  equal  cost  compared  with  a 
hand  engine,  as  i  to  113.  Each  IIP  requires  about  112  weight  of  engine. 

Floating  Bodies,  Velocities  of.  At  low  speeds  resistance  increases 
somewhat  less  than  square  of  velocity.  In  a  Canal,  at  a  speed  of  5  miles  per  hour, 
a  large  wave  is  raised,  which  at  a  speed  of  9  miles  disappears,  and  when  speed  is 
superior  to  that  of  the  wave,  resistance  of  boat  is  less  in  proportion  to  velocity,  and 
immersion  is  reduced. 

Length  of  Vessel. — The  proper  length  for  a  vessel  in  feet  (upon  the  wave-line 
theory)  is  fifteen  sixteenths  of  square  of  her  speed  in  knots  per  hour. 

Flow  of  Air.  67  -^/h  =  Velocity  per  second  X  C.  h  representing  column 
of  water  in  ins.,  and  C  a  coefficient  ranging  from  56  to  100. 

Circular  orifices,  thin  plate 56  to   .79 

Cylindrical  mouth-pieces,  short 81  "    .84 

do.  da  rounded  at  inner  end 92  "    .93 

Conical  converging  mouth-pieces 9    "  i 

Conoidal  mouth-piece,  alike  to  contracted  vein 97  "  i 

Fl vies,  Corrugated.  (Wm.  Parker.)  -  M  *  ~ 2)  =  Working  stress  in 
Ibs.  per  sq.  inch.  T  representing  thickness  in  i6ths  of  an  inch,  and  D  diameter  in  ins. 

Steel,  corrugations  1.5  ins.  deep.  Experiments  upon  a  furnace  31.875  ins.  in 
diam.,  6.75  feet  in  length,  and  with  13  corrugations. 

Foundation  Files.  When  piles  are  driven  to  a  solid  foundation,  they  act 
as  columns  of  support,  and  are  designated  Columns,  and  when  they  derive  their 
supporting  power  from  the  friction  of  the  soil  alone,  they  are  termed  Piles. 

Authorities  differ  greatly  as  to  the  factor  of  safety  for  Piles,  varying  .1  to  .01  of 
impact  of  ram.  ( Weisbach. ) 

As  columns,  their  safe  load  may.be  taken  at  from  750  to  900  Ibs.  per  sq.  inch. 
Authorities  give  a  higher  value  (Rankine  and  Mahon,  1000);  but  it  is  to  be  borne 
in  mind  that  when  piles  are  driven  to  a  solid  resistance,  they  are  frequently  split, 
and  consequently  their  resistance  is  much  decreased. 

As  a  rule,  the  following  coefficients  for  ordinary  structures  are  submitted: 

When  the  piles  are  wholly  free  from  vibration  consequent  upon  external  impulse, 
.35  to  .4,  and  when  the  structures  are  heavy  and  exposed  to  irregular  loading,  as 
storehouses,  etc.,  .15  to  .2. 

Ordinarily,  the  bearing  of  a  properly  driven  pile  not  less  than  10  ins.  in  diam.  may 
be  taken  at  10  tons. 

Friction  of*  Bottoms  of  "Vessels.  At  a  velocity  of  7  knots  per 
hour,  a  foul  bottom  requires  2.42  IP  over  that  for  a  clean  bottom. 

Friction  of  Planed  Brass  Surfaces  in  muddy  water  is  .4  pressure. 

Q-as,  Steam,  and  Hot-air  Engines.  Relative  costs  of  gas,  steam, 
and  air  engines  per  IP:  Otto  Gas  engine,  8.75;  Steam  engine,  3.5;  and  Hot-air 
engine,  4. 

Heat.  Available  heat) 16431535  __ 

expended  per  IIP  per  hour}  ~  Total  heat  of  combustion  x  Coefficient  for  fuel  ~" 
consumption  of  coal  per  IIP. 

Coal  14000  X  772  units  =  10  808  ooo.    Theoretical  evaporative  power  =  15  Ibs. 

water.     Efficiency  of  furnace  = .  5 ;  then  10  808  ooo  x  .  5  =  5  404  ooo,  and      43' 535 

5404000 
=  3.04  Ibs.  per  IIP  per  hour. 

Ice  Boats,  Speed  of.  Maj.-Gen.  Z.  B.  Tower,  U.  S.  A.;  assigns  the  speed  of 
Ice  boats  at  twice  that  of  the  wind,  and  the  angle  of  sail,  to  attain  greatest  speed, 
to  be  less  than  90°. 

Japan  Coal.  Analysis  of  Bituminous.— Specific  Gravity,  1.231.  Carbon, 
77.59.  Hydrogen,  5.28.  Oxygen,  3.26.  Nitrogen,  2.75.  Sulphur,  1.65  Ash,  8.4^ 
and  loss,  98. 

Its  evaporative  effect  =  4. 16  Ibs.  water  per  Ib.  of  coal 


910 


MEMORANDA. 


Lee- way.  A  full  modelled  vessel,  with  an  immersed  section  of  i  to  6  of  her 
longitudinal  section,  and  with  an  area  of  36  sq.  feet  of  sails  to  i  of  immersed  sec- 
tion, will  drift  to  leeward  i  mile  in  6.  A  medium  modelled  vessel,  with  an  im- 
mersed section  of  i  to  8,  and  with  like  areas  of  sail  and  section,  will  drift  i  in  9. 

Light,  Standard,  of.  Photometric,  English.— Spermaceti  candles,  6  per 
Ib. ;  120  grains  per  hour.  Carcel  burner  =  9. 5  candles. 

Locomotive  Axles,  Friction  of.  .016  of  weight.  Hence,  if  radius  of 
wheel  =  .1,  axle  friction  at  periphery  — ^-j-  10  =  3.73  at  periphery. 

Mercurial  Gauge.  To  prevent  freezing,  apply  or  introduce  Glycerine  on 
top  of  column. 

Metal  Products  of  TJ.  S.,  18SS.     Value,  $222000000. 

Mississippi  River,  Silt  in.  Near  St.  Charles  the  volume  of  silt 
borne  per  day  in  1879  was  475  457  cube  yards,  and  on  one  day,  July  3,  it  was 
4 113600.  At  times  the  volume  equals  3  ozs.  per  cube  foot  of  water. 

Motive  Power.  A  sailing  vessel  having  a  length  6  times  that  of  her 
breadth,  requires,  for  a  speed  of  10  knots  per  hour,  an  impelling  force  of  48  Ibs.  per 
sq.  foot  of  immersed  section. 

Mowing  Machine.  Kirby^s  (Auburn,  N.  Y.)— 670  Ibs.,  2  horses,  i  acre 
heavy  clover  in  46  min. 

Ordnance,  Energy  of.  In  a  competitive  test  ot  a  9-inch  Woolwich 
gun,  and  a  5. 75-inch  Krupp,  the  energy  per  inch  of  circumference  of  bore  was  re- 
spectively 118  and  123  foot-tons;  their  penetration  therefore  by  the  wrought  iron 
standard  being  about  the  same,  but  their  total  energies  were  respectively  16400 
and  5800  foot-tons. 

At  Mepper  a  shot  of  no  Ibs.,  with  a  velocity  of  1749  feet  per  second,  and  a  strik- 
ing energy  of  2300  foot-tons,  passed  through  a  target  composed  of  two  plates  of  soft 
wrought  iron  7  ins.  thick,  with  10  ins.  of  wood  between  them,  and  passed  800  yards 
beyond. 

Petroleum.  One  Ib.  crude  oil  heated  i  Ib.  water  315.75°  —  28.21  Ibs.  water 
at  60°  converted  to  steam  at  212°.  Relative  evaporative  effects  of  Oil  and  Anthra- 
cite coal  as  i  to  3.45. 

Population,   Comparative    Density   of,  and    N"um"ber    of 
Persons    living   in    a   House    in    different    Cities. 

Chicago,  4  ;  Baltimore  and  Naples,  4. 5 ;  Philadelphia,  6  ;  London,  Boston,  and 
Cairo,  8;  Marseilles,  9;  Pekin,  10 ;  Amsterdam,  n  ;  New  York,  13.5;  Hamburg, 
17.07;  Rome  and  Munich,  27;  Paris,  29;  Buda  Pesth,  34.2;  Madrid,  40;  St.  Peters- 
burg, 43.9;  Vienna,  60.5 ;  and  in  Berlin,  63. 

Power  of  a  'Volcano.  An  eruption  of  that  of  Cotopaxi  has  projected 
a  mass  of  rock  of  a  volume  of  100  cube  yards  a  distance  of  9  miles. 

Power    Required   to   Draw  a  'Vessel   or  Load    up  an  In- 
clined Hydrostatic  Rail  or  Slip  Way.    ( Wm.  Boyd,  Eng. ) 
W  I  =  R ;    C  d  W  -i-  D  =  F ;  and  P  d'  c  =f.     W  representing  weight  of  vessel,  or 
load  and  cradle,  I  inclination  of  ways,  as  length -r- rise,  R  resistance  of  vessel  or  load, 
Y  friction  of  cradle  and  rollers,  and  f 'friction  of  plunger  in  stuffing-box,  all  in  tons, 
C  and  c  coefficients  of  friction  of  cradle  and  stuffing-box,  d  diameter  of  axle  of  rollers, 
d'  product  of  circumference  of  plunger  and  depth  of  collar  or  stuffing,  all  in  ins.,  and 
f  pressure  per  sq.  inch  on  plunger,  in  Ibs. 

Hence,  W  ^ -=-  =  I,  and    R  -}-  F  -}-/= power  in  tons. 

ILLUSTRATION.  —  Assume  weight  of  a  vessel  and  cradle  2000  tons,  pressure  on 
plunger  2500  Ibs.  per  sq.  inch,  inclination  of  ways  i  in  20,  diameters  of  axle  of  roll- 
ers and  of  rollers  3  and  10  ins.,  depth  of  collar  2  ins.,  and  circumference  of  plunger 
50 ;  what  would  be  the  power  required  ?  C  = .  2,  and  c  =  .6. 

2000                          .2  X  3  X  2000  2500  X  2  X  50  X  .6 

Then =  100  tons  ;    =  1 20  tons  ;    — = =  67  tons ; 

20                                 xo  2240 
and  too  -f-  i2o-f  67  =  287  tons. 


MEMORANDA. 


911 


Propeller    Steamer,   Ordinary    ^Distribution    of    !Po\ver 

in  a.     Power  developed  by  engine,  88  IIP;  Power  expended  in  its  operation,  12. 


Per  cent. 

Friction  of  load 7. 5 

"       of  propeller 7.5 


Per  cent 

Power  expended  by  slip  of  propeller. ...  14 
"          in  propulsion 71 


I*ump,  Centrifugal,  has  lifted  water  28  to  29  feet,  drawn  it  horizontally  800 
feet,  and  then  lifted  it  15  feet.  Also  drawn  it  24  feet,  and  projected  it  50  feet 

Railway  'Trains.  Power  and  Resistance.  —  A  railway  train  running  at 
rate  of  60  miles  per  hour  =  88  feet  per  second,  and  velocity  a  body  would  acquire 

in  felling  from  88  feet  =  88-^8.02  =  120.3  feet  Consequently,  in  addition  to  power 
expended  in  frictional  and  atmospheric  resistance  to  train,  as  much  power  must  be 
expended  to  put  it  in  motion  at  this  speed,  as  would  lift  it  in  mass  to  a  height  of 
i2i  feet  in  a  second. 

If  the  train  weighed  100  tons  =  224000  Ibs.,  then  224000  X  120.3  —  26747200 
foot  Ibs.,  and  if  this  result  was  obtained  in  a  period  of  5  minutes,  it  would  require 
120.3-7-5  X  224000-^-33000=163.3  IP  in  addition  to  that  required  for  frictioual 
resistances. 

To  raise  the  speed  of  a  train  from  40  (58.66  feet  per  second)  to  45  (66  feet  per  sec- 
ond) miles  per  hour,  the  power  required  in  addition  to  that  of  friction  would  be  as 

58. 66  -f-  8. 02  =  53. 44  feet  is  to  66  -f-  8. 02  =  67. 57  feet  =  67. 57  —  53. 44  =  14. 1 3  feet. 

Assume  a  train  of  100  tons,  running  at  rate  of  60  miles  per  hour,  and  total  retard- 
ing power  at .  i  its  weight  ioo-=-  10  =  10.  Then  224  ooo  x  10  X  120.3  =  26  947  200-:- 
22400  =  1203  feet,  which  train  would  run  before  stopping.  If,  however,  train  was 
ascending  a  grade  of  i  in  100,  the  retarding  force  =  .n  (n  -f-  100)  of  weight  = 
24640,  distance  in  which  train  would  come  to  rest  would  be  26  947  200  -4-  24640  = 
1093. 6  feet 

Relative   Non-conductit>ility   of  Materials. 


MATERIAL. 

Per  cent. 

MATERIAL. 

Per  cent. 

MATERIAL. 

[Percent. 

Hair  felt 

Mineral  wool  No  i 

67     e 

T  '            It! 

i     .0 

Mineral  wool,  No.  2 

83.2 

Charcoal  

0/-5 
63.2 

Asbestos 

36  ? 

"         "  and  tar 

71    •? 

Pine  wood 

Coal  ashes 

68  J 

Loam  .  .  . 

qs 

Air  snace.  2  ins.  . 

1     34'I 
.      it.  6 

Resistance  to  a  Steam-vessel  in  Air  and  "Water.  In  air 
10  per  cent,  of  IIP,  and  in  water,  at  a  speed  of  20  miles  per  hour,  90  per  cent,  or  8 
IIP  per  sq.  foot  of  immersed  amidship  section. 

Saws,  Circular.  30  ins.  in  diameter,  are  run  at  2000  revolutions  per  minute 
=  3.57  miles. 

Spur  G-ear  has  been  driven  at  a  velocity  of  i  mile  per  minute. 

Sugar  Mill  Rollers.  5  feet  by  28  ins.,  at  2.5  revolutions  per  minute 
requires  20  IP,  and  18  feet  per  minute  is  proper  speed  of  such  rolls. 

Surface    Condensation,  Experiments  on.     (B.  G.  Nichol.) 

Tube  of  Brass,  .75  Inch  External  Diameter.     No.  18  B  W  G.     Surface  =  1.0656 

sq.feet.     Duration  of  Experiment,  20  Minutes. 


STEAM. 

Vertical. 

Horizontal. 

Temperature  

255°,K 

17-75    Ibs. 

18.5835  " 
52.32 

19.0625  •* 

256° 

18.25    Ibs. 
29-9585  " 
84-34      " 
30-4375  " 

253° 
1  6.  75     Ibs. 
24.0835  " 
67-8         « 
24-5625  " 

254°,K 

17.25  Ibs. 
43-0835  " 
121.29     ( 
43-5625" 

Pressure  per  sq.  inch  per  gauge. 
Condensation  by  tube  surface  
"  persq.  ft.  of      "    per  hour 
Condensed  during  experiment  .  .  . 

Steamers'  Engines,  Weights   of.     Engine,  Boiler.  Water,  and  all 
Fittings  ready  for  Service  per  IH». 

Mercantile  steamer 480  Ibs.  I  Light  draught. . .  . .  280  Iba 

English  Naval  »      36o   «    |  Torpedoes 60  " 

Ordinary  Marine  Boiler  with  Water 196  Ibs. 

Wind,  Pressure  of.    Estimate  of,  upon  Structures.  —  30  Ibs.  per  sq.  foot 
Per  lineal  foot  of  a  locomotive  train  =  10  feet  in  height,  300  Ibs.  per  sq.  foot 
A  Tornado  has  developed  a  pressure  of  93  Ibs.  per  lineal  foot 


MEMORANDA. 

Vi  a  S  xi  ez  C  an  al .  Passages  by  Steamers.  —1882,"  Stirling  Castle, ' '  Shang. 
hai  to  Gravesend,  in  29  days  22  hours  and  15  wu'w.,  including  i  day  22  Aours  and  30 
wiw.  in  coaling  and  detentions. 

"  Glenarc,"  Amoy  to  New  York,  N.  Y.,  in  44  days  and  12  Aowrs,  including  deten- 
tion at  Suez.  From  Gibraltar  in  n  days. 

Zino  Foil  in  Steam-boilers.  Zinc  in  an  iron  steam-boiler  consti- 
tutes a  voltaic  element,  which  decomposes  the  water,  liberating  oxygen  and  hydro- 
gen. The  oxygen  combines  with  fatty  acids  and  makes  soap,  which,  coating  the 
tubes,  prevents  the  adhesion  of  the  salts  left  by  evaporation.  The  mealy  deposit 
ean  then  be  readily  removed. 

Piles.     To  Compute  Extreme  Load  a  Foundation  Pile  will  Sustain. 

=  L.  R  representing  weight  of  ram,  P  weight  of  pile,  and  L  extreme 

load,  all  in  Ms.;  h  height  of  fall  of  ram,  and  s  distance  of  depression  of  pile  with  last 
blows,  both  in  feet. 

ILLUSTRATION. — Assume  a  ram  1000  Ibs.  to  fall  20  feet  upon  a  pile  of  400  Ibs., 
what  resistance  will  the  earth  bear,  or  what  weight  will  the  pile  sustain  when 
driven  by  the  last  blow,  from  a  height  of  20  feet,  .5  inch  ? 
s  = .  5  of  12  ins.  =  .0416. 

10002X20  20000000 

Then  - — ==  —       —  =  343  406  Ibs. 

(400  -f-  1000)  X  •  0416  58. 24 

Perimeter.    The  limits  or  bounds  of  a  figure,  or  sum  of  all  its  sides. 

Of  a  canal  it  is  the  length  of  the  bottom  and  wet  sides  of  its  transverse  section. 

Flood  Wave.  The  flood  wave  of  the  Ohio  River  in  March  (1884)  was  71 
feet  i  inch  at  Cincinnati,  being  higher  than  that  of  any  previous  record. 

Ice.  Crushing  Strength  of,  as  determined  by  U.  S.  testing  machine,  ranged 
from  327  to  looo  Ibs.  per  sq.  inch 

Atmosphere.  If  pure  air  is  exhausted  of  2. 5  per  cent,  of  its  oxygen,  it  will 
not  support  the  combustion  of  a  candle. 

Blasting  Paper.  Unsized  paper  coated  with  a  hot  mixture  of  yellow 
prussiate  of  potash  and  charcoal,  each  17  parts;  refined  saltpetre,  35;  potassium 
chlorate,  70;  wheat  starch,  10,  and  water,  1500. 

Dry,  cut  into  strips,  and  roll  into  cartridges 

Circular  Saws.  Speed,  9000  feet  per  minute.  Thus,  for  an  8  ins.,  4500 
revolutions,  and  progressively  up  to  a  72  ins. ,  500  revolutions.  (Emerson.) 

Foods,  Relative   Value   of,  compared    with   1OO   I_/bs.  of 
Q-ood    Hay. 

Additional  to  page  203. 


Lbs. 

Acorns 68 

Barley  and  Rye,  mix'd  179 

Barley  straw . .  180 

Buckwheat 64 

Buckwheat  straw. ...  200 


LlM. 

Linseed 59 

Mangel-wurzel 339 

Pease  and  Beans 45 

Pea-straw 153 

Potatoes 175 


Lttf. 

Rye 54 

Turnips 504 

Wheat 46 

Wheat,  Pea,  and  Oat- 
chaff 167 


Depth   of  the   Ocean.    Mean  depth  is  estimated  by  Dr.  Krummel  at 
1877  fathoms  =  1.85  geographical  miles. 


Q-a 

houi 


3-as-engine.    A  gas-engine  1.5  actual  IP  will  cost,  with  gas  at  8  cents  per 
ir,  10  cents  per  hour  for  10  hours.     (Am.  Engineer.) 

Locomotive.  Average  daily  run  100  miles  at  a  cost  of  $  12.80  for  driver, 
fireman,  fuel,  and  repairs.  (A'.  J.  Central  R.  R.  Co.) 

Consumption  of  Fuel  per  Milt.  Passenger,  25  to  30  Ibs.  coal.  Freight,  45  to  55 
Ibs.,  or  one  cord  wood  per  40  miles. 


MEMORANDA.  913 


Masonry.  In  laying  stones  in  mortar  or  cement,  they  should  rest  upon  the 
course  beneath  them,  more  than  upon  the  material  of  joint. 

Steel  Q-vua  (Krnpp's).  Bore,  15.75  ins.;  length  of  bore,  28.5  feet;  of 
gun,  32.66  feet.  Weight,  72  tons.  Charge,  385  Ibs.  prismatic  powder;  projectile, 
chilled  iron,  1660  Ibs.,  with  an  explosive  charge  of  22  Ibs.  of  powder. 

Moment  of  shot  at  muzzle,  estimated  at  31  ooo  foot-tons,  and  range  15  miles. 

Saw-Mill.    7722  feet  of  i  inch  Poplar  boards  in  One  Hour. 

Engine  (Non-condensing).     Cylinder. — 12  by  24  ins.  stroke  of  piston. 

Boilers.— Two  (cylindrical  flued),  38  ins.  in  diam.  by  26  feet  in  length,  two  14  ing. 
return  flues  in  each.  Heating  Surface.— 780  sq.  feet.  Grates.—  42.5  sq.  feet. 

Pressure  of  Steam.— 125  Ibs.  per  sq.  inch,  cut  off  at  16.5  ins. 

Revolutions.— 250  to  350  per  minute.  Saws.— Two  circular,  60  and  66  ins.  in 
diam. 

NOTE.—  Grates  set  28  ins.  from  under  side  of  boilers,  without  bridge- wall,  and  a 
combustion  chamber  under  boilers,  4  feet  in  depth.  Fuel,  sawdust. 

Steatn  Keating.  62  50x5  cube  feet  of  space  requires  6000  sq.  feet  of  heat- 
ing surface  to  attain  a  temperature  of  70°  in  the  vicinity  of  the  city  of  New  York 
in  its  coldest  weather. 

Or,  One  sq.  foot  of  iron  pipe  will  heat  10.5  cube  feet  of  space  in  an  ordinary  build- 
ing, temperature  of  exterior  air  70°.  (Felix  Campbell ) 

"Velocity  of*  Steam.  Steam  at  a  pressure  of  60  Ibs.  -f- atmosphere  has 
a  velocity  of  efflux  of  890  feet  per  second,  and  as  expanded,  a  velocity  of  1445  feet. 

Blasting.  In  small  blasts  i  Ib.  powder  will  detach  4.5  tons  material,  and  iu 
large  blasts  2.75  tons.  (See  page  443.) 

Delta  Metal  (Iron  and  Bronze).  Specific  gravity  8.4.  Melting  point  1800°. 
(See  page  384.) 

Jarrah  Wood  of  Axistralia.  Impervious  to  insects  and  the  Teredo 
Pfavalis. 

Natural  and  Artificial  O-as.  Relative  water  evaporating  powers 
differ  in  localities,  but  are  assumed  at  900°  and  600°  heat  units  (B. T.  U.)  Nat- 
ural compared  'with.  Bitviminovis  Coal  is  effective  in  the  ratio 
of  2.38  to  i. 

Free  Board  of  Vessels.  For  each  foot  of  depth  of  hold  (from  ceiling 
to  under  side  of  main  deck), .  i  inch  added  to  i.  5  ins.  for  a  depth  of  8  feet.  Thus, 
for  24  feet  depth  i.  5  -f- .  i  x  8  0024  =  3.  i  ins.  (American. ) 

Or,  2  ins.  for  8  feet  depth  and  .  i  for  each  foot  in  addition  thereto.     (Lloyd' 9.) 

Colors   for  "Working   Drawings. 


Brass Gamboge. 

Bricks Carmine. 

Clay Burnt  Umber. 

Concrete Sepia  with  dark  markings. 

Copper Lake  and  Burnt  Sienna. 

Granite India  Ink,  light 

Iron,  cast . .  .Neutral  tint. 

"  wrought .  Prussian  Blue. 
Lead Ind.  Ink  tinged  with  P.  Blue. 


Steel Neutral  tint, 

Water Cobalt. 

Wood Burnt  Sienna. 

Burnt  Umber. 


Stones 

and 
Earths  . 


Yellow  Ochre. 


and  Black. 
"         "    and  B't  Umber. 
Red  and  Indigo. 
Burnt  Sienna  and  Indigo. 


Stowage   of  Chain    Cat>le.    Square  of  diameter  of  chain  in  ins.  mul- 
tiplied by  .35  will  give  volume  of  space  required  to  stow  i  fathom. 

Asphalt  Mortar.    Asphaltum  i  part,  powdered  asphaltic  limestone  7.5 
parts,  residuum  oil  .28  parts,  sand  .6  parts. 
Melt  asphaltum  and  add  the  rest  in  order  named. 
Asphalt  Concrete.    Asphalt  mortar  n  parts  and  broken  stone  9  parts. 

Asbestos  is  a  fibrous  variety  of  Actinolite  orTremolite,  composed  of  silica, 
alumina,  magnesia,  oxide  of  iron,  and  water.     It  resists  heat,  moisture,  and  many 

acids. 


914  MEMORANDA. 

Daily  IToocl  of  an  :Ksqrzima\a.  Flesh  ot  a  sea-horse  8.5  and 
Bread  1.75  Ibs.,  Soup  1.25,  Spirits  i,  and  Water  .9  pint.  (Sir  W.  E.  Parry.) 

Ooignet'a  Concrete.  For  walls  that  resist  moisture. — Sand,  Gravel,  and 
Pebbles,  7  parts;  Argillaceous  Earth  3  parts,  and  Quicklime  i  part. 

Hard  and  quick  setting.—  Sand,  Gravel,  and  Pebbles,  8  parts;  Earth,  burned  and 
powdered  Cinders,  each  i  part,  and  Unslacked  hydraulic  Lime  i  5  parts.  For  a  very 
hard  mixture,  add  cement  i  part. 

Transmission  or  Conductivity  of*  Temperature  in  the 
Earth..  At  Edinburgh  thermometers  set  at  a  depth  of  16  feet  in  the  earth  at- 
tained their  maximum  and  minimum  at  about  six  months  after  the  corresponding 
maximum  and  minimum  of  the  surface,  being  lowest  or  coldest  in  July. 

The  average  rate  of  transmission  of  heat,  as  observed  at  Schenectady,  N.  Y.,  was, 
downwards,  2.9  feet  per  month,  and  upwards  3.4  feet.  (Olin  H.  Landreth.) 

Shafts.  When  loaded  transversely,  the  diameters  of  the  journal  should  first 
be  determined,  its  dimensions  then  at  any  other  point  can  be  deduced  from  those 
diameters.  It  being  observed  that  the  diameters  at  any  two  points  should  be  pro- 
portional to  the  cube  roots  of  the  stress  at  those  points. 

Journals.  —  For  operation  at  high  speed  a  greater  length  is  required  than  for  low 
speed.  The  less  their  length,  the  less  may  be  its  diameter  for  a  given  stress,  and 
consequently  the  friction  will  be  less. 

When  in  constant  operation,  a  large  surface  is  required  to  reduce  heating,  and 
as  friction  increases  with  diameter,  not  with  length,  for  like  stress,  it  is  best  to 
lengthen. 

Wrought  Iron.  — For  50  revolutions  length  to  diameter  as  1.2  to  i,  and  for  every 
50  revolutions  additional  .2  should  be  added.  Thus,  for  1000  revolutions  the  length 
to  diameter  should  be  5  times.  Cast  Iron.  —  Length  to  diameter  as  .9,  and  Steel 
as  1.25  of  above  value.  (W.  C.  Unwin.) 

Non-conducting  Materials.  By  the  investigations  of  Prof.  J.  M. 
Ordway  of  New  Orleans,  he  determined  the  relative  non-conducting  values  of  the 
following  materials,  compared  with  a  naked  pipe,  to  be: 


Hair-felt,  burlap i 

Asbestos  paper,  hair-felt,  duck 1.18 

Pine  charcoal 1.26 

Air  space 4 

(Engineering,  vol.  39,  page  206.) 


Cork  in  strips 2 

Rice-chaff. a.a 

Clay  and  vegetable  fibre 2.8 

Naked  pipe 31 


Marine  Transportation  of  Troops.  Height  between  decks 
(deck  to  under  side  of  beam),  men  6  feet,  horses  7  feet.  Hatchways.—  Horses  at 
least  10  by  10  feet.  Vessels.  —Horses,  beam  not  less  than  30  feet.  Men,  all  ranks, 
2  to  2.5  tons  capacity;  horses,  10  tons.  Rations. — If  biscuit  in  bags,  10000  require 
950  cube  feet  of  volume;  if  it  is  in  barrels,  1350  cube  feet. 

Cabins.  — Officers,  30  sq.  feet  and  195  cube  feet  of  volume,  two  men  42  sq.  feet,  and 
270  cube  feet  of  volume,  and  for  each  additional  man  10  sq.  feet,  exclusive  of  bed 
space  of  6  by  2  feet. 

Hammocks.— To  compute  number  that  can  be  swung  under  a  deck. 

-Hi  x  —  =«•    I  representing  length  under  deck  in  feet,  and  b  breadth  in  int. 
6         16 

(Sir  G.  Wolseley.) 

Horse -TPower  of  Boilers. —  30  Ibs.  water  evaporated  into  dry 
steam,  from  feed  at  100°,  under  a  pressure  of  70  Ibs.  per  sq.  inch  mercurial  gauge 
per  hour.  (Centennial  Exhibition,  1876.)  34.5  Ibs.  water  as  above  from  feed  at 
212°  into  steam  at  212°.  (Am.  Soc.  Mechanical  Engineers.) 


MEMOB^DOA, 


915 


Penetration  of  Light  in   \Vater.     Mediterranean,  clear  sunlight 
in  March,  at  a  depth  of  1200  feet;  in  winter,  600  feet.     (M.  M.  Fol  and  Sararin.) 

Railroad.    Horse.    First  in  operation  in  1826-7. 
fins.    First  in  use  in  England  about  14501 
Iron   Steamers.    First  build  in  1830. 
Lxicifer   Match..     First  made  in  1819. 
Watches.    First  constructed  in  1476. 

Xjoad.  on   Stone  per  sq.  foot.     Church  of  All-Saints  at  Angers,  86000  IDS. 
Pantheon  at  Rome,  60000  Ibs. 


Flexible    Faint    for    Canvas. 

Water  i.     Grind  while  hot  with  .83  parts  oil  paint. 

Fuel.    Evaporation  of  9  Ibs.  water  from  212°: 

1  Ib.  good  coal. 

2  Ibs.  dry  peat 
3.25  "  cotton  stalks. 
3.75  "  wheat  straw. 


Yellow  soap  1.66  parts.     Boiling 


.75  Ib.  petroleum. 
2.5  Ibs.  dry  wood. 
3.5  "    brush  wood. 
4      "   megass,  or  cane  refuse. 


Tramways   or   Street   Tfcailroacls. 

Resistance  on  straight  and  level  tracks  15  to  40  Ibs.  per  ton,  or  an  average  of 
30  Ibs. 

Power  required  on  a  good  track  to  start  a  car,  as  determined  by  A.  W.  Wright, 
M.W.S.E.,  116.5  Ibs.,  and  to  maintain  it  in  motion  17.2  Ibs.  C.  E.  Emery,  Ph.  D., 
rnade  it  13  Ibs.  On  a  bad  track,  the  power  is  134.6  Ibs.  to  start,  and  35  to  maintain 
it  in  motion. 

Power  required,  as  determined  by  Mr.  Wright,  to  start  a  car  is  33.53  IP,  with  an 
average  load  and  day's  work,  and  133.22  to  maintain  it  in  motion. 

Average  work  of  a  car-horse  5.75  hours  per  day  for  a  term  of  service  of  6  yeara 
Strong  draught-horses  will  exert  a  power  of  143  Ibs.  @  2.75  miles  per  hour  for  22 
miles,  and  an  ordinary  one  121  Ibs.  for  25  miles.  (G^yffier.) 

Cable  Railway.    Mr.  Wright  gives  the  power  required  per  ton  *  at  1.92  E?. 

*  All  tons  here  and  elsewhere  are  given  at  2240  Ibs. 

Result  of  Experiments  on  Motors  for  Street  Railroads. 

(1885.) 
At  Antwerp,  by  Capt.  D.  Gallon,  FR.S.,  etc. 

i.  Locomotive  Engine  and  Car.  Ordinary  type  of  steam-engine,  surface  condenser 

(Krauss). 

a.  Surface  condenser,  vertical  boiler,  escape  super- 

heated (Black  and  Hawthorn). 

3.  **  Compound  engine,  compressed  air,  water  -  tube 

boiler  (Beaumont). 

4.  "  and  car  combined.    Ordinary  type  of  steam-engine,  water- 

tube  boiler  (Rowan). 
'  $.  "  "       "      combined.     Electric  Fausse  Batteries. 


Weight  of  Train  per 
Passenger. 

Fuel  con 
Per  Mile  of  Course. 

gamed 
Per  Seat  per 
Mile  of  Course. 

Oil,  Tallow, 
etc. 

Water 
per  Mile  of 
Course. 

Lbs. 

5.  Electric.  ...1.78 
4.  Steam  2.3 
3.  Comp'dairr2.55 

Lbs. 
4.  Rowan  5.22 
5.  Electric  6.16 
2.  Black  and    )  Q  Q 
Hawthorn.  .  J  b'B2 
i   Krauss  9.1 

Lbs. 
.1 
•23 
•23 
.25 
.66 

Lbs. 
.038 
.038 

•073 
.101 
•255 

Gallons. 

Rowan  75 
Comp'd  air.  i.  06 
Black  and  >     Rg 
Hawthorn  P'by 
Krauss  6.52 

3.  Comp'd  air.  .39.48 

NOTE.  —  The  economy  of  the  Rowan  motor  occurred  mainly  from  the  extent  of  its  condensing 
fewer,  by  which  warm  water  was  supplied  to  the  boiler. 


916 


MEMORANDA. 


Corrosive   Effects   of  Salt-water  on  Steel   or  Iron. 

( J.  Farquharson. ) 
Loss  of  Plates  Submerged  for  Six  Months.    Area  12  Sq.  Feet. 

(   .07   Ib. 
"  I   -445  " 

Fractional   Resistance  of  a  Railway  Train.    (C.  H.  Hudson.] 
Resistance  per  ton,  due  to  atmosphere  at  maximum  speed,  .132  Ib. ;  to  start, 
17.27  Ibs. ;  and  to  maintain  in  motion,  5.1  Ibs. 

Blasting  Q-elatine.    (G  McRoberts,  F.C.S.) 

Is  composed  ( Nitro  glycerine 93  parts )  FffppHvp  nnwfir  ™  rft0f  iha 

by  weight. ...  1  Nitro-cotton 7    «     }  M       ve  P°wer-  •  •  •  J4°°  fo(   •«*• 

It  freezes  hard  at  a  low  temperature  (35  to  40°).  At  ordinary  temperature  above 
freezing,  it  does  not  explode  by  shock,  but  when  frozen  it  readily  explodes.  It  is 
insoluble  in  water  Specific  gravity  i  55  to  i  59. 

Effective  Power  of  some  other  Explosives. 

Nitro-glycerine,  1270  foot-lbs. ;  Dynamite,  No  i,  900;  Gun-powder,  extra  strong. 
as  Curtis  and  Harvey's,  272;  Dynamite,  No.  2,  of  18  nitro-glycerine,  71  nitrate  or 
potash,  10  of  charcoal,,  and  i  of  paraffin,  531,  and  Fulminate  of  Mercury,  367. 

Bolts   of  Wrought   Iron   as   Affected,   by  the   Thread. 

(D.  K.  Clark.) 

Strength  per  Square  Inch  of  Metal. 


Diam. 
of  Bolt. 

Tool. 

Strength 
when  cut. 

Loss. 

Diam. 
of  Bolt. 

TooL 

Strength 
when  cut. 

Lbs. 
44845 
51005 
436l3 
41888 

Loss. 

Per  cent. 
28 

H 

26 

33 

Ins. 
1.25 
1.25 

I 
X 

Dies.* 
Chaser. 
Old  Dies. 
New  Dies. 

Lbs. 
40812 
38528 

55H9 
42650 

Per  cent. 
25 
29 
ii 

30 

Ins. 

'.625 
.625 
.625 

Chaser. 
Old  Dies. 
New  Dies. 
Chaser. 

*  Die  not  giren,  evidently  new. 

Approximate    Bottom   "Velocities   of   Flow-    of   "Water  in 
Channels,  at  which  following  Materials  begin  to  Move. 

(Haupt.) 


Feet. 

Miles. 

Feet. 

Miles.  | 

S«c. 

Hour. 

Sec. 

Hour. 

•25 
.5 

•17 

•34 

Microscopic  sand  and  clay. 
Fine  sand. 

3 

2.04 

{Small  stones,  1.75  inch 
in  diam. 

I 

.68 

Coarse  sand  and  fine  gravel. 

(  Flint  stones,  size   of 

1.75 

1.19 

Pea  gravel. 

3-33 

2-3 

|     hen's  eggs. 

(  Rounded  pebbles,  i  inch  in 

{2-inch    square   brick- 

9 

1.39 

\     diam. 

5 

3-41 

bats. 

Scouring  force  of  the  current  is  proportioned  to  the  square  of  its  velocity. 
Transporting  capacity  varies  as  sixth  power  of  the  velocity.    Hence  the  impor- 
tance of  Increasing  lottom  velocities,  both  to  effect  a  scour  and  to  prevent  deposits. 

Chimney.     (MetternicJc  Lead  Mining  Co.) 

Foundation  36  feet  square  by  11.5  in  height;  base  circular  24.6  feet  by  39.37  in 
height;  shaft,  397.5  feet  in  height,  24.6  feet  at  base,  and  12.48  at  top;  flue  12.48  and 
9.84  feet  in  diameter.  Total  height  441.6  feet. 


Evaporation   of  "Water. 

Ins. ,  Ins. 

January 9  I  April 3.1 

February. i.a  I  May 4.61 

.  5.86 


Mean,  as  observed  at  Boston,  Mass. 


March i .  8  I  June . 


July 6.28 

August 5.49 

September...  4.09 


October 

November  . . 
December. . . 


Ins. 

*:tl 


Total 39.11  ins. 


MEMORANDA.  917 

Central   Widtn   of  a   Roadway   in.   a   Cut. 

Feet.  Feet. 

Railway,  single  line  ...........  18  to  20  I  Public  road  ...................  281030 

"       double  line  ..........  30  "  33  |  Turnpike  road  ................  38  "  40 

Hydraulic   Ram. 

Efficiency  under  Heads  of  Supply  from  2  to  24  Feet,  and  Delivery  of  Discharge  at 
Elevations  from  15  to  loofeet. 

Measurements  from  Valves  of  Ram. 

To    Compute    I?er  Cent,  of*  Total  "Volume  of  Water  Ex- 
pended. 

—  .  C  =  Per  cent.    H  representing  head  of  supply,  and  E  elevation  of  discharge, 
E 

both  in  feet,  and  C  =  .8. 

ILLUSTRATION.  —What  is  volume  of  water  delivered  with  a  head  of  21  feet  to  an 
elevation  of  60  feet? 

—  X  .8  =  .28  per  cent.    Hence,  if  the  volume  of  discharge  is  100  cube  feet,  vol- 
60 

ume  elevated  is  100  X  .28  =  28  cube  feet. 

Inversely.    By  formula  of  E.  B.  Weston,  M.  Am.  Soc.  C.  E. 

—  -  —  =  V.    S  representing  number  of  cube  feet  expended  in  ram  per  minute,  h  dif- 

100  h 

ference  in  elevations  of  ram  and  delivery  in  feet,  and  V  volume  raised  in  cube  feet. 
C  =  65  to  70. 
Assume  as  preceding,  H  =  21  feet,  E  =  60  feet,  and  S  =  100  cube  feet 

100X65X21      136500 
Then,  --  =  —  —  =  35  cube  feet 

100X60-21        3900 
Norm.—  To  conform  to  the  preceding  formula  C  should  be  52. 

To   Compute   Elements  of  a   Screw  Propeller. 


and 


=  = 

p  p  R  33  ooo  p  R 

P  representing  mean  pressure  on  piston  per  sq.  inch  in  Ibs.  ,  a  area  of  piston  in  sq. 
ins.  ,  p  pitch  of  propeller  and  I  length  of  stroke,  both  in  feet,  R  number  of  revolutions 
per  minute,  and  T  thrust  of  propeller  in  Ibs. 

ILLUSTRATION.—  The  elements  of  operation  of  a  steam-engine  are:  Mean  pressure 
on  piston,  having  an  area  of  1000  sq.  ins.,  is  30  Ibs.  ;  length  of  stroke  2  feet;  revolu- 
tions of  engine  130  per  minute;  and  pitch  of  propeller  12  feet.     What  is  the  thrust 
of  the  propeller,  and  what  the  power  of  the  engine? 
30X2  X2Xxooo_i_7^Qn,3oX2X2XioooXi3o_  15600  OOP 

12  33000  33000 

Centrifugal   Pump. 

(Southwark  Foundry  and  Machine  Co.    ^on-condensing.) 

'    Pumps.—  two  of  42  ins.,  with  runners  68  ins.  in  diameter;  discharge  pipe  42  ins. 
Engines.—  Two  of  28  ins.  in  diameter  of  cylinder,  and  24  ins.  stroke  of  piston. 
Boilers.  —12  Horizontal  tubular.     Heating  surface,  8568  sq  feet;  Grate,  330  sq. 
feet;  Combustion  natural. 

Pressure  of  Steam  70  Ibs.  per  sq.  inch,  cut  off  at  .625. 
Revolutions,  130  to  160  per  minute. 
Height  of  Delivery,  o  to  36  feet. 

Weight.—  Pumping  plant  exclusive  of  boilers  300000  Ibs. 

Discharge.—  From  Dry-dock  from  a  depth  of  water  of  o  to  36  feet,  mean  per  min- 
ite  1  12  92  2  gallons, 

4H* 


91 8  MEMOBANDA. 

FViction  of  a    Non-condensing   Engine.     (Prof.  R.  H.  Thurston.] 

Friction  of  a  non  condensing  engine  is  given  at  from  2  to  4.75  Ibs.  per  sq.  inch  of 
piston,  being  least  at  low  pressure.  The  conclusions  drawn  from  a  series  of  exper- 
iments are  as  follows: 

1.  It  is  sensibly  constant  at  any  given  speed  of  engine  at  all  loads. 

2.  It  is  variable  with  variation  of  speed  of  engine,  increasing  with  the  speed. 

3.  It  increases  with  increase  of  pressure  of  steam. 

NOTE. — This  per  cent,  of  friction  is  somewhat  less  than  that  given  ante  at  p.  733. 

Visibility  of"  "Vessel's  Sidelights.  The  minimum  distance  of 
visibility  assigned  by  the  International  regulations  for  green  and  red  lights  is  2 
nautical  miles. 

"Weight  of  Anvils.  The  weight  of  an  anvil  for  forging  iron  should  be 
8  times  that  of  the  hammer,  and  for  steel  12  times.  (Prof.  Friedrick  Rich.) 

Temperature  of  Mines.  Temperature  of  copper-mines  of  Lake  Su- 
perior increases  i°  for  every  100.8  feet  of  depth.  The  usual  gradient  is  from  50  to 
55  feet.  (H.  A.  Wheeler.) 

Horse.  In  transportation  by  sea  occupies  the  space  of  10  tons  measurement, 
and  requires  that  of  300  cube  feet  of  air. 

Stalls  6  feet  in  length  in  the  clear  of  padding  and  haunch  piece,  2  feet  2  ins.  in 
clear  width  between  padding,  10  per  cent,  of  this  width  2  ins.  narrower,  and  5  per 
cent,  of  it  6  ins.  longer. 

Mule.    A  pair  will  draw,  including  cart,  1500  to  2000  Ibs. 
Aes.    Will  carry  100  to  200  Ibs.  15  miles  per  day. 

Camel.  The  Arabian,  or  Dromedary,  has  one  hump  on  back,  the  Bactrian  has 
two.  Large  animals  will  carry  1500  Ibs.  for  3  or  4  days,  or  1000  Ibs.  for  several 
days,  and  450  to  600  Ibs.  for  a  long  march. 

One  has  travelled  115  miles  in  n  hours. 

EJlephant.  Weight,  3  to  5  tons;  weight  one  can  carry  about  1450  Ibs. ;  2000 
Ibs.  have  been  carried.  Occupies  55  sq.  feet;  will  travel  on  a  good  road  at  a  rate  of 
2.5  miles  per  hour  for  6  hours. 

"Whales.  Greenland  Right,  length  50  to  60  feet.  Finner,  80  feet.  Speed,  10 
io  12  miles  per  hour.  Extreme  weight,  74  tons.  IP  estimated  at  145. 

Chimneys.  Late  experiments  as  to  the  draught  of  chimneys  have  developed 
the  result  that  an  increase  of  its  area  near  to  the  top  increases  the  draught. 

Cost   of  Maintenance   of  Street    Railroads,  1876. 

Average  of  16  roads,  102  miles  in  length,  with  1297  cars  and  10300  horses. 
(H.  Haupt.) 

Cost  per  horse,  and  average  number  to  a  car  eight. 

Repairs  of  harness $    4.06      Shoeing $22.77 

Feed 124.39      Stall  expenses. 42.13 

Replacing  horses 22.1 

Total $215.45 

Cost  per  month  of  each  horse,  $  18.  On  one  of  the  longest  railroads  in  the  City 
of  New  York,  on  the  least  populous  route,  the  daily  cost  per  passenger,  exclusive  of 
general  expenses,  was  2.88  cents,  and  inclusive  of  general  expenses  4.1  cents. 

Magnesia    Covering   for    Steam-pipes   and   Boilers. 

Experiments  made  by  Bureau  of  Steam  Engineering,  U.S.N.,  developed  the  fol- 
lowing comparative  results: 
Felt  (as  standard) ...     100  |  Sectional  magnesia....  103.07  |  Sawdust 90.5 


APPENDIX. 


919 


APPENDIX. 
River   Steamboat.     Wood   Side   'Wheels. 

freight   and    ^Passenger. 

"  BOSTONA.  "—  HORIZONTAL  LEVER  ENGINES  (Non-condensing).— Length  on  deck, 
302  feet  10  ins.;  beam,  43  feet  4  ins.;  hold,  6  feet.  Tons,  993.52. 

Immersed  section  of  light  draught  of  26  ins.,  83  sq.feet.  Capacity  for  freight,  120* 
tons  (2000  Ibs. ). 

Cylinders.— Two  of  25  ins.  in  diam.  by  8  feet  stroke  of  piston. 
Boilers.—  Four  of  steel,  47  ins.  in  diam.  by  30  feet  in  length,  6  flues  in  each. 
Heating  surface,  903  sq.  feet.     Grate  surface,  98  sq.  feet. 
Pressure  of  Steam,  154  Ibs.  per  sq.  inch,  cut  off  at  .625. 

Revolutions, per  minute.    Speed,  10  miles  per  hour  against  current  of  upper 

Ohio,  3  to  5  miles. 

To   Compute    Meta-centre   of  Hull   of  a  Vessel. 

Operation  of  Formula  in  Naval  Architecture,  page  660. 

Assume  a  sharp- modelled  yacht,  45  feet  in  length,  13.5  feet  beam,  and  9.5  feet 
hold,  with  an  immersed  amidship  section  of  42  sq.  feet,  and  a  displacement  of  900 
cube  feet  at  a  mean  draught  of  water  of  6  feet 

lf^~D^  =  Meta-centre.  .  See  pages  650,  659, 
Ordinates  (dx)  taken  at  intervals  of  2.5  feet  are  as  follows: 
y 

y2 


=   o     =       .0 
=   .63=       .216 
=  i.33=     2.197 
=  23    =     8 
=  2.83  =   21.952 
=  3.  63=:   46.656 

y8 

y9 

ylO 

y" 
y12 

y* 
yi5 

f  cubes 

=  6.  s8  =  287.496 
=  6.73  =  300.763 
=  6.75  =  307.547 
=  6.5   =287.496 
=  6.25  =  244.14 
=  5.8   =  195.112 
~  5      ~~  *^5 

yI73  =  2.4  =13.824 
y18  =i-5  =  3-375 
y'9a=  .8  =     .512 

y»3=    o     =      .0 

2272.814 

2-5 

=  5  83  —  195.112 
imation  of  function  o 

of  ordinates  for  value 

5082.035 
of  /y3  d  x  =  5682.035. 

And  1  of  5682.035  =  ^  of  6      =        feet 

3  900  3 

NOTE.— The  other  elements  of  this  vessel  are: 

Area  of  load-line,  401. 12  sq.feet ;  Displacement  in  iveight,  27.974  tons ;  do.  at  load- 
draught,  .955  tons  per  inch;  'Depth  of  centre  of  gravity  of  displacement  below  load- 
line,  i. 49  feet;  Volume  of  displacement,  fa  volume  of 'immersed  dimensions,  26.8 
per  cent. 

To   Compute   Height   of  Jet   in   a   Conduit   3?ipe   from    a 
Constant   Head.     (Weisbach.) 

— 7 /\  /d'\4  ~  —  =  *'*  and  ~~  =  k"     *»  *'» an^  ^"  rePresen^nff  heights 


due  to  velocity  of  efflux,  loss  of  head  and  of  ascent,  I  length  of  pipe  or  conduit,  and  d 
and  d'  diameters  of  pipe  and  jet,  all  in  feet,  v  velocity  of  efflux  in  feet  per  second,  G 
and  C'  coefficients  of  friction  of  inlet  of  pipe  and  outlet,  and  z  a  divisor  determined 
by  experiment  with  diameters  of  .5  to  1.25  ins.,  ranging  from  1.06  to  1.08. 

ILLUSTRATION.— If  conduit  pipe  for  a  fountain  is  350  feet  in  length,  and  2  ins.  in 
diameter,  to  what  height  will  a  jet  of  .5  inch  ascend  under  a  head  of  40  feet? 

Assume  C  and 0'. 8  and  .5,  h  =  2sfeet,  d  —  z  tns.  =  .i66,  and  .5  =  .5  — 12  =  .0416. 

Then 2J 


02O  APPENDIX. 

To  Compute  Head  and  Discharge  of  Water  in  Pipes  of 

Gfreat    Length. 

It  becomes  necessary  first  to  determine  the  velocity  of  the  flow,  which  is  =* 
4  V  V 

—    6d2  =  »  =  1.273  — ,  independent  of  friction.    V  representing  volume  of  water 

in  cube  feet,  and  d  diameter  of  pipe  in  inf. 
When  head,  length,  and  diameter  of  pipe  are  given, — 


Coefficients  of  friction  C,  for  velocity  of  flow,  range  from  .0234  to  .0191  for  veloci- 
ties  from  3  to  13  feet  per  second,  and  c  that  for  the  pipe  as  a  mean  at  .5.  See  Weis- 
bach's  Mechanics,  Vol.  i.,  page  431. 

ILLUSTRATION.  —  What  head  must  be  given  to  a  pipe  150  feet  in  length  and  5  ins. 
in  diameter,  to  discharge  25  cube  feet  of  water  per  minute,  and  what  velocity  will 
it  attain  at  that  head?  C  =  .o24  and  c  =  .S. 

2c  X  I22 

Then  1.273   6oXsa  =  x-273  X  2.4  =  3.055/6^  velocity  per  second,  and 


viP  /TV' 

Or,  4.  72  =  V  in  cube  feet  per  minute,  and  .  538  |  /  -=—  =  d  in  ins. 

yl-T-fl  V       H> 

ILLUSTRATION.  —Assume  elements  of  preceding  case. 


f  150-7-  1.42 
p=  -538  X  -^69607  =  .538  X  9.301  =  5  ins. 

To  Compute  Fall  of  a  Canal  or  Open  Conduit  to  Con- 
duct and  Discharge  a  G-iven  "Volume  of  ^Water  per 
Second. 

Coefficient  of  friction  in  suck  case  is  assumed  by  Du  Buat  and  others  at 
.007565. 

C  _?  x  —  =  h.  h  representing  height  of  fall,  I  length  of  canal,  and  p  net  perime- 
ter, all  in  feet;  A  area  of  section  of  canal  in  sq.  feet,  and  v  velocity  of  flow  in  feet 
per  second. 

ILLUSTRATION  i.—  What  fall  should  be  given  to  a  canal  with  a  section  of  3  feet  at 
bottom,  7  at  top,  and  3  in  depth,  and  a  length  of  2600  feet,  to  conduct  40  cube  feet 
of  water  per  second  ? 

C  =  .oo76,  p  =  3  +  (V32-f  22 X  2)  =  10.21  feet,  A=7-^-~-2.=  15  sq.  feet,  and 
v  =  —  =  2.66  feet. 

,  2600X10.21      2.662  oft 

Then  .0076 X  ^ =  J3-45  X  .xx  =  1.48  feet. 

2. —What  is  volume  of  water  conducted  by  a  canal,  with  a  section  of  4  feet  at 
bottom,  12  at  top,  and  5  in  depth,  with  a  fall  of  3  feet,  and  a  length  of  5800  feet? 


i6.8/ee«. 

"* 


40  X  3.23  feet  velocity  —  129.2  cube  feet. 
For  Dimensions  of  transverse  profile  of  a  canal,  see  Weisbach,  page  492,  vot  i. 


APPENDIX. 


92I 


MAGNESIA   COVERING   FOR   STEAM   BOILERS,  HEATED  PIPES,  ETC. 
Robert    A..    Keas"bey9    Jersey    City    and.    New   York. 

This  covering  is  devoid  of  organic  matter,  hence  it  possesses  great  rapacity 
to  resist  a  high  temperature,  combined  with  high  rank  in  the  order  of  noi> 
conductors. 

It  is  furnished  for  pipes  in  the  form  of  hollow  cylinders  divided  longitu- 
dinally, and  covered  with  canvas ;  for  boilers,  in  blocks ;  and  for  covering 
odd  fittings,  filling  floors,  etc.,  in  dry  mass  in  bags. 

Relative  Valne  of  Noi\  -  Conducting  Coverings  on 
\Vromgnt -Iron  Steam  I*ipe.  Determined  toy  Tests 
at  St.  I^oxiis  Water- Works. 

Condensation  in  Cube  Centimeters  per  Foot  per  Hour.— (John  A.  Laird,  M.  E.) 


Material. 

Analysis. 

C.  C. 

Magnesia,  Sectional  

(  Carbonate  of  Magnesia.  . 

.  92  .  20 

No. 

Magnesia,  Plastic  

(  Carbonate  of  Magnesia.. 

.  92  .  20 

33-53 

(Asbestos  

.   82  oo 

33-4 

Asbestos  Fire  Felt,  Sectional  .  .  . 

(  Carbonaceous   

.   18  oo  —  100 

36.75 

Asbesto-Sponge  Molded 

j  Plaster  of  Paris  

.  92.80 

|  Fibrous  Asbestos  

.     4.20  —  100 

37-13 

NOTE.— The  test  at  the  New  York  Post-Office  gave  Fire-felt  superior  to  Magnesia 

DISTANCES,  VELOCITIES,  AND  ACCELERATION. 
To    Compvite    Velocities    of  an.    Accelerated    Body. 

^/V2  _j_  (2  v'  S),  Or,  v  +  T v'  =  V.  v  and  v'  representing  original  and  accelerated 
velocities,  and  V  final  velocity,  all  in  feet  per  second;  S  distance  or  space  passed  over 

in  feet,  and  t  time  in  seconds.       --  •  •  =  V.    V  representing  average  velocity  in 
feet  per  second.    V  t  —  S,  and  2  V  —  V  =  v. 

ILLUSTRATION  i.—  A  body  moving  with  a  velocity  of  10  feet  per  second,  is  acceler- 
ated at  rate  of  4  feet  per  second,  per  second,  for  a  period  of  6  seconds;  what  are  its 
different  velocities? 

v  =  10,    v'  =  4,    t  —  6. 

Then,  10  +  6X4  =  34  feet  final  velocity. 


*-  =  22  feel  average  velocity. 


22  X  6  =  132  feet  distance  passed  over.      Vio2-f-(2  X  4  X  132)  =  VII56  =  34  feeti 
and  2  X  22  —  34  =  lofeet  original  velocity. 

.   V  —  v        .         V  +  v  ...      „          V2~~v2"1=:t>'S,         t>2-}-2t>'S  =  V2, 


ai 


V-v 


=  t,  and 


2.— A  body  is  projected  vertically  with  a  velocity  of  200  feet  per  second,  and  is 
retarded  at  tbe  rate  of  30  feet  per  second,  per  second;  wbat  height  will  it  have 
passed  through  when  its  velocity  is  reduced  to  80  feet  per  second,  and  in  what  time? 

v  =  200,    v'  —  30,    and    V  =  80. 


Then  2°°~8°  =  4  seconds. 


3._A  vehicle  being  drawn  with  a  velocity  of  25  feet  per  second,  is  accelerated 
5  feet  per  second,  per  second;  what  is  its  velocity  and  time  of  operation  at  the  end 


of  ioo  feet  ? 
v  =  25,    v'  =.  5,  and  V  =  ioo. 


Then 


—  —  — 


=  15  seconds. 


Q22  APPENDIX. 

4.— A  stream  of  water,  after  flowing  a  distance  of  120  feet,  is  ascertained  to  have 
a  velocity  of  40  feet  per  second,  with  an  accelerating  velocity  of  2  feet  per  second, 
per  second;  what  was  its  primitive  velocity  and  time  of  flow? 

S  =  120,    V  =  40,    v'  =  2. 

Then  \/4o2  —  2  x  2  x  120  =  33. 47  feet.  33'47  =  3. 26  seconds. 

Delivery   and.   Friction   in   Hose. 

(B.  F.  Hartford,  Am.  Soc.  C.  E.) 
Hose  2. 5  ins.  in  diameter.    Nozzles  not  exceeding  i.  5  ins. 

Rubber  or  Leather.    .0408  vd2    and    .497  cd2  VP  =  G;          /24'51  G    and 
and    ^-1^ —  =  v 


.003  175  6  c2  d4  P  I      and        .000021  6  I  v2  d*  =P\        P— p  =  P'; 
2.3o6(P—  p)    and 


1.123X20  bc*d*  bv2d* 


_  . 

-p£-—  H,      i  —  .003  175  6  c2  d*  I       and       —  —  x.      G  representing  gallons  dis- 

charged per  second,  v  velocity  in  feet  per  second,  P  pressure  of  stream  at  hydrant  or 
source  of  supply,  p  pressure  lost  in  hose,  and  P'  pressure  at  nozzle,  all  in  Ibs.  per  sq. 
foot,  d  diameter  of  nozzle  in  ins.,  H  head  of  supply  at  hydrant,  h  head  at  nozzle,  and 
I  length  of  hose,  all  in  feet,  x  fraction  ofP  at  nozzle,  b  coefficient  of  material  of  hose, 
and  cfor  nozzle. 

b  =  i  for  rubber  hose  and  1.167  for  leather. 

c  =  .82  for  smooth  nozzle  and  .64  for  ring. 

ILLUSTRATION.—  Assume  length  of  a  rubber  hose  200  feet,  pressure  at  hydrant  100 
Ibs.,  diameter  of  ring  nozzle  1.25  ins.,  and  volume  of  discharge  4.97  gallons  per 
second;  what  are  the  other  elements  to  be  obtained  by  preceding  formulas? 

•497  X  .64  X  1.252  X  V'°°  =  4-97  gallons.          24-  5^X4-  97  =  J7^feet 

M.5I  X  4.97  and      /2.0I2  x  4-97  =          .ns           4^484l±_972==-o  = 
V  v  V     -64-^/100  .642Xi.254          i 

c  oi3  857  X  i  X  4.  97  2  X  200  =  63.  52  Ibs.         100  —  63.  52  =  36.  48  Ibs. 

looX  .3648  =  35.48  Ibs.    2.306  (100  —  63.  52)  =  84.  i2feet.     ^  //  9        =94.12  feet, 

i  —  .003175  X  i  X-642X  i.  25*  X  200=  i—  .6352  =  .3648  =  0;. 
314.96(1  —  .3648)  _  200  46750.82X100(1—  .3648)  _        f. 

i  X  .64*  X  1.25*  ~    i  iX  77  962  X  1.25*  °°'/C6t 


For  vertical  Jets,  see  page  549. 

G-anging   of  Weirs. 


When  there  is  an  Initial  Velocity.  (H~+ft  f  —  h  f  )  ¥  =  H'.  H  and  H'  represent- 
ing depth  of  water  on  weir,  and  when  corrected  to  include  effect  of  initial  velocity  of 
approaching  water,  and  h  head  to  which  this  velocity  is  due,  all  in  feet. 

Velocity  in  Pipes.  C  V**  I  =  V.  r  representing  mean  radius  or  hydraulic  mean 
depth,*  I  sine  of  angle  of  inclination  equal  to  loss  of  head  per  unit  of  length,  V  velocity 
in  feet  per  second,  and  C  a  mean  coefficient  0/142. 

In  small  Channels.    C  =  30  to  50. 

NOTE.—  Sectional  area  of  a  pipe  or  conduit,  divided  by  perimeter,  Is  termed  mean  rarft«»,and  whef 
the  pipe,  conduit,  or  channel  is  but  partially  filled,  the  area  is  termed  hydraulic  mean  depth. 

*  See  also  page  553. 


APPENDIX.  923 

Metric   Factors.    In  addition  to  pp.  27-37. 
By  Act  of  Congress,  July,  1866.  By  French  Metric  Computation 

Measures. 
x  Liter  per  cube  meter  =  .007  48  gallons  per  cube  foot . . .  |       .007  48  gallons. 

\Veights   and    Pressures. 


i  Centimeter  of  mercury  per  sq.  inch  = .  192  91  Ib.  per ) 

sq.  inch j 

i  Atmosphere  (14.7  Ibs.)  =  6.6679  kilograms 

i  Inch  of  mercury  per  sq.  inch  =  2. 54  centimeters 

i  Pound  per  sq.  inch  =  453. 6029  grams 

i  Cube  foot  per  ton  =  .0275  cube  meter 


.1929117  Ib. 

6.6678  kilogram*. 
2. 54  centimetres. 
453-  5926  grammes. 
.0279  cubic  metre. 


Heat. 

i  Caloric  per  Kilogram  =  1.8  heat  units  per  Ib |     1.8  heat  units. 

Velocity. 
i  Meter  per  second  =  3. 280  833  feet  per  second |     3. 280  869  feet. 

I*o\ver   and    "Work. 


i  Kilogrammeter  (k  X  m)  =  2.2046  x  3.28083. . 

i  Foot-pound  = .  138  26  kilogrammeters 

i  Kilogram  per  cheval  =  2.2352  Ibs.  per  EP. . . . 
i  Sq.  foot  per  IP  =  .091 63  sq.  meter  per  cheval'. 


j.233 

.  138  25  kilogrametre. 
2-  2353  pounds. 

.091 63  sq.  metre. 


Miscellaneous. 


i  Avoirs  Lb.=       '453  6  kilogram. 
i  Ton  =      1.016057  tonne. 

i  Sq.  Inch     =  645. 161  29  sq.  miWrs. 


i  Sq.  Foot      =  .092903  sq.  meter. 
i  Cube  Foot   =  .028  317  cube  meter. 
[  Cube  Yard  =  .764559  cube  meter. 


i  Mile  per  hour  =  26.8225  meters  per  minute. 

i  Knot  "      "    (6086.44  feet)  =  30.9192      "       "        " 
i  Cube  Meter  per  minute       =    7.848  cube  yards  per  hour. 
i     "     Yard     "        "  =45.8718  "    meters  "      " 

v2  V2 

tiooomotive   Brakes. and  — -=  =  distance  in  which  a  train  is 

64- 4  /  3°/ 

Stopped,    v  and  V  representing  velocity  in  feet  per  second,  and  miles  per  hour,  and 
f  proportion  of  resistance  of  brakes  to  weight  of  train. 

Brakes,  self-acting,  on  all  wheels,/— .  14.     Ordinary  hand,/=  .023  to  .031.    As- 
cending i  in  .5  resistance  is/-f  2  ;  descending  i  in  .5  /—  2'. 

Hydraulic   Rams.    Efficiency  decreases  rapidly  as  height  to  which  wate? 
is  to  be  raised  increases  above  the  fall  or  head. 

Number  of  times  the  height  to  which  the  water  is  raised  exceeds  that  of  the  head 
of  the  supply  and  efficiency  per  cent.     ( Walter  S.  Button,  C.  and  M.  E.) 
Number  ...    4      5      6      7      8      9    10    n     12    13    14    15    16    18    19    20    25 
Efficiency . .  75    72    68    62    57    53    48    43    38    35    32    28    23    17    15    12      a 

Speed  of  water  in  pumps,  200  feet  per  minute. 

To   Compute    Weignt   of  Water  at   any   Temperature. 

-  =  W.    W  and  w  representing  weights  of  water  per  cube 


T  +  46i.2°  3W 

500         '  1  +  461.  2° 

foot  at  temperature  T,  and  at  maximum  density  0/39.  2°  =  62.  425  Ibs.,  and  46i.ac 
equal  absolute  temperature. 

ILLUSTRATION.—  Required  weight  of  a  cube  foot  of  water  at  temperature  of  60°. 


60  +  461.2 


t 


=62.37  E* 


500  60-4- 461.3 


924 


APPENDIX. 


Results 


of  Experiinents    or    Performances 
Steam-engines   and.   Boilers. 


of 


Cylinders,  Cut-off,  Vacuum,  and  Diameters  in  Inches,  Revolutions  per  Minute^ 
Pressure,  Water,  and  Coal  in  Lbs.,  and  Surfaces  and  Areas  in  Sq.  Ins. 


ELEMENTS  OF  ENGINE. 

HARRIS. 

Non-con- 
densing. 

COR 
Con- 
densing. 

LISS. 

Con- 
densing. 

BOILERS. 

Cylinder    

18X42 
74.29 
58.5 
4-74 
26.93 

105-47 
12.64 
92.83 

l8.59 

2.34 
2.07 

>oo  Ibs. 

18X42 
73-6 
76'37 
7-94 
29.47 

"5-43 
13.07 
102.36 

25.39 
3-18 
2.82 

1.83 

•753 
t  Steam  p 

24X60* 
59-62 
92.88 
18.02 

89.38 

270.58 
12.55 

1.98 

26.4 
1.83 

er  Ib.  of  cot 

Number       .      . 

2 
60 
12 

1536.92 
51-75 
1256.64 
29.7 

5-93 

"4-3° 
8.85 

8.21 

10.3 

9.64 
to  i. 

Revolutions  
Pressure  in  Pipe  

Diameter  

Length 

Cut  off  

Tubes  50  

Mean  effective  Pressure 
IIP  

Heating  Surface  

Grate           " 

Friction  EP  

Calorimeter  

Net  IP     

Heating  to  Grate  
Grate  to  Calorimeter.  . 

Temperature  of  Feed  . 
Steam  per  Lb.  of  ) 
Combustiblet  .  .  j  '  ' 
Steam  per  Lb.  of) 

Pnal                              1    •  • 

Water  per  net  IP  ) 
per  hour  j  "  *  *  * 

Coal  per  IIP  per) 

Vacuum  

Combustible  per) 
IIP  per  hour  .  J 
Relative  efficiency  .... 

*  Weight  of  engine,  40  c 

Coal  per  Sq.  Foot) 
of  Grate  per  hour  J  " 
Steam  per  Temp.  212° 

1  8.21  Ibt.,  and  evaporation  9 

WINDMILLS.      (Andrew    J.   Corcoran,  New  York.) 

(Improved.    Patented  June  and  August,  1888;  March  and  June,  1889.) 

"Volume   of"  "Water   "Pumped,   per   ^Minute. 

From  10  to  200  Feet. 


Diameter 
of 
Wheel. 

V 
10 

BRTICAL   Dl 
15 

STANCE   FRO 
25 

M  WATER  i 
50 

ro  POINT  01 
75 

DELIVERY 

100 

IN  TEST. 

'SO 

200 

Feet. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

8-5 

15.242 

10.162 

6.162 

3.016 

10 

48.262 

32-I75 

19.179 

9-563 

6.638 

4.25 

12 

86.708 

57-805 

33-941 

17.952 

11.851 

8.485 

5-68 

'4 

111.665 

74-443 

45-139 

22.569 

15.304 

11.246 

7.807 

4.998 

16 

155.982 

103.988 

64.6 

31.654 

19.542 

16.15 

9.771 

8.075 

18 

20 

249-93 
309.604 

159-954 
206.403 

97.682 
124.95 

52.165 
63.75 

32.513 
40.8 

24.421 
31.248 

17-485 
19.284 

12.  211 

15.938 

25 

3° 

532.517 
1080.112 

355-012 
728.828 

212.381 

430-848 

106.964 
216.172 

71.604 
146.608 

49-725 
107.712 

37-349 
74-8 

26.741 
54-043 

Factory  in  Jersey  City. 

Velocity    of  Wind. 

The  average  over  the  United  States,  as  determined  by  the  Signal  Service  of  the 
U  S.  Army,  is  5769  miles  per  month,  or  about  8  miles  per  hour. 

Experience  has  determined  that,  to  operate  a  windmill,  there  is  required  an 
average  velocity  of  wind  of  six  miles  per  hour. 


-       -  - 


=  pressure  oj  mind  per  sq.foot  of  surface  in  Ibs. 


Or,  —  ra  and  —  u'a.    v  representing  velocity  of  air  in  feet  per  second,  and  «' 
400  200 

in  miles  per  hour. 

None.—  For  useful  tables  and  formulas  see  "  Windmills  as  a  Prime  Mover."  by  A.  R.  Wolff,  J.  Wiley 
&  Sons,  New  York,  188$. 


APPENDIX. 


To  Compute  Head  in  l^bs.  per  Sq.  Inch  to  Resist  Fric- 
tion,  of  Air   in    Long   and.    Rectilineal    IPipes,  etc. 


V  1728 


V'L 


-  =  H; 


H(3.7d)S83.i 

V3 


/H(3.7d)»83.i 
\ L~~ 


V  representing  volume  discharged 


a  60"  "  (3.7  d)583.i  " 

s/-  -  =.  -T-  3.7  =  d,  and  —  >T—  —  ~^—  =  IP. 

y  83.1  H  12"  x  33000 

in  cube  feet  per  minute,  L  length  of  pipe  in  feet,  d  diameter  of  pipe  in  ins.  ,  H  head 
and  P  pressure,  both  in  Ibs.  and  per  sq  inch,  v  velocity  of  discharge  in  feet  per  sec- 
ond, a  area  of  discharge  in  sq.  ins.,  and  IP  horse-power  of  friction  of  air  alone. 

ILLUSTRATION.  —  Assume  volume  of  air  discharged  44000  cube  feet  per  minute, 
diameter  of  discharge  pipe  40.54-  ins.  (say  1280  sq.  ins.  net),  length  of  pipe  1000 
feet,  and  pressure  at  discharge  3.5  Ibs.  per  sq  inch. 


Then 


44  ooo  X  1728 


K  1280X60 

44  ooo9  X  looo  __ 
6310406250000" 

/i  936  ooo  ooo  X  looo 
V        83. i  X. 3068 
74i  5  IP- 


=  990  feet,  and  (3.7  X  40.5)*  X  83.3=6  310406250000. 


.3068—  Ibs.; 


/.  3068  X  6  310  406  250  ooo 


=  44000  cube  feet; 


3.7  =  40.5  in*.,  and 


1280X60X990X3.5+  3068 

12  X  33000 


Volume  of  Enclosed.  Air  at  O°  that  may  t>e  Heated, 
One   Sq.u.are  Foot  of  Iron  Heating  Surface. 


ENCLOSURH. 

He 
in 
Cellar. 

iter 
in 
Room. 

ENCLOSURE. 

H« 

Celfar 

iter 
ta 
rioom. 

Dwellings  

Cub.  ft. 

£ 

70 

Cub.  ft. 
50 

1° 
So 

Large  stores,  average  
Hotels  

Cub.  ft. 
90 
IOO 

mo 

Cuh  ft 
no 
125 

200 

Offices  „  

Close  stores  .  .  , 

Churches  .. 

Commercial  tf*  of  Chimney  for  a  G-iven  Diam,  of  Flue. 

Height  of  Chimney  in  Feet 


Diam.  ol 
SUue. 

50 

60 

70 

80 

90 

IOO 

no 

125 

135 

'So 

175 

200 

»5 

250  i  300 

Ins. 

H? 

H> 

H> 

IP 

IP 

H? 

IP 

FP 

K> 

HP 

IP 

HP 

H> 

IP  i  IP 

16 

18 

| 

30 

84 

P2 

f9 
too 

107 

113 

IIQ 

124 













45 

250 

270 

288 

30.S 

321 

345 

353 

3«5 

— 

— 

— 

—  i  — 

55 

— 

•  — 

— 

330 

370 

4os 

438 

46S 

4QO 

S30 

Sbo 

— 

— 

i  

75 

— 

.  — 

— 

860 

000 

PIS 

IOOO 

1036 

1090 

1185 

1270 

'34.S 

I4l5il48o 

90 

— 

»— 

•  - 

— 

— 

I3SO 

1410 

isoo 

1556 

1640 

1770 

1800 

2OOO 

2IOOI2I9O 

IOO 

TT»»  ;..*<> 

.f,,oJ!o 

*~ 

1665 

1725 

1820 

188011970 

2115 

,  «^j  i 

2255 

2390 

2520(2645 

F®r  intermediate  Diameters  and  Powers,  take  proportionate  Diameters  and  Powers. 
E  -  Square  Chimney  deduct  one«ninth  to  one  twelfth  of  Diameter  of  Round,  for  Side 

Friction   of  Water  in   I»ipes.     (Weisbach.) 

- — •*         C  =  h.     I  representing  length  of  pipes  in  feet,  v  =  — ^- — »  or  velocity 

in  feet  per  second,  V  volume  of  water  in  cube  feet  elevated  per  second,  d  diameter  oj 
pipes  in  ins. .  and  C  a  coefficient,  ranging  from  .069  when  velocity  —  .ifoot,  .0387^07 
-5  foot,  .0375  for  i  foot,  .0265  for  zfeet,  023  for  4  feet,  .02 14  for  6  feet,  .0205  for  8 
feet,  .0193  for  12  feet,  and  .01 82  for  20  feet. 

ILLUSTRATION.— Assume  volume  125  cube  feet,  raised  25  feet  per  hour,  through  a 
pipe  2  ins.  in  diameter  and  500  feet  in  length ;  how  many  feet  of  vertical  head  will 
the  friction  in  the  pipe  be  equal  to  ? 

Then  •836^0XI225  =  3- 1»  velocity,  and  C  =  .028. 


Hence, 


.1865X500X3.18* 


X  .028  =  14.6  feet,  and  25 -f  14.6  =  39.6/«efc 


926 


APPENDIX 


Marine    Boiler. 

Tests  of  the  United  States  Government  on  a  Boiler  built  for  the  U.  S.   Cruise} 
1  Alert."     Conducted  by  a  Board  of  Naval  Engineers,  April,  1899. 

Heating  Surface,  2125  sq.  ft.     Grate  Surface,  48  sq.  ft.     Ratio,  44  :  i. 


Tlie   Batococ 

Elements 

k  and   "Wil 

Cumberland  Min 
8th           nth 

cox  C 

e  Run 
i3th 

Anthra- 
cite, egg 
i4th 

any. 

Cumb 

20th 

erland 

2ist 

Cardiff 
24th 

Moisture  in  coal  per  cent  

5.25 

4.09 

2-77 

.0 

2 

1.63 

i 

Refuse  in  dry  coal  per  cent.  .  .  . 

7-39    IO-5 

12.33 

ii.  6 

IO 

7-88 

Boiler  pressure    Ibs 

218 

Temperature  of  feed-water  

«»3 

157-2 

219 
93-4 

~.y 
91 

110.5 

152.6 

160 

204 
160.5 

Draught  at  base  of  pipe  

.61 

.26 

Draught  in  furnace 

•43 

''.26 

' 

<5*3 

•55 

Blast  pressure  ash  pit    

•3 

-1-    e 

_|_  tjj 

.07 

4"  -53 

•J5 

.27 

•  J4 

Temperature  of  gases  in  flue  .  . 

498~ 

I   '  J 

595 

567 

520 

410 

470 

433 

(%  C02.  . 

— 

11.9 

10.9 

ii.  i 

IO.2 

9-7 

10.8 

Analyses  of  flue  gases  <%  0  

— 

7-8 
.5 

8.2 

.06 

8.7 

.0 

8.8 

9-3 

.2 

8.1 

Moisture   in    steam,    decimals 

of  one  per  cent  

OQ 

Dry  coal  per  sq.  ft.  grate  per 

'^y 

' 

.0 

.0 

.0 

hour  Ibs      

28.8 

Water  evap.  per  hr.  IV.  &  at  212°,  Ibs.  : 

4O  '4- 

41  .y 

15.4 

' 

10.2 

Per  Ib  dry  coal  

9.41 

10.66 

Per  Ib  combustible  . 

TO 

'93 

1  T     Rr» 

"•33 

Per  sq.  ft.  heating  surface  .  . 
Per  sq.  ft.  grate  surface  

12.  13 
5.23 
231.9 

9-65'     9-Si 
427.4    421.3 

6.94 
307.2 

12.30 

167.8 

11.89 

5-34 
236.3 

13.8 
4-13 

183-1 

NOTE.—  The  test  of  April  i3th  was  made  with  air  heated  by  the  flue  gases  to  168°. 

Proportions  of  G-rat*   and.    Heating    Surfaces    of  \Vater- 

rPvit>e  Boilers,  as  Determined  by  Tests  of 

I3at>cocli  and  "Wilcox  Boilers 

from  ISrS  to  1884. 

(Committee  of  U.  S.  Centennial  Exhibition  and  Individuals.) 
Water  evaporated  from  and  at  212°. 


Duration 
of 
Test. 

S 
Grate. 

ur  f  ace 

Heating. 

s. 
Ratio. 

Combus 

sun 

Per 
Grate. 

bible  con- 
icd. 
Per 
Heating 
S     f 

Coal  per 
Grate 

Hour. 

Evapoi 

by  Com- 
bustible. 

ation 
Coal. 

Ash 

per  cent. 

Hours. 

Sq.Ft. 

Sq  Feet. 

Sq.  Feet. 

Sq.  Feet. 

Sq.  Feet. 

Lbs. 

Lbs 

8 

44-5 

1676 

37-7 

8.88 

.256 

— 

12.131 

— 

120 

50.7 

1980 

39«i 

II.  21 

.26 

12.99 

11.62 

9.71 

13-7 

216 

54-7 

2148 

39.1 

12.22 

.292 

11.982 

24 

61.9 

2760 

44-6 

8.22 

.198 

— 

11.626 

10.09 

13.2 

22 

59-5 

2757 

46.3 

14.25 

.307 

9-93 

"«43 

9.96 

12.9 

13-5 

39-7 

1680 

42.3 

5-8 

6.26 

12.495 

".53 

4 

25 

1403 

56.1 

12.41 

!276 

13.44 

12.38 

11.52 

7>5§ 

10.25 

70 

3126 

44-7 

18.15 

.406 

20 

12.42 

11.32 

8.8* 

Coals  :  *  Anthracite,  American,      f  Bituminous,  Welsh,     t  Bituminous,  Scotch.      §  Bituminous, 
Powelton.        f  Test  in  London. 

A  Galloway  boiler  of  standard  efficiency,  at  this  exposition,  having  a  ratio  of  heat- 
ing surface  to  grate  of  25  to  i,  and  feed  water  at  a  temperature  of  56°,  gave  the  fol- 
lowing results: 

Consumption  of  coal,  8.87  Ibs.  per  hour  per  sq.  foot  of  grate.  Pressure  of  steam, 
70  Ibs.  per  sq.  inch.  Water  evaporated  per  hour  per  sq.  foot  of  heating  surface, 
3  Ibs. ;  water  evaporated  per  Ib.  of  combustible,  9.68  Ibs.,  and  per  Ib.  of  coal,  8.63 
Ibs. 


APPENDIX. 


927 


To    Compute  Area   of  Cylinder   of*  a    Steam-engine    and. 
G-rate    and    Heating    Surfaces    of   a    Boiler. 

When  Required  Power  is  Given.— II  is  assumed  that  IP  of  a  steam  engine  is  at- 
tained by  evaporation  of  33.6  Ibs.  water  per  hour,  at  a  temperature  of  212°  from  feed 
water  at  100°. 

NOTK.— This  Is  a  deduction  from  the  elements  of  the  estimate  as  given  by  the  Am.  Soc.  of  Mech'l 
Engineers,  in  order  to  put  temperature  of  the  feed  at  100°  instead  of  212°. 

Non  •  condensing  (Single  Cylinder}.     — — ^-^ 5-  X  1728  =  area  of  cylinder 

DO  X  2  K  X  v> 

in  sq.  ins.     V  representing  volume  of  i  Ib.  of  water  at  terminal  pressure  of  steam  in 
cube  feet,  R  number  of  revolutions  per  minute,  and  S  stroke  of  piston  in  feet. 

ILLUSTRATION.— Required  EP  of  an  engine  is  300,  initial  pressure  of  steam  70  Ibs. 
mercurial  gauge,  cut  off  at  .5  stroke  of  piston  of  4  feet,  and  number  of  revolutions 
60  per  minute.  What  should  be  areas  of  cylinder  of  engine  and  grate  and  heating 
surfaces  of  boiler? 

Clearance  in  cylinder  and  steam  passages  =  1.8  ins.  =.15  foot,  point  of  cutting 
off =4  -=-.5  =  2  feet. 


Then  (formula  p.  711),  70  X  (2 +  .15  -=-4  + .15)  =  36. 26  Ibs.  terminal  pressure,  and 
steam  at  this  pressure  has  a  density  or  volume,  which  is  its  reciprocal  (formula 
table  p.  708)  of  11.26  cube  feet  for  each  Ib  of  water  contained  in  it. 


Hence, 


ii  26  X  36.26  X  300  ^ 


122  486 


=  851X1728=14705    cube 


60X60X2X2*'  14400 

which  -=-48  ins.  stroke  =  306. 35  sq.  ins.,  to  which  is  to  be  added  for  friction  of  en- 
gine and  load  and  waste  of  steam  15  per  cent.  —  45-95  -f-  3°6*35  —  352-3  ins- 
Grate  Surface — Evaporation  of  fresh  water  in  an  efficient  marine  boiler,  from  a 
temperature  of  feed  of  100°,  is  assumed,  with  a  proportion  of  heating  surface  to 
grate  of  30  to  i,  to  be,  with  a  combustion  of  20  Ibs.  coal  per  sq.  foot  of  grate  per 
hour,  213  Ibs.  per  sq.  foot  of  grate,  and  10.3  Ibs.  per  Ib.  of  coal. 

E  T-P 

Hence,  — —  =.  area.    L  representing  evaporation  per  sq.  foot  of  grate  per  hour. 

ILLUSTRATION.— Assume  elements  of  preceding,  with  evaporation  as  above. 
33.6X300. 

213 

Heating  Surface.— -Then  47.32  X  30=  1419  sq.feet  area. 
For  the  several  types  of  boilers  the  following  units  should  be  used: 

Ratio  of  Heating  Surface  to  Gratt 


-  =  47.32  sq.fset. 


30  to  i 


II 


50  to  i 


Coal  consumed  per  Sq.  Foot  of  Grate  per  Hoar 
inLbs. 


is 

2O 

30 

»5 

20 

30 

Marine  

164 

214 

314. 

l8* 

242 

3-3Q 

ICQ 

2O7 

2OQ 

182 

241 

333 

Portable  

I<iS 

187 

271 

ISO 

IQ7 

2QO 

T6< 

211 

305 

Coke... 

131 

I7O 

247 

IsQ 

IQ7 

^76 

Units   of  Heat  in.   Fuels. 


Anthracite 14  500 

Bituminous 14  200 

Petroleum,  light 22  600 

"        heavy 19440 


Petroleum,  refined 19  260 

crude ••••  19210 

Coal  Gas 20  200 

Water  Gas 8  500 


To    Resist    Oxidation    in    Cast-iron    Pipes. 

A  coating  of  hot  lime,  which  is  much  preferable  to  tar. 
*  Hall  r.troke  or  point  of  cutting  off. 


928 


APPENDIX. 


To  Compute  Relative  Velocities  of  Steam  Yachts,  from 
JCleineiits    of  their    Construction,   Capacity,  and    Op- 
eration. 
RULE.  —  Multiply  area  of  their  grate  surfaces  by  Constant  due  to  the 

character  of  the  combustion  of  their  furnaces,  divide  product  by  cube  root 

of  square  of  their  gross  tonnage  (U.  S.),  and  cube  root  of  quotient  will  give 

their  relative  velocities. 

Or,  3/— —  =  V.    G  representing  area  of  grate  surface  in  sq.  feet,  T  gross  tonnage, 


and  C  a  constant,  viz.  natural  draught  i.    Jet  or  exhaust  1.25,  and  blast  1.6. 

In  the  application  of  this  rule,  as  alike  to  all  others  when  there  is  material  differ- 
ence in  the  elements,  as  with  large  and  small  vessels,  those  that  approach  each 
other  in  general  dimensions  or  capacities,  as  determined  by  certain  ranges  or  limits 
of  tonnage,  should  be  classed  together. 

ILLUSTRATION.  —  The  grate  surface  of  a  yacht  is  27.5  sq.  feet,  her  tonnage  71.24, 
and  the  combustion  in  her  furnaces,  jet. 


7I.24  17-185 

This  result  is  an  -index  of  the  capacity  of  the  vessel,  when  compared  with  another 
in  like  manner. 

Thus,  assume  one  to  be  a  fair  exponent  of  her  class,  as  from  40  to  60,  60  to  80, 
or  80  to  ioo,  etc.,  tons,  and  her  speed  to  be  12  knots  per  hour,  or  60  minutes. 

If  then  a  competitor  possessed  the  elements  that  by  the  above  formula  would  give 
a  result  of  1.3,  their  relative  capacities  over  a  like  course  would  be  as  1.26  :  1.3  :: 
60  :  61.9,  and  61.9  —  60  =  1.9  minute  —  i  minute  54  seconds,  which  is  the  time  the 
yacht  of  greatest  capacity  would  have  to  allow  the  other. 

If  the  course  was  for  a  greater  distance,  as  for  80  knots,  than  —  x  1.9  =  «  win. 
4  sec.  the  allowance. 

For   Large   Steamers. 
/TP 
3.96  Jy  —  j  =  V.    S  representing  area  of  immersed  amidship  section  in  sq.  meters. 

NOTE.—  A  sq,  meter  is  10.764  sq.  feet. 

This  formula  is  used  in  Europe,  and  is  applicable  only  for  vessels  of  great  capaci- 
ty and  with  a  blast  combustion. 

Simple    Water    Tests. 

For  Hard  or  Soft  Water.—  Dissolve  a  small  quantity  of  soap  in  alcohol.  Put  a 
tew  drops  of  it  in  a  vessel  of  water.  If  it  becomes  milky,  it  is  hard,  if  not,  it  is  soft. 

For  Earthy  Matters  or  Alkali.—  Dip  litmus  paper  in  vinegar,  and  if  on  immersiou 
in  water,  the  paper  returns  to  its  true  shade,  the  water  is  free  from  earthy  matte* 
or  alkali.  Syrup  added  to  a  water  containing  earthy  matter  will  turn  it  green. 

For  Carbonic  Acid.—  Take  equal  parts  of  water  and  clear  lime-water.  If  com- 
bined or  free  carbonic  acid  is  present,  a  precipitate  is  produced,  to  which,  if  a  few 
dropi  of  muriatic  acid  be  added,  an  effervescence  occurs. 

For  Magnesia.  —  Boil  the  water  to  a  twentieth  part  of  its  weight,  drop  a  few  grains 
of  neutral  carbonate  of  ammonia  and  a  few  drops  of  phosphate  of  soda  into  it,  and 
if  magnesia  is  present  it  will  precipitate  to  the  bottom. 

For  Iron.—  (i.  )  Boil  a  little  nutgall  and  mix  it  with  the  water;  if  it  turns  gray  o> 
slate  black,  iron  is  present.—  (2.)  Dissolve  a  little  prussiate  of  potash,  and  mix  it 
with  the  water;  if  iron  is  present,  it  will  turn  blue. 

For  Lime.  —Into  a  glass  of  water  put  two  drops  of  oxalic  acid  and  blow  upon  it. 
If  it  becomes  milky,  lime  is  present. 

For  Acid.—  Immerse  a  piece  of  litmus  paper  in  it.  If  it  turns  red,  it  is  acid.  If 
it  precipitates  on  adding  lime-water,  it  is  carbonic  acid.  If  a  blue  paper  is  turned 
red,  it  is  a  mineral  acid. 


APPENDIX. 


929 


TOBIN  BRONZE. 

(Trade-mark  registered.) 

The  Aiisonia   Brass  and.    Copper  Co.,  Ne-w  York,  N.  Y. 
Sole  Manufacturer. 

Specific  gravity,  8.379.  Weight  of  a  cube  inch,  .3021  of  a  Ib.  Tensile 
strength  i-inch  round  rod,  79  600  Ibs.  per  sq.  inch.  Elastic  limit,  54  257  Ibs. 
per.  sq.  inch.  Elongation  in  a  rod  i  inch  in  diameter  and  8  ins.  in  length, 
15.4  per  cent.  Reduction,  37.26  per  cent. — Fairbanks. 

Is  readily  forged  into  bolts  and  nuts  at  a  dark-red  heat,  Torsional  strength 
and  Elastic  limit  equal  to  machinery  steel. 

Torsional    Strength. 

Bolt  .5  inch  in  diameter  and  i  inch  in  length,  load  at  end  of  lever  ifoot. 
Torsion,  2.67°.     Elastic  limit,  328  Ibs.     Rupture,  633  Ibs.     Torsion  point 
of  rupture,  92.2°. — J.  E.  Denton,  Stevens' s  Institute. 

Crushing  Strength,  maximum,  181  ooo  Ibs.  per  sq.  inch.— Fairbanks. 

Plates. 

Weight  per  Square  Foot. 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Ins. 
.0625 
.125 
.1875 
•25 
.3125 

•375 

Lbs. 
2.72 

5-44 
8.16 
10.88 

13-59 
16.31 

Ins. 

•4375 
•5 
•5625 
.625 
•6875 
•75       * 

Lbs. 
19.03 

21-75 
24.47 
27.19 
29.91 
32.63 

Ins. 
.8125 
.875 

•9375 
i 
1.0625 
1.125 

Lbs. 
35-35 
38.06 
40.78 

43-5 
46.22 
48.94 

Ins. 
1.1875 
1.25 
I.3I25 

1-375 
1-4375 
1-5 

Lbs. 
51-66 
54.38 
57-1 
59.82 

62.53 
65.25 

Bolts    and    Rods. 
Weight  per  Lineal  Foot. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weigbt. 

Diam. 

Weight. 

Ins. 

Lbs. 

Ins, 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.25 

.177 

•75 

1.6 

1-5 

6.42 

2-5 

I7.8 

4 

45.57 

•3J25 

.279 

.8125     1.88 

1.625 

7.5 

2.025 

19.6 

4.25 

51-44 

°375 

•399 

.875 

2.18 

1-75 

8.7 

2-75 

21-53 

4-5 

57.64 

.4378 

•544 

•9375     2.5 

1.875 

10 

2.875 

23.52 

4-75 

64.24 

•5 

.711 

i             2.84 

2 

11.38 

3 

25-53 

5 

71.16 

•5625 

.899 

1.125 

3-6 

2.125 

12.87 

3-25 

30.05       5.25 

78.46 

.625 

I.  II 

1.25 

4-46 

2.25 

14.43 

3-5 

34-86 

5-5 

86.11 

•6875 

1.34 

1-375 

5-36 

2-375 

16.06 

3-75 

40.01 

6 

102.45 

Owing  to  its  great  strength  and  non-corrosive  properties  the  rods  are  ex- 
tensively used  for  bolts,  forgings,  etc.,  Marine  and  Naval  Machinery,  Sugar- 
houses,  Breweries,  Pump  Piston-Rods,  and  Yacht  Shafting.  The  plates  are 
used  for  Pump  Linings,  Condenser  Heads,  Hulls  of  Yachts,  Centreboards, 
and  Rudders. 

Drop  Forgings  and  Nails  of  every  description  can  be  made  of  it. 

Weights   of  Steam-engines   and    Boilers  with    Water. 
Per  Indicated  IP  in  Lbs. 

Merchant  Steamer 480 

Royal  Navy    "      360 

Steamboats 280 


4i* 


Torpedo  Boats 60 

Marine  Boilers 196 

Locomotive"   60 


930 


APPENDIX. 


Weight    and    Strength    of  Ordinary    Stud  -  Link    Chain 
Catole. 


Dime 
Diana. 

isions  of  1 
Length. 

.ink. 
Width. 

Weight 
per 
Fathom. 

Admiralty 
Proof- 
stress.* 

Dimei 
Diam. 

isioij  of  1 
Length. 

ink. 
Width. 

Weight 
per 
Fathom. 

Admiralty 
Proof- 

stress.* 

Ins. 

Ins. 

Ins. 

LbB. 

Lbs. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lba. 

•375 

2.25 

1-35 

7-55 

— 

•375 

8.25 

4-95 

101.6 

76160 

4375 
•  5 

2.625 
3 

;:l75 

"•3 
13-4 

7840 
10080 

'.625 

9 
9-75 

& 

121 
I42 

90720 
96400 

•5625 

3-375 

2.025 

17.2 

12320 

•75 

10.5 

6-3 

164.6 

1*4320 

.625 

3-75 

2.25 

21 

15680 

.875 

11.25 

6-75 

l89 

141  680 

.6875 

4-125 

2-475 

25.4 

19040 

12 

7.2 

215 

161  280 

•75 

4-5 

2.7 

30.2 

22680 

.125 

12.75 

7-65 

242.8 

182000 

•&75 

5-25 

3-15 

41.2 

30800 

•25 

13-5 

8.1 

276.2 

204  1  20 

6 

3-6 

53-8 

40320 

•375 

I4-25 

8-55 

303-2 

227  360 

1.125 

6-75 

4-05 

69 

50960 

•5 

15 

9 

336 

252000 

25 

7-5 

4-5 

84 

63000 

•75 

l6.5 

9-9 

406.6 

304  940 

*  Adopted  by  Lloyds. 

NOTE  i.  —  Safe  Working-  stress  is  taken  at  half  the  Proof-stress. 

2.  —  Proof-stress  and  Safe  Working-  stress  for  close-link  chains  are  respectively 
two  thirds  of  those  of  stud-link  chains. 

3.—  Average  Proof-stress  is  72  per  cent,  of  ultimate  strength,  or  17000  Ibs.  per  sq. 
inch  of  section  of  both  sides.  Safe  working-stress  is  half  the  proof-stress,  or  8500 
Ibs.  per  sq.  inch  of  section. 

Weight  of  close-link  chain  is  about  three  times  the  weight  of  the  bar  from  which 
it  is  made,  for  equal  lengths. 

4.  —  Ultimate  Strength  per  sq.  inch  of  section  of  metal  is  35  ooo  Ibs. 

Comparing  the  weight,  cost,  and  strength  of  the  three  materials,  hemp,  iron  wire, 
and  chain  iron,  the  proportion  between  the  cost  of  hemp  rope,  wire  rope,  and  chain 
is  as  2  :  i  :  3  ;  and,  therefore,  for  equal  resistances,  wire  rope  is  only  half  the  cost 
of  hemp  rope,  and  a  third  of  the  cost  of  chains.  (Karl  von  Ott.) 

Height  and   Retrocession  of  Niagara  Falls.     (J.  Pohlman.) 
1842.    Height.—  American  ...........  167  feet. 

Horseshoe...  .......  158    '         Jas  HalL 

Width.—  American  ...........  600    ' 

Horseshoe  ...........  1800 


1886.—  Average  retrocession  2.5  feet  per  annum  (  Woodward). 
200  feet  in  ii  years,  and  9  feet  per  year  in  42  years. 
Descent  of  the  river  below  15  feet  per  mile. 
Bridge.—  Over  Oxus  on  Caspian  sea,  6230  feet  in  length. 

LENGTHS    OF    ENGLISH    RACE-COURSES. 


Course. 

Miles. 

Course. 

Miles. 

Course. 

Miles, 

NEWMARKET. 

Across  the  Flat  

1.292 

DONCASTER. 

Circular  

'•9*5 

GOODWOOD. 

Cup  Course  

2.5 

4.206 

Fitzwilliam 

Cambridgeshire  

1.136 
2.266 

Red  House  
St.  Leger  

'I" 

1.825 

New  Course  

i-5 

Round  
Rowley  Mile  

3-579 

I.  OOQ 

Cup  Course  

2.634 

NEW  CASTLE  

1.796 

2 

Summer  Course  .... 

2 

Craven  

I.  2? 

YORK. 

Two-year  old,  new  .  . 
yearling.....  

.702 
,277 

Derby  and  Oaks.  .  .  . 
Metropolitan  

i-S 
2.25 

Stakes  Course  
Two-mile..,..  

i-75 
1.923 

Railway   Speed   in   England. 

1887.— North  Western  Railway.    To  Crewe,  158.5  miles  in  178  minutes  without 

a  stop. 

Caledonian.    Carlisle  to  Edinburgh,  100. 75  miles,  including  10  consecutive 
miles  of  elevation  of  i  in  80,  in  104  minutes. 


APPENDIX. 


931 


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932 


APPENDIX. 


Niagara   River 

The  canal  through  which  the  water  is  to  be  drawn  commences  at  a  distance  of 
1.5  miles  above  the  Falls.  The  water-storage  of  the  river  is  computed  at  328855 
square  miles,  viz.  ,  87  620  of  lake  and  241  235  of  shed.  The  annual  rainfall  being  37 
inches. 

Assuming  the  rainfall  to  be  but  3^  inches,  the  flow  over  the  Falls  would  be  213000 
cube  feet  per  second.  The  Lake  survey  computes  it  at  265  ooo  cube  feet. 

The  BP  designed  to  be  used  by  the  company  constructing  the  canal  is  120000. 

A  late  and  corrected  determination  of  levels  gives  Lake  Ontario  246  and  Lake  Erie 
578  feet  above  mean  tide  at  oity  of  Nevr  York. 

Height   of*  Towers,   Spires,  etc. 

(Additional  to  page  180.) 

Eiffel  Tower,  Paris  ..........  984.3  feet.  I  Cathedral,  Strasburg  ..........  465  feet 

Cathedral,  Rouen  ...........  492       "     |  City  Hall,  Philadelphia  ........  535    " 


Zenith    and    Meridian    Distance 
at   New    York.     C.  H. 


and    A-ltitn.de 

(Lat.  40°  42'  44".) 


of   Sun 


June  2ist,  Zenith  distance.  .  .  17°  15'  44"  I  Dec.  2ist,  Zenith  distance.  .  .  64°   9'  44" 
Meridian  altitude.  72°  44'  16"  |  Meridian  altitude  .  25°  50'  16" 

"Water-pump.—  First  in  use  283  years  B.C.  Rotating  introduced  in  i7th 
century.  Plunger  pistons  invented  by  Morland  (England),  1674.  Double  acting  by 
De  la  Hire  (France). 

Symbolic   Hatching  and   Designations. 
As  adopted  by  Engineer  Department,  U.  S.  Navy. 


The   folio-wing   are   designed  and    added  "by  the  A-uthor 


BABBITT.  NICKEL. 

See  Colors  for  Drawings,  p.  IQ& 


GUM. 


ZINO. 


CONCRETE. 


APPENDIX. 


933 


.Results   of  Experiments  on  Operation  of  Steam- 
Engine.    (C.  E.  Emery,  M.  E.) 


Condensing. 
Cylinder  t  .  .  .  16  X  42  ins. 

Non-Condensing. 
18  X  42  ins. 
73-37  IDS. 
29-47 
73-6 

Condensing. 
Cutoff.  189 
IP                  78  79 

Non-Condensing. 
.189 
"5-43 
I3-07 

102.  36 
29.231 
3-25 
3-66 
.8685 
.8676 

Pressure,  81.69  Ibs. 

"      mean  effective.  31.06    " 
Revolutions  per  min.  .  .  60.3 
Water  per  IIP  per  hour  in  Ibs 

Friction  H?  10.  09 
Net  IP....  68.  7 

Coal        «      "         «            «    .".".".""." 

Coal  per  net  BP  per  hour  in  Ibs.  

Relative  efficiency  per  steam  

Relative  efficiency  per  coal.  .  . 

Safety  Valves   of  Steam-Boilers. 

Boilers  operated  at  a  low  pressure  of  steam  require  proportionately  larger  safety 
valves  than  when  operated  at  a  high  pressure.  Thus:  If  steam  at  20  Ibs.  pressure 
per  sq.  inch  is  raised  to  30  Ibs.,  a  valve  nearly  one  half  more  capacity  is  required; 
but  if  raised  from  100  Ibs.  to  no  Ibs.,  a  valve  of  nearly  one  tenth  more  capacity  is 
required. 

Belting.  Double  belts  will  transmit  one  and  one  half  times  the  power  of 
single. 

Wide  belts  are  less  effective  per  unit  of  area  than  narrow.  Long  belts  are  more 
effective  than  short.  Driving  belts  may  be  driven  at  a  velocity  of  3500  feet  per  min. 
ute.  Lathe  belts  from  1500  to  2000  feet.  Economy  of  wear  requires  less  velocities. 

Non-Conductors   of  Temperature. 
Their   Comparative   Efficiency. 

Materials  in  Italics  are  whotty  free  from  Carbonization  or  Ignition  from  slow  con- 
tact with  Boilers  or  Steam-pipes. 

The    following   Materials    were    used,    as   Covering    to    a 
Steam-pipe    2    Ins.   in    Diameter. 

Pounds  of  Water  Heated  10°  per  Hour  through  One  Square  Foot  of  the  Material 
(J.  M.  Ordway.) 


Material  i  Inch  in  Thickness. 

Lbs. 

Relative 
Solidity. 

Material  i  Inch  in  Thickness. 

Lbs. 

Relative 
Solidity. 

i    Wool  loose  

8.x 

.56 

13.  Anthracite  coalpow-  \ 

2  Feathers  of  live  geese  .  . 

Q.6 

.  e 

der  j 

35-7 

5.o6 

3.  Lamp  black,  loose  
4  Felt  of  hair  

fa 

10.  3 

.56 

1.85 

14.  Magnesia,  calcined) 
and  compressed     ) 

42.6 

2.85 

48 

6.  Lamp  black,  compressed 

10.6 

3.44 

16.  Asbestos,  fine  

4Q 

81 

7.  Charcoal  of  cork  
8.  Magnesia,  calcined) 
and  loose    ) 

11.9 
12.4 

•53 
•23 

17.  Sand  
18.  Stag  wool,  best  (fine) 
threads  of  brittle  glass)  J 

62.1 
13 

5-27 

9  Magnesia,  carbonate  ) 

19.  Paper  

14 

of  and  light     .   .    j 

13-7 

.6 

20  Rice-chaff       .... 

18  7 

to.  Charcoal  of  white  pine. 
ii.  Magnesia,  carbonate) 
of,  and  compressed  ) 
12.  Plaster  of  Paris... 

13-9 
iS-4 
30.  a 

1.19 

i-5 

7.68 

21.  Bit.  coal-ashes,  loose.  . 
22.  Asbestos  paper,  tight. 
23.  Anth.  coal-ashes,  loose 
24.  Clav  and  veg'ble  fibre. 

21 
21.7 
27 
10.  0 

- 

Flow   of  Several    Rivers.     Minimum    Dry    Weather. 

In  Cube  Feet  per  Minute. 


St.  lAwrence,  at  BrockviJ'.e,  Ont.,  18000000. 

Mississippi,  at  St.  Pauls,  Minn 2000000. 

Connecticut,  at  Holyoke,  Mass. ...     300  ooo. 
Okio,  at  Pittsburgh,  Penn. . . 


Illinois,  at  La  Salle,  III 36  ooo. 


Seine,  at  Paris,  France looooo. 

Mohawk,  at  Cohoes,  N.  Y 58  800. 

Thames,  at  London,  England. .  360001 
Chicago,  at  Chicago,  111 36  ooo, 


934 


APPENDIX. 


Standard   TJ.  S.  ^Weights   and   Measures. 

(U.  S.  Coasi  Survey.) 

Lineal. 


Inch  to 
Millimetres. 

Foot  to 
Metre. 

Yard  to 
Metre. 

Mile  to     II   Metre  to 
Kilometres.        Inches. 

Metre  to 
Feet. 

Metre  to 
Yards. 

Kilometre 
to  Miles. 

25.4 

Chain  = 

.304801 
20.1169111 

.9144 

etres.      P 
I 

1-60935   ||     39.37 

athom  =  1.829  metre 
.not  =  1853.27  metres 

3.28083 

s.     So.,  no 

1.093611 

ile  =  259  h 

.62137 

ectares. 

Inches  to          Feet  to 
Ceutimetre.      Decimetre. 

Yard  to 
Metre. 

Acre  to 
Hectare. 

Centimetre 
to  Inch. 

Metre  to 
Feet. 

Metre  to 
Yards. 

Hectare 
to  Acres. 

6.452              9.29 

.836 

.4047 

•155 

10.764 

1.196 

2.471 

Volume.     (Fluid.) 

Dram  to 
Milli- 
litres.* 

Ounce  to 
Milli- 
litres. 

Quart  to 
Litre. 

Gallon  to 
Litres. 

Milli- 
litref  to 
Dram. 

Millilitre 
to  Ounce 

Litres 
to 
Quarts. 

Deca- 
litre to 
Gallons. 

Hecto- 
litre to 
Bushels. 

3-7 

29.57 

.94636 

3-  78544 

.27 

.338 

1.0567 

2.6417 

2-8375 

Cube. 


Inch  to 
Centimetres. 

Foot  to 
Metre. 

Yard  to 
Metre. 

Bushel  to 
Hectoliter. 

Centimetre 
to  Inch. 

Decimetre 
to  Inches. 

Metre  to 
Feet. 

Metre  to 
Yards. 

16.38^     f   .02832 

.765 

•35242 

.061 

61.023 

35-3I4 

1.308 

Weight. 


Grain  to 
Milligrams. 

Av.  Ounce 
to  Grains. 

Av.  Pound  to 
Kilogram. 

Tr.  Ounce 
to  Grains. 

Milligram 
to  Grain. 

Kilogram 
to  Grains. 

Hectogram  J 
to  Av.  Ounces. 

Kilogram 
to  Pounds 

64.7989 

28.3495 

•45359 

31.10348 

•01543 

15432.36 

3-5274 

2  .  204  62 

Quintal  to  Av.  Pounds,  220.46.      Tonnes  §  to  Av.  Pounds,  2204.6.      Grams  to  Tr. 
Ounce,  .03215.    Av.  Pound  =  453. 592  427  7  grams.    Kilogram  — 15  432. 356  39  grains. 

NOTE. — The  U.  S.  yard  is  equal  to  the  British  yard.     British  gallon  =  4. 543  46 
litres.    Bushel  =  36.3477  litres. 

Value   of  tlie   .Metre  in  terms  of  the  British  Imperial  Yard,  and  of  the 
Committee  Metre  (C.M.)  of  the  U.  S.  Coast  and  Geodetic  Survey.     (0.  H.  Tittman.) 

Authority. 


Hassler 39.380917 

Kater 370  79 

Bailey 369  678 

Clarke 370  432 

Comstock 369  85 


39.36994 

39-3699 

39-36973 

39-369 

39 


-3697 
-36984 


Mean....  39.3698 

Dead    Sea  and   Valley   of*  tlie   Jordan.     Portions  of  these  are 
1300  feet  below  the  level  of  the  sea.     (R.  E.  Peary.) 

Value  of*  GJ-old.    From  1501  to  1889  the  ratio  of  gold  and  silver  varied 
from  1 1. 1  to  22. 

Dnr  ability  of*  Woods.    Wood  columns  or  posts,  set  in  earth  opposite 
to  course  of  its  growth,  are  more  durable  than  when  set  with  it. 


*  Cube  centimetres. 


f  Cube  centilitre. 


too  Grams.  $  Millien. 


APPENDIX. 


935 


Miscellaneous    Operations. 

To  Remove  Paint.      Apply  chloroform. 

To  Restore  Color  of  a  Fabric.  When  destroyed  by  an  acid  ap- 
ply ammonia  to  neutralize  it,  and  then  chloroform. 

Silverware.  Warm,  and  cover  with  a  mild  solution  of  collodion  in 
alcohol,  applying  it  with  a  soft  brush. 

Grilt  Frames.  To  restore,  rub  with  a  sponge  moistened  with  spirits  of 
turpentine. 

Egg  Stain.    On  silver,  rub  with  salt. 

Iron  Rust.  To  remove  from  white  fabrics,  saturate  the  spots  with 
lemon-juice  and  salt,  and  expose  to  the  sun. 

Ink:  Stains.  Wash  with  pure  fresh  water,  and  apply  oxalic  acid.  If  this 
changes  the  stain  to  a  red  color,  apply  ammonia. 

Clinkers  on  Brick:.    Apply  oyster  shells  on  the  top  of  a  clear  fire. 
Antidotes   for   Poisons. 
Additional  to  page  185. 

Antimonial  Wine  or  Tartar  Emetic. — AArarm  water  to  induce  vomiting. 

Arsenic  or  Fowler's  Solution.  —  Emetic  of  mustard  and  salt,  a  tablespoonfuL 
Then,  butter,  sweet-oil,  or  milk. 

Bed  Bug.— Oil  of  vitriol,  corrosive  sublimate,  sugar  of  lead. 

Caustic  Soda  or  Potash,  and  Volatile  Alkali.  —  Drink  freely  of  lemon  -juice  or 
vinegar  in  water. 

Carbolic  Acid.—  Flour  and  water,  and  glutinous  drinks. 

Carbonate  of  Soda,  Copperas,  or  Cobalt. — Administer  emetic;  soap  or  mucilagi- 
nous drinks. 

Chloroform.— -Apply  cold  water  to  head  and  face,  artificial  respiration,  and  gal- 
vanic battery. 

Laudanum,  Morphine,  or  Opium. — Administer  strong  coffee,  mustard  flour,  butter 
or  oils  in  warm  water,  and  exercise. 

Muriatic  or  Oxalic  Acid.  — Give  magnesia  mixed,  and  soap  dissolved  with  fresh 
water. 

Nitrite  of  Silver.— Bali  in  water. 

Sulphate  of  Zinc  or  Red  Precipitate.— Give  milk  or  white  of  eggs  copiously. 
Sulphuric  Acid. — Aqua  fortis. 
Strychnine.  —Emetic  of  mustard  or  sulphate  of  zinc,  aided  by  warm  water. 

Motive   iPower   of  the   "World. 
Steam-engines.     In  Horse-Power. 

United  States. ...  7  500 ooo  I  Germany 4  500000  I  Austria 1 500000 

England 7000000  |  France 3000000  |  Other  countries.  19000000 

Steam-boilers    in    Foreign    Countries, 

France,  including  Locomotive. .  51  390  |  Germany 60700  |  Austria 12000 

Locomotives    in    Foreign    Countries. 

France 7000  |  Germany. . .  10000  |  Austria 2800  |  Other  countries. . .  85  200 

The  steam-engines  of  the  world  represent  the  power  or  work  of  i  ooo  ooo  ooo  men. 

(Bureau  of  Statistics,  Berlin,  1887.) 

Destructive   Stress   of  Belting.     (Horace  B.  Gale.) 
In  Lbs.  per  Sq.  Inch. 


Material. 

Maxi- 
mum. 

Minimum. 

Exten- 
sion.* 

Material. 

Maxi- 
mum. 

Minimum. 

Exten- 
sion.* 

Best  Leather. 
Raw  hide  

Lba. 
8000 
6750 

Lbs. 

2850 
3000 

Inch. 
.018 
.18 

Rubber  
Cotton  belt'g 

Lba. 
3888 
2913 

Lba. 
3000 

2000 

Inch. 
.059 
.037 

*  At  400  Ibs.  per  eq.  inch. 


APPENDIX. 

Largest  Constructions  and.  INTatural  FormationSo 

New  Opera-House,  Paris.— Covers  3  acres,  and  has  a  volume  of  4287000  feet. 

Popocatapetl,  Highest  active  Volcano,  Mexico.— Has  a  crater  one  mile  in  diametei 
and  looo  feet  in  depth.  (See  p.  182.) 

Telegraph  Wire  over  river  Kistnah,  India.  —  6000  feet  in  length  and  1000  feet  in 
elevation.  (See  p.  179.) 

Chinese  Wall,  Built  220  B.C.     (See  p.  179.) 

Lambert  Coal  Mine,  Belgium. — 3490  feet  in  depth. 

Mammoth  Cave,  Kentucky.— Some  of  its  chambers  are  traversed  by  navigable 
branches  of  the  subterranean  river  Echo. 

St.  Gothard  Tunnel— Its  summit  is  900  feet  below  the  surface  at  Andermatt,  and 
6600  feet  below  the  peak  of  Kastlehorn.  (See  p.  179.) 

Bibliotheque  Nationals,  Paris.— Founded  by  Louis  XIV.,  contains  1400000  vol- 
umes, 300000  pamphlets,  175000  MSS.,  300000  maps  and  charts,  and  150000  coins 
and  medals.  Engravings  i  300000,  contained  in  1000  volumes,  and  100000  portraits. 

Desert  of  Sahara,  Africa.— Length  3000  miles,  average  breadth  900  miles,  and 
area  2000000  sq.  miles. 

Pyramid  of  Cheops,  Egypt.— Volume  of  masonry  89028000  cube  feet;  weight  of 
stone  computed  at  6  316  ooo  tons.  (See  p.  174. ) 

Bell,  Moscow.— Circumference  at  base  68  feet,  height  21  feet.     (See  p.  181.) 

Bridges.*    Rialto,  Venice.— A  single  arch  of  marble,  98.5  feet  in  length. 

Clifton  Suspension,  Bristol,  Eng.— Span  703  feet,  elevation  245  feet. 

Niagara  Suspension,  U.  S.— Cantilevers,  of  steel,  length  810  feet.  Elevation  above 
the  rapids  245  feet. 

Britannia,  England. — 1512  feet  in  length,  and  elevation  103  feet. 

Forth,  Frith  of  Forth,  Scotland.—  Length  8098.5  feet,  exclusive  of  approaches  of 
5349.5  feet.  Two  Cantilever  spans  of  1710  feet  each.  Piers  360  feet  above  water. 
Roadway  150  feet  in  the  clear  above  water.  Iron  and  steel  54000  tons.  Masonry 
250040  tons. 

Tay,  Scotland.— Length  2  miles,  85  piers,  and  elevation  77  feet. 

Colnmns    or    !Pillars. 

When  a  column  or  pillar  is  without  its  vertical  line;  one  with  slightly  rounded 
ends  becomes  capable  of  greater  resistance  than  one  with  square  ends. 

Experiments  at  the  U.  S.  Arsenal  at  Watertown,  Mass.,  developed  that  the  ver- 
tical resistance  of  timber,  to  transverse  compression  or  crushing,  was  about  one 
third  of  its  resistance  to  longitudinal  compression,  and  hence,  that  the  area  of  the  cap 
or  the  head  of  a  timber  column,  should  proportionately  exceed  that  of  the  column. 

Steam-engine    Notes. 

Horse- power, t  Nominal.  — Is  usually  computed  from  the  volume  of  steam  dis- 
charged from  the  cylinder.  Its  measure  for  an  ordinary  non-condensing  engine  is 
about  .4  of  its  actual  power.  It  refers  more  to  the  dimensions  of  an  engine  than 
its  capacity. 

Indicated.— Is  the  measure  of  the  force  exerted  by  an  engine,  and  from  this  is  to 
be  deducted  for  leaks,  friction  of  its  parts!,  and  of  its  connecting  parts,  about  10  per 
cent. 

Feed  Water. — Ordinarily  2  to  3.5  gallons  or  17  to  30  Ibs.  of  water  are  required  for 
each  IIP. 

Fuel— The  ordinary  consumption  of  fuel  may  be  taken  at  3  Ibs.  per  IIP  for  a 
non-condensing  engine,  and  2  Ibs.  for  a  condensing. 

Boilers.— 12  to  15  sq.  feet  of  heating  surface,  or  .4  to  .5  of  grate  surface,  with 
natural  draught,  will  give  one  IIP. 

Flow  of  Steam.— The  velocity  of  it  in  feet  per  second,  may  be  determined  by  the 
formula,  6oVT  -f-  460°  =  V;  or,  60  times  the  square  root  of  the  sum  of  the  tem- 
perature of  it  in  degrees,  and  460.  Thus  for  a  pressure  of  100  Ibs.  per  sq.  inch  a 
velocity  of  900  feet  may  be  obtained.  ( John  Richards,  Phila. ) 

*  Additional  to  page  181.        t  See  also  pp.  733,  734. 


APPENDIX.  937 

Atlantic  and  Pacific  Oceans.  There  is  not  any  difference  in  the 
mean  levels  of  these  Oceans  at  Aspinwall  and  Panama,  as  determined  by  Geo.  M. 
Totten,  who  constructed  the  Panama  Railroad. 

Origin  and.   IPeriocl   of  Grreat   Inventions. 

See  also  Chronology,  pp.  71,  72,  915. 

Air-engine. — Amonton,  1699.    Stirling,  1827.     Ericsson,  1855. 

Air-pump. — Otto  Gueriche,  1650.      Anemometer. — Walflus,  1709. 

Balloon. — First,  Lyons,  France,  1783.      Barometer.* — Torricella,  1643. 

Battery.—  Electric,  1745;  claimed  by  Kleist,  Cunseus,  and  Muschenbroch. 

Bridges  (Suspension). —Of  chains,  China,  100  B.  C. 

Bayonets.— At  Bayonne,  1670.     Socket  bayonet,  1699. 

Bells.—  In  Christian  church,  400;  in  France,  550.      Bellows.  —Egypt,  1490  B.CX 

Bessemer  Steel— Sir  Henry  Bessemer,  1856.     Blankets.*— England,  1340. 

Blasting. — Germany,  1620.      Bullets. — Of  stone,  1418;  of  iron,  1550. 

Calico  Printing. — Egypt;  introduced  in  England  1696. 

Camera  Obscura. — Roger  Bacon,  1214;  Newton,  1700;  Daguerre,  1839. 

Candles.—  Of  tallow,  1290.      Cannon.—  m8;  England,  1521. 

Carriages — Vienna,  1515;  England,  1580. 

Clocks.*— To  strike,  by  Arabians,  800;  by  Italians,  1200. 

Com.*— 1184  B.C. ;  China,  1200  B.C. ;  Rome,  576;  England,  uoi. 

Compass.*— China,  2634  B.C.      Cotton  Gin.—  Whitney,  1793. 

Dyeing. — 1490  B.C.     Prussian  Blue,  Berlin,  1710. 

Dynamite. — Sobrero,  1846;  Nobel,  1867. 

Electric  Discoveries.* — Leyden  Jar,  Cunaeus,  1746  ;  Electric  Light,  Davy,  1800; 
first  patent  of  it,  Greene  &  Staite,  1846. 

Electro- Magnetism. — Oersted,  Copenhagen,  1819. 

Electrotyping. — Jacobi  of  Russia  and  Spencer  of  England,  1837. 

Engraving.—  China,  1000  B.C. ;  on  metal,  1423;  line  or  steel,  1450;  etching,  1512. 

Gas.—  Murdock  Cornwall,  1792;  Meter,  Clegg,  1807;  Dry  meter,  Malam,  1820. 

Glass.*— Egypt,  1740  B.C.     Windows,  France,  i2th  century. 

Gold  Leaf.— Egypt,  1700  B.C.      Gunpowder.—  Unknown;  rediscovered  1324. 

Horseshoes. — 300;  'of  iron,  480. 

Hydraulic  Press. — Bramah,  1796.      Hydraulic  Ram. — Whitehurst,  1772. 

Hydrogen. — Isolated  by  Cavendish,  1766.  Iron  Vessels. — J.  Wilkinson,  England, 
1787;  Ship,  1821;  Steam-boat,  1830;  Shipbuilding,  1833. 

Kaleidoscope.— Sir  Daniel  Brewster,  1814-17.      Knives.—  Table,  England,  1550. 

Life-boat. — 1817.      Lithography.  —  Senefelder,  about  1796. 

Locomotive. — Watt,  1769  and  1784.     Cngnot,  1769. 

Matches. — Friction,  1829.      Medicine. — From  Greece,  in  Rome  200  B.  C. 

Mirrors. — Glass,  Venice,  i3th  century.      Newspaper. — First  authentic,  1494. 

Omnibus Paris,  1827. 

Organs.—  755.     England,  951.      Oxygen.—  Priestley,  1774. 

Paper.—  From  silk,  China,  120  B.C. ;  from  rags,  Egypt,  1085. 

Pens.— Of  steel,  1803;  gold,  1825.      Pencils.—  Of  lead,  50.     England,  1565. 

Pianoforte.—  Italy,  1710.      Phonograph.—  Edison,  1877. 

Photograph.—  England,  1802;  perfected,  1841.     Pottery.  —Oldest,  Egypt,  20006.  CL 

Post-Office.— Vienna  and  Brussels,  1516.     Stamps.—  England,  1840. 

Printing.*— Types,  L.  Coster,  1423. 

Railroad.  *— Passenger,  England,  Sept.  27,  1825. 

Sewing-machine. — Patented.  England,  1755. 

Sleeping-car.— 1858;  Pullman,  1864.      Soap. — England,  i6th  century. 

Spectacles.—  Italy,  isth  century. 

Telephone.—  A.  G.  Bell  and  C.  J.  Blake,  Boston,  1874. 

Torpedo.—  Credited  to  D.  Bushnell,  1777. 

«  Indicates  that  the  subject  it  also  given  at  pp.  71,  72. 

4  K 


938 


Cobalt.., 
G9ld.... 
Iridium . 


APPENDIX. 

Values   of  some   Precious   Metals. 
Per  Pound,  Troy. 


250 
295 


Osmium 

Platinum 

Potassium 

^Variable. 


102 
25 


Rhodium . .  $  415 

Ruthenium 075 

Silver* ia 


Missions 

Tea,  Coffee,  etc  

.  20 

Sugar  

Fuel  for  Households 

\\ 

Milk  

Linen  and  Cotton.  .  .  . 

20 

Butter  and  Cheese.  . 

•  35 

Expenditure  in.  England,  for  Various  Purposes  and  of 
Articles  Compared  with,  that  of  Spirituous  Liquors- 

In  Millions  of  Pound  Sterling. 

Woollen  Goods 46 

Bread 70 

Rents 130 

Liquors 136 

Aluminum.. 

Elastic  limit  of  bars  in  tension  14000  Ibs.  per  sq.  inch.   Specific  heat  .2185.  Melt 
at  1400°.   Malleable  at  from  200°  to  300°. 

Tensile  strength,  ultimate,  26000  Ibs.     Modulus  of  elasticity,  12000000. 

Shrinkage  .022  per  linear  foot.    It  is  comparatively  unaffected  by  exposure  to 
air  or  water.     Cube  inch  weighs  .0926  Ib.    A  cube  foot  weighs  160.013  «*• 
( Continued  on  page  976.) 

Bushels   of  Seed   Required   per   Acre. 


Barley  
Beans    

.1.5  t 

t 
•   -75 

:74 

.16 

i 
•25 

02.5 

2 

•33 
•33 

i 

In  Bushels  per  A 
Flax  5    t( 

ere. 

)  2 

•875 
2.25 
•5 

i 

Oats  

2            t 

•  5 
2.5 
5 

2 
.06 

i-5 

04 

•7 

'     3-5 

'    10 

1     2.5 

1       2 
"          .16 
'      2 

Grass,  blue.  .  .625 
"    orchard.  i.  5 
"    Herds1..  .375 
"  Timothy  .5 
Hemp             i 

Parsnips.. 
Pease  
Potatoes.. 
Rice  . 

Buckwheat.  .  . 

Clover,  red... 
white. 
Corn,  brown  . 
"     Indian... 

Rye 

Millet  i 

'.'fes 

Turnips... 
Wheat.... 

Mustard  25 

See  also  page  193. 

Domestic   Remedials. 

Colors.— Discharged  by  an  acid,  can  be  restored  by  Ammonia. 

Flies.— Carbolic  Acid  (20  drops),  evaporated  on  a  hot  surface,  as  a  shovel,  will 
drive  them  from  a  room. 

Ink.—  To  remove  stains  from  a  white  fabric,  wet  with  Milk  and  cover  with  Salt, 

Mildew  stains.— May  be  discharged  by  Buttermilk. 

Mosquito.— Camphor  Gum,  vaporized  over  the  chimney  of  a  gas-burner  or 
lamp,  will  drive  them  from  a  room. 

Mats.— To  drive  them  off,  apply  Chloride  of  Lime  to  their  locality. 

Sewer  Gas.—  The  noxious  effects  removed  by  Chloride  of  Lime. 

Snnstroke.  Remove  patient  to  a  cool  place,  administer  water  freely,  and 
Quinine  or  Salicate  of  Soda. 

Comparative   "Values    of  UTood.  for    Sheep. 

Wool  and  Tallow  Produced. 


FOOD. 

Wool. 

Tallow. 

FOOD. 

Wool. 

Tallow. 

Wheat  x 
Oats  94 
Barley        .     .         ..     .89 

Lb8. 
•97 

•77R 

Lbs. 
•99 
•7 

i 

Corn-meal,  wet  83 
Buckwheat  79 
Rye,  without  salt  58 

Lbs. 

•93 
•  7 
•97 

Lbs. 
.29 
•55 
.71 

Pease  88 

I 

.7 

Potatoes,  with  salt  ....  3 

•45 

.3 

Rye,  with  salt...           .87 

•  Q7 

•  S8 

"      without  salt.  .28 

•45 

.10 

APPENDIX.  939 

Croton   .A-queduct.    New  York,  1890. 
Dimensions,  Length,  and.    Capacity. 

^h.:::::::::::::  *!:S  ra'les}  30.75  m.*  m  umgth. 

Pipes  to  Central  Park  reservoir,  2.37  miles  in  length. 
Tunnel  under  Harlem  river,  307  feet  below  tide-water  level 

Course.  —From  Croton  Lake,  350  feet  above  the  Dam,  and  runs  generally  Southerly, 
through  Westchester  Co.  and  the  24th  Ward  of  New  York,  to  a  point  7000  feet  N.  on 
Jerome  Park,  with  a  uniform  inclination  of  .7  feet  per  mile;  its  general  form,  that 
of  a  horse  shoe  with  curved  invert ;  being  13.33  feet  in  height  and  13.6  feet  in 
width  ;  having  a  computed  capacity  of  318  millions  of  gallons  per  day.  From 
thence,  where  it  is  contemplated  to  construct  a  large  reservoir,  for  the  supply  of 
the  annexed  districts  of  the  city,  to  its  termination  at  issth  Street  and  xoth  Avenue, 
its  capacity  is  reduced  to  250  millions  of  gallons  per  day,  and  the  Aqueduct  which 
from  there  is  to  be  operated  under  pressure,  is  circular  in  its  section,  12.3  feet  in 
diameter,  with  varying  inclinations,  the  portion  under  the  Harlem  river  being 
10.5  feet 

From  135th  Street  it  is  connected  to  12  cast-iron  pipes,  48  ins.  in  diameter,  4  of 
which  connect  with  the  old  Aqueduct,  4  with  the  present  City  distribution,  and  4 
leading  through  Convent,  (gih)  and  8th  Avenues  to  the  Reservoir  in  Central  Park. 
The  operating  capacity  of  all  being  equal  to  that  of  the  Aqueduct,  250  millions  of 
gallons. 

The  Aqueduct  is  for  the  greater  portion  of  its  length  a  tunnel,  it  raising  to  the 
surface  but  at  four  points,  from  which  it  can  be  emptied  through  gates  into  the 
adjacent  rivers. 

Capacity.— The  water-shed  ef  the  Croton,  in  extreme  dry  weather,  with  sterage, 
is  250  million  gallons  per  day. 

The  present  storage  system  includes  Croton  Lake,  Reservoir  at  Boyd's  Corners, 
the  middle  branch  Reservoir  of  the  Croton  valley,  and  several  lakes,  with  a  total 
capacity  of  10000  million  gallons:  three  dams  being  in  progress  of  construction  and 
others  contemplated,  viz.,  one  at  Carmel  and  one  at  Quaker  Bridge. 

The  Capacity  of  the  Reservoir  in  Central  Park  is  computed  at  1000  million  gallons. 

Ice. 

Additional  to  p.  195. 

1.5  ins.  thick  will  support  a  man;  5  ins.,  an  84-lbs.  cannon;  10  ins.,  a  body  of 
men;  18  ins.,  a  railroad  train. 


Yield   of  Oil   in   Seeds. 

Per  Cent. 


Mustard,  white 37 

Hemp 19 

Linseed 17 

Corn,  Indian 7 


Oats 6.5 

Clover-hay 5 

Flour- wheat 3 

Barley a. 5 


Rape 55 

Almond,  sweet 47 

bitter 37 

Turnip 45 

Additional  to  page  189. 

Historical    Events    and    N"otatole    Facts. 
Australia. — Discovered  1622. 

Banana.—  Produce  per  acre  44  times  greater  than  potato,  and  131  times  greater 
than  wheat. 

Camels.—  Some  can  travel  800  miles  in  8  days. 
Catacombs. — Of  Rome,  remajns  of  6000000  bodies. 
China.— Authentic  history  of  it,  3000  B.C.      Crucifixion. — 37. 
Library  of  Alexandria — 47  B.  C.  contained  400000  books. 
Pens. — Steel,  consumption  4000000  per  day. 
Slavery.—  Abolished  in  Eng.  West  Indies,  1834;  Russia,  1861. 

N".  Latitude  reached  toy  Explorers.    1884.—  Adolphus  W.  Greely, 
U.  S.  Army,  83°  24'.    The  distance  from  this  to  the  Pole  is  456  02  miles. 


940 


APPENDIX. 


Drilling. 

Rand  Drill  Co.,  New  York. 


Drills. 

Cylinder. 
Diain. 

Usual 
Depth 
Drilled. 

Diam. 
of 
Bottom 
of  Hole. 

Depth 
Drilled 
in  10 
Hour*. 

Diam. 
of 
Hose. 

Diam. 
of 
Steel. 

Steam 
Boiler. 

Steam 
Pipe. 

No. 
Kid  

Ins. 
1.875 

Fest. 
1.5 

lus. 

Feet. 

Ins. 

•75 

Ins. 
.625 

If 
3 

Ins. 
•75 

2  .25 

4 

i  .0625 

CO 

.  ye 

•75 

5 

i 

2  and  2  A  
3  and  3  A  
3.25  and  3.  25  A 
4  and  4  A  

2-75 
3-125 
3-25 
3.625 

4.  5 

6  to  10 
10  to  15 

15 
20 
20  to  30 

i-5 
i-75 
i-75 

2 

2.25 

60 
70 
70 
70 
70 

•75 

•25 

.  5 

.125 
.125  to  1.25 

•375 
.  5 

7 

10 

10 

12 

15 

1.25 
i-5 
i-5 

2 
2 

7  

5.5 

'•& 

.75 

20  tO  21 

2-5 

Raiid.    Air    Compressors. 

Rand- Corliss  Class  "  B  B*." 
Compound  Steam  Condensing.     Compound  Air. 


Steam  Pressure  125  Ibs. 

Terminal 

Capacity  in 
Free  Air 

Cylinder 
Steam. 

Diameters. 
Air. 

Stroke. 

Revolutions 
per 

Air 

Pressure 

per  Minute. 

High. 

Low. 

High. 

Low 

Minute. 

at  80  Ibs. 

Cube  Feet. 

Ins. 

Ins. 

Ins. 

Ins 

Ins. 

No. 

IH5. 

670 

10 

18 

10.5 

17 

30 

85 

102 

1196 

12 

22 

13 

21 

36 

83 

l82 

1562 

'4 

26 

15 

24 

36 

83 

238 

1650 

'4 

26 

15 

24 

42 

75 

252 

1920 

16 

30 

17-5 

28 

36 

75 

293 

2242 

16 

30 

17-5 

28 

42 

75 

342 

2395 

16 

30 

17.5 

28 

48 

70 

365 

2520 

18 

34 

20 

32 

36 

75 

384 

2897 

18 

34 

2O 

32 

42 

75 

442 

3128 

18 

34 

20 

32 

48 

70 

475 

3960 

20 

38 

22.5 

36 

48 

70 

604 

4100 

22 

40 

24 

38 

48 

65 

625 

453° 

22 

42 

25 

40 

48 

65 

690 

5000 

24 

44 

26.5 

42 

48 

65 

763 

6000 

26 

48 

29 

46 

48 

65 

9*5 

6820 

28 

52 

30 

48 

48 

65 

1040 

Rand  "Imperial"  Type  X. 

Duplex  Steam  Non-  Condensing.     Compound  Air. 

Steam  Pressure 

80  to  100  Ibs.                                            gteftm 

Capacity  In 
Free  Air 
per  Min. 

Duplex 
Steam 
Cylinders. 

Dian 
AirC 
High. 

icter  of                                       Revolutions        and  Air 
Blinders.                  Stroke.                per              Pressure, 
Low.                                    Minute.         »*  100  Ibs. 

Cube  Feet. 

Ins. 

Ins. 

[ns.                   Ins.                   No.                   IH>. 

145 

6 

6-5 

10                      8                   200                   25 

245 

7 

7-5 

12                   10                  190                  4* 

37° 

8 

9 

14                   12                  175                  63 

535 

10 

10 

16                   14                  165                 91 

7°5 

12 

ii 

18                  16                 150               120 

1050 

J4 

*3 

22                         l6                        150                     178 

Rand  "Imperial"  Type  XI.    Duplex  Air  Cylinders.    Belt  Driven. 

Capacity  in 
Free  Air 
per  Min. 

Air  Cylinders. 
Diam  of                gt    k 
each. 

per  Min. 

Air  Pressure 
60  Ibs. 

per  Sq.  Inch. 
100  Ibs. 

Cube  Feet. 

Ins.                        Ins. 

No. 

IH>. 

Iff. 

11.7 

4                          4 

200 

1  .7 

2-3 

22.7 

5                       5 

2OO 

3-3 

4-5 

38 

6                       6 

200 

5-5 

7-5 

62 

7                       7 

2OO 

9 

12 

93 

8                       8 

2OO 

13-5 

lg-5 

163 

10                              10 

1  80 

24 

30 

275 

12                              12 

175 

4° 

53 

APPENDIX} 


941 


Suspension.  Furnaces — IVTorison, 

The   Continental  Iron    Works,  Brooklyn,  N.  Y. 


La   for    Corrugated.   Furnaces. 

Board  of  U.  S.  Supervising  Engineers,  October  ioth,  1891. 

P  X  B 

=T.  P  =  working  pressure  in  Ibs.  per  sq.  inch.     D  mean  diameter  of  fur- 

15  600 

nace=inside  diameter+1,  and  T  thickness  of  metal,  both  in  ins. 

Corrugated  not  less  than  1.5  inches  in  depth,  and  flat  surface  of  ends  not  exceed- 
ing 6  inches  in  length. 

Thickness    of   Metal     in     Suspension    Furnaces    for    dif- 
ferent   Diameters    and    "Working    Pressures    in   Lt>s. 
Per    Sq..    Inch. 

As  Determined  by  the  Formula  in  the  Rules  and  Regulations  of  the  U.  S.  Board. 


Inside 
Diam. 
Ins. 

A 

H 

1  Itt 

iV 

H 

i 

if 

A  1  U  1  ff 

H 

H 

1 

28 

162 

178 

i95 

211 

227 

243 

260 

276 

292 

308 

325 

34i 

357 

390 

29 

157 

172 

1  88 

2O4 

220 

235 

251 

267 

283 

298 

314 

330 

345 

377 

30 

152 

167 

182 

198 

213 

228 

243 

258 

274 

289 

304 

319 

335 

365 

31 

14? 

162 

177 

192 

2O6 

221 

236 

251 

265 

280 

295 

310 

325 

354 

32 

143 

157 

172 

1  86 

2OO 

215 

229 

243 

258 

272 

286 

301 

315 

344 

33 

139 

153 

167 

181 

195 

208 

222 

236 

250 

264 

278 

292 

306 

334 

34 

135 

i48 

162 

176 

189 

20.J 

216 

230 

243 

257 

270 

284 

297 

325 

35 

131 

144 

158 

171 

184 

197 

210 

223 

237 

250 

263 

276 

289 

316 

36 

128 

141 

153 

166 

179 

192 

205 

218 

230 

243 

256 

269 

282 

307 

37 

125 

137 

150 

162 

175 

187 

200 

212 

225 

237 

250 

262 

275 

300 

38 

121 

134 

146 

158 

170 

182 

195 

207 

219 

231 

243 

255 

268 

292 

39 

118 

130 

142 

154 

166 

I78 

190 

202 

214 

225 

237 

249 

261 

285 

40 

116 

127 

139 

150 

162 

174 

185 

197 

208 

220 

232 

243 

255 

278 

41 

113" 

124 

136 

147 

158 

170 

181 

192 

204 

215 

226 

238 

249 

272 

42 

no 

121 

132 

144 

155 

166 

177 

188 

199 

210 

221 

232 

243 

265 

43 

108 

IIQ 

130 

140 

151 

162 

i73 

184 

195 

205 

216 

227 

238 

260 

44 

105 

116 

127 

137 

148 

158 

169 

1  80 

190 

201 

211 

222 

233 

254 

45 

103 

114 

124 

134 

i45 

155 

165 

176 

1  86 

197 

207 

217 

228 

248 

46 

101 

in 

121 

132 

142 

152 

162 

172 

182 

192 

203 

213 

223 

243 

47 

99 

109 

119 

129 

i39 

149 

159 

169 

179 

189 

198 

208 

218 

238 

48 

97 

107 

117 

126 

136 

146 

156 

165 

175 

185 

195 

204 

214 

234 

49 

95 

105 

114 

124 

133 

143 

152 

162 

172 

181 

191 

200 

210 

229 

50 

93 

103 

112 

121 

131 

140 

150 

159 

168 

178 

187 

196 

206 

225 

Si 

9i 

IOI 

no 

119 

128 

137 

147 

156 

165 

174 

183 

193 

2O2 

220 

52 

90 

99 

108 

117 

126 

135 

144 

153 

162 

171 

I  80 

l89 

I98 

216 

53 

88 

97 

1  06 

US 

124 

132 

141 

150 

159 

168 

177 

186 

195 

212 

54 

87 

95 

104 

H3 

121 

130 

139 

147 

156 

165 

174 

182 

191 

208 

55 

85 

94 

102 

III 

119 

128 

136 

145 

i53 

162 

171 

179 

1  88 

205 

56 

84 

92 

IOO 

IO9 

117 

126 

134 

142 

151 

159 

168 

176 

184 

201 

57 

82 

90 

99 

107 

US 

123 

132 

140 

148 

156 

165 

173 

181 

198 

58 

81 

89 

97 

105 

H3 

121 

130 

138 

146 

154 

162 

170 

178 

195 

59 

79 

87 

95 

103 

III 

119 

127 

135 

143 

151 

159 

167 

175 

191 

60 

78 

86 

94 

102 

no 

117 

125 

133 

141 

149 

157 

165 

172 

188 

942 


APPENDIX. 


Influence    of  the    Rotation    of*  the    Earth    on    IMoving 
Bodies. 

The  Rotation  of  the  Earth  on  its  axis  effects  an  appreciable  displacement  of  the 
rails  in  a  line  of  railroad. 

In  the  case  of  an  express  train  weighing  400  tons,  running  N.  at  the  rate  of  50 
miles  per  hour,  the  pressure  on  the  right  hand  or  Eastern  rail  is  computed  at  501 
IDS.,  and  with  a  steamer,  alike  to  the  Inman  Line  "City  of  New  York,"  the  press- 
ure is  computed  at  936  Ibs.  This  lateral  force  increases  to  the  Poles.  (T.  Von  Bavier.j 

Bacteria   in    Earth-soil. 

In  Virgin  soil;  soil  from  beneath  Roadways;  from  Gardens;  adjacent  to  Factories; 
from  Courtyards  and  Cemeteries. 


-a  a 


pli* 

COc  $ 


No. 
124800 


til 

&* 


*!: 


No.  Meters 

750 
*  .061 022  cube  inches. 


til 


* 

CO  c  « 


No. 
64200 


No. 

590 


The  number  very  rapidly  decreases  in  the  deeper  layers  of  the  earth,  both  in 
virgin  soil  and  in  that  which  has  been  polluted.     (John  Reimers.) 

"Water-meters. 
"Worthington's.     New   York. 


Diam. 
of  Re- 
ceiving 
Pip.. 

Volume 
delivered  per 
Minute. 

Diam. 
of  Re- 
ceiving 
Pipe. 

Volume 
delivered  per 
Minute. 

Diam. 
of  Re- 
ceiving 
Pipe. 

Volume 
delivered  per 
Minute. 

Diam. 
of  Re- 
ceiving 
Pipe. 

Volume 
delivered  per 
Minute. 

Ins. 
.625 

•75 

Cube  ft. 
3S 

Galls. 
11.25 

22.5 

Ins. 
i 
i-5 

Cube  ft. 
6 

Galls. 
37-5 
45 

Ins. 
2 

3 

Cube  ft. 
8 
23 

Galls. 
60 
172 

Ins. 
6 

Cube  ft. 
58 

120 

Galls, 
435 
900 

NOTE  i.— .The  volume  of  delivery  here  given,  for  each  meter,  can  be  exceeded. 

2.— Extreme  velocity  of  a  meter  produces  incessant  and  improper  resistances; 
hence,  in  order  that  the  instrument  may  operate  only  within  a  perceptible  reduction 
of  the  head  of  the  supply,  it  should  be  of  a  capacity  to  effect  its  duty  at  a  moderate 
velocity  of  operation. 

Telescopes. 

Galileo's  first  telescope  magnified  but  three  times;  but  by  the  addition  of  a  con- 
cave eye  and  convex  object  glass  he  attained  a  magnifying  power  of  30  times. 

The  construction  of  large  lenses  is  at  present  limited  by  the  chromatic  aberration, 
or  separation  of  light  in  a  telescope. 

Euler  was  the  first  to  discover  the  principle  governing  this  aberration  and  the 
method  of  abolishing  it. 

Diameters   of  the   Principal   Objective   Grlasses. 
United  States. 


Location. 

Diameter. 

Focal 

Length. 

Ins. 

12 

Feet. 

Wesleyan  University  .  . 

12 

I2.S 

15 

Madison.  Wia  .  . 

12.56 

20.2 

Location. 

Diameter. 

Focal 
Length. 

Ins. 

Feet. 

Washington          .  •  •  . 

26 

32  472 

University,  Va  
Lick  Observatory.  .  .  . 

26 
36 

ft 

University  of  Southern  California  contemplates  the  construction  of  one  of  40  ins 
The  largest  telescopes  outside  the  U.  S.  are,  Gates  Head,  England,  24  ina  ;  Vienna 
Austria,  27  ins. ;  Nice,  France,  28  ins. ;  Pulkowa,  Russia,  30  ina 
*  To  have  four  lenses  of  24  inches. 


APPENDIX. 


943 


Manufacture   of  Ice. 
Machinery   and.    Apparatus. 


Produc- 
tion of 
Ice  in  24 
Hours, 

Steam-Engine. 

Com- 
pressors. 

Blocks  of  Ice. 

Water 
required 

M^ute. 

Coal. 

Op 

Engi- 

eratora. 
Fire- 
men. 

Lab- 

Weight 

of  Engine 
and 
Plant. 

Tons. 

Ins. 

Rev. 

Ins. 

Ins. 

Gallons. 

T's.* 

Lbs. 

I 

7X  9 

00 

SXiot 

8X  8X28 

5 

•5 

2 

2 

— 

2OOOO 

3 

8X16 

80 

5X15 

8X15X28 

IS 

i 

2 

2 

2 

58000 

5 

10X20 

75 

6X18 

8X15X28 

20 

1-5 

2 

2 

2 

69000 

10 

12X30 

70 

8X20 

j  11X22X28  ( 
1  11X11X28) 

30 

2 

2 

2 

3 

101  000 

12.5 

14X30 

65 

8X25 

j  11X22X28  ( 
|  11X11X28  j 

35 

2-5 

2 

2 

3 

129000 

15 

14X30 

65 

10X20 

j  11X22X28  | 
1  11X11X28) 

40 

3 

2 

2 

4 

167  ooo 

20 

16X30 

55 

10X30 

)  11X22X28  1 
}  11X11X28  ) 

So 

4 

2 

2 

5 

190000 

30 

- 

- 

- 

j  11X22X28  ( 
|  11X11X28) 

60 

5 

2 

2 

6 

225000 

40 

18X36 

So 

12X30 

11X11X28 

90 

6-5 

2 

2 

7 

260000 

45 

20X36 

5o 

15X30 

11X11X28 

2 

2 

8 

— 

60 

24X36 

45 

I2X30J 

11X11X28 

—  • 

— 

2 

2 

9 

— 

80 

26X48 

45 

20X36 

11X22X28 

100 

13 

2 

2 

10 

360000 

*  2000  Ibs.  f  One  compressor.  J  And  one  16x36  ins.  additional. 

All  others  two  compressors,  and  all  single  acting. 

Pressure  of  Steam. — For  all  75  Ibs.  per  square  inch. 

Out  -  off.  —  For  the  three  first,  which  are  slide  valves,  three  eighths.  For  the 
others,  as  Corliss  engines,  one  fifth. 

The  volumes  of  ice  above  given  cover  that  lost  in  thawing  the  molds  to  release  it. 
The  coal  given  as  that  required  is  inclusive  of  that  required  to  distil  water  from 
which  to  make  the  ice. 

NOTE.  —  In  order  that  the  proper  dimensions  of  engine  and  plant  may  be  arrived 
at  for  a  required  volume  of  ice,  it  is  necessary  that  the  quantity  and  temperature 
of  the  water  supply  should  be  furnished. 

2.— The  ice  is  produced  from  water  of  distillation;  hence,  it  is  clear  and  trans- 
parent. 

"When    a   Machine   is   operated,   "by  "Water-power.    As  the 

water  from  which  the  ice  is  made  is  not  distilled  from  steam,  as  in  the  case  where 
steam  is  the  motive  power,  the  ice  produced  is  less  clear  or  transparent,  and  is 
known  as  "  white  ice. " 

Refrigerating. 

Engines  for  Refrigerating  are  in  all  respects  alike  to  those  for  Ice-making,  with 
two  thirds  more  capacity.  As  distilled  water  is  not  required  in  refrigerating,  the 
saving  of  fuel  in  consequence  is  fully  thirty  per  cent.  Refrigerating  by  compression 
involves  a  much  less  expenditure  of  water  than  vhen  it  is  attained  by  absorption. 

Elements  of  a  Test  of  Operation  and.    Capacity  of  n 
Refrigerating  Machine. 

Ice  Liquefied  in  24  Consecutive  Hours,  78.41  Tons  of  2000  Ibs. 

Steam-engine.— Non-condensing,  18X36  ins.     Compressors  12.375X30  ins. 

Pressure  of  Steam,  86  Ibs.    Revolutions  per  minute,  56.5. 

IHP.— Steam-cylinder,  84.3.     Of  compressors,  67.78. 

Temperature  of  condensing  water,  76.2°.     Of  condenser  room,  62.5°. 

Volume  of  condensing  water  per  minute,  21.19  gallons.  Of  brine  per  meter, 
35  670  cube  feet  =  2  496  900  Ibs. 

Evaporating  pressure,  25.22  Ibs.     Condensing  pressure,  157.12  Ibs. 

Anthracite  coal,  consumed,  6108  Ibs.     Conibvsfible,  83.63  per  cent. 

Coal  per  IHP  per  hour,  3.02  Ibs.  Consumption  equivalent  to  the  liquefaction 
of  one  ton  of  ice.  77.88  Ibs. 


944 


APPENDIX. 


and.    .Asplialt    lavement. 

Barber  Asphalt  Paving  Co.,  New  York. 

Rock  Asphalt  is  amorphous  limestone  impregnated  with  asphaltum,  whereas 
Trinidad  asphalt  pavement  is  a  mixture  of  sand,  pulverized  limestone,  aud  asphal- 
tic  cement.  The  asphaltic  cement  is  composed  of  refined  Trinidad  asphalt,  with  a 
little  residuum  oil  of  petroleum,  the  pavement  being  an  artificial  asphaltic  sand- 
stone. 

In  the  cities  of  Europe,  where  asphalt  pavement  has  been  laid,  the  practice  is  to 
spread  from  1.5  to  2.5  inches  of  it  on  a  bed  of  concrete. 

The  process  of  preparing  the  material  for  use  is  to  crush  the  rock  to  powder,  heat 
it  to  about  280°,  spread  it  on  the  concrete,  and  then  compress  it  by  rammers. 

The  use  of  natural  asphaltum,  found  in  the  United  States,  as  Albertite  and 
Grahamite,  was  resorted  to,  but  without  success,  when  the  pitch  or  asphalt  lake  in 
Trinidad,  W.  I.,  was  discovered;  by  combining  this  material  with  highly  refined  pe- 
troleum, a  satisfactory  cement  was  produced,  which  being  mixed  with  a  sharp  sili- 
cious  sand  and  powdered  limestone,  a  desired  sandstone  was  formed;  a  compound 
possessing  the  necessary  firmness  and  resistance  to  the  changes  of  temperature  and 
durability,  under  the  wear  of  loaded  vehicles,  combined  with  smoothness,  cleanli- 
ness, and  comparative  freedom  from  noise;  without  danger  from  the  slipping  of 
horses'  feet,  usual  with  pavements  with  smooth  surface.  So  evident  was  the  useful 
application  of  this  construction,  that  in  1870  an  essay  of  its  merits  was  made  in 
Newark  and  New  York,  and  in  1876  it  was  further  essayed  on  an  extended  scale  in 
Washington,  its  merits  being  evidenced  by  a  Board  of  U.  S.  Engineers.  Since  which 
time  it  has  been  laid  in  over  100  other  cities  in  the  U.  S.  to  an  extent  of  about 
20,000,000  square  yards. 

The  advantages  of  such  a  pavement  are  the  reduction  of  the  resistance  to  traction, 
economy  of  transportation,  and  freedom  from  jolting  in  travel,  added  to  cleanliness 
and  public  health,  as  it  is  without  seams  or  joints  wherein  filth  may  be  collected. 

Its  durability  in  wear  is  less  than  granite,  and  greater  than  sandstone,  wood,  or 
macadam. 

As  regards  the  cost  of  its  maintenance,  it  is  less  than  that  of  any  other  material 
maintained  in  like  condition  of  repair. 

Origin    and.    Development. 

The  utility  of  asphalt  for  covering  of  a  road  was  not  discovered  until  1849.  As- 
phalt rock,  broken  up,  was  laid  in  the  manner  of  a  macadamized  road,  aud  the  re- 
sult was  such  that  in  1854  a  street  in  Paris  was  laid  with  compressed  asphalt  on  a 
foundation  bed  of  concrete. 

In  1869  it  was  first  laid  in  London,  and  is  now  extensively  laid  in  the  cities  of 
Europe  to  an  extent  in  excess  of  3,000,000  square  yards. 

SyiTostitntes.  —  Tar.  As  a  substitute  for  it  it  was  essayed  to  use  the  inex- 
pensive tar,  obtained  from  gas-works;  but  as  it  is  deficient  in  the  required  cement- 
ing qualities,  susceptible  of  being  rendered  viscid  by  the  heat  of  summer,  and  brittle 
by  the  cold  of  winter,  the  use  of  it  was  abandoned. 

Wood.—  Wood-pavement  is  laid  in  London  and  Paris  on  a  foundation  of  concrete, 
and  it  lasts  from  4  to  6  years. 

Stone-blocks  filled  in  with  asphaltum  water-proof  filling  has  been  practised  with 
success.  In  some  of  the  principal  cities  of  Europe,  the  uniformity  in  the  dimen- 
sions and  shape  of  the  blocks  contribute  to  their  durability.  The  cost  of  such  a 
pavement  is  in  excess  of  all  others. 

Macadam. — Macadam  pavement  is  unsuited  for  cities  from  the  wear  of  heavy 
Tehicles,  and  the  great  cost  of  maintenance. 

Brick.—  Brick,  hard  burned,  laid  in  two  courses  on  6  inches  of  sand,  the  first 
course  on  its  face,  and  the  second  on  its  longitudinal  edge,  has  been  used  in  Holland, 
Ohio,  and  Illinois.  The  duration  of  such  a  pavement  depends  wholly  on  the  uni- 
formity of  the  material  and  its  burning.  In  general  practice  it  was  found  to  be 
neither  enduring  nor  economical. 

Gen'l  Gillmore,  U.  S.  engineer,  in  his  report  (1879)  submits  the  following: 
Requisite  of  a  Good  Pavement.  — A  good  pavement  must  be  smooth,  aud  to  promote 
easy  draught  must  give  a  firm  and  safe  foothold  for  animals,  and  not  polish  or  become 
slippery  under  wear;  must  be,  as  nearly  as  possible,  noiseless  and  free  from  dust  or 
mud,  and  made  of  durable  material,  laid  upon  a  firm  foundation,  and  be  susceptible 
of  repairs  at  moderate  cost  at  all  seasons  of  the  year. 


APPENDIX. 


945 


Suitable    Foundations   for   Pavements. 

A  firm  and  unyielding  foundation  is  quite  as  necessary  for  stability  and  endurance 
of  a  pavement  as  for  any  other  structure. 

Following  are  suitable  foundations  for  street-pavements,  in  order  of  value,  pro- 
vided their  thickness  is  adapted  to  character  of  subsoil  and  nature  of  traffic,  viz. : 
i.  hydraulic  concrete  5  to  8  ins.  in  thickness;  2.  rubble-stone  set  on  edge  side  by  side, 
but  not  in  close  contact,  with  interstices  filled  in  with  hydraulic  concrete ;  3.  an  old 
coal-tar  pavement  properly  brought  to  slope  and  grade;  4.  rubble-stone  set  on  edge 
and  wedged  closely  in  contact  like  sub-pavement  of  a  Telford  road;  5.  an  old  pave- 
ment of  stone-blocks,  cobble,  or  rubble  stone ;  and  6.  an  old  Macadamized  or  gravel 
road,  or  a  compost  layer  of  broken  stone  or  gravel,  8  or  10  inches  thick. 

The  best  pavements  now  prominently  before  the  public,  classified  with  respect  to 
the  materials  of  which  they  are  made,  are  Asphalt,  Stone  block,  Wooden  block,  and 
Coal-tar  pavements.  The  wooden-block  pavement  is  not  entitled  to  a  place  in  the 
list. 

Stone  Pavements.— The  best  is  formed  with  rectangular  blocks  from  3. 5  to  4. 5  ins. 
thick,  10  to  13  in  length  on  wearing  surface,  and  8  to  9  inches  deep,  set  upon  frheir 
longest  edge  across  the  street,  upon  a  foundation  of  hydraulic  concrete. 

Asphalt  Pavements. — Best  asphalt  is  one  having  for  a  foundation  a  bed  of  hy- 
draulic cement,  or  something  equivalent  thereto  in  firmness  and  durability,  and  for 
its  wearing  surface  either  the  natural  bituminous  limestone  known  as  asphalt  rock, 
derived  from  the  Jurassic  region  on  the  confines  of  Switzerland,  or,  preferable  thereto, 
an  artificially  compounded  mixture  of  refined  asphaltum  and  silico- calcareous  sand, 
in  which  the  calcareous  ingredient  is  finely  pulverized  limestone.  As  the  material 
for  first-named  pavement  comes  principally  from  vicinity  of  Neufchatel,  the  pave- 
ment is  known  as  the  Neufchatel.  Asphaltum  for  the  other  pavements  referred  to 
comes  from  Island  of  Trinidad,  and  the  pavement  is  sometimes  called  the  Trinidad 
asphalt. 

Neufchatel  pavement. — Has  been  extensively  laid  in  London,  Paris,  and  other  Eu- 
ropean cities. 

Although  these  two  pavements  represent  the  best  type  of  street  surface,  there  is 
a  characteristic  and  somewhat  important  difference  between  them,  due  to  the  fact 
that  the  Trinidad  contains  nearly  75  per  cent,  of  sharp  silicious  sand,  and  does 
not,  therefore,  become  polished  and  slippery  by  wear;  while  the  Neufchatel,  being 
composed  entirely  of  bituminous  limestone  (a  species  of  amorphous  pulverulent 
chalk,  without  grit,  impregnated  with  bitumen),  is  by  no  means  free  from  this  fault. 
A  variety  of  asphalt  pavement  adapted  to  streets  of  exceptionally  steep  grade,  is  one 
Sbrmed  with  rectangular  blocks  of  compressed  asphalt  concrete. 

Comparative   Merits   of*  the    Several    Pavements. 

1.  Their  First  Cost.— In  cost  of  construction,  wood  is  the  cheapest;  Coal  tar  com- 
position second;  Sheet  asphalt  like  the  Trinidad  third;  Stone-blocks  fourth,  and 
Asphalt  blocks  fifth. 

2.  Their  Durability. — Assuming  each  of  the  four  pavements  named  to  be  the  bes* 
of  its  kind,  stone  and  asphalt  will  possess  the  longest  life,  and  wood  and  coal-tar 
very  much  the  shortest.    Between  the  first  two  and  the  last  two  there  is  a  wide  gap. 
Unless  the  stone  be  of  good  quality,  asphalt  will  take  first  place  and  stone  second. 

3.  Cost  of  Maintenance.  —  Order  of  merit  under  this  head  would  place  stone  and 
asphalt  first,  and  wood  and  coal-tar  last.     If  the  asphalt  is  good,  well  mixed  and 
laid,  the  stone  must  be  both  tough  and  hard  in  order  to  maintain  the  first  place. 

Relative    Loads    for    Roadways    and    Pavements. 

At  Low  Speed.     (J.  W.  Howard,  C.  E.) 
Loads  which  a  Horse  can  draw  on  a  level,  each  day  of  10  hours,  on  following  roads. 


Roadway. 

Lbs. 

Resistance 
of  Load. 

Roadway. 

Lbs. 

Resistanc 
in  Term 
of  Load. 

Asphalt 

6oCK 

Hard  Earth 

Stone  Block     

uoy^ 
3006 

•U37 
.076 

Worn  Stone  Block 

.191 

Ordinary  Stone  Block. 

1828 

.  124 

Cobble  Stone 

.2 

Hard  Macadam  
Hard  Gravel  

I391 

I27Q 

.164 
.178 

Ordinary  Earth  
Sand...' 

456 
228 

•5 
i. 

946 


APPENDIX. 


Sn"b-M:arin.e   Torpedoes. 

Formula  for  Determination  of  Pressure  per  Square  Inch  of  Various 
Explosives  at  Different  Distances. 


3//6636(A+E)C\2 
V\   (D  +  .oi)2-'  / 


=  P.    A  representing  angle  with  the  vertical  passing  through 

the  centre  of  the  charge,  made  by  a  line  drawn  from  it  to  the  surface  exposed  to  the 
shock,  determined  from  the  nadir,*  in  degrees;  E  a  constant  for  the  explosive,  as  de- 
termined by  experiment;  C  weight  of  the  explosive  in  Ibs.;  D  distance  from  centre  of 
the  explosive  to  the  surface  exposed,  in  feet ;  and  P  the  mean  pressure,  corresponding 
to  that  which  would  be  transmitted  to  a  disc  of  copper,  by  a  Rodman  indenting -tool, 
per  square  inch  of  surface  exposed  to  the  shock,  in  Ibs.  (Brev.  Brig.  General  H.  L. 
Abbott,  U.S.A.,i88i.) 

Value  ofE,or  Relative  Strength  of  Explosives  Fired  under  Water. 


4 

I'o 

K 

l°n 

•£< 

o  S 

| 

«S0' 

=.. 

i.- 

Explosive. 

4 

W 

i11 

P 

5  oo 

l! 

Explosive. 

£S 
bi 

w 

fc   0 

§11 

l-r 

1  ii 

X 

^  < 

Dualin  



232 

116 

III 

108 

Forcite  No.  i  .  . 



333 







Dynamite  No.  1  1 

186 

118 

No.  2  . 

75 

36 

1  20 

75 

83 

88 

Rackarock  .  .   . 

— 

220 

— 



_ 

Explosive  Gelat. 

89 

259 

125 

117 

"3 

Nitro  glyc'ne  . 

100 

in 

71 

81 

86 

Gun-cotton  

81 

87 

Rendrock...   . 

20 

IOI 

67 

78 

84 

Electric  No.  i  .  .  . 

33 

67 

Si 

69 

77 

"       ...   . 

40 

1  60 

91 

94 

95 

u       No.  2... 
Hercules  No.  i  .  . 

28 
77 

43 

211 

38 
109 

62 

106 

72 
105 

Vulcan  No.  i  . 

60 
30 

1  66 
99 

§ 

95 
78 

f3 

"        No.  2  .  . 

42 

118 

74 

83 

87 

"      No.  2  . 

35 

114 

72 

82 

86 

ILLUSTRATION.— Assume  the  distance  between  the  line  of  the  centre  of  a  charge 
of  dynamite  No.  i  and  the  bottom  of  a  vessel  to  be  5  feet,  the  angle  between  the  line 
of  centre  of  the  distance  and  the  bottom,  measured  from  the  nadir,  to  be  180°,  the 
constant  for  the  charge  186,  and  its  weight  100  Ibs.  What  would  be  the  mean 
pressure  on  the  object  in  Ibs.  per  sq.  inch? 

A  =  180°,    E  =  i86,    C  =  100,  and  D  =  5. 


3// 
V\ 


6636  (180+186)  i 


2428  736  Xioo\    _  ; 
) 


29.489 

*  A  point  of  the  globe  directly  under  our  feet,  or  that  opposite  the  zenith. 

f  Standard  of  comparison. 

J  For  (5  +  .oi)2-1,  see  p.  310.    Thus,       '  —  =  —  X  log.  5.01  =  2.1  X  .699 837  =  Numbtr  29.489 

When  the  Object  is  not  in  a  Vertical  line  with  the  Explosion. 
ILLUSTRATION.  —Assume  a  charge  of  gun-cotton  weighing  882  Ibs.,  set  in  water,  at 
a  horizontal  distance  of  24,  and  a  vertical  of  86  feet  from  the  object;  what  would 
be  the  effect? 

To  obtain  A,  or  angle  of  divergence,  180°  —  Tan.      —  =  15°  25',  and  180°  — 

86 

i5o  35'  =  164°  35'  =  164.58°.    D  =  \/242-f862  =  89,  and  E  =  135.    Hence, 

p 


/ /6636(i64.58  + 135)  882\ 


Log.  of  6636  =  3. 821 906 

"  "  164.58  +  135  =  2.476513 
"  "  88a  =2.945469 

Product  =9.243888 

4.093822 

Quotient  =5.150066 


Log.  of  89  -f  oi  =  1.949439 

2.1 

'949439 
3898878 

Log.  of  89.01 2>x  =  4.0938219 


3110.300132 
Log  cnbe  root  of  Quotient  =  3.433  378  =  Number  2712.5  Z&x.ss 


APPENDIX.  947 

Efficiency  of  Water-Tribe   Steam-B  oilers. 

In  a  late  test  by  J.  J.  Thorneycroft  of  his  patented  boiler,  the  following  elements 
and  results  are  reported  to  the  Institute  of  Civil  Engineers.  See  Vol.  XCIX.,  1889. 

Engine.— Triple  expansion,  Cylinders  14,  20,  and  31.5,  by  16  ins.  stroke  of  piston, 
and  jacketed.  Independent  engines  for  Circulating  pump,  Blower,  Donkey,  and 
Sheering.  All  exhausting  into  engine  condenser. 

Results   of*  Trials. 

Furnace. 


Elements  and  Dimensions. 

Natural  Draught. 

Blast  Draught. 

26.2 

26.2 

Heating      "          "        
"        surface  to  grate  

3° 
1837 
61.2 

1837 

1837 
61.2 

1837 

6l.  2 

1837 

Pressure  of  steam  in  boiler,  p'r  sq.  in. 

200.8 

70.1 
196.3 

1  86 

164.2 

7O.I 
194.9 

"        blast  in  fire-  room  in  ins. 

— 

27 

49 

2 

Revolutions  of  engine  per  minute.  . 

192.8 

165.2 

234.2 

268.7 

318.4 

Coal  per  sq.  foot  of  grate  per  hour.  . 
Water  evaporated  from  and  at  2  1  2°  ) 
per  Ib.  of  coal,  ash  utilized  J 

II.  I 

7-74 

11.22 

18.6 
10.48 

29.8 

10.2 

66.8 
8.89 

Do.        do.      per  Ib.  of  carbon.. 

— 

13.08 

12.  18 

II.7 

10.04 

Do.        do.      per  sq.  foot  of) 

heating  surface  per  hour            j 

— 

1.24 

3-2 

4-7 

8.05 

Temperature  of  gases  in  chimney.. 
44           of  air  in  fire-room.... 

474° 

4210 

69.30 

540° 
71.4° 

610° 
^0-3° 

??'6 

Fuel  per  IIP  per  hour.  

2.28 

1.981 

2.28 

1.99 

' 

jjp             "        "       

150.3 

89^1 

2.03 
282.1 

2.04 

AAQ    2 

2.32 

Efficiency  of  boiler  per  cent  

i 

86.8 

81.4 
.84 

% 

.42 

.-38 

Water  used  for  jacket  per  IIP  per) 
hour  in  Ibs.  .  .                         ...  1 

Fuel.  Calorific  value  of  14900  thermal  units  per  Ib.,  equal  to  1.025  of  a  Ib.  of  car- 
bon. Each  Ib.  of  coal,  if  completely  consumed,  is  capable  of  evaporating  15.41  Ibs. 
water  from  and  at  212°. 

Barbed.    Steel-wire   JETencing.    (Galvanized  or  painted.) 

J.  A.  Roebling's  Sons  Co. ,  New  York. 
Four  points,  barbs  6  inches  apart,  15  feet  =  i  Ib. 

4<  44  44  2  44  44  ,2         44      __   j      t( 

On  Spools.— 15  feet  in  length  of  the  regular  measures  and  12  feet  of  the  thickset, 
weigh  each  one  Ib. 

Spool,  about  18  x  18  X  17  ins.,  measuring  3.5  cube  feet,  weighing  from  60  to  100 
Ibs.,  and  length  of  wire  ordinarily  1500  feet.  Thickset  or  Hog  weighs  .2  more. 

To  Compute  Volume  of*  Boards  that  can  toe  Sawed  out 
of  a   Round    Log.     (M.  J.  Butler,  C.E.) 

RULE.— From  diameter  of  log  in  inches  subtract  4,  multiply  remainder  by  one 
half  of  it,  multiply  proceed  by  length  of  log  in  feet,  and  divide  product  by  8 ;  result 
will  give  number  in  feet. 

d~^4  X  —  X  Z  -r-  8  =  V.  d  representing  least  diameter  in  inches,  I  length  of  log 
itfeet,  and  V  volume  in  feet  of  board  measure. 

ILLUSTRATION.—  Assume  a  log  30  ins.  in  diameter  and  15  feet  in  length. 
30  —  4  X  26 -T- 2  X  15  -5-  8  =  633.75  feet  B.M. 

Foot-Pound— When  for  Unit  of  Work— Is  i  Ib.  lifted,  thrust,  or  projected 
through  i  foot,  against  gravity  or  inertia,  and  is  expressed  in  pounds  or  tons,  with- 
out regard  to  the  period  of  its  action. 

When  for  Unit  of  Rate  of  Work— Is  i  Ib.  liftod»  etc.,  as  above,  i  foot  In  a  given 
period,  as  in  i  second  or  minute. 


948 


APPENDIX. 


TVire    Rope.* 

Galvanizing  decreases  strength  of  unannealed  wire  5  per  cent,  and  its  ductility 
15  per  cent. 

Breaking  Weight  of  No.  20,  B  W  G  (.035  in.)  crucible  steel  rope  of  6  strands,  1.75 
ins.  in  circumference:  Wires,  78  to  102  tons  per  square  inch,  and  Ropes  5.75  to 
10.47  tons. 

Annealed  Wire  is  not  affected  by  galvanizing,  but  its  ductility  is  reduced  from 
179  twists  to  58,  =  68  per  cent. 

Annealing  Wire  reduces  its  strength  45  per  cent.,  but  increases  its  elasticity 
77  per  cent. 

Tensile  Strength  of  crucible  steel  wire  averages  85  tons  (80  to  90)  per  sq.  inch. 

Permanent  set,  Bessemer  iron  wire  12  tons  per  sq.  in.,  or  .25  of  ultimate  tenacity. 

Variation  of  tensile  strength  of  like  pieces  of  steel  wire,  galvanized  or  plain,  is 
but  3  per  cent,  for  the  former  and  8  for  the  latter 

Modulus  of  Elasticity  (ME).  Iron  wire  22400000,  Steel  35000000,  and  crucible 
Steel  33  ooo  ooo. 

Bending.     Stress  due  to  it,  in  a  wire  of  the  material  and  dimensions  given. 
ME  =  32  ooo  ooo. 

Diam.  of  pulley 10.5        13-125        16.875        18.75        24  ins. 

Stress  per  sq.  inch 50  40  31.4          28 . 2        22  tons. 

Durability.  Life  of  steel  wire  ropes  over  iron  pulleys,  of  material  and  dimen- 
sions above. 

Number  of  times  rope  passed  over  the  pulleys  without  Breaking.    Load  1568  Ibs. 

Ins.  Ins.  Ins.  Ins.  Ins.  Ins.  Ins. 

Diam.  of  pulley  ...  5.25        7.875        10.5        13.125        16.875        18.75         24 
Number  of  times. .  6075       10300      16000       23400         46800        72700       74100 
—         53  ioot     85  2oof          —         392  soot    336  600 

Over  Pulleys  24  Inches  in  Diameter.     Load  1568  Lbs. 


Manufacture  of  T.  &  W.  Smith. 

BW  G 

Diameter. 

Number  of  Bends  be! 

Wire  and  strands  laid 
in  opposite  direction. 

1-24  Inch.       3-24  Inch. 

ore  Breaking. 
Wire  and  strandu 
laid  in  same 
direction. 
3-24  Inch. 

Ordinary  crucible  steel  
Patent  improved  steel  
Plough  steel  

No. 

20 
2O 
20 
19 

18 

22 

Ins. 
•035 
•035 
•035 
.042 
.049 
.028 

No. 
74100 
96000 
109000 
66000 
87000 

III  OOO 

No. 
51000 
57000 
54000 
32000 
47400 

J.8  700 

No. 
26000 
42800 
34400 
79000 
17  zoo 

2O7OO 

Iron  wire 

Crucible  steel  

Crucible  steel... 

NOTE. — By  author:  diameter  of  pulleys  should  be  =  10  circumferences  of  rope. 

Tenacity    of*  Dovetails. 

White  Pine,  6  inches  square.     Notch  in  Length  equal  to  Depth  of  Timber. 
S  and  D  each  representing  proportion  or  depth  of  cuts  to  width  of 


S 


Destruction. 
.25        3. 9  tons. 

•33        5-75    " 
.41        5.1 


Destruction. 

D  .  125        6  tons. 

.167        6    " 

.208        6    " 


Greatest  strength  in  a  double  dovetail  is  attained  when  D  =  .  167,  and  in  a  single, 
when  S  =  .33.  (Gen? I  O.  M.  Poe,  U.  S.  E. ) 

Shafting    for    Lathes    and    Mills. 

Diameter. — Should  be  given  in  inches  or  quarters  only.  Length. — Not  to  exceed 
20  feet.  Velocity.  —  Machinery,  125  to  150  revolutions  per  minute;  Woods,  200  to 
300.  Power.—  Applied  at  middle  of  length  of  shaft  whenever  practicable.  Hangers. 
— With  adjustable  boxes,  in  order  the  easier  to  maintain  a  shaft  in  line. 


*  From  a  p»p«r  by  A.  S.  Biggart.  Ins'n  C.E. 


f  Long's  patent  lay. 


APPENDIX. 


949 


Cost   of  Sawing   and.   Dressing   Stone. 

Sa-wing. 
Per  Cube  Foot. 

Bedford  Stone. —20  cents'.  At  Chicago,  Soft,  medium,  8  to  10  cents; 
Limestone,  Magnesian,  and  Oolites.  —Medium,  13  to  17  cents; 
Marble  and  Grranite,  Hard,  25  to  30  cents. 


Rate  in  10  Hours. 

Ins. 


Granite 

Bluestone 


8 
36  to  40 


Marble,  Tenn. . . 

"    Vermont. 

Brownstone 


I  Limestone 


20  tO  25  I 


magnesia, 
oolite 


Ins. 
10  to  15 

36 

40  to  70 


NOTE.—  Depth  of  cut  without  reference  to  its  length  or  number  of  saws.     (R.  J. 
Cooke.) 

Dressing. 

Per  Square  Foot.     Labor  $  3  per  Day. 

Hard   Limestone.—  Bush  hammered,  rough,  25  cents;   Medium  work, 
30  cents;  Fine  work,  35  cents. 

Cost    of  Raising   "Water. 
1OOOOOO   Imperial  or  1  SOO  OOO  TJ.  S.  Grallons  1  Foot. 

Average  0/15  Years. 

Low  Service.  I  High  Service. 

By  water,  1.23  cents.     By  steam,  13.2  cents.  |  By  steam  ____  25  cents. 


of  Drifted    Bolts. 

Steel,  One  Inch  in  Diameter.—  Hole  Six  Inches  in  Depth. 

Mean  Holding  Resistance  per  Lineal  Inch. 


Wood. 

Hoi* 
i5-i6tha. 

Hole 

i4-i6ths. 

Hole 
i3-i6ths. 

Hole 
i2-i6ths. 

Hole 
i5-i6ths. 

Ra 

Hole 
i4-i6ths. 

ios. 
Hole 
i3-i6ths. 

Hole 

i2-i6ths. 

Yellow  pine  
White  oak... 

Lbs. 
36i 
1  300 

Lbs. 
616 

1778 

Lbs. 
761 

2AQQ 

Lbs. 
400 
im 

Lbs. 
•47 

.^2 

Lbs. 
.8 
.71 

Lbs. 

i 

Lba. 
•53 

.As 

Hemlock  in  i5~i6ths  hole  415  Ibs.  per  lineal  foot  to  withdraw  it,  and  White  or 
Norway  Pine  i2-i6ths  hole  830  Ibs. 

To  obtain  maximum  holding  resistance  of  timber,  diam.of  hole  to  bolt  as  13  to  16. 

Relative  holding  resistance  between  driving  parallel  or  perpendicular  to  the  fibre 
is  as  i  to  2.  (J.  B.  Tschamer.) 

Resistance    of    tlie    Air   to    Falling    Bodies. 


Falling  Body 

Lead  Ball,  2  ins.  in 
Diameter,  Weight  i  Ib. 

Body  Falling  Horizontally. 
Weight  i  Ib. 

In  Vacuo. 

In  Air. 

One  Foot  Square. 

Two  Feet  Square. 

Final 

Retar- 

Final 

Retar- 

Final 

Retar- 

Veloc- 
ity. 

Fall. 

Veloc- 
ity. 

Fall. 

dation 
per  Sec. 

Veloc- 
ity. 

Fall. 

dation 
per  Sec. 

Veloc- 
ity. 

Fall. 

dation 
per  Sec. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

16 

32 

30 

15-5 

•5 

28 

14-33 

1.66 

13-33 

13-33 

2.66 

48 

64 

55 

43-5 

4-5 

35 

33 

15 

24 

37-3 

24 

80 

96 

77 

67.5 

12.5 

38 

38.5 

41-5 

61 

66.66 

(P.  H.  Van  Der  Weyde.) 

Retardation  is  Inversily  as  Density  of  Body.  Velocity  after  fall  of  one  second 
becomes  measureably  uniform;  the  increased  velocity  being  balanced  by  the  in- 
creased resistance. 

Resistance  of  the  air  at  moderate  velocities,  to  the  velocity  of  a  fall  ing  body,  is  as 
the  square  of  its  velocity. 

Thus,  when  the  velocity  is  doubled,  the  resistance  is  quadrupled,  and  when  trip- 
led, nine  times  greater.  Applicable  alike  to  a  cannon  ball  in  air  or  a  body  in  water. 

4 


950 


APPENDIX. 


Cost   of  a   Horse-l?ower   "by    Steam.* 

Joule's  equivalent  (p.  504)  =  772  units  =  heat  required  to  raise  i  Ib.  water  i°,  = 
elevation  of  i  Ib.,  i  foot  high. 

Unit  of  evaporation,  to  evaporate  i  Ib.  water  to  steam  at  the  pressure  of  the  at- 
mosphere =  966.  i  British  thermal  units. 

Horsepower  33  ooo  Ibs. 

*?  —  42.75  heat  units  •=.  i  IP  i  °,  and  42.75  x  60  min.  •=.  2565  per  IP  per  hour. 


25  5  r=  2.655  Ibs.  water  required  to  generate  i  EP. 

:  \\  £  *?  "^  }  *»  other  fueU  see  p.  486. 


—— 


Then  I4*°°  =  15-01  I 
900.1 


.  water  evaporated  per  Ib.  of  coal. 


Hence,  ——  =  .1769  Ibs.  coal  per  IP  per  hour,  or  '  =  5.65  IP  per  hour  per 
Ib.  of  coal. 

Assuming  in  all  of  the  above,  the  normal  condition,  that  there  is  neither  expen- 
diture of  water  or  temperature  in  the  operation. 

Operalively.  —  From  elements  furnished,  in  part  by  Thos.  Pray,  C.E.,  the  cost  of  a 
IIP  at  the  pressures  and  expansions  given  is  as  follows: 


Coal  at  $  3  per  2000  Ibs. 


Engine. 


Non-condensing. 
i8"X42" 

58.5   Ibs. 

4.  74  ins. 

8.  12  Ibs. 

9.99   " 
18.59   " 

2.07   " 

6.  21  cents. 


Initial  pressure  of  steam  .....................  83.6   Ibs. 

Cut  off  ......................................   —     ins. 

Terminal  Pressure  ...........................    8.71  Ibs. 

Evaporation  per  Ib.  of  coal  ...................  10.31    " 

*'  "    IP  ..........................  16.84   " 

Coal  per  IIP  per  hour  ........................    2.28   " 

Cost  per  IIP  per  10  hours  ....................    2.6  cents. 

A  Condensing  Pumping  Engine  has  been  operated  at  a  cost  of  2.28  cents. 

*  For  Horse-powtr  tee  pp.  441,  733,  758,  and  014. 

Cost   of  Water   IPower   on   Driving   Shaft. 

Per  IP. 

Power  is  variable,  depending  upon  variation  in  head  of  water,  as  when  it  is  de- 

In order 


Jlvereu  ill  a  river  nuujeui.  i/u  noc  uy  ncoucto,  uu&i/  ui  watei,  Him  ui  piauu.       iu  uiuer 

to  attain  an  average  daily  power,  the  power  must  be  increased  to  meet  the  loss  of 
head  by  back  water  in  freshets. 

*  a  £~* 

& 

Cost 

ijs-s 

o?T5 

Cost 

LOCATION. 

j.p 

3  a  >«J 

E  03 

per 

LOCATION. 

H? 

5  ri  >>  Ja 

£  « 

per 

si"3* 

IK 

fr 

0^^ 

^  = 

H' 

No. 

Feet. 

Feet. 

$ 

No. 

Feet. 

Feet. 

$ 

Manchester.  N.  Y.  . 

890 

30 

30 

44 

Lowell,  Mass.  .  .  . 

IOOO 

575 

13 

IOO 

Lawrence,  Mass.  .  . 

IOOO 

490 

28 

42 

"          "      •  .  . 

IOOO 

290 

18 

57 

Cost  of  a  looo  IP  Plant  independent  of  cost  of  water  about  $  45  per  H?. 


Head. 

Co 

From 

TOO 

at   O] 

Supply 

f  a  1 

to  Disci 
3°o 

ike 

large  in 
400 

Pla 

Feet. 

500 

txt   n 

600 

nd.e 

Head. 

r   di] 

From 

IOO 

ffere 

Supply 
200 

nt   I 

to  Disci 
300 

leac 

large  in 
400 

Is. 
Feet. 

500 

600 

Feet. 
10 
20 

$ 

9I 
38 

1 

no 
46 

125 
54 

140 
62 

155 
70 

170 
77 

30 
40 

26 
20 

32 
25 

39 
3i 

45 

37 

5i 
42 

11 

At  Lawrence  and  Lowell,  a  Mill  Power  —  30  cube  feet  of  water  per  second,  with 
a  head  of  25  feet     At  Manchester  it  is  38  cube  feet,  with  a  head  of  20  feet. 

(Chas.  T.  Main,  M.E.} 
For  Power  of  a  Mill  Wheel  see  pp.  565,  566. 


APPENDIX. 


951 


Steam    Plant. 
Daily    and.    Yearly    Cost    of   Coal    and.    Hia"bor    in 

Operating   a   Plant   of  1OOO    H>. 
Year   of*  308    days   of  10 .  35    hours,  coal    at   &3    per   ton 

of  2OOO   Ibs. 
Deduced  front  Reports  ofChas.  T.  Main,  M.E. 


ENGINB. 

•Exhaust 
steam 
used. 

Coal  per 
HP  per 
hour. 

Attenc 
per 
Boiler. 

ance  and 
HPper  I 
Engine. 

Stores 
>ay. 
Stores. 

Coal 
per  IP 
per  day. 

fDaily 
per  HP 

fDaily 
for 
looo  H? 

Yearly. 

Per  cent. 

Lbs. 

c 

c 

e 

9 

9 

9 

9 

(    ° 

1-75 

•53 

.60 

•25 

2.14 

3-52 

3520 

10841,60 

Compound.. 

25 

1-5 

•45 

.60 

•25 

1.84 

3-16 

9732.80 

1.25 

•38 

.60 

•25 

2.76 

2760 

8500.80 

0 

2-5 

•75 

.40 

.22 

3.06 

4-43 

4430 

13644.40 

Condensing  . 

(25 

2.06 

.62 

.40 

.22 

2.52 

3-76 

3760 

11580.80 

(So 

1.63 

•49 

.40 

.22 

2 

3-" 

3110 

9578.80 

Non-Con- 

$  ° 

3 

.90 

•35 

.20 

3-67 

5-12 

5  120 

15769.60 

densing.  .  . 

25 

(50 

i.'88 

•73 
.56 

•35 
•35 

.20 
.20 

2-99 

2-3 

4.27 
3-47 

4270 

347° 

13151.60 
10687.60 

*  For  heating.                                                                           f  Including  coal. 

Yearly   Cost   of  1OOO   EP   and   of  a   EP. 

Year   of  308   days    of  1O.2S    hovirs.    Coal   at  $3   per  ton 

of  SOOO   IVbs. 

Deduced  from  Reports  of  Chas.  T.  Main,  M.E. 


ENGINE. 

*Exhaust 
Steam 
used. 

Engine 
and 
House. 

fOperat- 
ing  Ex- 
penses. 

Boiler- 
house  and 
Shed. 

fOpera- 
tor's 
Expense. 

tCoal  and 
Labor  and 
Stores. 

STotal  per 
IP 

Compound... 

Condensing  .  .  . 

Non-Condens- 
ing. .  . 

Per  cent. 
(° 

r5 
(so 

0 

r5 

(50 

0 
{25 

* 
40 
40 
40 
33 
33 
3i 
29.50 
29.50 
29.50 
1 
able. 

$ 

5-02 

5.02 

5-02 

4.14 
4.14 

3-95 
3-7° 
3-70 
3-7° 
Injector,  r 
§  Not  inc 

18.36 
16.16 
13.90 
24.80 
21.12 
17-33 

24.28 
19.46 
Depreciation 
uding  Cost 

I 

2-50 
2.20 

1.89 
Hi 

2.36 

3-95 
33' 
2.65- 

,  Taxes,  Int 
of  Plant  in 

10841.60 
9732.80 
8500.80 

13644-4° 
11588.80 
9578.8o 
15769.60 
13151.60 
10687.60 
erest,  and  Insu 
column  3  and 

18361.60 
16952.80 
15410.80 

21  164.40 
18608.80 
16888.80 

23  419-  6° 
20161.60 
17037.60 
ranee. 
5- 

(50 
*  For  Heating. 
t  As  per  previous  T 

Sugar   in   Mortar. 

It  has  been  demonstrated  that  the  addition  of  saccharine  matter  to  lime-mortar 
is  very  beneficial,  as  it  enables  it  to  be  laid  in  frosty  weather. 

It  is  claimed  also  that  it  causes  the  mortar  to  set  very  soon  and  strengthens  it, 
and  that  it  can  be  laid  with  dry  bricks. 

As  sugared  water  dissolves  lime,  it  is  necessary  to  dissolve  the  sugar  first,  and 
then  add  the  water  to  the  lime  slowly  and  cautiously.  The  mortar  should  be  very 
stiff. 

Proportions. —  For  mortar,  coarse  brown  sugar,  2  Ibs.;  lime,  i  bushel;  sand,  e 
bushels. 

If  sugar  is  added  to  mixed  mortar,  it  renders  it  too  thin.  (Manufacturer  and 
Builder.) 

Belting.  Speed  of  belts,  single  and  double,  i  inch  in  width,  should  not  ex- 
ceed for  the  first,  800  feet  per  minute,  and  for  the  second  500  feet,  each  =  one  H*. 

Railroad.    Speed.. 

London,  North  Western,  and  Caledonian. — London  to  Edinburgh,  400  miles 
Speed,  55.4  miles  per  hour  ;  3  stops  50.9  miles.  Engine,  tender,  and  car?, 
348  ooo  Ibs. 

Chicago,  Burlington,  and  Quincy.—i+.8  miles  in  9  minutes. 


952 


APPENDIX. 


Cost   of  Irrigation,    per    Acre. 

California. —  From  $7.18  to  $53.33.      Colorado. — $3.7510  $10.80.      Utah 
France.—  Average  of  several,  $  58.     India.—  Average  of  several,  $  i.  75  to  $  10. 


Alloy 
That  expands  in  cooling:  Lead  9  parts,  Antimony  2,  Bismuth  i. 

Extremes    of  Temperature. 

Artificial,  135°  (Faraday}.     Atmosphere,  77°  (Back). 

Extension   of  Woods   "by    Water,     (de  Volson  Wood.) 
Elongation.     Pine 065        Lateral.     Pine 2.6 


Oak 085 

Chestnut..   .165 


Oak 3.5 

Chestnut..  3.65 


Smokeless  Powder.  Gun  6  ins.  in  diam.  Charge  17.64  Ibs.  Energy 
at  muzzle,  4609  foot-tons.  Per  Ib.  of  powder  139.7,  and  per  weight  of  gun  720. 

"Volume    of  "Water   Flo\ving   over    Niagara   Falls. 

270000  cube  feet  per  second.  Since  1842,  Horseshoe  Fall  has  receded  140.5  feet, 
and  American  36.5  feet.  (J.  Bogart,  S.  E.) 

ROOFS. 

To  Compute  Stress  on  Roofs. 
Velocity     and     Pressure     of    Wind. 

RULE.— Multiply  square  ofvelocity  of  wind  in  feet  per  second  by  .0023,  or  square 
of  its  velocity  in  miles  per  hour  by  .005,  and  product  will  give  pressure  in  pounds 
per  sq.  foot. 

Or,  v2  X  .0023  =  P,  and  V2  x  .005. 

Also,  .0023  v2  sin.  x  =  P.  P  representing  pressure  per  sq.  foot  in  Ibs.,  x  angle 
of  incidence  of  wind  with  plane  of  surface  in  degrees,  V  velocity  of  wind  in  miles  per 
hour,  and  v  velocity  in  feet  per  second. 

Direction  of  wind  usually  makes  an  angle  of  10°  with  the  horizon,  hence  10°  is 
to  be  added  to  horizontal  plane  of  direction  of  the  wind. 

ILLUSTRATION  i. — Assume  wind  with  a  velocity  of  100  feet  per  second  to  impinge 
upon  a  plane  roof  set  at  an  angle  of  45°;  what  would  be  the  pressure  per  sq.  foot? 
Sin.  45°+  10°  =  .819.  .0023  X  ioo2  X  .819  =  18.837  #>»• 

2. — Assume  the  wind  to  have  a  velocity  of  150  feet  per  second,  and  angle  of  roof 
60°;  what  would  be  the  pressure  per  sq.  foot? 


Sin.  60°  +  10°  =  .94. 


.0023  x  150°  x  .94  =  48.75  Ibs. 


Pressure    of  Snow. 

This  pressure  decreases  per  square  foot  in  Ratio  of  half  space,  to  length  of  rafters, 
or  height  divided  by  space. 

Pressures   for   "Various   .A.ngles   or    Ratios. 

At  15  Pounds  Weight  per  Square  Foot. 


h-4-8 

Degrees. 

Lbs. 

h-i-S 

Degrees. 

Lbs. 

h-i-s 

Degrees. 

LU. 

•  5 
33 
•25 

45° 
33°  40 
26°  34' 

10.  6 

12.6 

i3-4 

.2 

.14 

17°  45' 
15°  39' 

13-9 
14-3 

14.4 

•125 
.11 

.10 

14°  2' 

12°  31' 
11°  19' 

14-5 
14.6 
14.7 

"Weights   on    Roofs. 

Single  tiles  20 
Slates,  ordinary  15 
Asphalt  on  slabs  20 
Paper,  tarred  6 

Per  Sq 
[ron,  shee 
Zinc,  shee 
Slates  on 
Iron,  shee 

jtare  Foot  in  Lb 
t  8 
t  8 

y. 
Iron,  corrugated,  on  iron    4.3 
Zinc         "             "            4.7 
Snow  20 
Wind  10 

ron  10 
t  on  iron  .  .     5 

APPENDIX. 


953 


Comparative  Operations  of  a  Simple    and  a   Compound 
Locomotive. 

Brooklyn  and  Union  Elevated  Railway  of  Brooklyn,  N.  T.     Forney  Type. 


ELEMENTS. 

Simple. 

Compound. 

ELEMENTS. 

Simple. 

Compound. 

Cylinders,  ins  
Drivers  diam    . 

11X16 

42  ins 

11.5    18X16 
42  ins 

Coal  per  car  mile. 
Water  

11.05  IDS- 

26  070  lbs. 

6.88  lbs. 
19  862  lbs. 

u        revolu-  1 

Gain  in  fuel 

07  7^ 

tions  per  mile  } 
Boiler  diam  

480 
42  ins. 

480 
42  inc. 

Evaporation       ) 
from  212°) 

8.09  lbs. 

9.  97  lbs. 

Flues  O  D 

i  5  ins 

Gain  in  water 

23  8£ 

Number  .... 

124. 

124 

Water  per  car     ) 

Exhaust  tip,  diam. 
Grates,  water  
Area,  sq.  feet.  .  . 
Heating  surface,  ) 
sq  feet             J 

3.  25  ins. 

15^6 
289.46 

289.46 

mile} 
Pressure  of        ) 
steam,  ave'  J 
Revolu's  per  min. 
Miles  per  hour.  .  . 

73.85  lbs. 
136  lbs. 

56.27  lbs. 
136  lbs. 

222 
27.73 

Ratio  of  do.  to) 

IP  

223.6 

grate                 J 

18.5 

18.5 

Weight  loaded    . 

45  35° 

45  850 

Coal  .  .  . 

1800  lbs. 

24^0  lbs. 

Miles  run..  . 

122 

122 

High.    Explosives. 
Firing    l*oint    and    Relative    Strength. 


DESIGNATION. 

Firing 
Point. 

Order  of 
Strength. 

DESIGNATION. 

Firing 
Point. 

Order  of 

Strength. 

Expl  Gelat  (Vou^e's) 

Degree, 
•jfic 

106  IT 

Tonite  

Degrees. 

68.24 

Helluofrite 

106.  17 

Bellite  

65.7 

Nitre-glycerine  (old) 

-,6c 

IOO 

Rack-  a-  rock  . 

61.71 

'  '           fresh 

Q2-37 

Atlas  powder.  

60.43 

"           French  . 
Sm  'less  Powder  (Nobel) 
Gun-cotton  1889  

346 

81.85 
92.38 
83.12 

Ammonia,  dynamite.  . 
Volney's  powder  No.  i 
"             "       No.  2 

— 

60.25 
58.44 
53-  18 

"          laboratory 

Si.qi 

Melinite  

50.82 

Dynamite  No.  i  
Emmensite  No.  i  
Oxinite  fr.  Pieric  acid. 
Amide  powder 

301 

81.31 
77.86 

%% 

Fulminate,  silver.  
I    4*f  ,       mercury.  . 
Mortar  powd.  ,  Dupont 
Forcite  No  i. 

315 
500 

•30Q 

50.27 
49-9* 
23-  *3 

(Lieut.  W.  Walt 

cc,  U.  S  < 

A.rmy.) 

Centrifugal    IPuimp. 
To   Compute   tlie    Required  Velocity  of  tlie  Outer  ICdge 

of   the    Blades. 
When  the  Height  of  the  Required  Lift  of  Water  is  Given. 

The  edge  of  the  blades  must  have  a  velocity  at  least  equal  to  that  acquired  by 
body  falling  from  the  given  height. 
Then,  to  lift  water  | 
and  sand  20  i 


water  )       , / • 

>  feet.  )    V  2  g  h  =  V 64. 4  X  20  =  35. 89  feet. 


Comparison    of   Operation    and    Cost    of   a    Q-as    and, 
Steam     lEngine. 

(In  addition  to  page  587. ) 


Elements. 

Gas.* 

Steam. 

Brake  IP         

76 

75 

(     Generator,  70.5$ 

Boiler,  72^ 

"      "    or  per  cent,  of  heat  in) 
power,  to  total  heat  generated.,  j 
Mechanical  efficiency  of  motor  
Power  to  operate  engine        

(do.  and  engine,  12.7^ 
18* 

69* 

and  engine,  7^ 
9-75# 
75* 

os-a; 

Coal  for  B  BP  per  hour  

31/* 
i  34  lbs. 

2  6  lbs 

^pace  occupied,   including  gen-  ( 
erator  or  boiler   ) 

470  sq.  feet. 

360  sq.  feet 

Professor  YVitz. 

Bryan  Don, 

954 


APPENDIX. 


STEAM-ENGINES. 
Compound. 

Duration  of  Operation  2  Hours. 

Cylinders. — 5.5,  9,  and  15.5  ins.  in  diameter.     Stroke  of  piston 

Revolutions. — 150  per  minute     IIP  40. 

Boilers.—  Fire  tubular.     Tubes,  38  of  2  ins. ;  6.25  feet  in  length. 

Heating  Surface.— 158  sq.  feet.     Grates.—  5.7  sq.  feet. 

Pressure  of  Steam. — 175  Ibs.  per  sq.  inch. 

Water.  —Weight  consumed,  1 140  Ibs.  Evaporation  per  Ib.  of  coal,  9. 8  Ibs.  Drawn 
from  jackets,  84  Ibs.  Consumption  per  IP  per  hour,  1.425  Ibs.  Temperature  of 
feed,  55°. 

Consumed  116  Ibs.— per  IIP  per  hour  1.45  Ibs. ;  per  sq.  foot  of  grate  10.2  Ibs. 

Indicator  Diagrams.—  Mean  IP  54=13.31  IIP;  Intermediate  18=  12  IIP;  Con- 
densing 7. 5  =  14. 7  IIP  =  40. 

Fly-wheel.— 5.5  feet  in  diameter  and  10.5  ins.  in  width. 

Weight  of  Engine  and  Boilers,  without  water,  14  560  Ibs. 

Builders.—  Marshall  &  Co.,  Kreigly,  Eng. 

PUMPING  ENGINE. 
Vertical    Compound. 

Cylinders. — 34  and  66  ins.  in  diameter  by  60  ins.  stroke  of  piston. 
Pressure  of  Steam. — 74.81  Ibs.  per  sq.  inch.     Vacuum,  26.25  ins. 
Revolutions.  —  25. 51  per  minute.     Grate  Surface.  —  70  sq.  feet. 
Pressure  of  Water  by  Gauge.  — 62.02  Ibs.      Head,  including  lift,  155. 17  feet  = 
67.62  Ibs.     Fuel.  —  675  Ibs.  per  hour. 
Duty.— 104 820 431.     Stack,  in  height,  125  feet. 
Constructors.— The  Edward  P.  Allis  Co.,  Milwaukee,  Wis. 

ELECTRIC  DYNAMO  ENGINE. 
Triple    Expansion. 

Arc  Lights. — 500.     Water  entrained  in  steam  7.39$. 

Cylinders.— 14,  25,  and  33  ins.  in  diam.  by  48  ins.  stroke  of  piston. 

Condenser. — Separate.  Circulating  Pump,  16  X  16  ins. ;  Air-pump,  single-acting, 
24  X  16  ins.  Cylinders,  12  x  16  ins.,  operating  both  pumps.  Revolutions,  61.29. 
IIP  16.4. 

Pressure  of  Steam. — 125  Ibs.  per  sq.  inch;  Revolutions,  engine,  99.12;  Steam  per 
IIP  per  hour,  12.94  lbs-  Iff  5l6-  Injection  Water.— 72°.  Reservoir,  90°. 

Constructors.—  The  Edward  P.  Allis  Co.,  Milwaukee,  Wis. 

Railroad.   Signals   and.    Significations. 


Stop,"  one  pull  of  bell-cord. 
Go  ahead,"  Two  pulls. 
Back  up,"  three  pulls. 
Down  breaks,"  one  whistle. 


" Off' breaks,"  two  whistles. 

"Back  up,"  three  whistles. 

"Danger,"  continued  whistles. 

"A  cattle  alarm,"  rapid  short  whistles. 


Go  ahead,"  a  sweeping  parting  of  the  hands,  on  level  with  the  eyes. 

Back  slowly,"  a  slowly  sweeping  meeting  of  fhe  hands,  over  the  head. 

Stop,"  downward  motion  of  the  hands  with  extended  arms. 
'Back,"  beckoning  motion  of  a  hand. 
'  Danger,"  a  red  flag  or  light  waved  up  the  track. 
'Stop,"  red  flag  raised  at  a  station. 
'Start,"  lantern  at  night  raised  and  lowered  vertically. 
'Stop,"  lantern  swung  at  right  angels  across  the  track. 

Back  the  train,"  lantern  swung  in  a  circle. 

Metropolitan    Opera    House,    Ne-w   York. 

Capacity. — Seating,  3600.    Standing,  400.     If  the  saloons  attached  to  the  private 
boxes  were  removed,  the  total  capacity  would  be  5000. 


APPENDIX. 


955 


Distillation,  of  Fresh.   TVater. 

Process  of  G.  W.  Baird,  U.  S.  Navy,  New  York. 

Marine  Steamers  for  long  voyages,  operated  under  a  high  pressure  of  steam,  are 
necessarily  provided  with  Evaporators,  to  replace  the  water  expended  in  leaks  and 
vents,  and  to  provide  for  the  ordinary  requirements  for  fresh  water. 

This  process  is  an  improvement  upon  existing  methods,  inasmuch  as  it  furnishes 
the  water  potable,  and  it  is  as  follows: 

The  Evaporator  contains  a  series  of  tinned  metallic  coils  and  a  volume  of  sea- 
water;  which  is  designed  to  be  evaporated  by  the  passage  of  steam  from  the  engine 
boilers  through  the  coils.  The  water  condensed  in  them  is  returned  to  the  boilers; 
the  water  vaporized  from  the  sea- water,  external  to  the  coils,  is  either  led  to  the 
Engine  condenser,  to  replenish  that  lost  by  leaks  and  vents,  as  from  gauge  cocks, 
etc. ;  or  if  required  for  potable  purposes,  is  led  to  a  Distiller,  where  it  is  aerated, 
condensed,  and  filtered,  from  which  it  is  drawn  for  use. 

As  the  sea- water  is  evaporated  in  vacuo,  vaporization  occurs  at  a  temperature 
below  that  at  which  much  scale  is  precipitated.  Hence  the  shell  and  coils  are  both 
measurably  free  from  it. 

Results    of*  £tn    Experiment. 

Pressure  in  coils,  20  Ibs.  above  atmosphere;  temperature  of  steam  in  coils,  259.3°; 
temperature  of  feed  water,  131.66°;  temperature  of  the  water  vaporized,  212°;  water 
vaporized  per  hour,  103.33  Ibs. ;  water  condensed  in  the  coils  per  hour,  112.12  Ibs.; 
total  heat  in  the  steam,  1193.7°,  and  in  the  water  "vaporized,  1178.6°. 

Capacities    of*  Kvaporators    and.    Uistillera. 
Gallons  per  day  0/24  hours. 


No. 

i 

2 

Evapo- 
rator. 

Dis-     II  „ 
tiller.       No- 

Evapo 
rator. 

Dis- 
tiller. 

No. 

4 
4-5 

Evapo- 
rator. 

Dis- 
tiller. 

Gallons. 
600 
I2OO 

Gallons. 
600       3 

1200     U   3.5 

Gallons. 
2OOO 
2000 

Galloni. 
1600 
1600 

Gallons. 
3000 
3000 

Gallons. 
2OOO 
2500 

I 


Evapo-        Dis- 
rator.        tiller. 


Gallons. 
4000 
6000 


Gallons. 
2500 
3000 


Coal   iProdiaotion   and.   Consumption 

Of  the  World  Per  Diem. 
Production. — Estimated  at  3  360000000  to  3  696000000  Ibs. 


ooo  ooo  Ibs. ;  Smelting 
20000000  Ibs. ;  Do- 


Consumption. — Generation  of  steam,  Land  and  Marine,  62400 
Iron  Ore,  28800000  Ibs. ;  other  metals,  23000000  Ibs. ;  Forges, 
mestic  use,  57  600000  Ibs.  Total,  2  700000000  Ibs. 

Corrosion    of*  "Wrought    Iron. 

The  purer  the  water,  the  more  active  it  is  in  corroding  and  pitting  Wrought  iron 
plates.  This  arises  from  the  greater  presence  of  air  in  pure  water,  and  hence  a 
greater  proportion  of  Oxygen.  (Locomotive. ) 

Earth.   Boring   and.   Heat   of  Mines. 

Sperentoerg,  near  Berlin.  Bore,  4172  feet  in  depth,  about  1000  feet  in  ex- 
cess of  Artesian  well  at  St.  Louis. 

In  lower  levels  of  some  of  the  shafts  in  the  Omstock  mines,  prior  to  the  draining 
into  the  Sutro  Tunnel,  the  water  was  at  a  temperature  of  120°. 

Preservatives    of  Iron. 

Pitch,  Black  Varnish,  Asphalt  and  Mineral  waxes  are  among  the  best,  provided 
the  acid  and  ammonia  salts,  which  frequently  occur  in  tar  and  tar  products,  are 
removed. 

If  in  addition  these  substances  are  applied  hot  to  warm  iron,  the  bituminous  and 
asphaltic  substances  form  on  the  surface  of  the  iron  an  enamel,  which,  unlike  to 
other  coatings,  is  not  microscopically  porous,  and  consequently  it  is  impervious  to 
water 

Spirits  and  Naptha  varaishe*  are  injurious.    (Prof.  Lewis.) 


956 


LIGHTNING-CONDUCTORS. 


Code    of   R-ules    for    the    Erection,    of   Lightning-Con- 
d.  victors. 

Lightning-rod  Conference. 

Points.  —Point  of  terminal  should  not  be  sharp — not  sharper  than  a  cone  of  which 
the  height  is  equal  to  radius  of  its  base.  A  foot  lower  down  a  copper  ring  should 
be  screwed  and  soldered  on  to  the  upper  terminal,  in  which  ring  should  be  fixed 
three  or  four  sharp  copper  points,  each  about  6  inches  in  length.  It  is  desirable 
that  these  points  be  platinized,  gilded,  or  nickel-plated. 

Upper  Terminals. — Number  of  conductors  or  points  to  be  specified  will  depend 
upon  size  of  the  building,  material  of  which  it  is  constructed,  and  comparative 
height  of  the  several  parts.  No  general  rule  can  be  given  for  this.  Ordinary 
chimney-stacks,  when  exposed,  should  be  protected  by  short  terminals  connected 
to  the  nearest  rod. 

Insulators. — Rod  is  not  to  be  set  off  from  building  by  glass  or  other  insulators, 
but  attached  to  it  by  metal  fastenings. 

Attachment. — Rods  should  be  led  down  the  side  of  building  which  is  most  ex- 
posed to  rain.  They  should  be  secured  firmly,  but  the  holdfasts  should  not  pinch 
the  rod,  or  prevent  contraction  and  expansion. 

Factory  Chimneys.— Should  have  a  copper  band  around  the  top,  and  stout,  sharp 
copper  points,  each  about  i  foot  in  length,  at  intervals  of  2  or  3  feet  throughout  the 
circumference,  and  the  rod  should  be  connected  with  all  bands  and  metallic  masses 
in  or  near  the  chimney. 

Ornamental  Iron-work. — All  vanes,  ridge-work,  etc.,  should  be  connected  with 
conductor,  and  it  is  not  absolutely  necessary  to  use  any  other  point  than  that 
afforded  by  such  ornamental  iron- work,  provided  the  connection  be  perfect  and  the 
mass  of  iron  considerable. 

Material. — Copper,  weighing  not  less  than  6  ozs.  per  foot  in  length,  and  the  con- 
ductivity of  which  is  not  less  than  90  per  cent,  of  that  of  pure  copper,  either  in  the 
form  of  tape  or  rope  of  stout  wires,  no  one  wire  being  less  than  No.  12  B.W.G. 
Iron  may  be  used,  but  should  not  weigh  less  than  2.25  Ibs.  per  foot  in  length. 

Joints.— Bad  joints  diminish  the  efficacy  of  the  conductor;  therefore  every  joint, 
besides  being  well  cleaned,  screwed,  scarfed,  or  riveted,  should  be  thoroughly  sol- 
dered. 

Protection. — Copper  rods  to  the  height  of  10  feet  above  the  ground  should  be 
protected  from  injury  and  theft  by  being  enclosed  in  an  iron  pipe  reaching  some 
distance  into  the  ground. 

Painting. — Iron  rods,  whether  galvanized  or  not,  should  be  painted;  copper  ones 
may  be  painted  or  not. 

Curvature. — Rods  should  not  be  bent  abruptly.  In  no  case  should  the  length  of 
it  between  two  joints  be  more  than  half  as  long  again  as  the  line  joining  them. 
When  a  string-course  or  other  projecting  stone- work  will  admit  of  it,  the  rod  should 
be  carried  through,  instead  of  around,  the  projection.  In  such  a  case  the  hole 
should  be  large  enough  to  allow  for  expansion,  etc. 

Masses  of  Metal. — As  far  as  practicable  it  is  desirable  that  the  conductor  be  con- 
nected to  extensive  masses  of  metal,  such  as  hot- water  pipes,  etc.,  both  internal 
and  external;  but  it  should  be  kept  away  from  all  soft  metal  pipes,  and  from  in- 
ternal gas-pipes.  Bells  inside  well  protected  spires  need  not  be  connected. 

Earth  Connection. — It  is  essential  that  the  lower  extremity  of  the  conductor  be 
buried  in  permanently  damp  soil  ;  hence  proximity  to  rain-water  pipes  and  to 
drains  is  desirable.  It  is  a  very  good  plan  to  bifurcate  the  conductor  close  below 
surface  of  the  ground,  and  adopt  two  of  following  methods  for  securing  escape  of 
the  lightning  to  earth.  A  strip  of  copper  tape  may  be  led  from  the  bottom  of  the 
rod  to  t?he  nearest  gas  or  water  main — not  merely  to  a  lead  pipe — and  be  soldered 
to  it;  or  a  tape  may  be  soldered  to  a  sheet  of  copper  3  feet  x  3  feet  and  .0625  inch 
thick,  buried  in  permanently  wet  earth,  and  surrounded  by  cinders  or  coke;  or 
many  yards  of  the  tape  may  be  laid  on  a  trench  filled  with  coke,  taking  care  that 
the  surfaces  of  copper  are,  as  in  previous  cases,  not  less  than  18  square  feet.  Where 
iron  is  used  for  the  rod,  a  galvanized  iron  plate  of  similar  dimensions  should  be 
employed. 

Inspection. — The  conductor  should  be  satisfactorily  examined  and  tested  by  a 
qualified  person,  as  injury  to  it  often  occurs  up  to  the  latest  period  of  the  works 
from  accidental  causes  and  carelessness. 

Collieries. — The  head-gear  of  all  shafts  should  be  protected  by  proper  lightning- 
conductors  to  prevent  explosion  of  fire  damp  by  sparks  from  atmospheric  elec- 
tricity being  led  to  the  mine  by  the  wire  ropes  of  the  shaft  and  iron  rails  of  the 
galleries. 


STONE    BREAKER. — CRUSHER. STEAM    HEATING.       95 f 


Stone    Breaker    and.    Ore    Cruslier. 


Stone  Breakers  and  Ore  Crushers  are  used  in  mak- 
ing Macadam  for  construction  of  roads ;  material  for 
concrete;  ballasting  railroads,  crushing  ores,  quartz, 
corundum,  and  all  brittle  substances ;  they  can  be  ad- 
justed to  pass  a  mass  from  the  size  of  a  pea  to  larger 

-   diameters,  depending  upon  the  capacity  of  the  machine. 

Crushed  to  Cubes  0/2  Inches.     Per  Hour. 


No. 

Receiver. 

Volume. 

Extreme 
Weight 
of  Stone. 

Weight 
Produced. 

] 

Length. 

)imension 
Breadth. 

c     ' 

Height. 

Pulley. 

Speed. 

IP 

Ins. 

Cub.  yds. 

Lbs. 

Lbs. 

Ft.  ins. 

Ft.   ins. 

Ft.  ins. 

Ins. 

No. 

i 

3X    1.5 

— 

40 

IOO 

!•    I 

.   6 

.10 

5X  i 

250 

•5 

2 

6X  2 

I 

56o 

I  200 

2.10 

2.    I 

2-    3 

nX  5 

250 

4 

3 

ioX  4 

3 

1800 

4900 

4 

3-   3 

3-  9 

2oX  6 

250 

6 

4 

ioX  7 

5 

3800 

7800 

5-  i 

3-  9 

4-  5 

24X  7-5 

250 

8 

5 

i5X  9 

8 

7400 

15500 

6.  6 

5 

5-ii 

3oX  9 

250 

15 

6 

15X10 

9 

7800 

16000 

6.  6 

5-  5 

5-" 

30X10 

250 

i5 

7 

2oX  6 

10 

53°° 

II  200 

5-  3 

2.  II 

4.  6 

30X10 

250 

15 

8 

20X10 

10 

8100 

18  300 

6.10 

5  -.9 

5." 

36X12 

250 

20 

9 

12X30 

16 

14200 

33000 

7.10 

8.  4 

6.  4 

36X12 

250 

30 

10 

15X30 

20 

14200 

35000 

7.10 

8.  4 

6.  4 

36X12 

250 

3° 

NOTE. — The  30X15  and  the  36X24  are  preparatory  Crushers,  the  former  breaking 
500  cube  yards  in  10  hours  to  4  ins.,  and  the  latter  800  cube  yards  to  8  ins. 

Crusher    -with.    Revolving    Screen. 


Dimen- 

Volume. 

Extreme 
Weight 
of  Stone. 

Weight 
Produced. 

'J'l 

Length. 

Mrnensions 
Breadth. 

U"'  '_ 
Depth. 

Pulley. 

Speed 
per 
Min. 

IP 

Ins. 
ioX   7 
i5X  9 
15X10 
20X10 

Cub.  yds. 
5 
8 

9 

10 

Lbs. 
3800 
6800 
7300 
7700 

Lbs. 

10200 

17700 
18  loo 

21  500 

Ft.  ins. 
5.    i 
6.  6 
6.  6 
6.10 

Ft.   ins. 
3-9 
5 
5-5 
5-9 

Ft   ins. 
4-    5 
5-ii 
5-n 
5-n 

Ins. 
2    X  7-5 
2.6X  9 
2.6X10 
3    X   i 

Rev. 
250 
250 
250 
250 

No. 
8 
i5 

12 

M 

Steam   Heating   and.   Boilers. 


Steam  Heating. — Is  effected  Directly  or  Indirectly.  In 
the  first  case,  the  steam  is  conveyed  through  a  pipe,  or  to  a 
cluster  of  them,  at  whatever  point  they  are  required,  termed 
a  Radiator ;  air  being  heated  by  contact  with  the  exterior  sur- 
face of  the  pipes,  and  the  water  of  the  condensed  steam  flows 
back  (by  gravity)  through  the  return  pipes  discharging  into 
the  boiler. 

In  the  second  case,  steam  is  conveyed  in  like  manner  to  a 
cluster  of  pipes  enclosed  in  a  chamber,  in  the  lowest  part  of  the 
building,  usually  the  cellar,  the  air  within  the  chamber,  upon  being  heated, 
ascends  by  its  rarefaction,  and  is  led  to  the  space  or  apartment  required  to 
be  heated. 

Hot-water  -Heating. — This  system  consists  of  circulating  hot  water 
in  the  radiators  instead  of  steam.  The  boiler,  pipes,  and  radiators  are  fully 
filled  with  water — the  flow  or  circulation  pipes  attached  to  the  top  of  the 
boiler  and  the  return  pipes  to  the  bottom.  The  water  in  the  boiler,  when 
heated,  rises  and  circulates  through  the  pipes  and  radiators,  and  parting  with 
a  portion  of  its  heat  it  becomes  denser,  and  gravitates  through  the  return 
pipe  to  the  boiler,  where  it  is  again  heated. 

This  system  requires  a  much  greater  proportion  of  radiating  surface  than 
that  of  steam. 


958 


BOILEES. HYEDAULIC    CEMENT. 


ELEMENTS.     No> 

B< 

2 

>ile: 

Wrou 

rs.    j 

ght-iror 
4 

rn  continual 
Water  Legs. 

5      |      6 

ion. 

7 

Cast-Iro 

O                I 

u  Legs. 

2      |     3  • 

Shell  di;un  Ins 

oe 

28 

1 

"     over  jacket.  ..  " 

3* 
35 

si 

41 
45 

J! 

51 

55 

54 
57 

24 
30 

3i 

32 

35 

38 

"     height               " 

"     extreme  " 

§ 

j/ 
72 

80 

s' 

43 

9° 

45 
92 

2 

64 

33 
67 

33 
69 

A/ 
72 

Furnace,  diam  " 

21 

24 

3° 

32 

38 

40 

18 

J9 

21 

24 

Tubes,  No.,  do.  2  ...   u 

44 

56 

84 

91 

124 

160 

30 

36 

44 

56 

u      length  " 

QQ 

34 

34 

42 

42 

42 

27 

OQ 

QO 

34 

Steam-outl'ts  2,  diam.  " 

j" 

2 

2 

2-5 

2-5 

3 

3 

*»5 

J" 

i-5 

o" 

2 

2 

Chimney  flue,  diam.   " 

8 

8 

10 

10 

12 

12 

7 

7 

8 

8 

Water-  line  from  base  " 

55 

59 

63 

70 

73 

74 

5' 

54 

55 

59 

Heating  surface..  Qfeet 

75 

105 

140 

185 

260 

320 

45 

60 

75 

i°5 

Direct  radiating  )      u 
surface  supplied  j 

450 

630 

830 

1050 

1500 

i  goo 

260 

350 

450 

630 

For  Direct  radiation,  each  nfoot  of  radiating  surface  will  heat  from  50  to  loocube 
feet  of  air  space,  and  for  Indirect,  from  25  to  50  cube  feet;  the  range  depending  upon 
the  conditions  of  construction  of  building  and  its  exposure  to  external  air. 

HYDKAULIC    CEMENT. 

IPortland. 

In  addition  to  pp.  589-590. 

Some  limestones  when  burned,  ground  finely,  and  made  into  paste,  attain  the 
element  of  hardening  in  water,  and  are  termed  Hydraulic. 

Cements  are  classed  as  Natural  and  Artificial.  The  stone  from  which 
Portland  or  Hydraulic  Cement  is  made  in  the  United  States  is  found  in  stratified 
beds  of  aqueous  deposits,  which  in  extent  cover  about  one-third  of  the  area  of  the 
State  of  New  York,  the  western  part  of  Vermont,  and  also  in  New  Jersey,  Penn- 
sylvania, Maryland,  Virginia,  and  East  Tennessee. 

Analysis  of  Glens  Falls  and  best  German  cement  are  nearly  identical,  both  In 
their  quality  and  volumes,  and  all  advantage  claimed  for  the  former  is  that  it  is 
finer  grained,  and  that,  in  common  with  this,  that  it  sets  slowly,  usually  requiring 
from  4  to  5  hours.  Consequently,  the  mixture  can  be  made  in  a  larger  volume 
without  being  rendered  useless  by  setting  before  all  of  mass  is  required. 

It  is  only  in  the  stopping  of  joints  leaking  water  under  a  pressure  that  the 
quick  setting  of  cement  is  better. 

Tensile   Strength..    Cement  1,  Sand  3.    Per  square  inch. 


Sieve. 

Sets. 

17  Days'  Test. 

28  Days'  Test. 

No.50. 

No.  100. 

tial. 

Hard. 

Max. 

Min. 

age. 

Max. 

Min. 

age. 

Min- 

Min. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs 

Lbs. 

Aqueduct  No.  3 
Vertical  Wall.  . 
Swing  Bridge.  . 
Vertical  Wall.  . 
Swing  Bridge.  . 
Vertical  Wall.  . 

Fort  Miller 
Glens  Falls 
Waterford  . 
Glens  Falls 
Waterford  . 
Glens  Falls 

ioo  T 

IOO 

ioo  T 

IOO 

ioo  T 
ioo  T 

98.25 

98875 
99-  125 
99 
99 

50 

5? 
140 

120 

230 
230 
780 

347 
404 

369 
379 
382 
346 

254 
305 
302 

?8o 

309 
337 
342 

350 
331 
321 

394 
442 
385 
398 
407 
372 

300 
320 
330 
326 
295 

350 
373 
350 

341 

343 

Swing  Bridge.  . 
Vertical  Wall.  . 

Waterford  . 
Glens  Falls 

ioo  T 

IOO 

99.125 
99 

155 

160 

300 
255 

368 
^ 

323 
273 

343 
337 

404 
442 

322 
326 

Vertical  Wall.  . 

Glens  Falls 

ioo  T 

99.125 

105 

190 

374 

313 

332 

440 

374 

405 

Wm.  P.  Judson,  Deputy  State  Engineer, 
Cru.sh.ing   Strength.    Per  Square  Inch. 
Tests  of  strength  made  at  New  York  and  Brooklyn  Bridge. 


.  T. 


Day. 


Lbs. 
490 


In 
Air. 

Lbs. 
948 

Water. 

Period. 

Air. 

In 
Water. 

Period. 

h, 
Air. 

Water. 

libsT 
1408 

Lbs. 
653 

Weeks. 

2 

Lbs. 
750 

Lbs. 
515 

Weeks. 
3 

Lbs. 

ELECTRIC    MOTOE 


959 


Klectrio  Motor.     The  Crochtr- Wheeler ,  New  York. 

This  Motor  has  been  designed  to  remove  difficulties 
which  experience  has  developed  to  be  attendant  upon 
other  instruments  of  like  purpose. 

Care  has  been  taken  in  its  design  and  construction. 
The  bearings  are  oiled  automatically,  and  magnetic 
circuit  is  made  as  perfect  as  practicable.  Its  centre 
of  gravity  is  low,  machine  strongly  built,  weight  of 
it  comparatively  low,  and  its  efficiency  high. 
Designed  to  run  at  low  speed,  in  order  to  reduce  wear,  heating  journals,  etc. 


EP 

Weight 

Veloc- 
ity. 

Pulle 
Diam. 

y. 

Face. 

D 
Length. 

intension 
Breadth 

s. 
Height. 

Betwec 
ho 

Length. 

n  Bolt- 

68. 

Breadth 

Sh 
Diam. 

ifts. 
From 
Base  of 
Motor. 

No. 

~LbI~ 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Via 

18 

2IOO 

I?1.:! 

•375 

i 

7-375 

5-5 

7-875 

4.625 

2-75 

•25 

3-375 

.125 

26 

2000 

{?.:? 

•375 

i 

9-75 

7-5 

8-5 

6-375 

3-375 

•375 

3-6875 

.166 

26 

l8oO 

2-5 

1-25 

9-75 

7-5 

8-5 

6-375 

3-375 

•375 

3-6875 

•25 

65 

1500 

3 

2 

14-75 

9-5 

io-75 

%9-75 

4-3125 

•5625 

4-75 

•5 

100 

1350 

3-5 

3 

18.25 

i 

13  '• 

i-5 

5-5 

•6875 

5-75 

i 

*57 

1050 

4 

3-5 

!9 

3-25 

15-5 

2.25 

7 

.875 

7.0625 

2 

290 

1050 

6 

3 

25 

2.625 

i8.375 

8.25 

9-25 

I 

8.25 

3 

300 

IOOO 

8 

4 

26.25 

5-625 

i8.375 

8.25 

9-25 

i 

8.25 

5 

485 

IOOO 

7-5 

4-5 

28 

8-75 

21 

8.75 

9-75 

1.125 

9-25 

G  Grooved.     F  Flat. 

Application  of  the  Motor. — For  Printing-rooms  and  mechanical  Shops  of  medium 
capacity  and  Elevators,  one  of  5  IP  is  sufficient. 

To    Compute    3?o>ver    required,    for   Elevators. 

RULE. — Multiply  twice*  product  of  weight  to  be  raised  in  Ibs.,  and  height  of  as- 
cent in  feet  per  minute;  divide  by  33  ooo,  and  the  quotient  will  give  the  number  of 
IP  required. 

Small  Motors,  of  .  166,  .  125,  and  .0833  IP  are  adapted  for  operating  Fans, 
Blowers,  Sewing-machines,  Small  Lathes,  Presses,  Tools,  Models  in  operation,  Ro- 
tating Advertisements,  Organ  blowing,  Raffing  wheels,  Knife  sharpeners,  Cloth  and 
Paper  cutting,  Experimental  models,  etc.,  etc. 

Electric    Fans. 

For  Ventilation  of  Offices,  Restaurants,  Kitchens,  Sick-rooms,  etc.,  etc. 

Constructed  in  various  styles,  Plain  and  Nickel-plated. 
Fans  12  Inches  in  Diameter. 

Regular,  .0833  IP  motor. —  Fast,  .125  IP  motor. — Double  (a  Fan  at  each  side), 
.1666  IP  motor,  or  one  i6-inch  Fan. 

La  Rue  Construction. — Variable  speed,  Fan  24  ins.in  diam. — 20  Inch, .  125  IP  motor. 
Electric    3?nmps. 

Pump,  .166  IP,  will  elevate  500  gall,  water  per  day  ot  10  hours  100  feet  in  height. 

These  pumps  are  arranged  to  operate  automatically,  so  that  when  a  receiving  tank 
is  tilled,  the  pump  is  arrested. 

Capacity    of  I*vimp    per    Hovir. 


FP 

Gallons.        || 

IP 

Gallons. 

FP 

Gallons. 

.166 
•25 

IOO 

250 

•5 

i 

37° 
750 

3 

5 

1670 
2600 

-A.ro    Circuit   Motors. 

Arc  Motors' totter  from  all  others.    The  manner  of  connecting  the  circuit  and  of 
their  operation  varies  from  that  of  other  motors. 
They  are  furnished  from  .125  to  5  IP. 

They  should  always  be  connected  to  the  arc  circuit  by  a  competent  lineman. 
*  The  twice  is  taken  to  cov«r  loss  of  power  by  friction  of  all  the  part*. 


960 


DYNAMO    LEATHER    BELTS. — WIRES    AND    CABLES. 


IDynamo    Leather  Belts. 

Belting,  —  For  Dynamos  and  Electric  Light  Machinery  should  be 
double  and  endless,  and*  not  over  .33  inch  thick,  when  run  at  a  velocity  of 
4000  feet  or  more  per  minute ;  should  be  perforated  to  prevent  air-cushion- 
ing, the  perforations  may  be  .093  75  inch  in  width,  .281  25  inch  in  length, 
and  placed  1.5  inches  apart;  furnishing  about  50  openings  per  sq.  foot  of 
belt,  without  material  injury  to  the  tensile  or  operating  strength  of  it. 

In  order  to  protect  the  surface  of  a  Dynamo  Belt  it  should  be  rendered 
impervious  to  the  mineral  oil  used  on  it,  which  is  destructive  to  the  fibre  of 
the  leather. 

LEATHER  LINK  BELTS. 

Where  Belts  are  run  at  right  angles  and  at  short  distances  apart,  Leather 
Link  Belts  are  recommended,  as  they  are  very  pliable  and  have  uniform 
oscillation. 

Link  Belts  made  .6875  inch  thick,  forming  when  combined  two  full  cir- 
cles, assure  the  required  uniformity  of  oscillation. 


WIRES  AND   CABLES. 

Telegraph,  Telephone,  and    Electric    JL.io.ht    Wires 

and.    Cables. 

For    Aerial,   Su.lb-marine,   arid.    "Underground.. 
The  Okonite  Company,  Ltd.,  New  York. 

Insulation. — In  consequence  of  the  decomposing  fnfluence  of  the  elements 
upon  insulated  wires  exposed  to  them,  it  is  necessary  that  their  insulation 
should  be  as  perfect  in  construction  and  enduring  in  material  as  it  is  practi- 
cable to  attain.  In  order  to  effect  an  enduring  insulation  this  Company  uses 
a  compound,  termed  Okonite,  a  material  possessing  both  tenacity  and  resist- 
ance to  abrasion,  while  it  is  equally  unaffected  by  extremes  of  temperature, 
with  insulation  of  a  high  order. 

Telegraph    and    Telephone   AVire. 

Size  of  Insulation  and  Diameter  per  B  W  G;    External  Diameter  of  Insulation  in 
32<*« ;  Weights  per  Mile  in  Lbs. ,  and  Insulation  Resistance  per  Mile  in  Megohm*. 


Insulation. 

Insulation. 

• 

Insulation. 

Insulation.. 

^ 

^ 

PLAIN. 

BKAIDKD 

L," 

PLAIN. 

BBAIOKD 

No. 

1 

1 

If 

If 

|j 

io 

Is 

No. 

1 

1 

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J.  • 

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ii 

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0 

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la 

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fc 

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29 

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43 

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55 

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6 

l64 

1600 

1600 

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24 

13 

3 

33 

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48 

1200 

56 

16 

6 

6-5 

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198 

2OOO 

42 

22 

13 

3 

45 

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60 

IOOO 

57 

16 

5 

7 

200 

2400 

223 

2400 

43 

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12 

3-5 

56 

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71 

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58 

1  6 

3 

8 

240 

2000 

267 

2600 

44 

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II 

4 

65 

I2OO 

80 

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59 

16 

2 

9 

288 

2800 

3i8 

2800 

45 

18 

II 

4 

81 

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96 

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60 

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8 

5 

I76 

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194 

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18 

10 

— 

91 

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107 

I2OO 

61 

14 

7 

5-5 

192 

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211 

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18 

8 

5 

119 

1600 

I600 

62 

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-  6 

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1400 

221 

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48 

18 

— 

6 

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175 

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63 

14 

5 

7 

236 

l6oO 

239 

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49 

18 

5 

7 

197 

2400 

220 

2400 

64 

T4 

3 

8 

273 

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300 

1800 

50 

18 

3 

8 

244 

2600 

271 

2600 

65 

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351 

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18 

2  9 

299 

3000 

329 

5000 

66 

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53 

1.6 

8 

5 

140 

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304  i  looo 

54 

16 

7  i  — 

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1500 

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1500  [',  69 

10 

4  '7  5 

387 

1200  ii  413   1200 

VOLTMETERS    AND   AMMETERS. 


961 


"Voltmeters    and    Ammeters. 

\Veston    Electrical    Instrument    Co.,   Newark,    N".   J. 

Instruments  are  Direct  Reading. — A  multiplying  constant  being  unnecessary, 
except  with  voltmeters  of  high  range,  as  a  simple  inspection  of  position  of  pointer 
on  scale  indicates  value  in  amperes  or  volts,  and  as  pointer  immediately  becomes 
fixed,  or  "dead  beat,"  it  reduces  period  for  reading,  added  to  which  there  does  not 
exi.st  the  "magnetic  lag"  which  induces  different  deflection  with  like  current  or 
potential,  as  in  instruments  in  which  moving  parts  are  of  iron. 

All  readings  of  the  scale  commence  at  o,  and  the  uniformity  of  the  divisions 
facilitates  the  visual  subdivision  of  them. 

Correction* for  Temperature  are  unnecessary,  being  not  in  excess  of  .25  percent, 
for  a  range  of  35°  above  or  below  70°,  and  for  the  ammeter  less  than  i  per  cent,  for 
a  like  range  of  temperature. 

Circuit.— These  instruments  can  be  retained  in  circuit  without  injury,  as  the 
beating  effect,  except  in  the  high  ranges,  is  inappreciable. 

Calibrating  Coil.— In  Voltmeters  provided  with  one.  changes  in  the  scale  value 
from  accidental  injury  are  readily  checked. 

All  the  respective  parts  are  made  to  a  uniform  gauge,  and  are  interchangeable. 

Standard    Voltmeters. 


No. 

Scale  in 
Volte, 
o  up  to 

Single- 
Scale 
Division 
Volte. 

Readable 
to 
Volte. 

Remarks.  ( 

No. 

Scale  in  j  S^jJ«- 
™ta/    1  DiviEion 
0  "P  *°  i    Volte. 

Readable 
to 

Volte. 

Remarks. 

i  i 

150 

i 

O.I 



7 

600 

5 

0.5 



1  s 

150 

I 

O.I 

C.  K.  C.  C. 

8 

600 

5 

0-5 

C.  K,  C.  C. 

*  3 

I 

°'J 

O.K. 

9 

(600 

4 

0-4} 

O.K. 

f  ' 

l/30 

Vsoo) 

i 

0.1  J 

!*- 

{150 

(      3 

Yso 

0.1  i 

VMO) 

O.K. 

9# 

J750 
(150 

5 

X 

0.1  f 

C.  K. 

I  4 

(15          Vio 

0.1  i 

ViooJ 

O.K. 

10 

(600 

4 

0-4} 

0.2  J 

C.  K. 

M 

300                2 

450  i    3 

0.2 

o-3 

C.  K.  C.  C. 
C.  K.  C.  C. 

ii 

12 

750           5 
1500     1     10 

0-5 

X 

— 

NOTE  i.— Calibrating  coil  (C.  C.)  and  contact  key  (C.  K.)  are  attached  to  instru 
ments  only  when  so  stated. — 2.  Instruments  up  to  No.  6,  provided  with  contact 
key,  can  be  kept  in  circuit  by  depressing  it  and  giving  it  a  quarter  turn. — 3.  Volt- 
meters of  other  ranges  furnished  as  required. 

High-Range    "Voltmeters    No.    1. 

In  these  instruments  part  of  the  resistance  is  contained  in  a  separate  box,  de- 
signed to  obtain  high  insulation.     Without  calibrating  coil  or  contact  key. 
Scale  from  o  to  150  volts  aud  a  resistance  box,  multiplying  valve  of  scale  divisions. 


No. 

Volte  to  150.  f       Scale. 

13 
'4 

2250 

3000 

By  15 

"    20 

Volte  to  150. 

Scale. 

J|  Ho. 

Volte  to  150. 

Scale. 

3750 
4500 

By  25 

"  30 

j|  18 

5250 
6000 

By  35 

"   49 

\l 

NOTB  i. — Above  are  graduated  to  read  in  lamps  in  addition  to  graduation  in  volts, 
indicating  by  simple  inspection  number  of  arc  lamps  open  on  the  circuit. — 2.  Other 
multipliers  giving  intermediate  ranges,  applicable  to  these  high-range  voltmeters, 
are  furnished. — 3.  Reversing  Key.  When  many  and  rapid  tests  of  distribution  of  po- 
tential or  electric  currents  of  unknown  polarity  are  required,  this  key  enables  oper- 
ator to  open  the  circuit  by  turning  the  milled  head  through  an  angle  of  45°,  or  to 
reverse  direction  of  current  in  the  instrument  by  an  additional  movement  of  45°. 

Milli- Voltmeters. 


Scale  in 

Single- 

Readable 

• 

Scale  in 

S«[°gle-  ;    Readable 

Volte. 

Scale 
Division 

to 

Remarks. 

.  No.    Volte. 

DiSrioni       Vo?u 

Remarks. 

o  up  to 

Volte. 

VlltB. 

o  up  to 

.02 

0.0002 

O.OOOO2 

fo.oi 

(O.I 

0.0002  <    0.00002) 

O.OO2      I    O.OOO2     ) 

Zero  in   centre. 
Contact     key 
redur.  sensib. 

(Zero  in   c«n- 

i     i 

10  times. 

.01 

0.0002 

0.00002 

1      tre.    Right 
(     and  left. 

(0.02 
i          i     (0-2 

O.  OOO2       O.  OOOO2  ) 
O.OO2      j    O.OOO3     f 

Higher  range 
compensated  for 

1          1     V 

temperat:ue. 

962     VOLTMETERS   AND  AMMETERS. — RAILROAD   CRANE. 


No.  x.  150  volts  in  single  volts  readable 

to  tenths. 

1.5  amp.  in  o.oi  amperes  readable 
to  tenths. 


"Volt- Ammeters. 

Scale. 


No.  2.  150  volts  in  single  volts  readable 

to  tenths. 

3  amp.  in  0.02  amperes  readable 
to  tenths. 


JVXilli-  Ammeters 

No. 

Scale  in 
Milli-ampere. 
o  up  to 

Single-Scale 
Division 
Milli-ampere. 

Readable 
to 

No. 

Scale  in 
Milli-ampere. 
o  up  to 

Single-Scale 
Division 
Milli-ampere. 

Readable 
to 

o 

J50 

I 

O.I 

500 

5 

0-5 

X 

300 

2 

0.2 

5 

50 

0-5 

0.05 

2 

600 

5 

°-5 

6 

500 

5 

o-5 

3 

IOOO 

10 

IO 

O.I 

O.OI 

4 

1500 

10 

I 

7 

500) 

10  j 

with  resistance  box  to  give 
readings  to   10  and  100 

volts. 

Ammeters. 

No. 

Scale  in 
Ampere. 

Single-Scale 
Division 

Readable 
to 

No. 

Scale  in 
Ampere. 

Single-Scale 
Division 

Readable 

o  up  to 

Ampere. 

0 

up  to 

Ampere. 

i 

5 

0.05 

0.005 

7 

200 

2 

O.  2 

2 

15 

O.I 

O.OI 

8 

250 

2 

0.2 

3 

25 

0.2 

0  02 

9 

300 

2 

0.2 

4 

50 

0.5 

O.O5 

10 

400 

5 

0-5 

5 

100 

I 

O.  I 

ii 

500 

5 

o-5 

6 

150 

I 

O.I 

Portable    Direct- Reading    Voltmeters    and    ^Vattmeters 
for    Alternating    and.    Direct-  Current    Circuits. 

Voltmeters,  22  ranges,  from  7. 5  to  3000  volts. 
Wattmeters,  12  ranges,  from  150  to  30000  watts. 

S-witchtooard    Ammeters     and    Voltmeters    for    Central 
Stations    and     Isolated    Plants. 

Illuminated  Dial  Instruments,  "  Round  Pattern  "  Instruments,  have  substantially 
the  same  characteristics  as  the  Portable  Standard  Instruments.  Are  " dead  beat," 
have  uniform  scales,  can  be  kept  in  circuit  continuously. 

Railroad.    Crane. 

The  Farrell  Foundry  and  Machine  Co.,  A  nsonia,  Conn. 

Post.  —  Of  cast  iron,  in  one  piece,  fitted  to  deck-plate,  with 
faced  joints  and  secured  by  bolts  running  through  a  stone  foun- 
dation, set  up  on  anchor  plates  on  its  under  side. 

Jib. — Of  two  wrought-iron  beams,  bolted  at  head  and  foot  to 
a  bonnet  and  shoe,  with  tie  bolts  between  them,  and  secured  to 
the   post  by  bolts   which  lead  from  its  head  to  a  yoke,  which 
turns  on  a  pin  in  the  hub. 
Hub. — With  a  pin  is  fitted  into  head  of  post,  on  which  the  jib  turns. 
Yoke.— Is  secured  by  two  bolts,  which  lead  down  through  and  are  secured 
at  the  deck-plate  on  the  foundation. 

Gearing.— Double  and  set  for  both  fast  and  slow  motions,  and  detachable, 
to  admit  of  lowering  load  by  a  brake. 

Chain.— Triple  "  B  Crane,"  and  all  sheaves  have  roller  bushes. 
Capacity. 


Radius. 

Capacity. 

Weight. 

Radius. 

Capacity 

Weight. 

Radius. 

~Feet. 
16 
20 

structLug  ca 

Capacity. 

Weight. 

Feet. 
12.5 
10 

Tons. 
2* 

4 
\ 

Lbs. 
5500 
6500 
Designed  f 

Feet. 
15 
15 
or  operation 

Tons. 
6 
10 
on  Wrecki 

Lba. 
10400 
14600 
ng  and  Cou 

Tons. 
15 

20 

r§. 

Lbs. 
17400 
23800 

VACUUM    PUMPS. 


"Vacuum    IPumps. 


acuum  Pumps. —  Air  pumps  are  so  termed  when  they  are  used 
in  connection  with  vacuum  pans,  multiple  effects,  or  niters. 

It  is  impracticable  to  define  a  general  rule  for  their  capacity,  as  the  cir- 
cumstances of  their  operation  vary  in  different  cases. 

Vacuum  Evaporators.— Their  dimensions  depend  upon  the  temperature  to  which 
they  are  submitted,  the  evaporation,  character  of  the  liquid  concentrated,  vacuum 
desired,  and  type  and  efficiency  of  the  condenser. 

Dry  Exhaustion. — When  air  alone  is  withdrawn. 

f  j    M  =  Q.     V  and  V  representing  volumes  Oj  cylinder  and  receiver,  M 

volume  of  air  in  receiver  at  commencement  of  operation,  both  in  cube  feet,  n  number 
of  strokes  of  piston,  and  Q  volume  of  air  remaining  after  n  strokes  of  piston. 

Condensation.— There  are  two  systems  in  operation  for  vacuum  pans  and  multiple 
effects,  viz.— 

Dry  System. —  Where  the  condenser  is  fitted  with  a  leg  pipe  or  barometric  tube, 
through  which  the  injected  water  passes  off  by  gravitation. 

Wet  System. — When  the  pump  receives  and  discharges  the  condensing  water,  in 
addition  to  its  maintaining  a  vacuum.  » 

In  either  system  the  pump  is  required  to  discharge:  ist.  The  air  contained  in  the 
injection  water,  in  the  liquid,  and  in  the  pan,  pipes,  and  condenser.— 2d.  The  incon- 
densable gases  evolved  from  the  liquid  in  operation. 

Notes.— The  Pan  and  its  immediate  connections  are  made  of  iron,  copper,  bronze, 
or  alloys. 

In  designating  the  design  and  construction  of  pump  required,  the  liquor,  the 
volume,  the  degree  of  concentration  required,  and  the  time  in  which  the  operation 
must  be  completed,  should  be  furnished. 

To  facilitate  transportation,  the  bed  plates  of  the  large  sizes  are  cast  in  two  parts 
and  bolted  together. 

An  order  for  a  pump  should  state :  ist.  What  liquor,  and  volume  of  it,  is  to  be  evap- 
orated in  a  given  period,  as  an  hour?  2d.  What  the  diameter  of  pan  or  evaporating 
vessel,  and  what  that  of  vapor  pipe  when  it  enters  condenser?  3d.  What  the 
heating  surface  of  pan,  and  has  it  a  steam  jacket  and  coils,  and  if  coils,  what  is  their 
diameter  and  length  ?  4th.  If  heating  surface  is  of  iron,  brass,  or  copper  ?  sth. 
What  the  average  temperature  of  condensing  water,  and  what  the  volume  of  it  ? 


Duplex  "Vacuum   IPumps. 
Fly-wh.ee!    Type    for    "Dry"    or    "Wet' 


System. 


Di 

Diameter 
of  Vacuum 
Cylinders. 

MKNSIONS. 

Diameter 
of  Steam 
Cylinders. 

Stroke. 

Volume 
per  Rev- 
olution. 

Displace 
Feet  Pist< 
Mi 
Per  Min. 

rnent,  at  75 
m  Speed  per 
nute. 
Per  Hour. 

Dia 

Suction 
and 
Discharge. 

meter  of  Pi 
Steam. 

pes. 
Exhauit. 

Ins. 

Ins. 

Ins. 

Cub.  feet. 

Cub.  feet. 

Cub.  feet. 

Ins. 

Ins. 

nt. 

6 

5 

6 

•589 

29.45 

1767 

.25 

8 

6 

6 

1.047 

52-35 

3Mi 

t 

-25 

•5 

10 

7 

6 

I-635 

81.82 

4909 

g 

-25 

10 

8 

6 

1-635 

81.82 

4909 

1 

-5 

12 

8 

9 

2-356 

117.81 

7068 

"3 

•5 

12 

9 

9 

2-356 

117.81 

7068 

•5 

14 

9 

9 

3.207 

160.35 

9621 

g 

•5 

in 

10 
10 

9 
9 

3.207 

4.188 

160.35 
209.4 

9  621 
12564 

| 

•5 
•  5 

-5 
•  5 

16 

12 

9 

4.188 

209.4 

12564 

\ 

-5 

18 

12 

9 

5.301 

265 

15898 

"C 

•5 

18 

14 

9 

5.301 

265 

15898 

1 

•5 

20 

12 

9 

6-545 

327-25 

19635 

2 

•5 

20 

H 

9 

6-545 

327-25 

19635 

•5 

22 

fi 

9 

7.919 

395-95 

23757 

•5 

•5 

22 

16 

9 

7.919 

395-95 

23757 

-5 

3 

24 

16 

9 

9.424 

471-25 

28275 

•< 

•  5 

3 

«4 

18 

9 

9.424 

471.25 

28275 

•5 

3 

964         DRAWING,  TRACING,  SECTION,  ETC.,  PAPER. 

Drawing,    Tracing,    IProfile,    Cross-section,    IPlioto- 
printing  ^Papers  ancl   Clotlis. 

Keuffel  &  Esser   Co.,  New  York,  Chicago,  St.  Louis,  San  Francisco 

In  Sheets.—  Whatman's  Hand-made  in  all  sizes,  H  P,  C  P,  and  R. 

Universal,  For  general  drawing  and  water-colors,  six  sizes,  14X17  ins.  to  27X40 
ins.— Normal,  Not  Hand-made,  but  very  similar  to  the  Not  Hot  Pressed,  in  Royal, 
Imperial,  and  Double  Elephant.—  Duplex,  Cream  color,  for  fine  detail  and  general 
drawings,  in  Royal,  Imperial,  and  Double  Elephant. — Duplex,  Drab  color,  heavy, 
Double  Elephant  only.— Paragon,  Medium  rough,  in  Royal,  Imperial,  and  Double 
Elephant  smooth  in  Double  Elephant  only.— Bristol- Board  (Reynolds's),  tive  sizes, 
12.5X15.25  ins.  to  21.5X28.75  ins.,  2,  3,  or  4  sheets  in  thickness.  K.  &  E.  Patent 
Office  Bristol- Board,  10X15  ins.  and  15X20  ins.  K.  &.  E.  Bond  Paper,  light  and 
very  tough,  three  sizes,  19X24  to  27X4°  ins- 

Tracing  Papers.—  Vegetable  (French),  five  sizes,  13X17  to  29X42  ins.— 
Cupola,  very  tough  and  transparent,  28X39  ins. — Hermes  (slight  grain),  20X30  and 
30X40  ins. — Ceres,  tough,  20X27  and  27X40  ins. — Corona,  thick,  27X40  ins.  Of 
these  the  Vegetable,  Ceres,  and  Corona  are  natural  Tracing  paper  (not  prepared). 

In  rolls:  —  Parchment,  Thick  Parchment,  Abacus,  Patera,  Colonna,  thin  and 
medium,  30,  36,  42  ins.  in  width  (can  often  be  substituted  for  tracing  cloth), 
Corinthian,  Gothic,  Doric,  Alba  (for  transferring),  Lotus,  and  Libra. 

Drawing  Papers  in  Rolls.— Duplex,  medium,  cream  color,  30,  36, 
42,  56,  and  62  ins.  in  width. — Do.,  thick,  drab  color,  36  and  56  ins.  in  width. — 
Universal,  for  general  drawing,  water-colors,  etc.,  36,  42,  56,  and  62  ins.  in  width. — 
Lava,  similar  to  Universal,  pearl  gray. — Anvil,  medium  and  thick,  surface  and 
appearance  similar  to  Whatman's  Not  Hot  Pressed,  medium,  36,  42,  and  62  ins. 
in  width  ;  thick,  62  and  72  ins.  in  width. — Paragon,  pebbled  surface  (similar  to 
egg  shells),  thin,  medium,  thick,  and  extra  thick.  All  58  ins.  in  width,  except 
medium  rough,  which  is  also  36  and  42  ins. — With  smooth  surface  (similar  to  What- 
man's N.  H.  P.  on  one  side,  smooth  on  the  other),  medium  and  thick,  both  58  ins.  in 
width,  except  medium,  also  36  and  72  ins.  All  can  be  had  by  the  yard,  in  ro-yard 
lengths,  or  in  rolls  of  about  35  Ibs. 

Detail. — Economy,  medium  and  light,  5o-yard  rolls,  36  and  60  ins.  in  width. 
—Simplex,  light,  medium,  and  heavy  (Manila),  36,  42,  48,  and  54  ins.  in  width,  in 
50  and  100  yard  or  ioo-lb.  rolls. 

ATonnted.  —  Universal,  Duplex,  Lava,  Anvil,  and  all  the  Paragon  Papers 
are  mounted  on  muslin,  in  all  the  widths;  by  the  yard,  or  in  10,  20,  or  30  yard 
rolls.  All  the  sheet  papers  are  also  to  be  obtained  mounted  up  to  20X30  feet. 

Photo-printing  Papers. — Helios  Blue  Print,  medium  and  thick,  pre- 
pared (sensitized),  24  to  54  ins.  in  width.  E.  T.  Paper,  thin,  for  mailing,  prepared, 
24»  3°»  36,  and  42  ins.  in  width. — Columbia,  Blue  Print  Papers,  medium,  thick,  and 
thin  (mailing),  24  to  42  ins.  in  width.  Prepared  paper  is  in  10  or  50  yard  rolls; 
unprepared  in  50  -  yard  rolls.  Blue  Process  Cloth,  prepared  and  unprepared,  in 
lo-yard  rolls,  30,  36,  and  42  ins.  in  width.—  Nigrosine  (Positive  Black  Process),  10- 
yard  rolls,  30,  36,  and  42  ins.  in  width,  prepared  only.  —  Umbra  (Positive  Black 
Process)  requires  no  developing  bath. — Maduro  (Negative  Brown  Process)  Paper 
and  Cloth,  requiring  only  a  flxing-bath;  30,  36,  and  42  ins.  in  width,  prepared  only. 
From  Maduro  prints  on  thin  paper  positive  blue  or  brown  prints  of  the  original 
can  be  taken.  Maduro  is  the  latest. 

Profile  Cross- Section  and  Tracing  Papers. —Tracing  and 
Drawing  Cloths  in  red,  green,  orange,  or  blue. 


NOTB.— A  complete  catalogue  of  Drawing  Materials  and  Surveying  Instruments,  500  pp.,  mailed 
on  application. 


REFRIGERATION.  965 

Mechanical   Refrigeration. 

The  De  La  Vergne  Refrigerating  Machine  (7o.,  New  York. 

Mechanical  Refrigeration  is  effected  by  Compression,  Condensa- 
tion, and  Expansion  of  a  liquefiable  gas. 

The  Refrigerating  or  Heat-absorbing  agents  are  Ammonia,  Ether,  Sul- 
phurous Oxide,  Carbonic  Acid,  etc.,  which  undergo  the  operations  above 
given.  The  De  La  Vergne  Machine  is  operated  with  Ammonia. 

Compression. — The  gaseous  agent  is  compressed  if  Ammonia  is  used 
to  from  125  to  175  Ibs.  per  sq.  inch ;  during  which  operation  heat  is  devel- 
oped in  proportion  to  the  pressure  exerted  upon  the  gas,  or  the  relative  vol- 
ume to  which  it  has  been  reduced. 

Condensation. — The  heat  developed  in  the  operation  of  compression 
is  withdrawn  from  the  compressed  gas,  which  is  forced  through  coils  of 
metal  pipe,  surrounded  with  cold  water.  As  soon  as  the  condition  of  satura- 
tion is  reached,  the  gas  assumes  a  liquid  state. 

Expansion. — The  liquefied  gas  is  also  passed  through  coils  of  metal 
pipe,  suspended  or  seated  in  a  space  where  the  substance  to  be  cooled,  as  air, 
water,  brine,  beer,  etc.,  is  introduced ;  the  pressure  in  the  interior  of  the  coils 
being  at  a  lower  point  than  that  required  for  the  maintenance  of  the  gas 
in  the  liquid  state. 

The  liquefied  gas,  upon  entering  these  coils',  again  expands,  and  extracts 
from  them  and  the  substance  around  them  the  same  quantity  of  heat  that 
was  previously  given  up  by  the  gas  to  the  water  of  condensation. 

The  gas,  having  passed  through  this  routine  of  operation  of  refrigerating, 
is  now  in  a  condition  to  be  used  in  a  repetition  of  it. 

The  gas  is  forced  through  these  coils  by  the  pressure  in  the  condenser,  which,  in 
the  use  of  Ammonia,  is  generally  from  125  to  175  Ibs.  per  sq.  inch.  Under  this 
pressure  and  the  cooling  action  of  the  water,  liquefaction  occurs,  and  the  resulting 
liquefied  gas  flows  to  a  stop-cock,  having  a  minute  opening,  by  which  the  pressure 
is  reduced  from  10  to  30  Ibs.  per  sq.  inch  in  the  expansion  coils,  and  where  the 
liquid  through  reduction  in  pressure  is  again  transformed  into  a  gas.  By  the  ex- 
hausting operation  of  a  gas  pump,  this  pressure  is  maintained,  and  then  the  gas  is 
forced  by  compression  into  the  condenser  again. 

Thus  the  expansion  coils,  although  similar  to  those  for  condensation,  are  operated 
for  the  reverse,  which  is  the  absorption  of  heat  by  the  liquefied  gas,  instead  of  the 
extraction  of  heat  from  it. 

In  Operation,  heat  is  transmitted  from  the  outside  through  the  walls  of  the  ex- 
pansion or  cooling  c/>ils,  and  is  absorbed  by  the  expanding  liquefied  gas  within  such 
coils.  This  heat  is  borne  by  the  gas  through  the  pump  into  the  condenser,  where 
it  is  in  turn  transferred  to  the  cooling  water  through  the  walls  of  the  condenser 
coils,  and  ultimately  carried  away  by  this  water. 

NOTE.  —  Liquefied  ammonia  in  a  gaseous  condition  at  atmospheric  pressure  and  temperature  of  60°, 
expands  about  1000  times,  and  upon  its  expansion  re-absorbs  a  quantity  of  heat  equal  in  amount  to  that 
originally  held  and  evolved  from  it  during  liquefaction. 

The  liquefied  gas,  entering  the  coils  through  the  minute  opening  in  stop-cock,  is 
immediately  relieved  of  a  pressure  of  125  to  175  Ibs.,  that  requisite  to  maintain  it 
in  a  liquid  state,  when  it  boils  and  expaLds  into  gas.  To  obtain  this,  heat  is  re- 
quired, and  which  alone  can  be  supplied  from  the  substance  surrounding  the  coils, 
such  as  air,  brine,  water,  etc. 

As  a  result,  the  surrounding  substance  is  reduced  in  temperature,  the  quantity 
of  heat  withdrawn  by  the  gas  being  the  same  as  that  which  was  withdrawn  from  it 
during  its  liquefaction  in  the  condenser. 

Consequently,  if  the  expansion  coils  are  set  in  an  insulated  space,  it  will  be  re- 
frigerated; and  if  brine  or  any  liquid  surrounds  the  coils,  it  will  be  reduced  in  tem- 
perature, and  brine,  in  this  condition  led  into  a  space  through  a  pipe  or  open  con- 
duit, will  refrigerate  it. 


966  REFRIGERATION. — FORCITE    POWDER. 

Results   of  Operation   of  Refrigerating   Machines 
of  SOO*    Tons. 

At  Lion  Brewery,  New  York.     Duration  of  Test  n  h  zomin. 

Steam  Cylinders.—  Diameter,  36  ins. ;  Stroke  of  Piston,  36  ins. Pressures 

of  steam  (mean),  48.4  Ibs. 

Gas  Compressors.  — Two  double  acting;  diam.  18  ins. ;  Stroke  of  Piston,  36  ins. ; 
back-pressure,  28.22  Ibs. ;  condenser,  180.78  Ibs.  persq.  inch;  Revolutions,  39.55  per 
minute. 

Test  for  cooling  made  by  running  water  of  a  mean  temperature  of  100.95°  over 
wort,  Baudelot  Cooler,  and  cooling  same  to  a  mean  temperature  of  50.77°. 

Refrigeration,  equal  to  melting  of  210  tons  Ice  per  day  of  24  hours. 

Horse  Power.  —  IBP  =  313,  and  assuming  consumption  of  coal  at  3  Ibs.  per  hour 
per  IB?,  ratio  of  refrigeration  =:  20. 84  Ibs.  ice  per  Ib.  of  coal. 

If  operated  under  ordinary  condensing  pressure  of  156  Ibs.,  the  IB?  would  be  278, 
and  ratio  23. 47  Ibs.  ice  per  Ib.  of  coal ;  IB?  per  ton  of  ice  per  day  =  1. 183. 

Of  a  26-Ton    Machine.      At  Bohlen-Huse  Machine  and  Lake  Ice  Co., 
Memphis,  Tenn.     Duration  of  Operation  20  Days. 

Steam  Cylinder:  Diameter,  22  ins. ;  Stroke  of  Piston,  28  ins. ;  Steam,  93.49  Ibs.  per 
sq.  inch.—  Gas  Compressors,  Two  single-acting:  diam.,  14  ins. ;  Stroke  of  Piston,  28 
ins. — Revolutions,  40  13  per  min. —  Temperatures  :  Cooling  water  63°,  brine  18.62°; 
coal  consumed,  180597  Ibs. ;  Ice  produced,  i  221 172  Ibs.— Ice-making,  26.83  tons  per 
day  of  24  hours. — Steam-boiler  evaporated  5.5  Ibs.  water  per  Ib.  coal. 
*  All  tons  are  given  at  2240  Ibs.  See  foot-note,  p.  xxvi. 

FORCITE   POWDER. 

American  For  cite  Powder  M'fg  Co.,  New  York. 

Foroite. —  Is  an  improvement  in  Nitro-glycerine  compounds,  and  it 
presents  the  following  elements : 
It  is  less  sensitive  to  shock  than  other  explosives. 

Assuming  Dynamite  No.  i  as  the  Standard—  100. 

Forcite  No.  X,  95  per  cent,  Nitro-glycerine,  133  per  cent,  intensity. 

i*75       "  "        125        « 

3t4o       "  "          95        « 

•  »5  per  cent,  stronger  than  Dynamite  No.  i.      f  Within  5  per  cent,  the  strength  of  No.  i,  75  per  cent. 

It  is  more  powerful  than  any  other  known  explosive  in  our  market. 
See  Report  of  Henry  L.  Abbott,  Lieut. -Col.  E.  U.  S.  A. 

It  is  safe  in  handling  and  transportation,  quintuple  force-caps  being  ap- 
plied to  explode  it,  and  free  from  noxious  fumes.  Water-proof,  free  from 
the  absorption  of  moisture,  and  is  not  injured  by  submersion  in  water. 

Directions  in  Use. 

In  Blasting,  fill  the  hole,  and  thoroughly  tamp  the  charge. 
Thaw  it,  if  frozen,  as  frozen  powder  will  not  explode  with  its  proper  effect. 
Exploder  or  caps  should  be  maintained  dry,  and  are  not  to  be  stored  in  same 
buildings  as  the  powder. 
Powder,  ignited  by  weak  caps,  instead  of  being  exploded,  emits  noxious  vapors. 

I*er    Cent,  of  Nitre-glycerine    in    Brands   of  Foroite. 

Gelatine 95  I  No.  i 75  I  No.  2 50  I  No.  3 40  I  No.  3  B 2a 

No.iX 8o|    "2X....6o|    «3X....45|    "3A....35|    "3C 30 


SURFACE    CONDENSATION. REFRIGERATING.        967 

SURFACE    CONDENSATION. 
Wheeler  Condenser  &  Engineering  Works,  New  York. 

Construction. — The  Wheeler  Condenser,  alike  to  others  for  the  same 
purpose,  is  an  elongated  vessel,  cylindrical  or  cubical,  with  the  necessary  at- 
tachments for  Steam  and  water  connections. 

Its  distinguishing  features  are :  The  exhausted  steam,  upon  entering  the 
condenser,  impinges  upon  a  perforated  scattering  plate,  which  distributes  it 
generally  over  the  tubes  and  thus  diverts  the  deteriorating  effect  of  the  direct 
impingement  of  it  upon  one  portion  of  the  tubes ;  the  steam,  expanding 
in  a  void  above  the  tubes,  is  reduced  in  pressure,  and  consequent  temperature, 
before  it  flows  into  contact  with  the  surfaces  of  the  tubes. 

Each  pair  of  tubes  is  composed  of  an  external  and  internal  tube,  set  hori- 
zontally, the  inner  tube  having  an  open  end,  the  other  end  being  screwed 
into  a  removable  head  or  vertical  diaphragm,  which  is  set  at  a  space  of  a 
few  inches  from  a  like  head,  into  which  one  end  of  this  large  tube  is  screwed, 
the  other  end  being  closed  by  a  screw  cap. 

This  design  permits  the  tubes  to  expand  or  contract,  without  the  use  of 
tube  packings  or  ferrules  of  any  kind,  as  only  orfe  end  of  each  tube  is  fixed. 

The  tubes  are  tinned  both  externally  and  internally,  and  can  be  readily 
withdrawn  for  cleaning,  etc. 

Operation. — Tftie  tubes  are  divided  into  two  distinct  tiers ;  the  condens- 
ing water  flowing  through  the  small  tubes  in  the  lower  division  passes  out 
of  then*  open  ends  and  through  the  annular  space  between  their  external  sur- 
faces and  the  internal  surfaces  of  the  larger  tubes,  and  from  thence  into  the 
upper  division,  and  through  its  tubes  in  like  manner  to  the  space  between  the 
two  heads  referred  to,  and  finally  out  through  the  discharge  pipe. 

The  circulation  of  the  condensing  water  is  by  this  manner  of  flowing  ren- 
dered very  active,  and  consequently  a  less  volume  of  it  is  required,  and  there 
is  less  tube  surface  needed  for  a  required  volume  of  condensation. 

Results    of*  an    Operation    to    Determine    tne    Efficiency 
of  tliis    Condenser,  -with.    and.    -without    a   Vacuum. 

Steam  Condensed  per  Hour  per  Sq.  Foot  of  Condensing  Surface. 


Condenser. 

Vac- 
uum. 

Te 

jection 
Water. 

mperatut 
Dis- 
charge 
WateV. 

es. 

Reser- 
voir. 

Steam 
Con- 
densed. 

Condenser. 

Te 
In- 
jection 
Water. 

mperatui 
Dis- 
charge 
Water. 

es. 

Reser- 
voir. 

Steam 
Con- 
densed. 

Lbs. 
204.2 

With 
Vacuum 

Ins. 

}H.S 

Deg's. 

56.5 

Deg's. 
98 

Deg's. 

138 

Lbs. 

1  01.  8 

Without 
Vacuum* 

Deg's. 
J78-5 

Deg's. 
139 

Deg's. 
201 

*  As  a  simple  surface  condenser  without  air  pump  attached. 

REFRIGERATING    AND    ICE-MAKING. 

A  Refrigerating  Machine  is  one  that  produces  as  low  a  temperature  as  a 
given  volume  of  ice,  at  the  temperature  attained,  would  in  melting  from  the 
temperature  of  the  air,  or  void  to  be  refrigerated  =  142°  (142.6°)  of  temper- 
ature are  required  to  transfer  one  Ib.  ice  at  32°  to  one  Ib.  water  at  32°,  which 
difference  represents  the  Latent  heat. 

In  order  to  operate  such  a  machine  for  the  formation  of  ice,  there  will  be 
required,  instead  of  142°,  about  236°. 

Thus,  Assume  the  water  from  which  the  ice  is  to  be  formed  to  be  of  an  average 
temperature  of  72°;  then  to  reduce  it  to  32°,  before  ice  can  be  formed,  40°  or  40 
thermal  units  are  to  be  abstracted  from  each  Ib.  of  water;  then  142°  are  to  be  ab- 
stracted from  the  Ib.  of  water  of  32°  to  reduce  it  to  one  Ib.  ice  at  32°. 


968       EEFRIGEKATIXG   AND    JOE-MAKING,  ETC.,  ETC. 


If  the  ice  is  produced  at  the  general  temperature  of  18°,  and  the  Specific  heat  of 
it  is  taken  at  .5°;  then,  32  — 18  X-5  =  7°.  To  reduce  this  water  from  72°  to  32°, 
there  is  a  reduction  of  40°  or  thermal  units  from  each  Ib.  of  water. 

If  ice  is  produced  at  18°,  Then  7°  additional,  as  deduced  above,  are  required. 

In  practice  it  is  observed  that  the  average  loss  of  temperature  by  radiation  of  it 
from  the  freezing  tank,  melting  the  external  surface  of  the  ice,  to  withdraw  it  from 
the  molds,  etc.,  is  fully  20  per  cent,  of  the  total  capacity  of  the  machine.  Hence,  of 
the  236°  which  are  to  be  abstracted  from  the  water  per  Ib.  of  ice,  in  order  to  reduce 
it  to  ice,  47.2°  are  lost  by  radiation.  And  40+  i42H~7-h47  — 236°  are  to  be  ab 
stracted  from  each  Ib.  of  water  of  72°,  in  order  to  produce  i  Ib.  ice  at  18°. 

Consequently,  If  142°  are  required  in  Refrigerating  machine  and  236°  in  Ice- 
making,  the  relative  requirements  are  as  i  to  i  .66  or  as  6  to  10. 

Refrigerating  Capacity.  —  Of  a  machine  is  designated  by  the 
number  of  Ibs.,  or  tons  of  Ice,  which  it  is  capable  of  producing. 

One  Ib.  of  ice  at  32°  absorbs  142°  or  thermal  units  in  melting.  Hence,  one  ton 
of  ice  absorbs  142°  X  2240^318000°,  and  a  machine  of  50  tons'  capacity  absorbs 
318000°  x  50  —  15900000°  every  24  hours  of  its  operation. 

Ice-malting  Capacity. — Of  a  machine  is  also  designated  by  the 
number  of  Ibs.,  or  tons  of  Ice,  which  it  is  capable  of  producing. 

To  freeze  one  Ib.  of  water  at  72°  to  ice  at  18°,  it  requires  the  absorption  of  236°, 
viz.,  To  reduce  one  Ib.  of  water  at  72°  to  32°,  it  requires  the  absorption  of  40°,  to 
freeze  it  requires  142°;  to  reduce  ice  from  32°  to  18°  requires  14  x  .5  —  7°  (Specific 
heat  of  ice  =  .5).  Reduction  of  temperature  from  surface  of  freezing  tank  and 
withdrawing  the  ice  from  its  molds  by  the  application  of  heat,  about  20$  of  total 
capacity  of  machine  =  20$  of  236  =  47°.  Hence,  Total  heat  to  be  absorbed  per  Ib. 
of  ice  =  40  -f 142  -f-  7  -f  47  =  236  °. 

Ratio  of  Capacity  of  Refrigerating  to  Ice-making.  —  As  142  :  236  :  :  6  :  10,  as  pre- 
ceding, or  a  Refrigerating  machine  of  9.97  tons  capacity  will  produce  about  6  tons 
of  ice  in  the  same  period. 

Higliest    Klevation    of  a    l^alre. 

Colorado.— "  Green  Lake"  is  10252  feet  above  level  of  the  sea  and  300  feet  in 
depth. 

Magnifying. 
Bavaria,  Munich,  possesses  a  microscope  that  magnifies  16000  diameters. 

fower    of  Scre-vv    Bolts. 

Results  of  an  Experiment. 

Wrought-iron. — Diameter,  2  ins.  Thread,  V.  Pitch,  .22  ins. 
Mean  Power  applied  at  a  circumference  of  78.85  ins.,  213  Ibs. 
Loss  by  friction,  10. 19  per  cent. 

(Jas.  McBride,  M.  Am.  Soc.  M.  E.) 

Duration,   of  Railroad.   Cross-ties. 
IDnration   of  Following    \Voods. 


Wood. 

Years. 

Wood. 

Years. 

Wood. 

Years. 

White  Cedar 

8    7< 

Chestnut  

7  5 

Yellow  Pine  

6 

White  Oak 

R 

Red  Spruce 

Hemlock 

5  5 

Black  Cypress  

8 

Red  Oak  

5-5 

Tamarack  

4 

The  elements  of  durability  are  Resistance  to  decay  and  to  wear.  White  Oak  com- 
bines both  qualities  to  the  highest  degree.  Yellow  Pine  resists  wear,  but  not  decay. 
Red  Cedar  and  Black  Cypress  resist  decay,  but  not  wear. 

Ties  should  not  be  cut  when  the  tree  is  in  leaf,  and  should  be  well  seasoned  or 
preserved  by  some  antiseptic  process  before  being  laid. 

Proper  draining  of  a  road-bed  will  add  to  the  duration  of  ties,  and  all  indentations 
of  their  surface  by  tools,  etc.,  should  be  avoided,  and  all  spike-holes  plugged  to 
avoid  the  absorption  of  water.  (H.  W.  Real.) 


GAS  AND  ELECTRIC   LIGHTING. — RAILROAD  SPEED.    969 


GAS   AND   ELECTRIC   LIGHTING. 
(In  Addition  to  pp.  583-587).     Gras. 
Candle  3?oxver  and.   Consumption  of  Different  Burners. 


Candle  Power. 

Cons 

ump- 

Candle  Power. 

Consump- 

tic 

tion 

Burner. 

No. 

Per  Foot 
per  Hour. 

per  Hour 
per  Lamp. 

Burner.               No 

Per  Foot 
per  Hour. 

per  Hour 
per  Lamp. 

No. 

Fe 

et. 

| 

No. 

Fact. 

Batswing 

10 

2  . 

•3-3 

0 

Flat     )     In    (      60 

JC 

Flat     )  from  .  . 
Flame  j  to  

"•5 
13-8 

2. 

3 

JO 

5 

4 
4 

6 
8 

|  Clus-*j    90 
Flame)  ters.  (    150 

4 

5-5 
5 

20 
3<> 

Electric. 

A.rc    Lamps. 

Relative 

Current. 

C 

Horiz'tal.   Angle  7°. 

andle  Pow« 
Angle  10°. 

r. 
Angle  20a. 

Angle  40°. 

Watte 
Required. 

Units 
per 

Cotts*  of 
Gas.  Elec- 
tric—i. 

Ampert. 

No.              No. 

No. 

No. 

No. 

No. 

Hour. 

6 

92               175 

207 

322 

460 

300 

•3 

2.67 

8 

156              300 

350 

546 

78o 

400 

•4 

3-77 

10 

22O                  420 

495 

77° 

IIOO 

500 

•5 

4-83 

*  Per  Candle  Power  for  Batswing  Burner. 

Arc  Lights  should  be  set  high  and  for  the  following  causes: 

1.  Their  high  candle  power  and  distance  apart  being  in  excess  of  gaslights. 

2.  Light  radiating  at  a  depressed  angle  is  greater  than  when  cast  horizontally. 

3.  Horizontal  rays  are  not  as  steady  as  angular. 

NOTE.— The  greatest  intensity  with  continuous  currents  is  at  an  angle  of  40°  be- 
low a  horizontal  line. 

To   Determine    the    Coefficient    of  Minimum    Lighting 
I?ower   in    Streets. 

L  H  -r-  D3  =  Co.  L  representing  candle  power  of  lamps.  D  maximum  distance 
from  lamp,  and  H  height  of  lamp,  both  in  feet,  and  Co,  coefficient. 

Usual  standard  for  Gaslighting  is  assumed  for  a  unit  of  pavement  50  feet  dis- 
tant for  a  lamp  of  12  candle  power  9  feet  in  height.     Hence, 
12  X  9  -T-  so3  =  .000864. 

Adopting  this  coefficient,  the  following  capacities  of  arc  lights  will  give  the  same 
standard  of  light  at  the  following  freight  and  distance. 

A  minimum  standard  would  increase  the  coefficient  to  .001 728. 

NOTE.— One  arc  light  can  replace  from  3  to  6  gas-lamps,  according  to  locality  and 
standard  of  light  adopted. 

2.— Arc  lighting,  based  on  the  substitution  of  one  light  for  3.5  to  4  gas-lamps, 
would  double  the  minimum  standard  of  light;  while  the  average  standard  would 
be  increased  from  10  to  12  times. 

(Eliminated,  etc.,  from  Papers  of  Henry  Robinson,  M.I.C.E.) 

Railroad    Speed. 

1891,  Sept.  14.  N.  Y.  Central  and  Hudson  River  R.  R.—  From  Grand  Central 
Station  to  East  Buffalo,  N.  Y.  436  miles  in  426  minutes,  actual  running  time  = 
61 . 40+  miles  per  hour.  Weight  of  Train  230  tons. 

From  Station  to  Fairport,  361  miles  in  360  minutes,  there  delayed  by  a  hot  journal. 

1891.  Philadelphia  and  Reading  R.  R. — One  mile  in  39.75  seconds=the  rate  of 
90 . 54  miles  per  hour. 

"Flying  Scotchman,"  London  to  Edinburgh,  400  miles;  stops,  44  minutes  ex- 
cluded, in  8 . 5  hours  =  47 . 05  miles  per  hour. 

Weight  of  Train,  excluding  locomotive,  80  tons. 


970 


TENACITY    AND    RESISTANCE    OF    BOLTS. 


Tenacity  of  Round,  and   Sqxiare  "Wrougnt-Iron   JBolts, 
Holes  of  Different  Diameters. 

Round.— .75-inch,  driven  into  a  bole  of  .625  inch,  in  White  Pine,  for  12  ins., 
required  6875  Ibs.  to  withdraw  it. 

i-inch,  driven  into  a  hole  ot  .75  inch,  in  White  Pine,  for  12  ins.,  required  10612 
IDS.  to  withdraw  it;  and  in  Norway  Yellow  Pine,  10830  Ibs. 

i-inch,  screwed,  8-threads  per  inch  into  a  hole  .8125  inch,  in  White  Pine,  for  12  ins., 
required  15  125  Ibs.  to  withdraw  it,  and  one  of  12  threads  required  15  250  Ibs. 

i.i25-ins.,  driven  into  a  hole  of  .875  inch,  in  Hemlock,  for  12  ins.,  required  8875 
Ibs.  to  withdraw  it. 

Square — The  difference  between  that  and  Round,  under  like  conditions,  was 
essentially  different,  and  when  a  hole  was  bored  10  ins.  in  depth,  the  difference  was 
not  essential. 

Railway  Spilses. 


Length 
iu 
Tie. 

Chestnut. 

T 
Y.  Pine. 

o  Withdra 
W.  Cedar. 

w 
W.  Oak. 

Hemlock. 

Remarks. 

Ins. 
4.6 

Lbs. 
3264 

Lbs. 
3198 

Lbs. 
2305 

Lbs. 
4330 

Lbs. 
3485 

In  solid  wood,  sharp  pointed. 

Ship  Spikes.—  . 375  inch  square  and  7  ins.  in  depth,  driven  3  ins.  in  White 
Pine  and  drawn  back,  required  1617  Ibs. ,  their  edge  with  the  grain  of  the  wood,  and 
1317  Ibs.  with  it  across. 

Note.— The  above  are  deduced  from  Experiments  of  Gen.  Weitzel,  U.  S.  E., 
1874-77. 

Resistance  of  Bolls,  after  being  7  months  driven  =  10  per  cent,  greater  than  im. 
mediately  after,  and  when  driven  through  in  direction  of  fibre  it  is  but  60  per 
cent,  of  that  of  being  withdrawn. 

Smooth  bolts  have  greater  retention  than  ragged,  either  driven  or  withdrawn. 

Moderate  "  ragging "  reduces  their  power  25  per  cent.,  and  extreme  50  per  cent. 

Relation  between  diameters  of  bolt  and  hole  showed  that  the  resistance  of  a  bolt 
of  i  inch  in  a  .6875-inch  hole  was  greater  than  in  one  of  .75  or  .8125  inch. 

With  a  -75-inch  bolt  the  resistance  was  greater  in  a  hole  of  .625  inch,  and  was 
one  quarter  greater  than  in  one  of  a  sixteenth  greater  or  less. 

One-inch  square  bolt  in  a  .875-inch  hole  was  the  same  as  a  round  bolt  in  a  ,6875- 
inch  hole. 

Screw-bolts  are  about  50  per  cent,  more  effective  than  plain  round. 

Long  pointed  blunt  bolts  are  more  effective  than  short  pointed. 

Experiments  of  Mr.  F.  Collingwood  and  Wm.  H.  Paine,  made  in  connection  with 
construction  of  the  New  York  and  Brooklyn  Bridge,  gave  for  a  i-inch  round  bolt, 
driven  in  a  .9375-inch  hole,  in  best  Georgia  Pine,  a  resistance  of  15  ooo  Ibs.  per  lineal 
foot,  and  in  a  .875-inch  hole  12  ooo  Ibs.  In  lighter  woods  the  tenacity  was  less. 

Mr.  J.  B.  Tscharner,  in  the  laboratory  of  the  University  of  Illinois,  determined 
that  a  like  bolt  (i-inch  round),  under  like  conditions  in  White  Pine,  was  6000  Ibs., 
and  that  a  bolt  driven  parallel  to  the  grain  of  the  wood  has  but  half  of  the  resist 
ance  of  that  driven  perpendicular  to  it.  Further,  that  assuming  a  bolt  of  i  inch  in 
a  .0375-hole  as  i,  that  if  driven  in  a  .75-inch  hole  it  would  be  1.69,  and  in  a  .8125- 
inch  hole  2.13. 

Relative    Driving    Resistance    of"   Roxind    and    Sq.ixare 

Steel    Bolts. 
One  Inch  in  Diameter.    Drive  into  Pine  Wood.    Six  Inches  in  Depth. 


Square. 

Round. 

i 
3972 
662 

•9375 
4260 
710 

•875 
4660 

777 

.8125 
4050 
675 

•9375 
2250 
375 

•875 
3798 
633 

.8125 
4728 
788 

Power  applied  in  Lbs     .  ... 

Tenacity  per  Inch  of  Depth  

(J.  H.  Powell  and  A,  E.  Harvey). 

NOTE.— Inasmuch  as  the  amount  of  metal  in  the  Round  bolts  is  but  .7854  that 
of  the  square,  Round  drift  bolts  are  the  least  expensive. 


MORTAR. — SPEED    OF    VESSELS.  MTC. 


971 


Mortar. 

Brick.— Clean  and  Sharp  Sand,  3  parts;  Lime,  i  part;  laid  in  a  bed  suffi- 
ciently large  to  admit  of  the  composition  being  in  a  thin  layer. 

In  slaking  the  lime,  apply  sufficient  water  to  prevent  its  burning.  Stir  rapidly 
and  thoroughly,  in  order  to  enable  the  water  to  cover  each  lump  of  lime  as  it  deli- 
quesces; and  when  this  operation  is  fully  effected,  stir  the  substance  into  a  condi- 
tion alike  to  milk,  and  then  mix  it  with 'the  sand  in  the  bed,  with  water  sufficient 
to  render  the  mass  semi-fluid. 

In  this  condition  it  should  remain  for  a  period  of  at  least  24  hours — a  longer  pe- 
riod is  preferable. 

When  required  for  use,  add  and  thoroughly  mix  with  it  another  part  of  sand. 

Hair  Mortar. — Lay  the  hair  on  a  floor  and  beat  it,  in  order  to  break  the 
bunches  and  remove  foreign  substances.  Then  soak  and  wash  it  in  water  for  24 
hours,  to  remove  all  glutinous  matter. 

Spread  it  on  a  layer  of  sand  in  a  bed,  add  lime,  and  proceed  as  directed  for  Brick 
Mortar. 

J^arge    Trees    in.    Australia. 

In  Victoria,  Eucalyptas.— One  435  and  one  450  feet  in  height. 

Speed,  of  Vessels. 

To  Determine  the  True  Speed  of  a  Vessel  by  Consecutive 
and.  Alternate  Runs   over  a  Measured  Distance. 

Assume  the  Runs  as  follows: 


Run. 

Miles  or 

Knots. 

IBt 

Result. 

ad 

Remit. 

Result. 

4th 
Result. 

Mean 
of  Result*. 

I 

I5.6 

12.9 

2 

IO.2 

'    '  ' 

VJ 

12.6 

' 

12.3 

•>."'  g      , 

12.55 

3 

14.4 

12.7 

12.5                        \     12.5 
}    12-45    { 

12.45  =  True  Speed. 

4 

II 

:    v-' 

12.4     j                  }     12.4 

12.  1 

J    12.35    1 

5 

13.2 

r  •'  i^i 

12.3 

12.5 

6 

ii.  8 

62.s-f-«?  =  i2.«;  Ordinary  mean  Sneed. 

NOTE.— The  mean  of  second  result  is  sufficiently  accurate  for  ordinary  determi- 
nations. 

Velocity  of  tlie  Current. 

To  Determine  tne  Velocity  of*  tne  Current  in  Line  of* 
tne  Vessel's   Course. 

From  the  observed  speed  of  the  vessel  deduct  her  true  speed,  and  the  difference 
is  the  velocity  of  the  current. 
ILLUSTRATION. — Assume  preceding  runs. 


Runs. 

SJH 

Observed. 

ed. 
True. 

Difference. 

Miles  or  Knots. 

X 

5-6 

3-*5 

With  the 

vessel 

2 

3 

4 
5 

O.2 

4-4 

i 

12.45 

2.25 
1-95 

•75 

Against 
With 
Against 
With 

do. 
do. 
do. 
do. 

6 

I'.B 

•65 

Against 

da 

Relative    Corrosion    of  Wrought    Iron    in    Sea   Water. 

In  Air i. 

In  contact  with  brass 3-4  I  In  contact  with  lead s.5 

'     copper 4.9)          "          "     gun-metal 6.5 

In  contact  with  tin 8.7. 


972  PILE-DRIVING. RINGING    ENGINE. 

PILE-DRIVING. 

(Continued  from  page  672.) 

To  Compute    \Veiglit   of  Ram.     (Molesworth.) 

(hP          \ 
-_- —  i  j  =  R.     P  representing  weight  of  pile  in  Ibs.,  h  height  of  fall  of  ram, 

and  L  length  of  pile,  both  in  feet,  and  A  area  of  section  of  pile  in  sq.  ins. 

Piles  are  distinguished  according  to  their  position  and  purpose :  thus, 
Gauge  Piles  are  driven  to  define  limit  of  area  to  be  enclosed,  or  as  guides  to 
the  permanent  piling. 

Sheet  or  Close  Piles  are  driven  between  gauge  piles  to  form  a  compact  and 
continuous  enclosure  of  the  work,  and  are  driven  as  close  and  uniform  to 
each  other  as  practicable  of  attainment,  and  the  intervening  space  or  joint, 
however  close,  is  made  water-tight  by  the  introduction  of  a  "  feather  "  driven 
in  a  groove  on  the  sides  of  the  piles. 

Crushing. — Crushing  resistance  of  a  pile,  unless  of  very  hard  wood,  should 
not  be  estimated  to  exceed  a  range  of  from  500  to  1000  Ibs.  per  sq.  inch. 

Refusal  of  a  pile  intended  to  support  a  weight  of  13.5  tons  can  be  safely 
taken  with  a  ram  of  1350  Ibs.,  falling  12  feet,  and  depressing  the  pile  .8  of 
an  inch  at  final  stroke. 

Pneumatic  Piles. — A  hollow  pile  of  cast  iron,  2.5  feet  in  diameter,  was  depressed 
into  the  Goodwin  Sands  33  feet  7  ins.  in  5. 5  hours. 

Water  Jets.— A.  stream  of  water  is  ejected  under  pressure  at  the  point  of  a  pile, 
and,  rising  around  it,  removes  the  end  and  surface  resistance,  so  that  it  will  be  more 
easily  driven.  Suited  for  sand  or  fine  soil. 

Nasmyth's  Steam  Pile-hammer  has  driven  a  pile  14  ins.  square,  and  18  feet  in 
length,  15  feet  into  a  coarse  ground,  imbedded  in  a  strong  clay,  in  17  seconds,  with 
20  blows  of  ram,  making  70  strokes  per  minute. 

Shaw's  Gunpowder  Pile-driver  is  operated  by  cartridges  of  powder  on  head 
of  pile,  which  are  ignited  by  fall  of  the  ram.  30  to  40  blows  per  minute 
have  been  made  under  a  fall  of  5  and  10  feet. 

Sheet   filing. 
Bevelling 120°    |    Shoeing 25° 

To    Compute    Coefficient  of  Resistance   of  the   Eartl*. 

— -  =  C.  R  representing  resistance  of  the  earth,  h  height  of  fall  of  ram,  and  d  ftnal 
depression,  both  in  feet. 

Ringing   Kngine 
Requires  i  man  to  each  40  Ibs.  of  ram,  which  varies  from  450  to  900  Ibs. 

To   Color  Brass    (Copper  and.    Zinc)   Blu.e. 

Mix  in  a  close  vessel  100  grains  =  6. 5  oz.  Troy,  of  Carbonate  of  Copper  and  750 
grains  — 4.06  Ibs.  Troy,  of  Ammonia;  shake  until  solution  is  effected  and  then  add 
distilled  water;  shake,  and  the  solution  is  ready  for  use. 

Keep  it  cool  and  effectively  stopped.     If  deteriorated,  add  a  little  Ammonia. 

Articles  to  be  colored,  to  be  perfectly  clean,  suspended  in  motion  in  the  solution; 
remove  therefrom  in  from  2  to  3  minutes,  wash  in  pure  water,  and  dry  in  sawdust 
or  like  effective  material. 

Expose  during  the  operation  as  little  to  the  air  as  practicable. 
Other  alloys,  as  copper  and  tin  and  argentine,  are  not  available. 

(Chemical  Journal.) 


STEEL   SPEINGS.  973 

STEEL    SPRINGS.     (Additional  to  page  779.) 
To    Compute    Safe    Elements    of  Springs.* 


D  representing  dejlection  and  t  thickness  of  plates,  both  in  i&h*  of  an  inch;  I  length 
of  span  or  bearings  when  weighted,  and  b  breadth  of  plates  of  springs,  both  in  ins.; 
n  number  of  plates,  and  L  load  or  stress  in  1000  Ibs. 

NOTE. — The  plates  are  assumed  to  be  similar  and  regularly  formed. 

ILLUSTRATION.— Assume  a  spring  of  the  following  elements  : 

I  =  20  and  6  =  3  in*.,  t  =  4  i6*A«,  n  =  5,  and  L  2400  Ibs. 

•  =  6-^  =  6.66  ,6*:          3/6.66x3xT3xl_,/64oo__, 


3/ 
V  6. 


. 8  X  2o3     _  q  76400 _  .8X2o3       _6400_ 

^     *  '          A    f.f.    XX    -,    \/    A  ?  ,r,8^  ^' 


_ 
6.66X3X5      V   ioo  '     6.66X3X43      1280 

•  8  X  zo3          6400  3  X  42  X  5      240 

-—  —  -  5  —  —  =  -  =  3  -j-  ins.  •  -  -  -  -  -  =  —  —  =  2.  4  1000  Ibs. 

6.66X43X5      2133  5X20         100 

NOTE.  —  When  back  or  short  plates  are  added,  they  are  to  be  added  to  the  number 
of  plates  if  of  the  ruling  breadth  and  thickness. 

When  extra  thick  back  or  short  plates  are  adde.d,  they  are  to  be  represented  by 
plates  of  ruling  thickness  having  an  equivalent  resistance,  prior  to  computation  by 
formulas  for  D  and  L,  and  are  thus  ascertained:  multiply  number  of  additional 
plates  by  cube  of  their  thickness,  and  divide  product  by  cube  of  ruling  thickness. 

ILLUSTRATION.—  Assume  as  preceding,  thickness  of  plates  =  4  16^  number  of 
them  5,  and  3  extra  plates  of  5  i6ths  to  be  added. 

Then,  L^_  =  21*  =  5.86  =  no.  of  plates,  and  5  +  5.86  =  10.86,  the  no.  of  plates 

0/4  i6<A«  in  thickness.       10.86  X  43  =  695,  and  5  X  43  =  320)  fi 

3X53  =  375J    95' 

Hence,  3  plates  of  5  i6tfa  added  to  the  5  of  4  i6tt»  =  10.86  plates  0/4  i6*A«. 
Conversely,  695  -4-  5  3  =  5.  56  plates  of  5  16^*  are  equal  to  the  10.  86  of  4  i&k*. 

Helical   Steel    Springs. 

d^L_  /d3  L  D  /C<4D  CMP 

C  t*  ~~  V      C       nS5*rf-  m  V      L     :  d3    = 

An  addition  of  .125  to  .25  should  be  added  to  the  diameter  or  square  to  compensate  for  a  set  of  the 
jprings. 

Safe  Load.    3  /  -  =  t  jor  round,  and  3  /—  -  =  tfor  square. 

d  representing  diameter  or  distance  between  the  centres  of  the  rod  or  bar  of  the 
spring,  and  D  compression  of  the  spring,  both  in  ins.  ;  L  load  or  stress  applied  in  Ibs.; 
t  diameter  of  rod  or  side  of  square  of  bar  in  16^*  of  an  inch,  and  C  a  coefficient  =.  22 
for  round  rods  and  30  for  square  bars. 

ILLUSTRATION.  —  Assume  as  follows:  d  =  j  ins.  square;  11  =  3363  tbs.;  t  =  i6  six- 
teenths, and  C  =  22. 


I  441  792 


'      4  AsxlJ6jx^=  4  /»  44»  79-  =  l 

V  22  V  22 


3/"X'6*X.8  /,,S3434          <  «  X  ,64  X  .8  =  iI53434  =      6    ^ 

V          3363  V     3363  73  343 


The  load  and  deflection  obtained  for  one  coil  are  each  to  be  multiplied  by  the 
number  of  coils  for  the  respective  total  load  and  deflection  of  the  spring. 
A  square  spring  is  approximately  equal  to  a  round  of  like  area* 

*  EssentiaUy  from  D.  K.  Clark's  Manual. 


974 


MEMORANDA. 


Blast   Draught   in    -A.sh.pit   of*  a   Marine    Boiler. 

S.  8.  "Resolute." 


1 

Of  Blower 
Engine. 

hP 
Of 

Engine. 

C( 
Per  HP 

per  hour. 

al 
Consumed 
per  hour. 

Water 
Evaporated 
per  Ib;  of  Coal. 

Relative 
Efficiency. 

No. 

No. 

Lbs. 

Lbs. 

Lbs. 

Per  Cent. 

Natural  ) 

Draught  )  '  • 

57-5 

3-72 

214 

10.77 

I 

.96 

88.8 

3.26 

290 

8.82 

.186 

2 

100.5 

3.12 

3*4 

8 

•05 

a 

106.1 

3.04 

323 

7.82 

.086 

4.2 

118.8 

2-93 

348 

7.82 

.172 

5 

119.8 

3.12 

374 

7-53 

.179 

6 

127.9 

3-12 

399 

7 

.158 

7-4 

135-7 

3-1 

421 

7-03 

•  283 

When  the  Power  was  Doubled.— The  fuel  consumed  was  as  1.5  to  i,  the  watei 
evaporated  as  .73  to  i,  and  the  saving  of  coal  was  19  per  cent. 

An  average  of  the  above  results  gave  a  saving  of  15.8  per  cent. 

By  trials  in  the  R.  N.,  it  was  ascertained  that  a  blast  draught  increased  the  powei1 
of  the  engines  52.5  per  cent.,  and  the  boilers  65  per  cent,  per  ton  of  their  weight. 

First    Steam-Launch. 

"  SWEETHEART.  "—Was  built  at  the  Navy  Yard,  New  York,  in  1837. 

Length,  35  feet;  beam,  4.25;  depth,  1.83. 

Engine,  vertical  cylinder  beam,  4  ins.  in  diam.  by  12  ins.  stroke  of  piston. 

Water-wheels^  4  feet  by  10  ins.     Boiler,  horizontal  fire  tubular. 

On  her  trial  trip  she  was  saluted  by  steamboats  and  assemblages  of  people  in 
ferryboats  and  on  the  piers.  Designed  by  and  constructed  under  the  direction  of 
the  Author. 

Bearings   without   Lu.'bricants. 

Graphite  or  Plumbago — Is  the  essential  element  in  dry  bearings. 

"Fibre  graphite" — Consisting  of  finely-powdered  plumbago  mixed  with  moist 
wood  fibre,  is  pressed  in  a  mold  of  the  required  form,  then  saturated  with  a  drying 
oil  and  oxidized  in  a  hot  dry  air. 

NOTB. — This  bearing  *  has  been  favorably  reported  on  by  a  committee  of  the  Franklin  Institute. 

" Carboid "— Is  carbon  mixed  with  finely-powdered  steatite;  its  specific  gravity 
=  1.66,  that  of  carbon  being  1.48.  It  can  be  molded,  turned,  bored,  and  shaped  to 
any  form. 

NOTE.— The  coefficient  of  friction  with  dry  bearings  is  lower  than  that  of  many  oil 
bearings  in  good  condition. 

Tests   for    Water. 

(Additional  to  page  852.) 
To    Ascertain. 

If  Hard  or  Soft. — Into  a  clean  glass  tube  put  a  solution  ot  soap,  add  a  small  vol- 
ume of  the  water,  when,  if  hard,  the  mixture  will  become  milky. 

If  Alkaline.—  It  will  turn  red  litmus-paper  blue. 

If  Acid. — It  will  turn  blue  litmus-paper  red. 

If  Carbonic  Acid  is  present.  —Equal  volumes  of  it  and  lime-water  will  become 
milky.  Add  a  little  hydrochloric  acid  to  the  mixture  and  it  will  become  clear. 

If  Sulphate  of  Lime  (Gypsum)  is  present. — Add  to  it  a  little  chloride  of  barium: 
if  a  white  precipitate  is  formed,  which  will  not  dissolve  when  a  small  volume  or 
nitric  acid  is  added,  it  contains  the  sulphate. 

Anchoring    Bolts    in    Stone. 

A  test  of  the  relative  value  of  Lead,  Sulphur,  and  Portland  Cement,  for  the  re- 
tention of  iron  bolts  in  limestone  rock,  give  similar  results. 

*  Philip  H.  Holmes  patent. 


GATE    VALVES. — HYDRANTS. 
Crate  Valves.    Eddy  Valve  Co.,  Waterford,  N.  Y. 


975 


Gate  "Valves,  Double  Seated,  have  faces  set  at  a 
slight  angle  to  line  of  stem,  and  as  the  gates,  in  consequence 
of  their  angular  faces,  cannot  fill  the  space  between  the 
valve-seats  until  they  are  fully  down  to  their  position,  the 
adhesion  of  them  ito  the  valves  in  their  progress  down, 
from  the  interposition  of  sediment  or  other  obstructions', 
is  not  only  not  arrested,  but  they  are  impracticable  of 
arrest  before  being  fully  seated,  and  left  partially  open, 
under  the  impression  on  the  part  of  the  operator  that  they 
are  in  position  and  the  flow  of  the  fluid  arrested. 

The  valves  are  attached  to  the  stem  by  an  articulated  ball 
joint,  hence  they  are  rendered  free  to  revolve,  and  their 
faces  varying  with  that  of  their  valve  -  seats,  cutting  or 
grooving  is  measurably  avoided. 

The  valves  are  two  independent  pieces,  whereby  a  single 
defect  involves  the  repair  or  removal  of  but  one  of  them. 

The  stem  rotates  in  a  screw-collar  connected  to  the  ball 
joint,  and  hence  it  is  not  elongated  outside  its  glands  upon 
the  raising  of  the  valves. 


3OO  3?ou.nds'  Test  Pressure. 

Brass  Va 
Ends. 
«p< 

ves.     Screwed  and  Flanged 
Stationary  Stem   and  Quick- 
ening Rack  and  Pinion. 

Hub-end  Valves. 
All  Iro»  for  Gas. 
Iron  Body  Brass 
Mounted  for  Water. 

Iron  Body  with  Brass  or  Bronze  mounted 
Valves.    Screwed  and  Flanged  Ends. 

Diam- 
eter. 

End  to 
End  of 
Screw 

Sockets. 

Face  to 
Face  of 
Flange. 

Diameter 
of  Stan- 
dard 
Flange. 

Diam- 
eter. 

End  to 
End  of 
Hub 
Ends. 

Diam- 
eter. 

End  to 
End  of 
Screw 
Sockets. 

Face  to 
Face  of 
Flange. 

Diameter 
of  Stan-  ' 
dard 
Flange. 

Ins. 

las. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

•5 

2-375 

2-5 

3 

2 

7-5 

2 

5-5 

5.75 

6 

•75 

2-75 

3 

4-5 

3 

10.5 

2-5 

6 

6.5 

7 

i 

2.875 

3-25 

4-75 

4 

12 

3 

7-25 

7-75 

I'3 

1-25 

3-375 

4 

5 

5 

12.5 

3-5 

I'5 

7-75 

8.5 

*-5 

3-5 

4 

5-25 

6 

'3-5 

4 

8 

7.75 

9 

2 

4-375 

4.625 

5-75 

8 

I4.25 

4-5 

8.5 

8.5 

9-25 

2-5 

5-125 

5-5 

6 

10 

15-5 

5 

9 

9.25 

10 

3 

5-375 

5-875 

6-5 

12 

16 

6 

0.25 

9-75 

ii 

3-5 

6 

6.25 

7-125 

14 

17-25 

7 

i 

ii 

12.5 

4 

6.625 

7 

8 

15 

17-5 

8 

i 

ii 

'3-5 

4-5 

7 

7-2S 

9-25 

16 

17-75 

10 

3-25 

12 

16 

5 

7-25 

7-5 

9-5 

18 

18 

12 

4-75 

*3-S 

'9 

6 

7  75 

7-75 

ii 

20 

19-5 

M 



i5 

21 

7 

8.25 

8.125 

12.5 

22 

21 

15 



»5 

22.25 

8 

9 

8-5 

13-5 

24 

22 

16 



16.25 

23-5 

•  

30 

25 

18 



16.25 

25 









36 

30 

20 



17.75 

27-5 

•  











22 



'9 

20-5 













24 



20 

3^-5 













30 



22.5 

38 



•  









36 



26.5 

44-5 

Kddy  Hydrants.    Eddy  Valve  Co.,  Waterford,  N.  T. 


I 

Pipe  Con- 
nection. 

Mameters  o 
Stand 
Pipe. 

r 

Seat 
Ring. 

g 

2-5 
Ins. 

Noz 

£ 

ties. 

2-5 

Ins. 

fi£ 

Steam- 
er. 

Steam 

25 
Ins. 

er  and 

inf. 

Ine. 

Ins. 

Ins. 

No. 

No. 

No. 

No. 

No. 

No. 

No. 

No. 

3or4 

4-5 

3 

i 

— 

— 

— 

— 

— 

— 

— 

3°r4 

5-5 

4 

V.f» 

3 

— 

— 

I 

6 

5-5 

4 

<l>.  ik  ~A 

3 

— 

— 

I 

4  or  6 

6 

4-5 

— 

3 

— 

— 

I 

4  or  6 

6.625 

5 

— 

3 

4 

— 

i 

6 

7.625 

6 

— 

3 

4 

— 

I 

8 

7.625 

6 

— 

3 

4 

— 

i 

8  or  10 

9-75 

8 

— 

— 

6 

— 

— 

— 

Eddy  Valves  and  Hydrants  are  adopted  by  Jf ire  Insurance  Companies. 


MEMORANDA. 

.A.ruminnm. 

(Continued  from  page  938.) 

The  available  properties  of  Aluminum  are  its  relative  lightness,  freedom  from 
tarnish,  not  being  affected  by  sulphurous  fumes  and  being  slowly  oxidized  by  a 
moist  atmosphere,  its  extreme  malleability,  its  facility  of  being  cast,  its  high  speci- 
fic heat  and  electrical  and  heat  conductivity,  and  its  extreme  ductility. 

Its  transverse  and  torsional  resistances  are  very  low,  its  maximum  shearing  re 
sistance  for  castings  12000  Ibs.,  and  forgings  16000  Ibs.  per  square  inch. 

It  is  adapted  for  structures  under  water,  can  be  welded  by  electricity  and  an- 
nealed if  heated  and  gradually  cooled  just  below  a  red  heat.  The  tensile  strength 
of  its  wire  is  greater  than  that  of  its  rolled  metal. 

Its  properties  are  materially  changed  and  impaired  by  alloying  it  with  small  per- 
centages of  other  metals,  and  its  tensile  resistance,  relative  to  its  weight,  is  in 
plates  as  strong  as  steel  at  80000  Ibs.  per  square  incht  and  in  cold  drawn  wire  as 
strong  as  it  is  at  180000  Ibs.  (Alfred  E.  Hunt.) 

^Magnesium. 

Specific  gravity  1.74,  is  .33  lighter  than  Aluminium;  is  harder,  tougher,  and 
denser;  less  affected  by  alkalies,  and  takes  a  higher  polish. 

Staff. 

Staff  is  composed  of  Plaster  of  Paris,  water,  and  hemp  fibre,  the  latter  used  to 
bind  the  mass. 

For  ornamental  pieces,  matrices  of  hardened  gelatine  are  used. 
It  resists  the  weather  and  even  frost  after  being  saturated. 

Boiler    Setting. 

The  fire-brick  should  be  laid  with  very  thin  joints,  and  set  in  Kaolin*  or  pre- 
pared fire-clay,  so  thin  that  it  is  necessary  to  lay  it  with  a  spoon  instead  of  a 
trowel. 

Every  fifth  course  should  be  a  header  course.     ("  The  Locomotive.11) 

GUue.— Its  tenacity  varies  from  500  to  700  Ibs.  per  square  inch. 

ITriotion   of  Engines    and.    Grearing. 

(In  addition  to  pages  469-478,  etc.) 
Deduced  from  Experiments  of  Alfred  Saxton,  Manchester  Assn.  of  Engineers. 

Spur  Gearing. 25.9  per  cent.  I  Belt  Driving 28.6  per  cent. 

Rope  Driving 29.6       "         |  Direct  Acting 23.8       " 

Engines 6  and  10.3  per  cent. 

Spur  gearing  gave  the  best  result  when  not  complicated  with  rope  driving. 
Rope  driving  gave  best  results  at  high  speeds. 

Belt  driving  for  developing  large  power  is  only  equal  to  an  average  rope-driving 
engine. 

Relative    Value    of    various    "Woods,   their    Crushing 
Strength    and    Stiffness    feeing    Combined. 

Mahogany 3.7    Yellow  pine. ..  3 

Spruce 3.6    Sycamore 2.6 

Walnut 3.4    Cedar i 

Comparative  "Value   of  Long    Solid  Columns    of  various 

Materials.    (Hodgkinson.) 
Cast  Iron 1000  |  Cast  Steel. . . .  2518  |  Oak 108.8  |  Pine 78.5 

Hence,  To  compute  destructive  weight  of  an  Oak  or  Pine  column,  take  weight  for 
one  of  Cast  iron  of  like  dimensions,  and  if  for  Oak  divide  by  9,  and  for  Pine  by  12.7. 

*  A  variety  of  clay,  one  of  the  two  ingredients  in  Oriental  porcelain ;  the  other  ia  termed  in  China 
fttunts. 


Teak 9.4  I  Elm 5 

English  oak  ...  5.8    Beech 4.4 

Ash 5.  i  I  Quebec  oak. ...  4.  i 


SPIRALLY    RIVETED    IRON    OR   STEEL    PIPE. 


977 


Spirally    Riveted    Iron,    or    Steel    3?ipe. 

Abendroth  &  Root  Mf'g  Co.,  Neioburgh,  N.  Y. 

Spirally    Riveted    !3VEetal    I?ipe. 

Compared  with  Wrought  or  Cast  iron  Pipe,  has  the  advantage  of  low 
original  cost  and  expense  of  transportation,  maintaining  a  nearly  equal 
bursting  pressure  with  that  made  of  heavier  material.     It  is  made  of  Sheet 
Iron  or  Sheet  Steel,  varying  in  thickness  from  No.  20  to  No.  12  B.W.G.,  ac- 
cording to  diameter  and  pressure.     The  rivets  in 
the  seam  are  set  by  compression,  while  the  laps 
are  thoroughly  coated  with  hydraulic  cement  to 
make  it  water  tight. 

Connections. — When  a  moderate  pressure 
is  maintained,  these  pipes,  their  ends  being 
crimped,  are  usually  connected  by  a  cement  joint, 
as  shown  in  the  annexed  cut. 

When  the  pressure  is  excessive,  a  bolted,  joint  is  resorted  to,  as  also 
shown,  and  which  is  in  effect  a  stuffing  box  or  sleeve  joint,  dispensing 
with  lead  calking,  and  admitting  of  a  slight  flex- 
ure of  the  pipe.  £  •    j 

For  service  connections  the  collar  may  be 
tapped.  When  lead  calking  is  required,  the 
inner  ends  of  the  pipe  are  reinforced  by  an  iron 
collar. 

Bursting    Fressnre. 


ll 

Jl-a 

ll 

ll 

*•_• 

3  * 

I! 

Per  Sq.  Inch. 

§g 

Per  Sq.  Inch. 

1! 

Per  8q.  Inch. 

Per  Sq.  Inch. 

II 

Per  Sq.  Incb. 

55 

£5 

5.S 

5~ 

5*9 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lb». 

Ins. 

Lbs. 

Ins. 

Lbs. 

3 

900  to  1300 

6 

350  to  800 

10 

275  to  650 

16 

190  to  400 

22 

125  to  300 

4 

700  "  looo 

8 

350  "  825 

12 

225    "  550 

18 

150  "  375 

24 

no  "  275 

5 

550  "    800 

9 

300  "  750 

14 

200     "  470 

20 

140  "  325 

— 

In  order  to  enable  an  estimate  of  the  relative  cost  of  these  pipes,  com- 
pared with  cast  and  ordinary  wrought-iron  pipes,  the  weights  of  each  are 
submitted. 

'Weights. 


Heavy 
Spiral. 
•U 

ID*.     L+- 
a  a        5  a 

Heavy 
Spiral. 
«i 

l*n 

^ 

Heavy 
Spiral. 

«.• 

fe 

^ 

He 

Spi 

a 

«< 

ih 

^,4- 

i  o 

i 

§ 
I 

P 

°~ 

.§ 

22 

<sE 

5 

f 

S* 

o£ 

§ 
0 

1 

22 

<-^ 

Ins. 

Ltm. 

Lbs. 

LbB. 

Ins. 

Lbs 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Lb«. 

Lbs. 

Lb«, 

3 

2 

7-5 

13 

8 

8 

28 

40 

14 

20 

58 

94 

2O 

28 

1  80 

i 

2-5 

5 

iQ-75 
18.75 

20 

30 

10 
12 

10 

'3 

40 
49 

55 
70 

16 
18 

21 

"77" 

109 
160 

22 
24 

31 
33 

—  . 

200 
250 

978 


GRAPHITE    AND    DRAWING    PENCILS. 


GS-raph.ite    as    a    Lubricant. 

Joseph  Dixon  Crucible  Co.,  Jersey  City,  N.  J. 

Results    of     Comparative    Tests    of  its    Operation 
"VS^itli    best    Sperm.    Oil    and.    ^Perfected    Graphite. 


Lubricant. 

Weight 
of 
Lubricant. 

Pressure  on 
Bearing 
per  Sq.  Inch. 

Revolutions 
per 
Minute. 

Time  of 
Duration  of 
Test. 

Frictional 
Surface  in 
Sq.Feet. 

Best  Sperm  Oil  
Perfected  Graphite*. 

Grains. 
5-l6 
i-75 

Lbs. 

4S 

48 

No. 

2OOO 
2000 

Minutes. 
II 
30 

No. 
7198 
I9635 

*  Mixed  with  water  enough  to  distribute  it  over  the  bearing. 

NOTE. — Hence  it  appears  that  Graphite,  under  like  conditions,  pressure,  and  velocity  of  operation, 
was  2.72  times  more  effective  than  the  best  Sperm  Oil. 

With     best     Sperm    Oil,    Lubricating    Grease,    and     lilre 


(jt-rease,    cents 
ite. 

Lubricant. 

mining    1 

Weight 
of 
Lubricant. 

Pressure  on 
Bearing 
per  Sq.  Inch. 

Revolutions 
Minute. 

erlected 

Time  of 
Duration  of 
Test. 

Graph- 

Frictional 
Surface  in 
Sq.  Feet, 

Best  Sperm  Oil  

Grains. 

5-i6 

Lbs. 
60 
60 

60 

No. 

2000 
2OOO 

2OOO 

Minutes. 
Si 
51 

293 

No. 
3336o 
3336o 

194941 

Lubricating  Grease.  . 
Same  mixed  with  \ 
Graphite... 

NOTE.— The  grease  without  Graphite  gave  only  like  results  with  the  Sperm  Oil ;  but  when  the  per 
cent,  of  Graphite  was  added,  the  time  of  operation  of  the  bearings  was  5.82  times  longer  without  cut- 
ting and  at  the  same  velocity.  Prof.  R.  H.  Thurtton. 

To  introduce  in  Steam  Cylinders.  The  method  preferred  by  experienced  engi- 
neers is  to  mix  the  graphite  with  oil  for  the  valves  and  inject  by  a  small  hand  oil 
pump  attached  to  steam  pipe.  The  graphite  must  not  be  too  coarsely  ground. 

Pure  Graphite,  mixed  with  oil,  applied  to  a  scratched  or  cut  surface,  as  cylinder 
or  piston  rod  of  an  engine,  valve  seat  or  journal,  will  arrest  the  cutting  of  them. 

The  selection  of  the  perfectly  pure  involves  the  experience  of  an  Expert,  or  conn 
dence  in  the  manufacturer. 


Drawing    IPenoils. 

Engineers,    Architects,    Artists,    eto. 
Joseph  Dixon  Crucible  Co.,  Jersey   City,  JV.  J. 

Hexagonal    in    Section    and    ITu.rnish.ed    in    Ten    Grades 
of   Hardness. 


Trade 

Nos. 


Grade  Stamps. 


Character. 


Similar  grade  to  the  European 
stamp  of 


V  V  S Very,  very  soft 

V  S Very  soft 

S Soft 

S  M Soft  medium 

MB Medium  black 

M Medium \ 

M  H Medium  hard 

H Hard 

V  H Very  hard 

V  V  H Very,  very  hard 

-The  first  5  numbers,  210  to  214,  are  especially  designed  for  Artists. 
M  H  is  designed  for  sketching,  V  H  for  ordinary  drawings,  and  V  V  H  for  very  fine. 


(BBB.) 

(BB.) 

(Band  No.  i.) 

(H  B  and  No.  2.) 

(F.) 

(Hand  No.  3.) 

(HH.) 

(HH  Hand  No.  4.) 

(H  H  H  H  and  No.  5.) 

(H  H  H  H  H  H.) 


MEMORANDA. 


979 


.Absorption   of    Gfeological    Strata. 

Water  in  100  Parts  or  Per  Cent,  of  Volume. 


Material. 

From  Where. 

Per  Cent. 

Authority. 

Gabbro  

Duluth,  Minn  

.29 

Geo  P  Merrill 

Granite  Hornblende 

St  Cloud  Minn 

Limestone  

Quincy  111       

•  ^"? 

u 

Rockford,  111  
Bedford   Ind.   . 

2.1 

D.  W.  Mead. 

Red  Wing,  Minn  

2  e 

Geo  P  Merrill 

Sandstone  

Fond  du  Lac  Wis 

A      Si 

Fort  Snelling  Minn 

62 

« 

u 

Berea  0 

6  6 

D  W  Mead 

Dry  Clay  

Jordan,  Minn  

12.5 

Geo.  P.  Merrill. 
R  J  Hunter 

36.5 

Red  Sandstone    .   .  . 

Gloucestershire  Eng 

u  6 

E  Wetherell 

Oolite  Limestone  

Cheltenham  Eng  ... 

<( 

"     Sandstone... 

21.08 

u 

Supporting   IPower   of   Sand,   and    Clay. 

Per  Square  Foot. 

Sand  with  Loam,  4  to  5  tons.  Clay,  2  tons.  Friction  resistance  of  sides 
should  be  neglected. 

NOTE.  —The  walls  of  the  Capitol  at  Albany,  N.  Y. ,  at  some  points  settled  at  a 
weight  of  2  tons. 

To   Compute   the   13?   of*  Ropes. 

RULE. — Multiply  sectional  area  of  it  in  square  inches  by  its  velocity  in 
feet  per  minute  and  divide  product  by  330. 
EXAMPLE.— Area  of  rope,  4.2  sq.  ins.,  and  velocity  72  feet  per  minute. 
4.2  X  72  -r- 330  =  . 916  IP. 

Roofing    Slates. 

Strength    and.    Qualities. 

24  by  12  ins.  and  .1875  to  .25  ins.  in  thickness. 


Quarry. 

Modulus 
of 
Rupture. 

Specific 
Gravity. 

Porosity. 

Corrodibil- 
ity. 

Density. 

Deflection. 

Albion  

2^8 

208 

80 

.27 

Old  Bangor  

o  810 

2780 

*45 

.169 

128 

.312 

Peach  Bottom... 

y  oio 
ii  200 

280,1 

.224. 

.086 

QO 

•  2(H 

Modulus  of  Rupture,  in  Ibs. per  sq.  inch,  —  M.     W  destructive  load  in  Ibs., 

I  distance  between  the  supports,  and  b  and  d  the  breadth  and  depth,  all  in  inches. 
rorosity,per  cent,  of  water  absorbed  in  24  hours.  Corrodibility,per  cent,  of  weight 
lost  in  24  hours  in  an  acid  solution.  Density,  grains  abraded  by  50  revolutions  of  a 
small  grindstone  ;  and  Deflection,  on  supports  22  inches  apart,  in  inches. 

Slates  are  finished  in  varied  dimensions,  ranging  from  6  X  12  to  14  x  24  ins.  In 
Roofing,  with  a  slate  of  24  ins.,  10.5  ins.  are  exposed,  10. 5  covered  by  the  slate  above 
it,  and  the  balance  of  3  is  covered  by  the  two  above  it.  The  slates  required  to  cover 
an  area  of  ioX  10  feet  in  the  above  manner  is  termed  a  Square,  which  is  the 
Unit.  For  slates  12  X  24  ins.,  114  are  required  to  make  a  Square,  and  for  8  X  16 
ins.,  377  are  required.  (If.  JHerriman,  M.  Am.  Soc.  C.  E.) 


KNOTS,  HITCHES,  ETC. 
In.    Mechanical    and.    Engineering    Operation*. 


Slip-Knot.          Square  or  Reef  Knot.         Flemish  Loop.  Sheet  Bend,  or 

Weaver's  Knot. 


Bowline.  Marline-spike  Carrick  Bend.  Stevedore's 

Hitch.  Knot. 


Halt  Timber       Clove         Rolling  Timber  Hitch  Black- 

Hitch.  Hitch.        Hitch.         Hitch.  and  Round  Turn.  wall 

Hitch. 


Cat's-paw. 


Short  Splice. 


Racking,  or       Round  Turn  Round 

Trapping,      and  Half  Hitch.         Seizing. 


Cask  Sling. 


Selvagee 
Strap, 


SYMBOLS. 


SYMBOLS 

For    Elements    and.  Formulae,  proposed,  toy  tlie   Author. 

For  the  purpose  of  inducing  a  uniformity  in  their  expression  (1891). 


Angle  _N^ 

Sum   -s....                   Sm 

Of  Incidence  z 

Tangent                       Tan 

Kilojoule,  -s  Kj 

Cotangent  Cotan. 

ea'   s  a  ' 

Watt,  -s  wt 

Thrust  Tt 

Grate                            Gt 

Kilowatt,  -s  kwt 

Time  -s                  T    t    t' 

Heating  surface.  ...    Hs 
Section  .  .  Sen,  H,  L  or  1" 

Millihenrv  -s     .       mh 

Second,  -s.Sec.  sec.  or  " 
Minute,  -s  Min.  min.  or  ' 
Degree,  -s  Deg.  or  ° 
Hour  -s               Ho    ho 

Milliampere,  -s  ma 

Megohm  (Greek  c.oinega)  « 
Microvolt,  -s  Mv 

Square  foot,  feet.  .  .  Dft- 

Ohm,  -8.  .(Greek  omega)  o> 

Volt,  -s  Vo 

Day,  -s  Ds 

Month  -s                   Mo 

Atmosphere,  -s  At 
Barometric  Be 
Breadth  -s                 b    b' 

Evaporation,  -ive....  Evp 
Foot  pound,  -s,  tons  Fp.Ft 
Force                                F 

Year,  -s  Ys 

Triangle,  -s  A  A' 

Centrifugal  force            Cf 

Triple  Tpl 

Centre  of  gravity  Cg 

Friction  Fn 

Unit  -s     Heat              Hu 

Circumference,  -s..C.c.c' 
Coefficient  or  Factor.  .  Co. 
Compound    •  .       ....  Cpd 

Gravity  ,..*g 
Height,  -s  H  .  h  .  h' 
Horse-power  B? 

Calorific  or  French  .  .  Co 
Vacuum  Vm 

Velocity                 V    v    v' 

Cube                   Cub  or  |5£] 

Effective  EH* 

Versed  sine                 v-sin 

Cylindrical                   Cyl 

Indicated    IBP 

Vertical                            Vt 

Nominal  NB? 

Volume  -s            Vol  vol 

Depth,  -s  dp  .  dp' 
Departure   Dpt 

Inclination  In 

Joule's  Equivalent  jE 
Ijjiti  tilde    .......          Lat 

Chaldron,  -s  Ch 

Chord,  -s  Co 

Diameter,  -s.  ...  D  .  d  .  d,' 
Distances.     Inch,  -es.  ins. 
Feet                             ft 

Bushel,  -s  Bl 

Length  -s                L    1    1' 

Cube  foot  feet            Cf 

Barrel,  -s  bbl 

Yard    s                     Yds 

Hyperbolic  .  .  .  Hyp.  log. 
Mercurial  gauge  Mg 

Gallon,  -s  gl 

Chain,  -s  Chn 
Rood,  -s  Rd 

Microliter  (Greek  lambda)  X 

Milliliter,  -s  ml 

Knot,  -s  K 
Mile,  -s  .Ms 

Millimeter,  -s  mm 
Centimeter,  -s  cm 

Meridian  M 

Modulus  of  Elasticity  .  ME 
Moment    s                     Mt 

Deciliter,  -s  dl 
Liter,  -s  1 

Number,  -s  No 
Ordinate,  -s  O  .  o  .  o' 
Perpendicular  Pr 
Pitch  -s                 Ph    Ph' 

Dekaliter,  -s  dal 
Hektoliter  -s        ....  hi 

Kiloliter  or  Stere,  -s.  Kr 
Water-line  Wl 

Meter,  -s  m 
Dekameter  -s             dA 

Hektameter,  a  hk 
Kilometer,  s  km 

Pressure,  -s  P  .  p  .  p' 
Quadruple  Qpl 
Radius,  -ii  R  .  r  .  r' 
Revolution,  -s..  Rev.  rev. 
Secant            Sec 

Weight,  -s  W  .  w  .  w' 

Pound,  -s  Ib.  .  Ibs. 
Ton,  -s  (2240)  Tons 
"        (2000)  ...  —  Tons 
Milligram  -s              mg 

Centare  orsq.  meter,  -s  Ce 
A  re  -  s                           a 

Hectare  -s                  Ha 

Cosecant  Cosec 

Draught  of  water  Dw 

Sine  Sin. 

Centigram  -s               eg 

Cosine  Cosin. 

Decigram,  -s  dg 

Slip  Sp 

Gram,  -s  g 

Equivalent,  U.  S.  or  ) 
French  )    Eq< 

Solid  Sd 

Dekagram,  -s  dgra 
Hektogram,  -s  hgm 
Kilogram  or  Kilo,  s.  Kg 
Myriagram  -s    y 

Specific  gravity  Sg 

Span                                Sn 

Electric.    Ampere,  -s.  Am 
Farad,  -S.  (Greek  cap.  phi)  4> 
Microfarad,  -s  (Greek  phi)  <j> 

Stability     St 

Steam      ....            .    Stm 

Quintal  -s              .  .  .    q 

Stroke                          S    s 

Miller  or  Tonneau,  -8  Mr 

i 

TT  =  3  1416 

982 


MEMORANDA. 


Relative  Efficiency  of  a  Non-condensing  Steam-Engine 
and   a   Ki-Sulphide   of  Carbon   (C  82)   Engine. 

With  like  Engine,  Boiler,  and  Fuel,  as  developed  by  competitive  tests  of  both  at 
Riverdale,  near  Chicago,  1892-94. 

Cylinder,  16  x  42  ins.  Jacketed  and  with  automatic  Cut-off.  Boiler,  horizontal 
cylindrical  fire  tubular.  Grates,  21.6  sq.  feet,  and  Heating  surface,  1028  sq.  feet. 
Fuel,  anthracite;  Combustion,  natural  draught. 

Steam.—  Coal  consumed  per  i  IP  per  hour,  4.299  Ibs. 

C  S3.—  "  "  2.49  "  =42.08  per  cent.,  or  as  i  to 

1.72  -}-,  or  942.6  Ibs.  coal  in  each  ton. 

Mortar    for    Masonry    loelcrw    Freezing-point. 

Anhydrous  Carbonate  of  Soda,  u Sodium  carbonate"  (Na2CO3),  2.3  Ibs.  per 
gallon,  dissolved  in  water,  maintained  at  86°,  mixed  with  equal  volume  of  water. 
Mix  the  mortar  with  25  per  cent,  more  of  this  solution  than  if  pure  water  was  used. 
Hands  of  operatives  should  be  protected,  as  with  India-rubber  gloves.  Extra  cost 
35  cents  per  cube  yard  of  masonry.  Setting  of  mortar  accelerated.  It  will  set  at 
5°  below  freezing  point  as  readily  as  at  10°  above. 

(Caen  and  Vive- Saint  Lo  R'y.)    Mons.  Rabat. 

Freestone. 

Result  of  a  Series  of  Tests  of  Connecticut  Brownstone  to  Resist  Crushing. 
Per  Sq.  Inch  of  Cross  Section. 

Portland  stone 6222  to  10928  Ibs Colt's  Fire  arms  Mfg.  Co. 

New  England  Brownstone  Co. .  7843  "  13  297    '*    U.  S.  Ordnance  Dept. 

Portland  Shaler  and  Hall 9330  "  13  980   "    u  " 

Highest    Rail-way    in    Europe. 

Brienzer  Rothhornbahn.  —  Alps,  7886  feet  at  summit  level.  Rack  and  Pinion. 
Greatest  grade,  i  in  4. 

Elevation    in    Feet   of   Localities    in    the    Upper    Missis- 
sippi   and    \Vest   of   L.ake    Michigan. 


(In  addition  to  page  582.) 


Davenport,  la 615 

Dubuque,     " 665 

Ft.  Madison,  la 600 

Independence,  la 850 

Keokuk,  " 618 


Monticello, 


880 


Ashland,  Bay  City,Wis.  610 
La  Crosse,  "  744 
Milwaukee,  "  697 
Ft.  Ridgely,  Minn  1230 
Ft.  Snelling,  "  820 
Minneapolis,  "  ....  856 
St.  Paul,  "  ....  831 

Age    of   Trees. 
Lime  i  too  years 

Ft.  Ripley,  Minn  
Chicago,  111  

.1130 

Galesburg  111 

Ottawa,        "    .... 

Peoria,         "  

Rockford,    "  

.  800 

St.  Louis  Mo 

Oriental  Plane.  1000  yeara 
Spruce               1200     " 

Oak  1500  " 

Walnut                ooo 

" 

Olive  800  u 

Yew                   3200 

Orange  670  " 

Appleton,  Wis 653 

Cedar 2000  years. 

Cyprus 800     " 

Elm 300      " 

Ivy 335     " 

Larch 576      ' 

Monolith. 

(In  addition  to  page  179.) 

Wisconsin.—  At  World's  Fair,  10  feet  square  at  base,  115  in  height,  and  4  at  apex. 
Angle    of  Repose    of   Earth. 


Clay,  dry 29° 

damp,  well 

drained  45° 
wet !6° 


Earth,  vegetable  dry..  29° 


moist. 


Shingle. . , 39° 


Gravel,  clean  .........  48° 

with  sand.  ...  26° 

Sand,  wet  ............  26° 

34° 


R.  E.  Aide-Memorie. 


The  co-efficient  «f  friction  =  tan.  of  degrees,  as  co-efficient  of  shingle  ==  tan.  of 
39°  --Si- 


MEMORANDA.  983 


To    Compute    the    H*    of  a    "W>ought-iron    Shaft. 

RULE. — Multiply  cube  of  diameter  of  shaft  in  its  journal  in  inches,  by 
number  of  its  revolutions  per  minute  and  divide  product  by  80. 

EXAMPLE. — Diameter  of  journal  17  ins.,  and  revolutions  of  engine  20.  What  is 
its  IP? 

17  3  x  20  =  98  260,  which  -4-  80  =  1228. 25  H?. 

To  Determine  the  South  toy  the  Hour-hand  of  a  Watch, 
between   the  Hours   of  8  A.3VL.  and  4  P.M. 

When  the  Sun  is  Visible. 

OPERATION.— Point  the  hour  hand  to  the  sun,  and  half  the  distance  between  thai 
point  and  the  figure  12  is  the  South. 

XOTK. — The  greatest  error  is  in  latitude  38°,  and  is  about  15*  too  far  Ea«t  at  8  A.M.  and  15*  too  far 
West  at  4  P.M.  Hence  allowances  are  to  be  correspondingly  made. 

Q-reatest    Depths    and    Heights. 

(In  addition  to  pp.  179-184.) 

Greatest  depth  of  Ocean,  Pacific 28  ooo  feet. 

Deepest  Well  in  North  America,  Wheeling,  Va 4  560 

44  Mine  in  Comstock,  Nev 3000 

Highest  Mountain  in  North  America,  St.  Elias. .  : 18000 

"  Structure,  Washington  Monument 550 

"  Tide,  Bay  of  Fundy '.  .T.  .'KV. . . .  i * 50 

Tests    for    \Vater. 

(In    addition   to  page   851.) 

If  Hard—  When  mixed  with  a  solution  of  soap,  it  will  be  rendered  milky. 
If  Carbonic  Acid  is  present.—  When  mixed  with  lime  water  it  will  be  rendered 

milky. 
If  Sulphate  of  Lime  (Gypsum)  is  present. — Mix  with  a  little  chloride  of  barium, 

and  if  a  white  precipitate  is  formed,  which  will  not  dissolve  when  nitric  acid  is 

added. 

Comparative    Tenacity  of  Cut   and  "Wire    Iron    Nails. 
In  Lengths  from  1.125  t°  6  ins.,  and  Driven  in  Spruce  and  Pine  Timber. 

In  Spruce. — The  tenacity  of  ordinary  cut  nails  exceeded  that  of  wire  47.51  per 
cent.;  of  finishing,  72.22;  and  of  box,  50.88. 

In  Pine.— Box  nails  taper  perpendicular  to  grain  of  wood  135.2  per  cent;  paral- 
lel to  it,  100.23;  a°d  driven  in  end  of  wood,  64.38. 
Average  of  58  series  of  tests,  with  40  sizes  of  nails,  72.74  per  cent 

Maj.  J.  W.  Reilly,  U.  S.  O.  D. 
By  Wm.  H.  Burr,  C.  E. 
Hydro-GJ-eology. 

U.  S.  Census  Report,   Vol.   XVII. 

Upper  Missouri.— The  average  annual  Discharge  of  the  principal  tribu- 
taries of  it,  to  the  precipitation  on  the  basin  =  30. 8  per  cent. 

Upper  Mississippi.— Average  annual  Rainfall  in  it  and  in  the  valley 
west  of  Lake  Michigan,  33.7  inches. 

Average  flow  per  second  per  square  mile  of  drainage  area,  .703  cube  feet. 

Elevation  of  ordinary  low  water  at  its  extreme  source  above  the  Ocean,  1680  feet 

Drainage  area  above  mouth  of  Missouri  river,  169000  square  miles. 

J.  L.  Greenleaf,  C.  E. 

Cost    of  an    Electrical    EP. 

Available  IP,  84  per  cent,  of  Indicated.     Coal  at  $3.75  per  Ton. 
Coal,  64  cents.    Wages,  water,  etc.,  16  cents. 

Alex.  Siemens,  M.  I.  C.  E. 


984  MEMORANDA. 

Capacity    of   Gri.rd.ers   and.    Floor   Beams. 

Loaded  in  their  Centre. — Girders  of  single  span  of  Georgia  or  Yellow  Pine, 
10  ins.  in  breadth,  12  ins.  in  depth,  and  10  feet  in  length  between  its  sup- 
ports at  both  ends. 

The  mean  capacities  of  the  woods  in  ordinary  use  in  the  floors  of  buildings  for 
one  inch  square  and  one  foot  between  supports  are  as  follows: 


Lbs. 


Georgia  or  Yel.  Pine. .  850 

Hemlock 450 

Chestnut 480 


White  Pine. 

White  Oak 650 


Lbs- 


Spruce 550 

Canada  Oak 560 

N.E.Pine 5< 


Ash  ..................  900 

In  the  computation  of  the  strength  of  posts,  girders,  and  floor  beams  in  build- 
ings, it  is  impracticable  of  assured  safety  at  all  times  to  assume  their  strength  at 
their  mean  value  as  determined  by  experiment,  inasmuch  as  allowance  is  to  be 
made  for  defects,  as  knots  and  shakes,  fungus  growth  and  dry  rot.  Hence  a  Factor 
of  safety  or  a  deduction  of  capacity  in  each  case  must  be  resorted  to. 

In  the  case  above  given,  the  strength  of  the  pine  is  assumed  at  850  and  coeffi- 
cient of  capacity  at  4. 


Hence,  10  X  I  =  3o6oo  Ibs.  and  -         =  3060  Ibs. 

4  X  10  10 

If  Uniformly  Loaded,  this  result  would  be  doubled. 

Estimate    of  ."Paint    Required,    for    Open.    Iron    "Work    on 
Bridges,  etc. 

First  coat,  .625  gallon  per  ton;  second  coat,  .375  gallon. 

(71  J.  Swift,  C.  E.} 

To    Compute    T--P    of   a    Stream    of  Water. 

When  Maintained  at  a  Uniform  Height. 

ILLUSTRATION.  —  Assume  section  22  sq.  feet,  velocity  of  flow  5  feet  per  second,  and 
fall  25  feet. 

22X5  =  110  cube  feet  water  in  volume  per  second.  1  10  X  60  x  62.  5  =  41  2  500  Ibs. 
water  per  minute. 

Then,  ^-^  -  —  —  =  weight  oj  water  in  Ibs.  -7-33000  =  312.5  theoretical  IP  per 

minute,  from  which  is  to  be  deducted  loss  of  efficiency  of  instrument  of  applica- 
tion, and  which  are  given.  (See  pp.  561-580.) 

Assume  a  Turbine  wheel  at.  7  of  efficiency.    Then,  312.5  X  .7  =  218.  75  effective  IP. 

If  the  surface  is  maintained  (impounded)  at  a  uniform  height,  the  power  of  it. 
for  the  period  of  working  hours,  will  be  218.75  x  60  X  N.  N  representing  the  num- 
ber of  hours. 

The  result,  then,  for  a  period  of  10  hours  would  be  218.75  X  60  X  10  =  131  250  IP. 

If  a  stream  has  a  supply  equal  to  the  expenditure  of  it,  it  will  overflow  in  the 
intervals  between  working  hours;  but  if  it  is  unequal  to  the  operating  volume  or 
consumption  required,  there  must  be  a  storage  reservoir,  and  the  greater  the 
area  of  it  the  less  the  decrease  of  the  head  or  level  of  it,  when  being  drawn  from. 

To    Compvite    Elements    of  a    Flume. 

The  Volume  of  Water.    Area  and  Height  being  Given. 

/—  V 

-  A  v  zghG=.  volume.     I  2p»)  -*-  2  0  =  height.    V  representing  volume  in  cube 

\3        I 

feet,  A  area  in  sq.feet,  C  coefficient  of  discharge,  h  height,  and  I  length  in  feet. 

Assume  Volume  36.29  cube  feet,  Area  8  sq.  feet,  and  C  .6. 

36  .29 


or,  ~  =  !  = 


MEMORANDA. 


985 


Effect  of  Tapping   of  T^.ong-leaf  Pine. 

Late  tests,  conducted  by  the  U.  S.  Dep't  of  Agriculture,  of  32  trees,  have  conclu- 
sively evidenced  that  the  timber  of  the  Long  leaf  Pine  is  in  no  wise  affected  by  the 
tapping  of  it  for  turpentine.  See  Circular  No.  9,  Forestry  Division. 

Comparative  Strength  of  Tapped  and  Untapped  Long-leaf  Pine. 
In  Pounds  per  Sq.  Inch. 


Condition. 

Specific 
Gravity. 

Tensile. 

Transverse. 

Crushing. 

Detrusive. 

Tapped. 
25  pieces  green     .  .     .... 

Lbs. 

TC  448 

Lbs.* 
176 

Lbs. 
4755 

Lbs. 

dry  

687 

14  7S7 

177 

6627 

648 

115  tests  mean 

?6 

re  081; 

eil8 

6ofi 

Untapped. 
m  tests,  mean  

.71 

1  6  A.2Q 

i^6 

*66i 

•«• 

*  One  inch  square  and  one  foot  in  length,  weight  supported  from  one  end. 

See  Circular  No.  8. 

Tapped  and  untapped  is  known  as  "boxed"  and  "  unboxed." 
The  pores  of  wood  leading  upward,  or  in  the  direction  of  its  growth,  facilitate 

the  flow  or  passage  of  moisture  in  that  direction.     Hence,  timber  set  inclined  or 

vertical,  with  the  abut  end  uppermost  and  exposed  to  moisture,  will  decay  at  the 

top  more  readily  than  if  set  with  the  abut  down. 
The  effect  of  varying  the  set  of  wood  is  frequently  observed  in  the  staves  of  a 

cistern  or  tub,  etc.,  some  of  them  being  saturated  with  moisture  and  others  quite 

drv. 


To    Compute    Weight    of   Flue    and    Tvitmlar    Marine 
Steam    Boiler. 

To  weight  of  the  metal  plates,  as  determined  by  their  area  and  thickness,  add  as 
follows  : 

For  Laps  and  Rivets.— One  Ib.  per  sq.  foot  for  each  .  125  ins.  in  thickness  of  plate. 
"  Bolts,  Stays,  and  Braces.— 20  per  cent,  of  total  weight  of  the  plates  in  Ibs. 
"  Mean  and  Handhole  Plates.—  750  to  1000  Ibs. 


Notes  on.  Portland.  Cernent  and.  Cement  Mortars. 

(In  addition  to  pp.  515,  589,  871,  907,  958.) 

Cements  that  harden  rapidly  produce  a  brittle  texture ;  they  should  increase  in 
strength  with  uniformity.  The  strength  at  termination  of  one  day  should  not  ex- 
ceed 45  per  cent,  that  of  the  seventh  day. 

The  addition  of  Sulphate  of  Lime  and  Gypsum  to  American  Portland  cement  in- 
creases the  strength  of  the  cement;  with  3  per  cent,  of  the  former  it  increases  it 
64  per  cent. ;  from  that  it  diminishes  the  effect;  and  with  5  per  cent,  of  the  latter 
it  increases  it  33  per  cent. 

To  English  Portland  cement  2  per  cent,  of  Sulphate  of  Lime  increased  it  60  per 
cent. 

Dry  Sands.— Standard.— Weight,  92  Ibs.  per  bushel,  and  its  voids  are 
47. 5  per  cent.  Natural  or  Bar. — 103  Ibs.,  and  41.25  per  cent. 

Requirements  and  Specifications.— Tensile  Tests.— An  average  of  5  briquettes  in 
each  case. 

Fineness.  — 97. 5  per  cent,  through  No.  50  sieve,  and  87. 5  through  No.  100  sieve. 

Specific  Gravity.— To  exceed  3. 

Homogeneous.— Discs,  3.5  ins.  in  diameter,  and  .375  thick  at  centre,  tapering  to  a 
sharp  edge  at  the  circumference;  they  should  not  crack  or  warp. 

Samples.— Ten  per  cent,  of  the  quantity  selected  at  random. 

Tensile  Strength.—  Limit  of  results,  10  per  cent. 

For  Exposure  in  Salt-water. — Initial  set  with  fresh-water  not  to  exceed  ten  min- 


utes. 


40 


986 


MEMORANDA. 


Absorptive   Power   of*  Charcoal. 

Of  Fine  Boxwood.     By  Volumes. 


Ammonia 30 

Carbonic  acid 35 

Carbonic  oxide 9. 42 


Cavburetted  hydrogen  5 

Hydrogen 1.75 

Nitrogen. 6.5 


Nitrous  oxide 

Oxygen 

Sulp'd  hydrogen. . 


40 
9-25 
55 


3?op   Safety  and.   Relief  "Valves.    Crane  Co.,  Chicago. 

BRASS.  IRON   BODY.    BRASS  SEAT 


WT     u^     r\±j     j 

J         \        \  2 

t  I  — 1     2. 


•8 

s  "* 

H.  P. 

tio  of  Valve 
to  Grate. 

Centre  of 
alve  to  end 
of  Outlet. 

° 

& 

> 

Ins. 

No. 

Sq.  Ft. 

Ins. 

•75 

3  t°  6 

1.32 



i 

6    10 

— 

1.25 

10   20 

3-68 

— 

*-5 

20     30 

5-3 

— 

2 

30   40 

9.42 

— 

2-5 

4°    75 

14.72 

3-75 

H.  P. 

Ratio  of  Valve 
to  Grate. 

Centre  of 
Valve  to  end 
of  Outlet. 

No. 
40  to  75 
75    loo 
100    125 
125    150 
150    175 

175     200 

250   300 

Sq.  Ft. 
14.72 

21.2 

28.86 

47-7 
84.82 

Ins. 
3-75 
4-375 
4-75 

6-375 
6-375 
7-I25 

Approved  by  U.  S.  Board  of  Supervising  Inspectors  of  Steam  Vessels,  and 
will  be  approved  by  all  Local  Inspectors,  on  a  basis  of  One  square  inch  of  area  to 
three  square  feet  of  grate  surface. 

-A.meri.oan.   "Woods. 


With    the    Order  of  their  Strength. 


Order. 

Ash,  mountain,  Pyrus  Americana..  — 

Fraxinus  pistacicefolia  .......  234 

Oregon,  Fraxinus  Oregana...  210 

red,               "        pubescens.  .  105 

white,           "       Americanum  29 
prickly,  ^fanthoxylum  Ameri- 

canum .........................  — 

Basswood,  Linden,  Tilia  Americana  249 

Beech,  Fagus  ferruginea  ..........  24 

Butternut,  Juglans  cinerea  ........  205 

Button-wood,  Conocarpus  erecta.  ..  76 

Cedar,  white,  Libocedrus  decurrens.  200 

"      red,  canoe,  Thuya  gigantea.  — 

Cherry,  wild  red,  Prunus  Pennsyl- 

vania .........................  — 

Cherry,  wild  black,  Prunus  serotina  119 

Chestnut,  Castanea  vulgaris  .......  — 

Cotton  wood,  Populus  monilifera.  .. 
Cucumber,  mountain,  Magnolia  acu- 

minata  .................  «  ......  208 

Elm,  slippery,  Ulmusfulva  ........  — 

"    white,           "     Americana.  .  114 

Fir,  white,  Abies  grandis  ..........  280 

Gum,  sweet,  Liquidambar  styraci- 

flua  ............................  222 

Hemlock,  Tsuga  Mertensiana  ......  87 

Hickory,  shell-bark,  Carya  alba.  .  .  12 

41       nutmeg,         "      myristi- 

cceformis  .......................  x 


150 


Hickory,  pignut,  Carya  porcina 

Iron  wood,  Cyrilla  racemiftora 

Larch,  hackmatack,  Larix  Ameri- 
cana  

Laurel,  big,  Magnolia  grandiflora. . 

"  white  "  glauca 

Lignumvitae,  Guaiacum  sanctum. . . 
Lime,  wild,  JTanthoxylum  Pterota.. 

Locust,  Robinia  pseudacacia 

Maple,  mountain,  Acer  spicatum. . . 
"  sugar,  hard,  "  saccharinum 
"  silver,  soft,  "  dasycarpum. 

"  swamp,  "  rubrum 

Oak,  black,  Quercus  tinctoria 

"    live,  u       virens 

"  white,  "  alba 

Pine,  white,  Pinus  strobus 

"     yellow,    "     Arizonica 

"     pitch,      "     rigida 

"  scrub,  "  inops 

Poplar,  white  wood,  Liriodendron 

tulipifera 

Redwood,  Sequoia  sempermrens 

Satin  wood.  Jfanthoxylum  caribaum 

Spruce,  white,  Picea  alba 

Sycamore,  Platanus  occidentalis . . . 

Tulip,  yellow 

Walnut,  black,  Juglans  nigra 

Willow,  Salix  Icevigata 


Order. 
44 
305 

94 
139 
170 

143 


56 
126 


1 68 
214 

215 

246 

'57 
163 
231 


ELECTRICAL.  987 


Electrical. 

Compiled  by  Prof.  A.  E.  Kennelly. 
Units    in    Electrical    Engineering. 

The  following  units  have  been  legally  adopted  by  the  U.  S.  Government,  53^  Con- 
gress, 1894: 

Unit  of  Resistance.— The  International  Ohm,  represented  by  the  re- 
sistance offered  to  an  unvarying  electric  current  by  a  column  of  mercury  at  the 
temperature  of  melting  ice,  14.4521  grammes  in  mass,  of  a  constant  cross-sec- 
tional area,  and  of  the  length  one  hundred  and  six  and  three-tenths  centimetres. 

Unit  of  Cvirrent. — The  International  Ampere,  which  is  the  one  tenth  of 
the  unit  of  current  of  the  centimetre-gramme  second  system  of  electro  magnetic 
units,  and  is  the  practical  equivalent  of  the  unvarying  current  which,  when  passed 
through  a  solution  of  ni Irate  of  silver  in  water,  in  accordance  with  standard 
specifications,  deposits  silver  at  the  rate  of  .001 118  gramme  per  second. 

Unit  of  Electromotive  Force. — The  International  Volt,  which  is 
the  electromotive  force  that,  steadily  applied  to  a  conductor  whose  resistance  is 
one  international  ohm,  will  produce  a  current  of  an  international  ampere,  and  is 
practically  equal  to  1.434  times  the  electromotive  force  between  the  poles  or  elec- 
trodes of  the  voltaic  cell  known  as  Clark's  cell,  at  a  temperature  of  15°  C.,  and  pre- 
pared in  the  manner  described  in  the  standard  specifications. 

Unit  of  Quantity. — The  International  Coulomb,  which  is  the  quantity 
of  electricity  transferred  by  a  current  of  one  international  ampere  in  one  second. 

Unit  of  Capacity. — The  International  Farad,  which  is  the  capacity  of  a 
condenser  charged  to  a  potential  of  one  international  volt  by  one  international 
coulomb  of  electricity. 

Unit  of  \Vork.— The  Joule,  which  is  equal  to  io7  units  of  work  in  the 
C.-G.-S.  system,  and  which  is  practically  equivalent  to  the  energy  expended  in  one 
second  by  an  international  ampere  in  an  international  ohm. 

Unit  of  ]Power.—  The  Watt,  which  is  equal  to  10?  units  of  power  in  the 
C.-G.-S.  system,  and  equivalent  to  work  done  at  the  rate  of  one  joule  per  second. 

Unit  of  Induction. — The  Henry,  which  is  the  induction  in  a  circuit  when 
the  electromotive  force  induced  in  circuit  is  one  international  volt,  while  the  in- 
ducing current  varies  at  one  ampere  per  second. 

The  following  list  presents  these  units  with  their  derivatives,  and  also  other  elec- 
tro-magnetic units  which  are  in  use: 

Resistance,  O hm.  —Megohm,  one  million  ohms;  Begohm,  one  billion  ohms; 
Tregohrn,  one  trillion  ohms;  Microhm,  one  millionth  ohm;  Bicrohm,  one  billionth 
ohm. 

Current,  Ampere. — Bicro-ampere,  one  billionth  ampere;  Micro  ampere,  one 
millionth  ampere;  Milli-ampere,  one  thousandth  ampere;  Centi  ampere,  one  hun- 
dredth ampere;  Deci-ampere,  one  tenth  ampere;  Deka-ampere,  ten  amperes;  Heclo- 
ampere,  one  hundred  amperes;  Kilo-ampere,  one  thousand  amperes. 

E.  M.   IT.,  Tort.— Microvolt,  one  millionth  volt;  Kilovolt,  one  thousand  volts. 

Capacity,  Farad. — Bicrofarad,  one  billionth  farad;  Microfarad,  one  mill- 
ionth farad. 

Worlz,  Joule.  —  Kilojoule,  one  thousand  joules;  Megajoule,  one  million  joules. 

IPower,  Watt.—  Kilowatt,  one  thousand  watts. 

Induction,  Henry.  —  Microhenry,  one  millionth  henry;  Millihenry,  one 
thousandth  henry. 

Magnetic  Flux,  Weber.—  Kiloweber,  one  thousand  webers;  Megaweber, 
one  million  webers. 

Magnetic  Relnctance,  Oersted — Millioersted,  one  thousandth  oersted.  . 
'*  Intensity,  Gauss. — Kilogauss,  one  thousand  gausses. 

Magnetomotive  Force,  Gilbert.  —The  Q-ilbert  is  the  M.  M.  F. 
produced  by  .7958  ampere-turn. 

The  Oersted  is  the  reluctance  of  a  cubic  centimetre  of  air  measured  between 
opposed  parallel  faces. 

The  Weber  is  the  flux  produced  by  a  M.  M.  F.  of  one  gilbert  through  a  mag 
netic  circuit  in  which  the  reluctance  is  one  oersted. 

The  Q-anss  is  an  intensity  of  one  weber  per  normal  sq.  centimetre. 


988 


ELECTRICAL. 


Electrical.    (British  Association.) 


Resistance.— Unit  of  resistance  is  termed  an  Ohm,  which  represents  resist- 
ance of  a  column  of  mercury  of  i  sq.  millimeter  in  section  and  ^.0486  meters  in 
length,  at  temperature  o  °  C. 

i  eoo ooo  Microhms =  i  Ohm. 

i  Microhm =  1000  absolute  electro-magnetic  units. 

i  Ohm =         i  ooo  ooo  ooo 

i oooooo  Ohms =  i  Megohm  or  ioJ5    " 

Kleotro-xnotive  Force.— Unit  of  tension  or  difference  of  potentials  is 
termed  a  Volt. 

i  oooooo  Microvolts. .  =  i  Volt. 

i  Volt =  100000000  absolute  electro-magnetic  units. 

i  Megavolt . . .  =     i  oooooo  Volts. 

Current.  —Unit  of  current  is  equal  to  i  Ampere,  or  the  current  in  a  circuit 
which  has  an  electro  motive  force  of  i  Volt  and  a  resistance  of  one  Ohm. 

Capacity. — Unit  of  capacity  is  termed  a  Farad. 
i  oooooo  Microfarads  or  io~ 9  absolute  units  of  capacity =  i  Farad. 

Heat.— Unit  of  heat  is  quantity  required  to  raise  one  gramme  of  water  from 
00  C.  to  i°  C.  of  temperature. 

Quantity. — Unit  of  Quantity,  one  Coulomb,  and  is  the  quantity  of  Electricity 
transferred  by  one  ampere  during  one  second. 

rj?o    Determine    the    East    and.    "West    ^Meridian. 

Set  up  a  rod  vertically  on  a  level  area  or  plane  in  the  approximating  meridian 
and  describe  arcs,  with  a  radius  of  about  twice  the  height  of  the  rod.*  At  any  time 
before  M.  mark  the  point  in  the  arc  where  the  shadow  of  the  rod  touches  it,  and  in 
the  P.  M.,  at  the  same  length  of  time  of  before  M.,  mark  the  point  of  the  shadow  on 
the  arc;  remove  the  rod,  and  a  line  drawn  through  these  points  will  give  the  true 
bearing  of  E.  and  W. 

To    Compute    the    Increase    of  -A^rea    of  a    Circle    or 
"Volnme    of   a    Ctiloe. 

By  Differential  Calculus. 
CIRCLE,     n  x2  =  u  and  2  -n  x  d  x=du. 

u  representing  area,  x  radius  of  circle,  and  du  increase  of  area. 
ILLUSTRATIONS.— i.  Assume  x  10  ins.,  and  d  x,  or  difference  of  radius,  .05  inch. 
Then,  2X3. 1416  X  10  X  .05  =  3. 1416  sq.  ins. 
Circle  of  10  ins.  radius  =  3. 1416  sq.  ins. 

Hence,     /  —  —  - —  =  V  404  =  20.0997  ins. ,  the  increased  diameter. 

CUBE.     x3  =  u  and  3  x2  d  x  =  du. 

2.  Assume  x  side  of  a  cube  12  ins.  and  d  x  increase  of  side  .05  inch. 
x  representing  side  of  cube,  and  du  increase  of  volume. 

Then,  3  x  i22  X  .05  =  21.6  cube  ins. 
Hence,  -^123-1-21.6  —  12.05  ins.  the  increased  side. 

du      -idx      .15  ,  dx      .05  ,     .0125 

—  =  - =  —^.  —  .0125  and  — =  —^-=.004166  and- 77  =  3-    Hence,  the 

u          x         12  x        12  .004166 

cubical  expansion  is  three  times  that  of  the  linear. 

*  Varying  with  the  latitude  of  the  location,  a»  the  more  vertical  the  sun  the  greater  the  height  of 
thu  rod. 


ENERGY    AND    MOTION — KINETICS.  989 

Energy    and.    ^Motion. 

The  science  of  Motion  is  included  in  Mathematics,  and  is  termed  Kine- 
matics ;  the  science  of  Force,  Dynamics  or  Kinetics ;  and  the  investigation  or 
operation  of  forces  in  equilibrium,  Statics. 

All  standards  of  Energy  and  Motion  are  Units,  as  the  unit  of  Length  may  be  an 
inch,  foot,  yard,  or  mile,  but  usually  a  foot ;  that  of  Time  a  second,  minute,  hour, 
or  day,  usually  a  second;  and  \relocity  by  the  number  of  units  of  lengths  or  opera- 
tion in  a  unit  of  time. 

Uniform  Acceleration  is  the  uniform  increase  or  decrease  of  velocity  per  unit  of 
time  or  distance,  but  this  increase  or  decrease  of  velocity  is  that  which  the  force 

produces  in  a  unit  of  time  ;  hence  — —  =F.     m  representing  unit  of  force,  a  unit 
of  acceleration,  t  the  time,  and  F  the  force. 

ILLUSTRATION.— Assume  a  body  moving  at  the  rate  of  50  feet  per  second,  and  at 
the  end  of  10  seconds  it  has  acquired  a  velocity  of  75  feet  per  second,  the  increase 
of  velocity  is  25  feet  in  10  seconds,  equal  to  2.5  feet  per  second  in  each  second,  or 
2. 5  feet  per  second  per  second.  * 

2.  Given  a  uniform  acceleration  of  velocity  of  40  feet  per  second  per  second; 
what  is  the  acceleration  in  yards  per  minute  per  minute? 

40  feet  per)  _4oX  60  feet  per  min.  per  sec.   )  _4QX6o2  = 
sec.  per  sec.  J  ~  40  X  6o2  "  '    min.  (  3 

min.  per  min. 

3.  A  train  of  cars  2  minutes  after  starting  attains  a  uniform  velocity  of  15  miles 
per  hour  ;  what  is  the  acceleration  in  miles  per  minute  per  minute? 

15  miles  per  hour  =.  .25  mile  per  minute,  increase  of  velocity  = .  25  mile  per  minute 
which  occurs  in  2  minutes. 
Hence,  acceleration  = .  5  of .  25  mile  per  minute  = .  1 25  mile  per  min.  per  min. 

Kinetics. 

Force  and  Mass. 

If  two  bodies  of  equal  dimensions  and  unequal  weights  be  simultaneously  pro- 
jected with  like  Velocity,  the  heavier  one  will  go  farther  than  the  lighter;  but  if  these 
bodies  were  projected  with  like  Force,  the  lighter  one  would  go  the  farthest. 

The  difference  is  in  consequence  of  the  difference  of  the  Mass  or  Matter,  and  as  a 
result  it  requires  more  force  to  stop  the  heavier  body  when  started  than  the  light 
one. 

In  operation,  there  are  two  common  Units  of  Mass,  as  there  are  two  of  Force. 

The  Poundal,  or  British  absolute  unit  of  force,  is  that  which  the  action  on  a  mass- 
pound  for  one  second  produces  in  it  a  velocity  of  one  foot  per  second. 

The  Dyne  is  that  force  which,  acting  on  a  mass-gram  for  one  second,  produces  in 
it  a  velocity  of  one  centimeter  per  second. 

OPERATION. — If  15  poundals  bear  upon  a  mass  of  70  Ibs.,  in  what  time  will  it  pro- 
duce a  velocity  of  60  feet  per  minute? 

i  poundal  =  a  velocity  of  i  foot  per  sec.  in  i  Ib.  in  i  second. 

Hence,  15  poundals  ==  a  velocity  of  i  foot  per  sec.  in  i  Ib.  in  —  second. 
15  poundals  =  a  velocity  of  i  foot  per  sec.  in  70  Ibs.  in  —  seconds,  and  15  poundals 
=  a  velocity  of  60  feet  per  sec.  in  70  Ibs.  in  — =  280  seconds. 

Hence,  ^^  =  P.  m  representing  unit  of  force,  v  unit  of  velocity,  t  unit  of  time, 
and  F  the  force. 

*  Second  per  second,  Minute  per  minute,  etc.,  although  unusual,  is  proper.  Thus,  the  expression, 
"  The  train  went  with  a  velocity  of  60  miles  per  hour,''  is  indefinite,  as  it  may  have  gone  at  that  rate 
but  for  a  period  of  one  minute,  or  10  minutes ;  whereas,  60  miles  per  hour  per  'hour  indicates  both  the 
rate  of  the  velocity  and  of  that  per  hour. 


990  KINETICS.  —  GAS    ENGINES. 

Impulse  —  Is  when  a  force  acts  during  a  given  time. 

Thus,  if  a  force  of  5  Ibs.  bears  upon  an  object  during  3  seconds,  3  X  5  =  15  units 
of  effect,  and  —  —  =  F,  representing  the  number  of  units  of  impulse, 

Momentum,  or  Moment  —  Is  the  product  of  the  number  of  units  of  velocity  with 
which  the  mass  is  moving. 

ILLUSTRATION.  —  A  mass  of  200  Ibs.  is  moved  with  a  uniform  velocity  of  75  yards 
per  minute  ;  during  what  time  is  a  force  of  80  Ibs.  required  to  arrest  the  motion? 

75      3  =  3.  75  feet  per  second,  and  2°°^_75  =  750,  which  -4-  80  =  9.  375  seconds. 

2.  A  mass  of  6.75  Ibs.  is  acted  upon  by  a  force  of  .5  Ib.  during  5  minutes  ;  what  is 
the  velocity  acquired? 

5  minutes=3oo  seconds,  and  300-:-.  5  (half  pound)  =  150  units  of  impulse,  and 

^-  —  22.  2  feet  per  second. 
6-75 

3.  A  rod  8  feet  in  length,  weighing  8  Ibs.,  has  a  weight  of  205  Ibs.  suspended  from 
one  end  and  60  Ibs.  from  the  other  ;  at  what  point  in  the  bar  will  the  effect  of  the 
weights  be  equalized? 

By  rule  To  Compute  Position  of  Fulcrum,  p.  624, 


8  -r-  ^  4-  i  =  -  —  =  1.8113  feet=  distance  of  205  Ibs.  from  its  end.  and  8  — 

60  4-4167 

i.  8113=:  6.  1887  feet  =  distance  of  60  Ibs.  from  its  end. 
Then,  to  include  weight  of  rod  — 

205-]-  1.  8113-7-60-1-6.18874-  1  =4.1246,  and  8-7-4.1246  =  1.9396  —  feet  =  distance 
of  205  Ibs.  from  its  end,  including  its  weight  of  the  rod.  Hence,  8  —  1-9396  —  = 
6.  0604  -\-feet  =  distance  of  60  Ibs.  from  its  end. 

Verification.  205  -}-  1.9396  X  1-9396  =  401.  367  Ibs,  ,  and  60  -f-  6.064  -f-  x  6.064  ~f*  = 
401.376  Ibs. 

GAS   ENGINES. 
Grcts   Engines.    Are  divided  into  three  types. 

TyPes'  Theoretical.       Efficiency. 

1.  Engines  igniting  at  constant  volume,  without  previous) 

compression  ........................................  J 

2.  Igniting  at  constant  pressure,  without  previous  compres-  )  R 

sion  ................................................  J         2 

3.  Igniting  at  constant  volume,  with  previous  compression.  ..        3       =       .34 
In  the  first  two  types  the  cylindrical  conditions  are  most  favorable  to  cooling, 

and  a  practical  efficiency  of  .06  is  attained.  In  the  third  the  conditions  for  loss  by 
cooling  are  very  favorable,  and  an  actual  efficiency  of  .17  is  obtained. 

The  ordinary  heat  efficiency  is  17  per  cent,  of  all  the  heat  expended  in  an  en- 
gine, and  the  highest  obtained  25  per  cent. 

For  powers  up  to  20  IIP,  when  gas  is  cheap,  as  in  towns  and  cities,  it  competes 
with  steam,  as  it  is  more  economical  and  more  convenient,  and  is  most  usually  re- 
sorted to  for  a  power  of  from  4  to  6  horses.  Some  engines  have  been  constructed 
and  are  in  use  of  100  IP. 

Non-Compression   Kngines.    Are  principally  used  for  small  power, 
as  up  to  .5  IP. 
The  pressure  is  applied  only  during  a  portion  of  the  stroke. 

In  the  Leiioir  the  piston  is  moved  only  for  about.  5  its  stroke,  when  it  re- 
ceives a  mixed  volume  of  gas  and  atmospheric  air,  which  is  ignited  by  an  electric 
spark,  the  pressure  rising  to  about  45  Ibs.  per  sq.  inch  above  the  atmosphere.  The 
piston  then  is  driven  through  the  remaining  portion  of  the  stroke,  and  at  the  end 
of  it  the  pressure  falls  to  about  3  Ibs. 

The  mean  effective  pressure  being  usually  8.5  Ibs.  per  sq.  inch. 


GAS    ENGINES. 


991 


M.ean     rtesxilts    of    Operation    of*    Thirteen     10ngin.es    of 
Five    Different    Constructions. 


Remits. 

Iff 

BH? 

Gas 
I» 

per 
BHP 

Revolution 
Mnfute. 

Heat  Con- 
verted into 
Work. 

Mean. 
Least. 
Extreme. 

No. 
15-35 
3-42 
336 

No. 
12-55 
2-7 

27.75 

Cube  feet. 
22.21 

18.92 

3°-9 

Cube  feet. 
28.05 
23-58 
33-4 

No. 
180.7 
132 
223.8 

Per  cent. 
'7-43 
10.5 
21.2 

This  Type  of  engine  is  held  to  be  wasteful  of  gas,  as  it  consumes  over  90  cube  feet 
of  16  candle-power  gas  per  IIP  per  hour. 

The  Otto  «fc  JLanger  is  a  free-piston  or  atmospheric  engine,  admitting  of 
high  piston  speed  and  great  expansion,  hence  it  is  more  economical. 

An  explosion  of  gas  drives  the  piston  upward,  and  by  the  projectile  force  of  it 
and  the  reduction  of  the  temperature  of  the  gas  under  it,  a  partial  vacuum  is 
formed,  the  piston  returning  under  atmospheric  pressure. 

In  an  engine  with  a  cylinder  of  12.5  ins.  diameter  and  an  observed  stroke  of  pis- 
ton of  40  ins.,  25  cube  feet  of  gas  gave  a  maximum  gauge  pressure  of  54  Ibs.  per  sq. 
inch,  and  a  result  of  2.9  per  IIP  per  hour. 

Compression  Engines  possess  the  advantage,  of  furnishing  greater 
power  With  less  volume  and  weight,  as  well  as  economy  of  operation. 

The  Ot  to.— It  is  a  single-acting-piston  engine,  serving  alternately  as  a  pump  and 
a  motor,  and  one  explosion  of  gas  is  given  for  every  two  complete  revolutions. 

OPERATION.— The  piston  receives  a  volume  or  charge  of  gas  and  air,  then  returns, 
compressing  the  volume  into  a  space  at  end  of  its  stroke,  which  mixture  is  ignited, 
and  the  pressure  therefore  forces  up  the  piston,  when  it  returns  with  an  exhaust 
valve  open  to  free  it  from  the  force  and  products  of  combustion ;  at  the  termination 
of  the  stroke  it  is  in  position  to  receive  a  new  charge. 

Thus,  one  driving  stroke  of  the  piston  is  given  for  two  revolutions  of  the  engine. 

The  Mean  effective  pressure,  with  gas  of  16  candle-power,  is  about  55  to  60  IbB. 
per  sq.  inch,  the  maximum  pressure  of  the  explosion  being  from  140  to  160  Ibs., 
and  even  up  to  180  Ibs. 

In  an  engine  rated  at  6  IP  the  consumption  of  gas  was  21  cube  feet  per  IIP,  and 
for  brake  IP  29  cube  feet. 

The  Clerk  has  a  second  cylinder,  termed  the  charging,  the  function  of  which 
is  to  receive  a  charge  of  gas  and  air  at  each  stroke  and  deliver  it  into  the  motor 
cylinder.  The  charge  dispels  the  consumed  gas  of  a  previous  operation  and  fills  it 
with  mixture,  to  be  compressed  by  the  return  stroke  of  the  piston  and  ignited  at 
each  revolution. 

The  Mean  effective  pressure  in  an  engine  of  12  IP  is  65  Ibs.  per  sq.  inch,  pressure 
of  compression  57  Ibs.,  and  maximum  pressure  of  explosion  238  Ibs.  Gas  consumed 
24  cube  feet  of  24  candle  power  per  IIP  per  hour.  One  of  9-inch  cylinder  and  20 
ins.  stroke,  at  132  revolutions  per  minute,  developed  27.5  IIP  per  hour,  or  23.2  IP 
at  the  brake. 

The  CampToell  and  2VdLid.lan.xl  are  of  this  type,  and  are  alike  to  it  in  the 
use  of  a  charging  cylinder. 

The  Stodspnt  resembles  it  also,  but  the  operation  differs  in  the  passing  of 
each  charge  from  the  gas  pump,  which  is  a  combination  of  the  motor  piston,  into 
an  intermediate  reservoir,  whence  it  blows  out  into  the  motor  cylinder  and  dis- 
charges the  burned  gas. 

Three-Cycle  engines  are  like  the  Otto  in  their  operations,  but  give  only 
one  impulse  for  every  three  revolutions,  one  extra  double  stroke  being  used  in  re- 
ceiving a  charge  of  air  and  expelling  it  at  the  exhaust  port,  so  that  the  compres- 
sion space  is  cleared  at  each  operation. 

The  earliest  of  these  engines  was  known  as  the  Linford ;  those  now  in  use  are 
the  Griffin  and  the  Barker, 


992 


GAS   ENGINES. 


The  Atkinson  Cycle  gives  an  impulse  at  each  revolution  of  the  crank 
shaft,  and  the  piston,  by  a  system  of  links,  is  connected  so  that  it  makes  two  out 
and  two  in  strokes  for  each  revolution  of  the  crank-shaft,  and  one  explosion  is 
given  for  each  cycle  of  four  strokes. 

It  resembles  the  Otto  in  using  the  same  piston  alternately  for  pump  and  motor 
operations,  but  differs  from  it  in  making  unequal  strokes.  This  arrangement  en- 
ables the  exhaust  gases  to  be  swept  out  of  the  cylinder  at  every  operation  and  great 
expansion  is  obtained.  This  engine  is  held  to  be  very  economical,  as  in  recent 
trials  it  consumed  but  22  cube  feet  of  gas  per  brake  IP  per  hour. 

(The  Practical  Engineer.) 

The  Crossley  is  a  horizontal  engine,  with  a  single  cylinder,  and  of  nominal 
powers  from  .  5  to  30  IP,  indicating  from  2  to  85  IP  :  with  double  cylinders,  from 
16  to  170  IIP. 

A  12  IP  engine  has  developed  28  IIP  and  23  at  the  brake  =  82^  of  the  indicated, 
consuming  20  cube  feet  of  gas  per  IIP  per  hour. 

Results    of   Trials    of   Q-as    Engines. 


Type. 

IFF. 

Gas  per 
IH*. 

Heat 

converted 
into  Work 

Revolu- 
tions per 
Minute. 

Type. 

IFF. 

Gas  per 
IFF. 

Heat 
converted 
into  Work 

Revolu- 
tions per 
Minute. 

Otto. 

Clerk. 
Beck. 

No. 
22.56 
3-42 
27.46 
6.12 

Cube  ft. 
23-6 

30-9 
20.39 
20.67 

Per  cent. 
17-5 
14.46 

15-5 
21.  1 

No. 
158-7 
160.3 
132 
168.9 

Griffin. 
Forward. 
Fawcett. 
Atkinson. 

No. 
17.46 

5-54 
11.49 
11.15 

Cube  ft. 
18.92 
20.79 
18.4 
19.22 

Per  cent. 
21.2 
I9.2 
19.6 
22.8 

No. 
223.8 

(T.  L.  Miller.) 

Results     of    Trials     of    Crossley     Engine    -with     London. 
Coal    Q-as.* 

Pressures  and  Revolutions  per  minute,  Pressures  and  Water  in  Ibs.,  Gas  in  cube 
feet,  and  Efficiency  and  Heat  per  cent. 


Power. 

Full. 

Half. 

Power. 

Full. 

Half. 

Revolutions  per  minute.  .  . 
Explosions      "        u 
Mean  initial  pressure  
Mean  effective    "      
Brake  IP           

1  60.  i 
78.4 
196.9 
67.9 
14.74 

158.8 
41.1 
196.2 
73-4 

Water  for  cooling  per  hour 
Mean    pressure    during  ^ 
working  stroke  equiv-  ! 
alent  to  work  in  pump-  [ 
ing  stroke   .     .            J 

7i3 
2.19 

480 

Indicated  IP   

17.12 

9-  73 

Corresponding  IIP  

.58 

Mechanical  efficiency  

86.1 

76.2 
205  8 

Heat  converted  to  work..  . 
Heat  rejected  in  jacket) 

22.1 

20.9 

Gas  per  IIP  per  hour    .... 

355-3 
20  76 

water                             j 

43-2 

41.1 

Gas  per  brake  IP  per  hour. 

24.1 

27.77 

Heat  rejected  in  exhaust.  . 

35-5 

38 

(D.  K.  Clark.) 

Pressures    Produced     l>y    the     Explosion    of    Q-aseous 
Mixtures    in    a    Closed    "Vessel. 

Mixture  of  Air  and  Coal- Gas,  Temperature  64 


Mixture. 

Pressure! 
per 

Mixture. 

*y  I!        MM™. 

Pressure 
per 

Gas. 

Air. 

Sq.Vh. 

Gas. 

Air. 

Sq.  Inch.  ||      Gas. 

Air. 

Sq.  Inch. 

Volume. 

Volumes. 

Lbs. 

Volume. 

Volumes. 

Lbs.       1    Volume. 

Volumes. 

Lbs. 

I 

5 

96 

I 

9 

69                  I 

13 

52 

I 

7 

89 

I 

II 

63        II 

(D.  Clerk.) 

*  See  Report  on  Trials  of  Motors  for  Electric  Lighting  for  Society  of  Art*,  1889. 
f  Maximum  above  the  atmosphere. 


DIMENSIONS  OF    BOLTS,  NUTS.  —  TENACITY  OF  NAILS.  993 


Standard.   Dimensions    of  Iron  and   Copper  Bolts 

and    !N"\rts,  TJ.   S.  Navy. 
Square    and    Hexagonal    Heads    and    Nuts. 

Finished. 
From  .25  Inch  to  6  Inches  in  Diameter. 


Dl  AM 

Bolt. 

HTER. 

Effec- 
tive. 

Effective 
Area. 

DlAM 

Head  a 
Hexagonal. 

ETKB. 

nd  Nut. 
Square. 

WIDTH. 
Head  &  Nut. 
Hexagonal 

Square. 

DK 

Head. 
Hex. 

and 
Square. 

PTH. 

Nut. 
Hex. 

and 
Square. 

Threads. 

Ins. 

Ins 

Sq.  Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

No. 

•25 

.185 

.026 

9-16 

23-32 

•5 

•25 

•25 

o 

•3125 

•24 

•045 

11-16 

27-32 

19-32 

19-64 

•3125 

8 

•375 

.294 

.067 

25-32 

31-32 

11-16 

n-32 

•375 

6 

•4375 

•345 

•093 

29-32 

•   3-32 

25-32 

25-64 

•4375 

4 

•5 

4 

.125 

i 

•25 

-875 

•4375 

•5 

3 

•5625 

•454 

.162 

1-25 

-625 

31-32 

31-64 

9-16 

2 

.625 

•507 

.202 

i.  7-32 

•5 

i.   1-16 

17-32 

.625 

I 

•75 

.62 

.302 

i.  7—16 

•75 

1.25 

.1625 

•75 

O 

•875 

•731 

.419 

1.21-32 

2.   1-32 

i.  7-16 

23-32 

-875 

i  x 

-837 

1-875 

2.  s-,6 

i.  5-8 

13-16 

i 

1.125 

.94 

•694 

2.    3-32 

2.    9-16 

1.13-16 

29-32 

1.125 

7 

1-25 

1.065 

.891 

2.     5-l6 

2.27-32 

2 

i 

1-25 

7 

'•375 

1.16 

1-057 

2.17-32 

3-   3-32 

2.    3-l6 

i.   3-32 

i-375 

6 

1.284 

1.294 

2-75 

3-11-32 

2-375 

i.   3-16 

6 

1.625 

1.389 

I-5I5 

2.31-32 

3-625 

2.     9—16 

i.  9-32 

1.625 

5-5 

1.75 

1.491 

1.746 

3-  3-i6 

3.875 

2-75 

J.375 

1.75 

5 

'•875 

1.616 

2.051 

3-I3-32 

4-   5-32 

2.I5-I6 

1.15-32 

1-875 

5 

2 

1.712 

2.302 

3-19-32 

4-I3-32 

3-125 

i.  9-16 

2 

4-5 

2.25 

1.962 

3-023 

4-   i-S2 

4.15-16 

3-5 

i-75 

2.25 

4-5 

2-5 

2.176 

3-7I9 

4-I5-32 

5-I5-32 

3-875 

1.15-16 

2-5 

4 

2-75 

2.426 

4.622 

4-29-32 

6 

4-25 

2.125 

2-75 

4 

3 

2.629 

5-428 

5.11-32 

6.17-32 

4-625 

2.    5-l6 

3 

3-5 

3-25 

2.879 

6.5I 

5-25-32 

7.  1-16 

5 

2-5 

3-25 

3-5 

3-5 

3-  l 

7-  547 

6-  7-32 

7-19-32 

5-375 

2.  II-l6 

3-5 

3-25 

3-75 

3-3I7 

8.641 

6.625 

8125 

5-75 

2.875 

3-75 

3 

4 

9-99 

7.  1-16 

8.21-32 

6.125 

3-   1-16 

4 

3 

4-25 

3-798 

11.329 

7-5 

9.  3-16 

6-5 

3-25 

4-25 

2.875 

4-5 

4.028 

12.743 

7.15-16 

9-25-32 

6.875 

3-  7->6 

4-5 

2-75 

4-75 

4-256 

14.226 

8-375 

0.25 

7  25 

3-625 

4-75 

2.625 

5 

4.48 

15-763 

8.13-16 

o.  25-32 

7.625 

3.13-16 

5 

2-5 

525 

4-73 

I7-572 

9-25 

i.   5-16 

8 

4 

5-25 

2-5 

55 

4953 

19.267 

9.11-16 

1.27-32 

8-375 

4-  3-i6 

5-5 

2-375 

5-75 
6 

5.203 

21.262    !  10.   3-32 
23.098       10.17-32 

2-375 
2.29-32 

8-75 
9-125 

4-375 
4.  9-16 

I'75 

2-375 
2.25 

For  Rough  Bolts  and  Nuts,  add  .066  to  above  dimensions,  and  for  other  notes  see 

PP-  156-159- 

Relative    Tenacity    of    Wrougnt-Iron    Cut    and    Wire 
Nails. 

Per  Cent,  of  Cut  and  Wire  Nails. 


Dimen- 
sions. 

SPRUCE. 
Designa- 
tion. 

Per 

Cent. 

Dimen- 
sions. 

Designa- 
tion. 

PINK. 
Per 
Cent. 

Direction  of 
Penetration. 

Ins. 
1.125X6 

Ordinary. 

47-5 

Ins. 
1.25X4 

Box. 

135-2 

{Taper  perpendic 
ular  to  grain. 

1.125X4 

Finish. 

72.2 

1.25X4 

Box. 

100.2 

(Taper  parallel  to 
I           grain. 

1.125  X  4 

Box. 

509 

i  25X4 

Box. 

66.4 

In  end. 

2X4 

Floor. 

80 

1.25X4 

Box. 

99-9 

lln     the     three 
(           ways. 

In  Spruce  40  tests  of  cut  nails  averaged  60  per  cent. 

In  Spruce  and  Pine  combined  the  average  was  72.7  per  cent. 

t:«ao  n    ^~^  i  U 


See  p.  970. 


. 
(  Wm.  H.  Burr,  C  E.) 


994  COMPRESSION    OF   AIR. 

Compressed,    and    Compression,    of    Atm.osph.eric 

Computations    of  Flow,    Operation,   Kfiect,   .Po\ver,   etc. 

For  fuller  information,  see  "Compressed  Air,"  by  Fredk.  C.  Weber,  M.E.,  before 
the  Engineering  Society  of  Columbia  College,  April  22d,  1896  ;  Wm.  L.  Saunders, 
N.  Y. ;  also  by  Frank  Richards,  Mem.  A.S.M.E. ;  a  lecture  by  R.  A.  Parke  ;  I).  K. 
Clark's  Pocket- Book:  a  treatise  by  W.  C.  Unwin,  in  Vol.  CV.  of  Proceedings  of  In- 
stitution of  C.  E.  of  Great  Britain,  and  a  treatise  of  The  Norwalk  Iron  Works  Co., 
etc. 

Pressure    and.    Temperature. 

Under  constant  pressure  the  volume  of  air  varies  directly  as  the  Absolute 
temperature.  For  constant  volume  the  pressure  is  in  direct  proportion  to 
an  increase  in  temperature. 

Compression. 

Heat  and  Mechanical  energy  are  mutually  convertible ;  when,  therefore, 
the  piston  of  an  air-compressing  engine  is  in  operation,  heat  is  evolved 
(theoretically)  in  exact  proportion  to  the  work  performed,  in  the  ratio  of  one 
British  thermal  unit  (B  T  U)  for  every  772*  foot  pounds  expended. 

When  atmospheric  air  is  compressed,  the  degree  of  its  compression  may 
be  indicated  by  a  pressure  gauge. 

The  heat  evolved  by  the  compression  of  air  generates  by  expanding  it  an 
increased  resistance,  and  involves  increased  power  to  compress  it.  This 
loss  of  power  consequent  upon  the  expansion  of  the  air  by  the  heat  of  com- 
pression is  so  great  that  it  is  necessarily  essayed  to  reduce  the  heat,  and  a 
cooling  medium  is  resorted  to,  to  abstract  it  in  the  operation  of  compression. 

The  rate  of  increase  of  temperature  of  air  during  compression  is  not  uni- 
form, as  the  temperature  rises  faster  during  the  primitive  stages  of  compres- 
sion than  the  later.  Thus,  in  compressing  from  i  to  2  atmospheres,  the 
increase  of  temperature  will  be  greater  than  in  compressing  from  2  to  3 
atmospheres,  and  in  like  ratios.  The  rate  of  increase  also  varies  with  the 
initial  temperature,  as  the  higher  it  is  the  greater  will  be  the  rate  of  increase 
at  any  point  of  the  compression.  When  air  at  atmospheric  pressure  and 
o°  is  compressed  to  15  Ibs.  gauge,  the  final  temperature  will  be  100°,  or  an 
increase  of  100°.  If  at  60°,  it  will  be  175°,  an  increase  of  115°,  and  at  90° 
it  will  be  210°,  an  increase  of  120°. 

The  rise  in  temperature  due  to  the  compression  of  atmospheric  air  at  32°, 
when  it  is  reduced  to  one-fourth  its  volume,  is  given  by  Kimball  at  344°. 

The  great  reduction  of  the  temperature  of  compressed  air  when  it  is  dis- 
charged from  the  compressing  cylinder,  against  a  resistance,  as  the  cylinder 
piston  of  an  engine,  precludes  the  economical  operation  of  using  it  expan- 
sively alike  to  steam  or  any  similar  vapor.  The  available  energy  of  com- 
pressed air  is  that  which  it  exerts  against  a  resisting  medium,  in  its  increase 
of  volume  by  expansion. 

When  air  is  compressed,  if  it  neither  gains  or  loses  temperature  by  com- 
munication with  any  other  body,  the  heat  generated  by  compression,  re- 
maining and  adding  to  it,  the  operation  is  termed  Adiabatic  compression. 
When  pressure  is  removed  from  compressed  air  and  it  expands  without  re- 
ceiving heat  externally,  the  air  is  termed  to  have  expanded  A  diabatically. 

If  during  the  compression  of  air,  it  is  maintained  at  a  uniform  tempera- 
ture by  the  reduction  of  it,  coeval  with  its  generation,  the  compression  is 
termed  Isothermal.  Hence  when  the  air  remains  at  a  uniform  tempera- 
ture throughout  the  operation  of  compression  or  expansion,  it  is  designated 
as  Isothermal. 

*  Joule's.    Later  experiments  put  it  at  778. 


COMPRESSION   OP   AIR. 


995 


The  specific  heat  of  atmospheric  air  at  constant  pressure  is  .2  377,  hence 
the  unit  of  heat  that  would" raise  the  temperature  of  i  Ib.  of  water  i°  would 
raise  the  temperature  of  i  Ib.  of  air  (i  -f-  .2377)  =  4.207°. 

13.141  cube  feet  of  air  at  62°  (table,  p.  521)  weigh  i  Ib.,  and  air  at  60° 
compressed  to  half  its  volume  evolves  116°  heat,  and  the  specific  heat  of 
air  under  constant  pressure  is  .2377,  which  x  116  =  27.573  heat  units,  pro- 
duced by  the  compression  of  i  Ib.  or  13.141  cube  feet  of  free  air  into  one- 
half  its  volume :  Hence,  27.573  x  778  =  21 452  foot  Ibs.,  and  as  heat  and 

mechanical  energy  are  held  to  be  convertible  terms, =  .65  H*  pro- 

Tiooo 

duced  or  lost  by  the  compression  of  i  Ib.  of  air. 


33000 


"Volume,    Mean     Pressure,    and    Temperature    of    Com- 
pressed.   Air. 

From  i  to  200  Lbs.  and  from  60°  to  672°. 
Air  assumed  at  14.7  Ibs.  and  Temperature  60°. 


Pressure 
Lba. 

VOLUME 

Constant 
Temperature 
Isothermal. 

OF  AlB. 

Not 
Cooled. 
Adiabatic. 

MEAN 

Constant 
Temperature 
Isothermal. 

PRKSHURB 

Not 
Cooled. 
Adiabatic. 

AIR  PBB  STR 
During  Co 
onl 
Constant 
Temperature 

OKB 

mpreasion 
Y- 
Not 
Cooled. 

final  Tem- 
perature,* 
Air  not 
Cooled. 

0 

I 

I 

0 

0 

o 

0 

60° 

i 

•?363 

•95 

.96 

•975 

•43 

•  44 

71 

2 

.  8803 

.91 

1.87 

1.91 

•95 

.96 

80.4 

3 

8305 

.876 

2.72 

2.8 

1.41 

88.9 

7861 

.84 

3-53 

3-67 

1.84 

1.86 

98 

5 

7462 

.81 

4-3 

4-5 

2.22 

2.26 

106 

10 

5952 

.69 

7.62 

8.27 

4.14 

4.26 

'45 

15 

495 

.606 

10.33 

11.51 

5-77 

5.99 

178 

20 

4237 

•543 

12.62 

14.4 

7-2 

7.58 

207 

25 

3703 

•494 

'4-59 

17.01 

8-49 

9-°5 

234 

3° 

3289 

.464 

16.34 

19.4 

9.66 

10.39 

255 

35 

2957 

•42 

17.92 

21.6 

10.72 

".59 

281 

4° 
45 

2462 

•  393 
•37 

19.32 
20.52 

23.66 
25-59 

12.62 

12.8 

13-95 

302 
321 

So 

2272 

•35 

21  79 

27.39 

13.48 

15-05 

339 

i 

65 

1844 

•33' 
.301 

22.77 
23.84 
24.77 

29.11 

30-75 
31.69 

15-05 
I5-76 

15.98 
16.89 
17.88 

357 

i 

70 

1735 

.288 

26 

33-73 

16.43 

18.74 

405 

i5 

1639 

,276 

26.65 

35.23 

17.09 

19.54 

420 

So 

1552 

.267 

27-33 

36-6 

17.7 

20.5 

432 

85 

1474 

•257 

28.05 

37-94 

18.3 

21.22 

447 

90 

1404 

.248 

28.78 

39.18 

18^87 

22 

459 

95 

134 

•  24 

29-53 

40.4 

19.4 

22.77 

472 

100 

105 

1281 
1228 

.232 
.225 

30.07 
30.81 

41.6 
42.78 

19.92 
20.43 

23  43 
24.17 

I 

no 

1178 

.219 

3i.39 

43-91 

20.9 

24.85 

5°7 

"5 

"33 

.213 

31.98 

44.98 

21.39 

25-54 

518 

1  20 

1091 

.207 

32.54 

46.04 

21.84 

26.2 

529 

125 

1052 

.202 

33-07 

47.06 

22.26 

26.81 

540 

130 

1015 

.197 

33-57 

48.1 

22.69 

27.42 

55° 

135 

0981 

34-05 

49.1 

23.08 

28-°5 

560 

140 

095 

.'188 

34-57 

50.02 

23-  41 

28.66 

570 

'45 

.0921 

.184 

35-09 

51 

23-97 

29.26 

580 

I5° 

.0892 

.18 

35-48 

5189 

24.  28 

29.82 

589 

1  60 

.0841 

.172 

36.29 

24.97 

30.91 

607 

170 

.0796 

.166 

37-2 

55-39 

25-71 

32.03 

624 

180 

•0755 

.16 

37-  96 

57-01 

26.36 

33-04 

640 

190 

.0718 

•  154 

38.68 

58.57 

27.02 

34-o6 

657 

200 

.0685 

.149 

39-42 

60.14 

27.71 

35-02 

672 

*  Produced  by  compression. 


(Frank  Richardt.* 


99° 


COMPRESSION    OF   AIR. 


For  determination  of  absolute  pressure  add  14.7  Ibs.  to  gauge  pressure. 

Column  2  gives  the  volume  of  air  (initial  —  i),  assuming  that  its  temperature 
has  not  risen  during  the  compression,  or  that  if  the  air  has  not  been  wholly  cooled 
during  the  compression,  it  has  been  cooled  to  the  initial  temperature  after  the 
compression.  Or  volume  of  one  cube  foot  of  free  air  at  given  pressure. 

Absolute  isothermal  compression  is  not  attainable,  as  it  is  impracticable  in  the 
compression  of  air,  simultaneously  to  abstract  all  the  heat  evolved  in  the  compres- 
sion. This  column,  however,  does  give  the  volume  of  air  that  will  be  realized,  if  it 
is  transmitted  to  such  a  distance  from  the  compressor  or  in  any  manner  that  the 
heat  is  abstracted  before  it  is  used.  Air  radiates  its  heat  very  rapidly,  and  this 
column  may  be  taken  to  represent  the  volume  of  available  air  after  compression. 

Column  3  gives  the  volume  of  air  at  completion  of  the  compression,  assuming 
that  the  air  has  neither  lost  nor  gained  during  the  compression,  and  that  all  the 
heat  developed  by  the  compression  remains  in  the  air.  The  condition  represented 
by  this  column— adiabatic  compression— is  alike  to  that  of  isothermal  compression, 
never  actually  attained.  In  any  compression,  the  air  will  lose  some  of  its  heat,  and 
consequently  the  air  is  not  as  heated  at  any  period  of  the  compression  to  the  ex- 
tent that  theory  assigns  to  it.  Physically,  the  theory  is  correct,  but  practically  it 
fails.  The  slower  a  compressor  is  operated,  the  more  readily  will  the  air  radiate 
some  of  its  heat,  and  as  a  result,  the  less  will  be  its  volume  and  less  the  power  rc~ 
quired  for  compression. 

Column  4  gives  the  mean  effective  resistance  to  the  piston  of  the  air-compressor 
cylinder  in  the  stroke  of  compression,  assuming  that  the  air  throughout  the  stroke 
remains  uniformly  at  its  initial  temperature — isothermal  compression — but  as  the 
air  does  not  remain  at  constant  temperature  during  compression,  the  results  in  this 
column  are  to  be  essayed  to  be  attained  in  economical  compression. 

Column  5  gives  the  mean  effective  resistance  to  be  overcome  by  the  piston,  as- 
suming there  is  not  any  cooling  of  the  air  during  compression — adiabatic  compres- 
sion— inasmuch  as  there  is  always  some  cooling  of  the  air  during  compression,  the 
actual  mean  effective  result  will  be  somewhat  less  than  that  given  in  the  column. 
For  the  computation  of  power  required  for  the  operation  of  the  air-compressor 
cylinder,  the  results  given  may  be  taken,  with  a  per  cent,  added  for  friction*— o  to 
10  per  cent. — and  the  result  will  very  nearly  give  the  power  required  to  operate  the 
compression. 

Column  6  gives  the  mean  effective  resistance  for  the  compression  of  the  stroke 
of  the  piston  in  compressing  air— isothermally— from  that  of  14.7  Ibs.  to  any  given 
pressure. 

ILLUSTRATION. — Assume  an  air-compressing  cylinder  20  ins.  in  diameter  by  2 
feet  stroke  of  piston,  making  75  revolutions  per  minute,  with  an  adiabatic  pressure 

of  75  Ibs. • 

2o2X-7854X  35-23  (columns)  X  75  X  2  X  2 -=- 33  ooo  =  100. 6  IP. 

ILLUSTRATION.  —Assume  an  adiabatic  pressure  of  50  Ibs.  by  gauge,  the  volume  ot 
air  compressed  and  delivered  will  be  (column  3)  ,35  for  each  stroke  of  the  piston  in 
a  cylinder  full  of  free  air;  while  for  the  compressing  part  of  the  stroke  i  — 35  —  .65, 
the  mean  resistance  will  be  15.05  Ibs.  (column  7).  Thus,  15.05  x  .65  +  50  X  -35  = 
27.28;  corresponding  very  nearly  with  27.39  (column  5)  for  the  whole  stroke. 

Comparing  isothermal  compression  with  adiabatic,  to  50  Ibs.  as  above,  in  column 
6  is  13.48  which  x  i  —  2272  =  .7  728  (column  2)  -j-  50  X  -2  272  =  21.78  or  21.79,  as 
given  in  column  4. 

Columns  6  and  7  are  useful  in  the  computation  of  power  in  the  first  operation  ot 
compression,  as  the  function  of  the  first  cylinder  is  that  of  compression  only. 

The  results  given  in  columns  7  and  8  are  elements  of  computation  for  the  IP  of 
the  compressing  engine,  and  a  like  computation  applied  to  the  result  in  the  air 
engine  will  give  the  power  attained  in  the  compression  of  the  air.  Column  7  gives 
also  the  mean  effective  resistance  for  the  compression  of  the  stroke  in  compressing 
air — isothermally — from  a  pressure  of  14.7  Ibs.  to  any  given  pressure,  and  column 
8  gives  the  theoretic  temperature  of  the  air  after  compression — adiabatic — to  the 
given  pressure. 

*  In  some  operations  the  air  will  become  so  cooled  that  it  will,  by  the  resulting  decrease  of  require- 
ment of  power  of  operation,  fully  compensate  for  the  friction  of  the  compressing  machine. 


COMPRESSION    OF   AIB.  997 

To    Compute    IH?    "with,    the    Elements    of   the     Preced- 
ing   Table. 

Assume  a  cylinder  40  ins.  in  diameter,  with  4  feet  stroke  of  piston,  in  which  air 
is  compressed  by  75  revolutions  at  75  IDS.  pressure  per  sq.  inch.  Area  of  cylinder, 
less  .5  that  of  piston  rod,  1250.  sq.  ins.  and  mean  pressure  per  stroke  of  piston  as 
per  table  (column  5)  35.23. 

Then  1250  X  35.23  X  75  X  4  X  2-7-33000  =  800.6  IP. 

Efficiency  of  Engine  of  Operation. — The  efficiency  of  an  engine  is  the  per 
cent,  of  power  developed  by  it,  that  it  bears  to  that  required  to  compress  the 
air,  the  loss  by  leaks,  friction  in  pipes,  of  parts  and  heated  air  from  the  en- 
gine-room (varying  with  the  weather  and  the  season),  including  that  of  the 
driving  engine. 

Compressed  air  can  be  transmitted  with  great  facility,  provided  the  trans- 
verse area  of  the  conduit  is  proportioned  to  the  volume  and  pressure  of  the 
flow,  and  the  suitability  of  the  interior  surface  of  it  for  its  transmission. 
Under  such  conditions,  the  volume  of  the  external  flow  or  discharge  of  air 
may  be  computed  by  the  volume  of  the  cylinder  of  the  air  engine  and  the 
number  of  strokes  of  its  piston,  less  the  loss  and  friction  of  the  flow,  which 
may  be  estimated  at  5  per  cent. 

Theoretical  Efficiency  of  the  compression  and  delivery  pf  air  T-f-<=zE.  T  and 
t  representing  the  absolute  temperatures  of  the  air  at  its  entrance  into  the  operating 
cylinder  and  its  flow  from  the  compressor. 

In  order,  then,  to  increase  the  efficiency,  the  heat  evolved  during  compression 
of  the  air  must  be  abstracted,  or  by  operating  at  a  lower  pressure. 

Practical  Efficiency  is  the  difference  between  the  power  developed  by  the  dis- 
charged air  and  that  expended  in  its  compression,  and  in  operation  at  a  low  speed 
of  compressing  engine  and  under  a  pressure  of  but  from  60  to  75  Ibs.  an  efficiency 
of  .9  has  been  attained. 

Spray  injection  of  cold  water  into  a  cylinder  Is  more  effective  than  a  water 
jacket,  and  by  compressing  the  air  in  two  or  more  cylinders,  and  cooling  it  between 
them,  the  work  lost  or  expended  in  the  heating  of  the  air  by  its  compression  is 
much  reduced.  Hence  compound  compression  with  inter-coolers  has  been  intro- 
duced with  advantage.* 

If  air  is  flowing  with  uniformity,  a  like  weight  of  it  flows  through  each  trans- 
verse section  per  section.  Hence,  G  a  V  =  W;  G  representing  weight  of  a  cube  foot 
of  air  in  Ibs.  ;  a,  area  of  transverse  section  in  sq.feet;  V,  velocity  in  feet  per  second; 
and  W,  weight  of  air  in  Ibs.  per  second. 

Friction  of  Air  in  Long  Pipes. 
_V!L  /.coco  d.  C  *  _VM.    =  d.  MM,  d»  0  >  =  ^     y 

iooood*C         V  ^  10 ooo  h 

representing  volume  of  air  delivered  in  cube  feet  per  minute  ;  L  =  length  of  pipe  in 
feet ;  d==  diameter  of  pipe  in  inches ;  and  C  =  coefficient  as  per  following  table  : 
»"        -SSli-S"  .5*    2-5"     .6513.5"  -78715"      .934!  8"    I-I25JI2''     x.26|2o"     1.4 
1.25      .42|2'        .5653"        .7314'        -84l6'    i.       |io       1.2     |i6        i.34l24        MS 

For  fifth  power  of  d,  see  pp.  303,  304. 

ILLUSTRATION. — It  is  required  to  transmit  1200  cube  feet  of  free  air  per  minute, 
at  75  Ibs.  gauge  pressure,  through  a  pipe  4  ins.  in  diameter  and  1000  feet  in  length; 
what  is  the  additional  pressure  required  to  overcome  the  friction  in  the  pipe? 
1200  X- 1639  (col.  2,  Table,  p.  995)  =  196.68  cube  feet. 

'96.682x1000         =          lu 

looooX  1024  (45)  x.84 


Mr.  Unwin  gives  the  following :  .0027  i  -j-  3  -i- 10  d  =  C.    d  representing  diameter 
of  pipe  in  feet,  and  C  a  constant,  due  to  diameter  of  the  pipe. 

For  pipes  less  than  one  foot  in  diameter,  .5.    C  =  . 00435,  .656  =  .00393,  and  for 
.98  feet  =  .oo  351. 

*  x88z.    Norwalk  Iron  Works  Co.  claim  to  have  first  constructed  Compound  Compressors. 


COMPRESSION    OF    AIR. 

'J?o    Compute    Loss    of   Head    in    Flow    of  A.ir    in    Long 
JPipes. 

V2  4.  I 

—  C  X  ^-  =  h.  V  representing  velocity  of  air  in  feet  per  second,  C  as  above,  I 
length,  d  diameter  of  pipe,  and  h  head,  all  in  feet. 

Assume  a  pipe  having  a  diameter  of  .5  foot  and  a  length  of  1000  and  the  velocity 
of  the  air  10  feet  per  second. 

C  =  . 0027  ( i  -f"— )  =.00432.     Then,— x  .00432  x  -- '      —  =  53-71  feet. 

Assume  the  transmission  of  1200  cube  feet  of  free  atmospheric  air  per  minute, 
through  a  pipe  4  inches  in  diameter  and  1000  feet  in  length,  under  a  gauge  press- 
ure of  73.5  Ibs.  persq.  inch;  what  will  be  the  additional  pressure  or  head  required? 

1200  cube  feet  of  free  air -r- 73' •  '4'7  =  1200-^-6  =  200  feet  at  73.5  Ibs.  46  = 
1024  and  C  for  4  ins.  =  .84. 

2oo2Xiooo  /looooX  1024  X  .84  X  4-65 

Then =  4.65  Ibs.  head,  and     / — - — -  = 

10  ooo  X  1024  X  .84  V  1000 

200  cube  feet. 

If,  however,  this  volume  of  free  air  was  under  a  pressure,  the  volume  of  free 
during  its  transmission  would  be  due  to  the  pressure.     Thus,  if  it  was  58.8  Ibs. 

gauge,  the  volume  would  be  —        -^-7-  =i  5,  and  200  X  5  =  1000  cube  feet. 

Loss  of  Pressure  per  Mile  of  Pipe. 

II  V2    \ 

P1   /[i ,  )  =  P-     P1  =  conventional  pressures  as  given  below ;  V  repre- 

V  \        14  072  d/ 

senting  initial  velocity  in  feet  per  second,  d  diameter  of  pipe  in  feet,  and  P  terminal 
pressure  in  Ibs.  per  square  inch. 

Assuming  initial  velocities  of  25,  50,  and  100  feet  per  second  and  initial  pressures 
of  50,  loo,  and  200  Ibs.  absolute. 


e  air 
58.8  Ibs.  per 


Diameter  of  Pipe,  One  Foot. 


Diameter  of  Pipe,  Two  Feet. 


Initial 
Velocity. 

Terminal  Pressure  =  P. 

Initial 
Pressure 
lost  in 

Initial 
Velocity. 

Terminal  Pressure  =  P. 

Initial 
Pressure 
lost  in 

' 

P^So. 

Pi  =  100. 

PJ=200. 

One  Mile. 

PI  =  5o. 

P*  =  100. 

PI  =  200. 

One  Mile. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

Per  cent. 

Feet. 

Lba. 

Lbs. 

Lbs. 

Per  cent. 

25 

48.8 

97-7 

195-4 

2.4 

25 

49-4 

98.9 

197.8 

1.2 

50 
100 

45-3 
26.9 

90.0 
53-8 

181.2 
107.6 

9.4 
46.2 

50 

100 

47-7 

40.1 

95-4 
80.3 

190.8 

160.6 

4-6 
19.8 

ILLUSTRATION. — Assume  initial  pressure  50  Ibs.  per  sq.  inch,  velocity  100  feet  per 
second,  diameter  of  pipe  or  conduit  one  foot. 

50    /  f  i —  J  =  50  x  V  -29  =  26.92  Ibs.  terminal  pressure. 

Hence,  if  50  —  26.92  =  23.08,  100  =  46.2  per  cent,  loss  in  one  mile. 

The  per  cent,  loss  in  one  mile  is  the  same,  whatever  the  initial  pressure,  the 
velocity  increasing  and  the  density  decreasing  with  the  length  of  the  pipe. 

Results  observed  by  Prof.  A.  B.  W.  Kennedy,  M.Inst.C.E.,  in  the  operation  of  a 
plant  of  six  Compound  cylinder  engines,  each  operating  two  compressors,  having  a 
combined  capacity  of  2000  IP. 

For  a  distance  0/3.1  miles,  through  a  pipe  n.8  inches  in  diameter. 

At  the  termination  of  the  flow  of  air  as  it  was  about  to  enter  the  motor,  it  was 
heated  from  a  coke-burning  stove.  Compression  of  the  air  88.2  (73 .5-1-14.7)  Ibs. 
per  sq.  in.  at  a  temperature  of  150°  reduced  to  66.15  IDS->  and  delivery  of  the  com- 
pression cylinders  348  cube  feet  of  air  at  atmospheric  pressure  and  70°  tempera- 
ture per  IIP  per  hour. 

The  average  loss  was  3  per  cent,  velocity  of  air  1550  feet  per  minute,  with  an 
IIP  of  1250. 


COMPRESSION   OF   AIE. 


999 


Summary  of  Results  of  two  experiments,  each  with  cold  and  heated  air,  in  the 
Transmission  of  Compressed  air  at  Paris,  1889,  for  a  distance  of  4  miles.  Motor 
10  IP  and  pressure  of  air  reduced  to  66  Ibs.  One  IIP  gave  .845  IIP  in  compression, 
or  348  cube  feet  of  air  per  hour  from  atmospheric  pressure  of  88.2  Ibs. 

A  summary  of  other  results  showed  that  a  small  motor  at  a  distance  of  4  miles 
from  the  compressor  indicated  i  IP  for  2  IP  at  the  motor,  or  2.5  IP  when  the  air 
was  not  heated  before  entering  the  motor. 

Heating  the  air  caused  a  saving  of  225  cube  feet  of  it  per  IIP,  at  a  cost  of  4  cents 
per  IIP. 

The  exhausted  air  from  a  motor,  when  that  in  the  pipe  is  even  but  slightly  heated, 
will  be  so  much  reduced  in  temperature  as  to  be  available  for  cooling  and  even 
freezing  application,  so  great  is  the  effect  of  instantaneous  expansion  of  the  air 
when  exhausted  that  ice  is  formed  in  the  air-ports  of  the  cylinder,  and  hence  the 
operation  of  a  plant  at  high  pressures  or  above  90  Ibs.  is  held  to  be  objectionable. 

By  operating  at  full  pressure,  the  high  velocity  of  the  flow  mechanically  restricts 
the  deposit  of  ice  crystals,  but  inasmuch  as  the  useful  effect  decreases  with  an  in- 
crease of  pressure,  it  is  held  by  Robert  Zahner  and  others  that  60  Ibs.  is  the  limit 
unless  the  operating  air  is  reheated. 

When  air  is  operated  expansively  at  half-stroke,  the  temperature  falls  160°,  and 
at  one-fourth  stroke  284°. 

Compressed  air  is  the  only  power  of  general  application,  as  it  can  be  applied, 
extended,  and  distributed  without  restriction  to  distance,  course,  elevation,  and 
depression,  and  under  ground  or  water,  and  under  some  of  these  conditions  the 
only  power  at  all  practicable  of  operation.  Alike  to  water  it  can  be  stored,  which 
condition  is  unattainable  with  steam. 

Heating  Compressed  Air. — When  compressed  air  has  been  transmitted  to 
the  point  where  it  is  to  be  employed,  an  increase  of  power  is  attainable  by 
the  addition  of  heat  to  it,  before  it  is  applied. 

Absolute  temperature  is  461.2°.  Hence  when  the  air  is  60°,  the  absolute 
temperature  is  461.2  +  60  =  521.2,  and  when  it  is  —  30°,  it  is  431.2°  abso- 
lute. 

Loss  of  Efficiency.    Initial  Temperature  of  Air  62°. 


Pressure. 

Final  Tem- 
perature. 

Efficiency. 
Reduced.    I     Loss  of. 

Pressure. 

Final  Tem- 
perature. 

Effic 
Reduced. 

cncy. 
Loss  of. 

Lba. 
29.4 
44.1 
58.8 

Degrees. 
178 
258 
321 

Per  cent. 
82 
73 
67 

Per  cent. 
18 
27 
33 

Lbs. 
73-5 
147 

Degrees. 
373 
559 

Per  cent. 
63 
51 

Per  cent. 
37 
49 

Assuming  efficiency  of  Compression  and  also  that  of  the  Engine  at  80  per  cent, 
the  resultant  efficiency  of  the  combination  at  147  Ibs.  pressure  = °  X  51  = 


32.6  per  cent.    At  44.  i  Ibs.  the  efficiency : 


80X80 


X  82  =  52. 5  per  cent. 

(D.K.Clark.) 


Air  expands  at  constant  pressure  from  32°  to  212°  .002036  per  degree  of  tem- 
perature. 

Efficiency"   of  Cooling. 

Cooling  of  compressed  air  effects  a  saving  of  power  required  for  its  compression, 
and  aids  in  the  lubrication  of  the  piston.  It  is  most  effective  at  low  pressures. 
Thus  at  15  Ibs.  pressure  the  temperature  consequent  upon  compression  is  raised 
from  60°  to  177°  and  from  75°  to  90°,  but  39°. 

When  air  is  heated  by  compression  and  water  is  introduced  it  becomes  saturated, 
and  when  after  performing  its  work  it  is  exhausted  into  the  open  air,  it  expands  so 
rapidly  that  its  temperature  is  frequently  reduced  below  zero,  and,  as  a  result,  the 
moisture  in  the  air  gravitates  as  ice  in  the  exhaust  passage  of  the  engine,  and  its 
capacity  is  choked  and  even  closed.  Hence  it  is  imperative  that  the  air  of  com- 
pression should  be  maintained  as  dry  as  practicable. 


IOOO 


COMPRESSION    OF   AIR. 


Air    Receivers. 

The  operation  of  a  Receiver,  if  of  sufficient  volume,  is  to  reduce  the  effect  of  the 
pulsations  consequent  upon  the  stroke  of  the  compressor,  for  without  it  the  press- 
ure of  the  air  at  its  delivery  from  the  compressor  to  a  pipe  would  be  momentarily 
in  excess  of  the  average  pressure  of  operation.  This  effect  may  be  reduced  by  in- 
creasing the  length  of  the  pipe,  also  by  the  attachment  of  a  second  Receiver  at  the 
termination  of  a  long  pipe. 

As  the  presence  of  a  Receiver  checks  the  flow  of  the  compressed  air,  some  of  the 
water  which  is  in  the  air,  which  otherwise  would  be  borne  with  the  current,  is 
precipitated. 

Efficiency    of   Compressed    Air    Engines. 

At  the  ordinary  pressure  of  60  Ibs.  per  sq.  inch,  the  decrease  in  resistance  effected 
by  the  cooling  of  the  air  is  held  to  be  equal  to  the  friction  of  the  compressor.  This 
effect  is  greater  with  high  than  low  temperatures  of  the  air,  in  consequence  of  the 
higher  temperature  at  the  higher  pressures  of  the  air. 

Adiabatic  Expansion. 

The  more  air  is  in  compression  and  the  friction  of  its  passage  in  the  pipe  in- 
creased, the  efficiency  of  compression  is  increased. 

The  following  table  gives  the  Lowest  Pressures  which  should  be  operated  in  the 
Compressor,  with  varying  amounts  of  friction  in  the  pipe: 

& 


Lbs. 
20.5 
29.4 


70.9 
64-5 


it 

§| 

II 

if 

4 

4e 

£3 

n 

if 

£  c 
£.2 

s| 

o  ^ 

i  >* 

|| 

Lbs. 

38-2 
47 

P'r  ct 
60.6 
57-9 

Lbs. 
14.7 
17.6 

Lbs. 
52.8 
61.7 

P'r  ct. 

55-7 
539 

Lbs. 
20.5 
235 

Lbs. 
7°-5 
76.4 

P'rct. 
52-5 
51-3 

Lbs. 
26.4 
29.4 

Lbs. 
82.3 
88.2 

P'r  ct. 
50.2 
49 

Operation    and     Mean.     Results    of    a    Hardie    Motor    at 
Rome,   1ST.   Y.,   1895. 


Elements. 

One 
Run. 

Mean  of 
Screw. 

Elements. 

One 
Run. 

Mean  of 
Screw. 

Pressure  persq.  inch. 
Distance  run    ...    . 

1.41 

-3    eg 

1  01  Ibs. 

2.  6  1  miles 

Difference    in    tem- 
perature in  heater 

Temperature    of   air 
entering  heater 

6"?  2° 

68.2° 

at   start  and  fin- 
ish   . 

40° 

29  8° 

Temperature    of   air 

IIP  

leaving  heater  
Temperature    of   air 
at  exhaust  

240.3° 
130  7° 

219.6° 
123.5° 

Water  supplied  
Air  per  IIP  per  min- 
ute   

29-37 
6 

21.46  Ibs. 
6  5  cube  ft 

Heat    absorbed    in 
heater... 

I7S.I0 

117.1° 

Power  from  heater.. 

43-2 

^o.ipercwt. 

The  power  obtained  from  the  Reheater  was  about  45  per  cent. 

(Frederick  C.  Weber.) 

I»o\ver    Required    to    Compress    Air    at    tlie    Uniform 
Temperature    of   62°. 


Pressure 
per 
Sq.  Inch. 

HP  per 
Cube  foot 
of  Com- 
pressed Air 

Volume  of 
Compres'd 
Air  per 
inin.perH? 

Pressure 
per 
Sq.  Inch. 

H?per 
Cube  foot 
of  Com- 
pressed Air 

Volume  of 
Compres'd 
Air  per 
min.  per  H? 

Pressure 
per 
Sq.  Inch. 

H?per 
Cube  foot 
of  Com- 
pressed Air 

Volume  of 
Compres'd 
Air  per 

Lbs. 

No. 

Cube  feet. 

Lbs. 

No. 

Cube  feet. 

Lbs. 

No. 

Cube  feet. 

30 

.089 

11.25 

120 

1.07 

•938 

210 

2-37 

.422 

45 

.211 

4-73 

135 

1.27 

.788 

225 

2.61 

•384 

60 

.356 

2.88 

ISO 

1.48 

.667 

240 

2.84 

•352 

75 

.516 

1.94 

165 

1.69 

•591 

255 

3-°9 

•324 

90 

.69 

i-45 

1  80 

X.9I 

•523 

270 

3-34 

•3 

105 

.874 

1.14 

195 

2.14 

.468 

300 

3-84 

.26 

At  the  Mont  Cenis  tunnel,  64  cube  feet  of  compressed  air  per  minute  through  a  cast-iron  pipe  7.625 
ins.  in  diameter,  5325  feet  in  length,  and  under  a  pressure  of  838  Ibs. :  the  loss  of  the  head  including 
leaks  and  friction  was  but  3.5  per  cent.,  and  in  a  length  of  pipe  of  20  ooo  feet  the  loss  was  but  5  per  cent, 
of  the  initial  pressure. 

(D.  K.  Clark.) 


COMPRESSION    OF    AIR. 


1001 


Gauge  Press- 
ure at  en- 
trance to  the 
Pipe. 

H 

5'  2- 

VI    OO  OOvp  VQ 
*.          ON  N    00 

&- 

ra 

OOVI    ON*.    N 
vi    ON  00  M 

58 

*.    M    ON  H  *. 

8 

M  VI            M     ON 

8 

OJ    M  VO  VI  OJ 

8 

OJ    W    10    M 

O    ON  M    ON  TO 
OJ    ONOJvp^ 

8 

***h 

1 

IFf* 

5 

w 

r 

Gauge  Press- 
ure at  en- 
trance to  the 
Pipe, 

^ 
s 

1  Diameter 
Pipe  in  Ir 

*.    ON  TO  M 

JP* 

FR 

ss££s 

o 

Oovi  vi  vi  vi 

•••    TO  ONOI    M 

Q 

vl  ui  vO    N  Oi 

OJ  OJ    M     W     W 
M    O  VO    TOvi 

8 

TOV|  VI     ONO-I 
HI  *.    p    ON  00 
On    (0    ON  ONVI 

1 

M    O  vO    TO  ON 

<b  *•  bovb  ON 

8 

M    M    10    M    (0 
Ot           ON        *• 

1 

V|    ON  ON  ON  ON 
O  vi  tg  -»«.    M 

ON  00  W  01    TO 

b 

VI  v»  OJ    00 

8 

Gauge  Press- 

ure  at  en- 

trance  to  the 
Pipe. 


(ji  -^  OJ  S3  >- 
01  VI  VI  VO  ON 
N  M  00 


OJ    M    00  ON*. 
M    M    00  00  0 


w  to  M 

ON^    O^ 


N 

VIVIMVOOJ 


K3  vO  On    M    ON 
O^'ONbivO0*^ 


vO  OJ  vo  ^  \O 


1002 


COMPRESSION    OF   AIR. 


Volume     of*    Free     Air    in     Cvttoe     Feet     RecfuiredL     in. 
Motor    per    ItP    per    ]Miiixite.* 

Without  Reheating. 
Gauge  Pressure  in  Pounds  at  60°. 


Point 
of 
Cut-off. 

I 

•75 
.66 

•5 
-33 
•25 

* 

30 

40 

50 

60 

70 

80 

90 

IOO 

no 

125 

150 

17-05 
I3'I 

12.6 

10.85 
9-5 
9.1 

31.2 
25-6 
24.8 
25-8 
37 

23-3 

18.7 

17-85 
16.4 

20.6 

21.3 
17.1 

16.2 

14-5 
15-2 

15.6 

2O.  2 

16.1 
15-2 

12.9 
13-4 

19.4 

15-47 

12.8 

11.85 
13-3 

18.8 

14.2 
12.3 
11.26 
11.4 

18.42 
14.6 
13-75 
"•93 
10.8 
10.72 

18.1 
14-35 
13-47 
11.7 
10.5 
10.31 

17.8 

14-15 
13-28 
11.48 

IO.2I 
10 

17.62 

13.98 

13.08 

"•3 

IO.O2 

9-75 

17.4 
13-78 
12.9 
n.  i 
9.78 
9.42 

To  these  results  is  to  be  added  the  per  cent,  of  clearance  as  determined  in  each 
case. 

If  the  air  is  reheated,  the  volume  in  the  table  will  be  decreased,  depending  upon 
the  temperature  of  the  air  at  admission,  and  it  is  proportional  to  T-r-T",  T  repre- 
senting absolute  temperature  at  60°,  and  T'  460  -f-  temperature  of  air  at  admission  to 
motor. 

Hence,  if  the  air  is  reheated  to  300°,  the  volume  in  the  table  is  to  be  multiplied  by 

460+60   _  520^_ 

460  +  300      760 
To  Ascertain  the  Economical  point  of  Cut-off  for  the  Gauge  Pressures  in  the  Table. 

An  inspection  of  it  will  show.  Thus,  at  60  Ibs.  the  least  volume  of  free  air  per 
IIP  is  at  .33  cut-off,  and  at  80  Ibs.  at  ,25.  (Frederick  C.  Weber.) 

Loss  of  Pressure  toy  Friction  of*  Compressed.  Air  in  Pipes, 

In  Pounds  per  Square  Inch  for  1000  Feet  of  Pipe. 

Volume  of  Free  Air,  Compressed  to  a  Gauge  Pressure  of  60  Ibs.  per  Square  Inch, 
Delivered  per  Minute. 


Diana, 
of  Pipe. 

So 

75 

IOO 

125 

ISO 

200 

250 

300 

400 

600 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

10.4 

— 

— 

— 

— 

— 

— 

— 

1.25 

2.63 

5-9 

— 

— 

— 

— 

— 

— 

1.5 

1.22 

2-75 

4-59 

7-65 

ii 

— 

— 

— 

— 

3 

•35 

•79 

1.41 

2.2 

3-17 

5-64 

8.78 

— 

— 

2-5 

.14 

•32 

•57 

•9 

1.29 

2-3 

3.58 

5^8 

9.2 

— 

3 

— 

.11 

.2 

•  31 

•44 

.78 

1.23 

1.77 

3-H 

7-05 

3-5 

— 

— 



•  15 

.21 

•38 

•59 

•85 

3-4 

4 

— 

— 



— 

.2 

•3i 

•45 

'.Bo 

1.81 

5 

— 

— 

— 

— 

— 

— 

.1 

•i5 

.26 

•59 

Cube  Feet. 

Diam. 
•f  Pipe. 

800 

1000 

I2OO 

1500 

1800 

2000 

2500 

3000 

4000 

5000 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

3-5 

6.03 

— 

— 

— 

— 

— 

— 

— 

— 

— 

4 

3.22 

5.02 

7-23 

"•3 

— 

— 

— 

— 

— 

— 

5 

1.04 

1.63 

2-35 

3-66 

5-28 

6-5 

10.2 

— 

— 

— 

6 
8 

.41 
.1 

.64 
.16 

•93 
•23 

1.46 

•37 

2.09 
•53 

2.59 
•65 

4.06 
1.02 

1.47 

10.3 
2.61 

4~o8 

10 

— 

— 

— 

.13 

•17 

.21 

•33 

•47 

.84 

1  -3 

12 

— 

— 

— 

— 

— 



•13 

.19 

•34 

•53 

*  Copyrighted. 


(Rand  Drill  Co.,  F.  A.  Halsey.} 


COMPRESSION    OF    AIR. 


1003 


Dimensions  and  Elements  of  Air  Compression, 

Operated  by  Steam. 


DlAMK 

Air. 

TEH  OF  CYL 

Compres- 
sions. 

INDKR. 

Steam. 

Stroke 
of 
Piston. 

Revolu- 
tions per 
Minute. 

Volume 
dis- 
charged. 

Steam. 

DlAMBTEB 

Exhaust. 

OF  PlPl 

Air. 

8. 

Water. 

IP 

Ins. 

Ins. 

Ins 

Ins. 

No. 

Cube  ft. 

Ins. 

Ins. 

Ins. 

Ins. 

No. 

8 

5 

8 

10 

200 

116 

2 

2-5 

2 

•5 

18 

IO 

6.75 

IO 

12 

180 

195 

2-5 

3 

2-5 

•75 

30 

J4 

9-5 

14 

16 

150 

427 

3 

4 

4 

57 

16 

9-5 

16 

16 

I5° 

558 

3 

4 

4 

82 

20 

i3-5 

20 

24 

no 

960 

5 

6 

5 

•25 

145 

22 

22 

24 

no 

1160 

5 

6 

5 

.25 

175 

26* 

'7-5 

24 

30 

90 

1659 

6 

8 

6 

•25 

215 

28 

»7-5 

28 

30 

9° 

1924 

8 

10 

6 

•25 

300 

32 

21.5 

30 

36 

80 

2686 

8 

10 

8 

•5 

350 

Horse-P  ower  Required  to  Compress  One  Cn"be  IToOt 
of  Free  A_ir  per  Minute  to  a,  GHven  Pressure,  and. 
the  Power  Required  to  JJeliver  One  Cu/be  Foot  of 
A.ir  at  a  Given  Pressure. 


Compressing  to 
Given  Pressure. 

Delivering  to 
Given  Pressure. 

Compressing  to 
Given  Pressure. 

Delivering  to 
Given  Pressure. 

Gauge 
Pressure. 

Constant 
Temper- 
ature. 

Without 
Cooling. 

Constant 
Temper- 
ature. 

Without 
Cooling. 

Gauge 
Pressure. 

Constant 
Temper- 
ature. 

Without 
Cooling. 

Constant 
Temper- 
ature. 

Without 
Cooling. 

Lbs. 

H? 

IP 

H? 

IP 

Lbs. 

H? 

H? 

H? 

H? 

5 

.0188 

.0196 

.0251 

.0263 

55 

.0994 

.127 

.4711 

.6023 

IO 

.0332 

.0361 

•0559 

.064 

60 

.104 

.1342 

.5285 

.6818 

15 

•045 

.0502 

.091 

.1014 

65 

.1081 

.1403 

.5861 

.7608 

20 

-055 

.0628 

.1299 

.1483 

70 

.1124 

.1472 

.6481 

.8483 

25 

.0637 

.0742 

.1719 

.2004 

75 

.1163 

•'537 

.7095 

•938 

30 

.0713 

.0846 

.2168 

•2573 

80 

.1193 

•1597 

.7684 

.0291 

35 

.0782 

.0942 

.2644 

.3189 

85 

.1224 

•1655 

.8304 

•  1231 

4° 

-0843 

,  1032 

-3137 

.3842 

90 

.1256 

.171 

.8944 

.2176 

45 

.0895 

.1117 

•3637 

-4535 

95 

.1289 

.1763 

.9616 

.3148 

So 

.0951 

•1195 

.4185 

.526 

100 

.1312 

.1815 

1.0243 

.4171 

To  these  must  be  added  a  per  cent,  due  to  the  estimated  friction  of  the  com- 
pressor. 

Mean  and  Terminal  Pressures  of  Compressed  Air  at 
Several  Points  of  .Expansion  and  at  Given  Grange 
Pressures. 

When  the  Pressure  is  Less  than  Atmosphere  it  is  Given  Absolute. 


Cut 

Pressure  50. 

Pressure  60. 

Pressure  70. 

Pressure  80. 

Pressure  90. 

Pressure  100. 

off 
at 

Mean. 

Termi- 
nal. 

Mean. 

Termi- 
nal. 

Mean. 

Termi- 
nal. 

Mean. 

Termi- 
nal. 

Mean. 

Termi- 
nal. 

Mean. 

Termi- 
nal. 

Point. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.125 

4-5i 

3-47 

7-51 

4.01 

10.51 

4-54 

I3-5I 

5-o8 

16.52 

5.61 

I9-5I 

6.15 

.2 

13.84 

6.74 

17.7 

7-77 

22.06 

8.81 

26.6 

9-85 

30.78 

10.88 

35  H 

11.92 

•25 

18.45 

9-23 

23-6 

10.65 

28.74 

12.07 

33-89 

13-49 

39-  °4 

14.91 

44.19 

i-33 

•33 

25.84 

13-83 

32.13 

.96 

38.41 

3-09 

44.69 

5-22 

50.98 

7-35 

57.26 

9.48 

•375 

29.07 

i-34 

35-85 

3-85 

42.63 

6.36 

49.41 

7.88 

56.2 

11.39 

62.98 

13.89 

.5 

37-  J  2 

7-49 

45-14 

13.26 

53-  16 

17 

61.18 

20.  8  1 

69.19 

24.56 

77-21 

28.33 

.625 

42.99 

i8.53 

51.92 

23.69 

60.84 

28.85 

69.76 

34.01 

78.69 

39.16 

87.61 

44-32 

•75 

46.98 

28.34 

56-52 

35-01 

66.05 

41.68 

75-59 

48.35 

85.12 

55-02 

94.66 

61.69 

(Frank  Richards.  ) 

For  Volume  of  Air  Transmitted  in  Cube  Feet  per  Minute  in  Pipes  of  Diameters  from  i  to  24  inches, 

see  Frank  Richards,  p.  109. 

*  At  elevations  of  2000,  6000,  and  10000  feet  above  the  sea-level  the  capacity  and  IP  of  this  enj 
vrould  be  reduced  respectively  to  1560,  1373,  and  1200  feet  capacity  and  207, 195,  and  182  HP. 


1004 


COMPRESSION   OF   AIR. 


Heat    Prodnoed   toy    Compression    of  Dry    .Air. 

Without  Cooling. 


Pressure 
above 
Atmos- 
phere. 

Volume. 

Tempera- 
ture of 
the  Air 

Pressure 
above 
Atmos- 
phere. 

Volume. 

Tempera- 
ture of 
the  Air. 

Pressure 
above 
Atmos- 
phere. 

Volume. 

Tempera- 
ture of 
the  Air. 

Lbs. 

Cube  feet. 

O 

Lbs. 

Cube  feet. 

0 

Lbs. 

Cube  feet. 

0 

o 

i. 

60 

22 

.5221 

218.3 

88.2 

•  2516 

454-5 

1.47 

.9346 

74-6 

29.4 

.4588 

255.1 

102.9 

.2288 

490.6 

3-67 

•8536 

94.8 

36.7 

.4113 

287.8 

117.0 

.2105 

523.7 

7-35 

•7501 

124.9 

44-1 

•3741 

3I7.4 

132.3 

•1953 

554 

n.  ii 

.6724 

151.6 

58.8 

'3194 

369-4 

205.8 

.1465 

681 

14.7 

.6117 

175-8 

73-5 

.2806 

4I4.5 

279.3 

•"95 

781 

The  presence  of  moisture  will  increase  these  results  as  it  increases  both  the 
specific  heat  and  the  heat-conductive  capacity  of  the  air.  ( W.  L.  Saunders.) 

Kffioienoy  of  an  Engine.— With  perfect  expansion,  without  the  air 
receiving  any  increase  of  temperature,  the  efficiency  at  pressures  above  the  at- 
mosphere and  friction  in  pipes  are  estimated  as  follows  : 

Per  Cent. 


Friction 
estimated. 

14.7 

29.4 

44.1 

58.8 

73-5 

88.3 

Lbs. 

2.0 

5-8 
14.7 

Per  cent. 
70.44 
57-  *4 

Per  cent. 
68.  81 
64-49 
48.53 

Per  cent. 
64.87 
62.71 
55-13 

Per  cent. 
61.48 
60.  12 
55.64 

Per  cent. 
58.62 
57-73 
54-74 

Per  cent. 
66.23 
56.59 
53-44 

As  friction  increases,  the  most  efficient  and  economical  pressures  increase. 


Lowest  Pressures  at  Compression. 

Friction. 

Compres- 
sion. 

Efficiency. 

Friction. 

Compres- 
sion. 

Efficiency. 

Friction. 

Compres- 
sion. 

Efficiency. 

Lbs. 

tl 

8.8 

Lbs. 
20.5 
29.4 
38.2 

Per  cent. 
70.92 
64.49 
60.64 

Lbs. 
11.7 

& 

Lbs. 

£, 

61.7 

Per  cent. 
57.87 
55-73 
53.98 

Lbs. 
20.5 
23-5 
26.4 

Lbs. 
70.5 
76.4 
82.3 

Per  cent. 
53.52 
51.26 
50.17 

13.134  cube  feet  of  air  at  62°  (table,  p.  521)  weigh  i  lb.,  and  air  at  60°  compressed 
to  half  its  volume  evolves  116°  heat,  and  as  the  specific  heat  of  air  under  constant 
pressure  is  .2377,  which  x  116  =  27.573  heat  units,  produced  by  the  compression  of 
i  lb.  or  13.134  cube  feet  of  free  air  into  one-half  its  volume  :  Hence,  27.573  X  778  = 
21452  foot-lbs.,  and  as  heat  and  mechanical  energy  are  held  to  be  convertible 

terms,  "        =.65  IP  produced  or  lost  by  the  compression  of  i  lb.  of  air.    Inas- 
33000 

much,  then,  as  the  compression  of  air  develops  heat,  and  if  the  temperature  of  the 
compressed  air  is  reduced  to  that  of  the  atmosphere  from  which  it  is  drawn  before 
being  used,  the  mechanical  effect  of  this  difference  in  heat  is  work  lost. 

"Work    Lost    toy    Heat    of   Compression. 

Air  assumed  to  be  cooled  to  temperature  of  atmosphere  between  stages  of  com- 
pression and  without  effect  of  jacket  cooling. 


Gauge 
Pressure. 

First 
Stage. 

Second 
Stage. 

Third 
Stage. 

Fourth 
Stage. 

Gauge 
Pressure. 

First 
Stage. 

Second 
Stage. 

Third 
Stage. 

Fourth 
Stage. 

Lbs. 

Per  cent. 

Per  cent. 

Per  cent.  Per  cent. 

Lbs. 

Per  cent. 

Per  cent. 

Per  cent.  Per  cent. 

60 

23 

n.8 

— 

4-45 

800 

47-4 

26.3 

— 

14-3 

80 

25-3 

I3-I 

— 

4.8 

IOOO 

49-2 

28.1 

— 

14.4 

100 

27.6 

14.6 

— 

7.41 

1200 

Si.6 

28.6 

— 

14.8 

200 

34-4 

18.9 

— 

8.27 

I4OO 

52 

29.4 

— 

15 

400 

40.7 

22.9 

— 

u 

1600 

53-3 

3° 

— 

15.5 

600 

44.6 

24.6 

— 

13  i 

I800 

54 

30.6 

— 

16.1 

The  power  of  compressing  at  high  pressure  is  not  proportional  to  the  pressure. 

(Frederick  C.  Weber.) 


COMPRESSION    OF   AIR. 


1005 


Loss    of    IPressure    throngli    Friction    of  -A.ir    in. 
I^ipes. 

Per  100  Feet  of  Length  (Initial  Gauge  Pressure  80  Lbs.  at  Receiver). 


Equivalent 
Volume  of 
Free  Air 
Discharged. 

X 

1-25 

Ins. 
.12 

•45 

i 

„ 

2 

2.5 

DIJ 

3 

LMKTR 

4 

a  OF  I 

5 

>IPK. 
6 

7 

8 

10 

12 

14 

Per  minute. 
25 
50 
75 
zoo 

200 
300 
400 
500 
750 
1000 

I  500 

2OOO 
3000 
4000 
5000 
6000 
7500 
10000 

Ins. 
.24 

i 
2.4 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

.18 
•4 
•7 

•13 

- 

- 

- 

— 

- 

- 

- 

- 

— 

- 

- 



- 

2.15 
3-3 

.67 

2-5 

27 
4 

4 

.06 
.1 

.22 

•4 

i 
i.  60 
3-70 

.07 

.12 

•3 
•  5 

I  .2 
2 

.012 
•03 
•05 
.12 

.2 

i-3 

3 

.013 
.023 
•052 

.22 

1.4 
2-5 

.012 
.027 
.048 
•US 
.2 

•3 

1.25 

.017 
.036 
.07 
.1 
•15 
.22 

•4 

.015 
.026 

•°4I 
.06 

.09 
•'7 

.012 
.018 
.028 
.04 
•°75 

= 

5 

- 

i 

- 

- 

- 

(Frederick  C.  Weber.) 

ILLUSTRATION. — An  air  compressor  furnishes  500  cube  feet  of  free  air  per  minute 
at  a  pressure  of  80  Ibs.  per  square  inch  in  the  receiver.  If  this  air  is  used  at  the 
end  of  a  3-inch  pipe  1000  feet  in  length,  the  loss  due  to  friction  will  be  10  X  •4  =  4 
Ibs.  If  a  like  volume  of  air  were  supplied  by  the  same  compressor  at  a  like  press- 
ure and  passed  through  a  s-inch  pipe  1000  feet  in  length,  the  loss  would  be  only 
,03  x  10  =  .3  Ibs. ;  thus  illustrating  the  importance  of  using  pipes  of  large  diameter. 

Strictly,  the  loss  of  pressure  is  not  directly  proportional  to  the  length  of  the  pipe, 
but  for  all  practical  purposes  it  may  be  taken. 

Elbows  and  irregularities  in  pipes  increase  the  friction  in  excess  of  the  figures 
here  given. 

The  results  in  the  table  represent  the  loss  by  friction  in  the  pipes.  There  is  also 
a  slight  loss  due  to  friction  of  the  air  with  itself  at  the  mouth  of  a  pipe  when  it 
leaves  the  Receiver. 

Leakage. 

All  leaks  in  compressors  or  valves,  air  receivers  or  pipes,  should  be  strictly 
guarded  against  for  economy,  as  they  are  fully  as  expensive  as  steam  leaks.  When 
air,  at  60  Ibs.  pressure,  issues  from  a  leaky  joint  in  a  pipe  at  a  velocity  of  over  500 
feet  per  second  the  waste  of  it  will  become  apparent. 


IVlean  Effective  Pressures  in  the  Compressing  and. 
Delivery  of*  Free  Air  to  a  Griven  <3-axige  Pressure 
in  a  Single  Cylinder. 


Gauge 
Press- 
ure. 

Compr 
Adia- 
batic. 

ession. 
Isother- 

Gauge 
Press- 

Compr 
Adia- 

ession. 
Isother- 

Gauge  '    Compr 
Press-      Adia- 

ession. 
Isother- 

Gauge 
Press- 

Compr 
Adia- 
batic. 

ession. 
Isother- 
mal. 

Lbs. 

i 

2 

3 

4 
5 

10 

Lbs. 

1.41 

1.86 
2.26 
4.26 

Lbs. 
•43 
•95 

2.22 

4.14 

Lbs. 
'5 
20 
25 
30 

35 
40 

Lbs. 
5-99 
7-58 
9-°5 
10.39 

"•59 

12.8 

Lbs. 
5-77 
7-2 

10.72 

Lbs.        Lbs. 
45     |  13-95 
50    •  I5-05 
55    i  15-98 
60      16.89 
65    i  X7.88 
70    |  18.74 

Lbs. 
12.62 
13.48 
14-3 
I5-05 
I5-76 
16.43 

Lbs. 

85 
90 
95 

IOO 

Lbs. 
19-54 
20.05 
21.22 
22 
22.77 
23-43 

Lbs. 
17.09 

III 

18.87 
19.4 
19.92 

(Frank  Richards.) 


IOO6  COMPRESSION    OF    AIR. 

To   Compute    tlie    Steam    fressure   and."  IPoint  of  Cutting 
off   for    a    GUven    .A-ir    Compressor. 

Assume  steam  and  air  cylinders  each  22  X  24  ins.  and  temperature  of  initial  air 
62° 

Area:  378'23*x  24  — 5.26  cube  feet  per  stroke. 
1720 

- —  =  .4  —  Ibs.,  and,  if  compressed  adiabatically,  68755  (see  ante)x  .4  =  27502 

foot-lbs.,  and  assuming  friction  of  operation  at  12  per  cent.     ^-^  —  31252  foot- 

.88 
Z&s.  resistance  to  be  overcome  by  steam  pressure. 

Hence,       3I     ^—-=41.2  Z&s.,  which  corresponds  with  .2  cut-off  at  initial  press- 
ure of  80  Ibs.  gauge  pressure.  (F.  C.  Weber,  M.  E. ) 


Xo  Compute  Volume  of  One  Found  of  Dry  Air  in  Culoe 
Ifeet  and   Weight  of  One  Cxitoe   Foot  of  it  in   Pounds, 

At  Various  Temperatures  and  at  Atmospheric  Pressure. 

T  +  T'-r-  39. 819  =  volume.    T  representing  temperature  of  air  and  T'  absolute  tem- 
perature in  degrees. 


//rT  =  62°.     62° +  461° -=-39. 819=  13.134  cube  feet 

Inversely.     39. 819  -f-  62°  +  461°  = . 076  097  Ibs. 

NOTE. — For  Table  of  Volumes,  Pressures,  and  Density  at  62°  =  i,  and  for  Computation  of  Volume, 
Weight,  Pressure,  Density,  and  Elasticity  at  other  Temperatures,  see  pp.  521,  522. 

When  the  Pressure  and  Temperature  of  Air  both  vary. 

2-  7093  X  P  -i-  T  =  cube  feet  in  Ibs.     T  representing  absolute  temperature  and  press- 
ure in  Ibs.  per  sq.  inch. 

ILLUSTRATION. —What  is  the  weight  of  a  cube  foot  of  air  at  60  Ibs.  pressure  and 

ioo°?       .  

2.7093  X  60 +14. 7 -=-461° +100°  =  .3607  Ibs. 

461  °+  100°  -4-  60  -f- 14. 7 

Inversely. - - — —  =2.771  volume. 

2.7093 


Comparison    of    Single    and.    Compound    Compression. 

Assume  areas  of  cylinders  for  Single  and  Compound  compression  respectively 
loo  and  33.33  sq.  ins. ,  and  pressure  of  compression  100  Ibs.  per  sq.  inch.  Resistance 
to  cylinder  of  single  compression  =  100  x  100=  10000  Ibs.,  and  to  second  cylinder 
of  compound  compression  =  33. 33  x  100  =  3333  Ibs. 

The  resistance  upon  the  large  piston  is  its  area  multiplied  by  the  pressure  re- 
quired to  force  the  air  from  its  cylinder  into  the  less.  In  this  case  it  is  30  Ibs.  per 
sq.  inch ;  but  inasmuch  as  this  30  Ibs.  presses  upon  the  reverse  side  of  the  less  pis- 
ton, and  thus  assists  the  operation,  the  net  resistance  to  forcing  the  air  from  the 
large  into  the  less  cylinder  is  equal  to  the  difference  of  the  area  of  the  two  pistons, 
X  the  30  Ibs.  pressure,  —66.66  x  30  —  2000  Ibs. 

Hence,  the  resistance  to  forcing  the  air  from  the  larger  into  the  less  cylinder  is 
2000  Ibs.,  and  the  resistance  in  the  small  cylinder  to  the  compression  of  it  to  100 
Ibs.  =  3333  Ibs.,  the  sum  of  the  resistance  =  5333  Ibs. 

(The  Norwalk  Iron  Works  Co.) 

The  compression  of  air  develops  heat,  and  if  the  temperature  of  the  compressed 
air  is  reduced  to  that  of  the  atmosphere  from  which  it  is  drawn  before  being  used, 
the  mechanical  effect  of  this  difference  in  heat  is  work  lost. 

*  Deducting  area  of  piston-rod.  f  2  feet  stroke. 


COMPRESSION    OF   AIR. 

Isothermal    Compression. 

P  V  hyp.  log.  —  -  =  F.     P  representing  atmospheric  pressure  in  Ibs.  per  sq.  foot-=. 

14.7  X  144  =  2116.8,  V  volume  of  i  lb.  air  at  atmospheric  pressure  (62°)  =  13.  141 
cube  feet,  p  andp'  terminal  and  atmospheric  pressures  absolute  in  Ibs.  per  sq.  inch, 
and  F  foot  Ibs.  per  lb.  of  air.  Assume  p  =  80  Ibs.  per  gauge. 

Then,  2116.8  x  13-141  X  hyp.  log.  ^=27  814.7  x  1.8625  =  51  804  /ooMte. 

Acliat>atic    Compression.     One  Cylinder. 
P  V  ^-^  (j^\  ~S~—  i  =  F.    n,  omitting  cooling  of  jacket,  =  i  .  408. 

Then,  as  preceding,  27  814.7  X  3-45  X  (^\     —  i  ==  95960  X  6.44.29—1=95.960 
K.7i65  =  68755/oo«-H*. 

Compound    Air    Cylinders.      Two  Cylinders. 
Air  cooled  to  atmospheric  temperature  before  admission  to  second  cylinder. 


P  V  ^—^  (  ~-\     n  +  f  —  j     n    —2  =  F.    p2  andp3  representing  terminal  press- 

es in  ist  and  \d  cylinder** 

P  V  —  -  =  95  96o,  as  preceding,  and  p2  =  <V/Pi  X  J>3  =  37  25- 


'* 


Then,  95  960  X  f  5ZJ_£J  +  /^ilLj  —2  =  95960  X  1.30964 -1.3096-2  =  . 6192 
X  95  960  =  59  418  foot-lbs. 

For  N  Cylinders,  95  960  X  N  X  R t29  —  i  =  F.  N  representing  number  of  cylinders 
and  R  ratio  of  compression,  equal  in  each  cylinder. 

NOTB.— Initial  pressure  in  ist  cylinder  =  14.7  Ibt.;  terminal  37.25  Ibs.  absolute;  initial  in  ad  cylinder 
37.25,  same  as  terminal  in  ist,  and  at  terminal  in  21!  cylinder  =  94.7  Ibs.  absolute. 

To  Compute  \Vorlz  per  Pound  of  Air  in  Compressing 
it  to  SOO  Lt>s.,  Gfauge  Pressure,  from  an  Initial 
Temperature  of  62°  in  First  Cylinder. 

Cooling  to  Atmospheric  Temperature  before  Air  is  admitted  to  next  Cylinder,  and 
Jacket  Cooling  not  considered,  hence  n=  1.408. 

F  =  9596oXNxR-29  — i 
—  =^  ~=  —  =—  =  R.     p  representing  atmosphere  in  Ibs.  per  sq.  lnch=  14.7, 

pl  terminal  pressure  (absolute)  in   ist  cylinder  and  admission  to  zd=\ 
=  Vi4-7  X  86.8  =  35.7,  jp2  terminal  in  2d  cylinder  and  admission  to  %d  =  > 
V'4-7  X  514-7  =86.8,  jp3  terminal  pressure  in  $d  cylinder  and  admission  to  ^th  = 
Vpz  XJ>4  =  V86.8  X  SM-y^sn,!^  terminal  pressure  =  514.7  Ibs.,  and  N  =  4- 

Hence,  — —  =  2.43,  — —  =  2.43,  —  ^^=2.43,  and  — ^  =  2.44. 


95  960  X  4  X  2.43'2»  —  i  =  95  960  X  4  X  .2937  =  112  734  foot-pounds. 

kPo  Compute  the  Steam  Pressure  Required  in  tlie  Steam 
Cylinder  of  a  Sixnple  -A.ir  Compressor. 

When  the  Air  Pressure  and  Diameter  of  Both  Cylinders  are  Given. 
—  x  ( ~\  2=  PI-    P  and  Pt  representing  mean  effective  air  and  steam  pressures  in 

Ibs.  per  sq.  inch,  E,  mechanical  efficiency  of  Compressor,  and  d  and  di  diameter  of 
air  and  steam  cylinders. 


IOO8  COMPRESSION    OF   AIR. 

ILLUSTRATION. — Assume  pressure  of  air  60  Ibs.,  diameter  of  steam  and  air  cylin- 
ders respectively  12  and  14  ins.,  and  mechanical  efficiency  .85. 

Mean  eflective  air  pressure  of  air  for  adiabatic  compression  at  60  =  30.75  Ibs. 
(see  table,  p.  995). 

3^Z5  x (iiV:=  36. 18  X  1.36  =  49-2  Ibs.  per  sq.  inch. 

Corresponding  to  a  steam  pressure  of  70  Ibs.  gauge,  at  .375  cut-off. 

Temperature  is  a  direct  function  of  the  pressure,  hence  it  is  apparent  that  in  the 
multiple  stage  compression,  where  the  temperature,  by  the  application  of  inter- 
coolers,  is  reduced  back  to  that  of  the  atmosphere  before  admission  to  each  cylin- 
der, that  the  loss  in  radiation  is  reduced.  In  compound  compression,  in  order  to 
divide  the  work  equally,  the  ratio  of  compression  should  be  the  same. 

The  temperature  of  the  air  (theoretical)  in  the  single  stage  compression  her« 
given  is  about  400°,  and  that  at  the  end  of  each  compression  in  the  compound  case 
is  about  200°. 

The  mean  effective  pressure  or  resistance  of  the  air  of  compression  in  a  single 

C!  804 

cylinder,  and  for  the  given  pressure  and  temperature,  is  Isothermally — 

68  7«  '144X13-141 

=  27.38,  and  Adiabatically —  —  =  36.33,  and  51 804  -5-  68  755  =  75.35.  Hence, 

144  X  13-  *4r 
Adiabatic  compression  is  but  75.35  per  cent,  as  effective  as  Isothermal.* 

For  the  heat  evolved  and  given  to  the  air  by  Adiabatic  compression  is  diffused  to 
the  surrounding  media  before  the  air  is  admitted  to  the  Motor  cylinder  of  an 
engine,  the  extra  work  in  compression  is  lost,  and  in  the  case  here  referred  to,  the 
loss  is  100  —  75.35  =  24. 65  per  cent. 

In  a  water-jacketed  cylinder,  the  loss  is  not  so  much,  as  the  heat  of  compression 
does  not  rise  so  high.  (Frederick  C.  Weber.) 

To    Compute    tHe    Weight    of   Air   used    in    a    Motor   per 
Minute    for  a    Griven    -A.mou.nt  of  "Work. 

^~-  =      =  cube  feet.      N  representing  number  of  IP.      U  =  P  V       ~ 


P  T 

i I  M  —.     p  initial  and  Pt  exhaust  pressure  in  Ibs.  per  sq.  inch,  V  volume 

P         1 1 

of  air  in  cube  feet,  n  1.408,  and  T  and  Ti  absolute  temperatures  at  admission  and 
atmospheric  temperature ;  W  weight  of  air  per  minute  to  deliver,  N  IP  per  minute, 
and  w  weight  per  cube  foot  at  atmospheric  pressure  in  Ibs. 

N  assumed  12,  w  .076,  T  and  Tx  63  -}-  460  =  523,  and  300  -f-  460  =  761,  P  80  Ibs.  per 
gauge,  and  volume  at  62°  =  13. 141  cube  feet. 

33  °°°  X  12  X  523 


.408 


14.  7  X  144  X  13-  141  X  —        —  X  i  --  —   MOS      X  761  X  .0761 
1.400  —  i  94.7 

-    2°7*°8ooo  -  -  -  =  89.  2  cube  feet  per  minute. 
95  960  X  (i—  .i552'2fl)  .  4174X76!  X.  076!  =  2  319510 

89.2-^.686  —  129.9  cube  feet  without  reheating  and  129.9  X  .686  =  89.  l  cube  feei 
when  reheated. 


1.408-1      _      t  Log.  14.7  X.  29  =  1.167317  X-  29  —  .338521 
—i—.  1  552  •«».  94.  7  X.  29  =  1.97635    X  .29  =  .  573141 


94-7  —.234620 

i  —  .23462=1.76538  and  number  of  .76538  —  i  =.5826  —  i  =  .4i74. 

By  Logarithms.     959601=4.98209  33000  =  4.51851 

.4174=762055  12  =  1.07918 

761=^88138  523  =  2.71850 

.0761=2.88138  8.31619 

6.36540 

Log.  of  1.95079  =  89.28  cube  feet.  1.95079 

*  For  an  illustration  of  the  curves  of  pressure,  see  Frank  Richards.     Frontispiece  and  p.  43. 
t  By  Logarithm*. 


COMPRESSION    OF    AIR. 


Dimensions    of  Valves,    Pipes,    and.    Clearance    of  Air 
Cylinders. 

Pressure  of  Air,  75  Lbs. 


Cylinder. 

Area. 

Free  Air. 

Pressure, 
75  Lbs. 

INLET 
Diameter. 

PIP«. 
Area. 

DlSCHARG 

Number. 

•  VALVKS. 
Area. 

.  Ins. 

Sq.  ins. 

Per  cent. 

Per  cent. 

Ins. 

Sq.  ins. 

No. 

Sq.ins. 

10.25  X  12 

78 

.0098 

.047 

2 

3-'4 

2 

5-4 

12.25  X  H 

"3 

.0086 

•043 

2-5 

4-9 

2 

8.8 

14.25  X  18 

154 

.0066 

•033 

3 

7 

3 

13.2 

i6.25X  18 

201 

.0066 

.023 

3-5 

9.6 

13.2 

18.25  X  24 

255 

.0049 

.0225 

4 

12.5 

8 

35-2 

20.25  X  24 

3o4 

.0049 

.0225 

4-5 

'5-9 

8 

35-2 

22.25  X  24 

380 

.0049 

.0225 

5 

19.6 

10 

44 

30.25  X  60 

707 

.002 

.01 

6 

28.2 

18 

79  .2 

36.25X48 

1018 

.002 

.01 

7 

38.5 

20 

88 

Clearance,  .0625  inch  at  each  end  of  cylinder.  The  area  of  the  discharge  depends 
upon  the  speed  of  the  compressor;  for  a  speed  of  300  feet  per  minute,  ten  per  cent, 
of  area  of  cylinder;  for  a  speed  of  450  to  500,  fifteen  per  cent. 

(W.  L.  Saundcrs.) 


1010 


TIDAL    OR   FLUVIAL    EFFECT. 


Tidal    or    Fluvial    Effect    on    Speed    of  a    Steam    or    Lilie 
Propelled  Vessel. 

Deduced  from  the  Experiments  and  Notes  of  Edwin  A.  Stevens,  Associate,  and 
C.  P.  Paulding,  Junior,  Members  N.  A.  and  M.  E. 

To    Compute    Velocity    of  the    Tide    or    Current    in    Feet 
per   Minute. 

TJ   „ 

C =  V,  representing  the  velocity  in  feet  per  minute ;  C,  length  of  course 

R  t  -\-  r  T 

in  feet ;  R  and  r,  T  and  t,  respectively,  whole  number  of  revolutions  of  engine,  and 
times  of  run  in  minutes,  both  against  and  with  a  tide  or  current 

ILLUSTRATION Assume  C  one  mile  =  $  280  feet,  R  and  r  970  and  548  number  of 

revolutions,  and  T  and  t  8.45  and  4.77  times.     What  is  velocity  of  tide  or  current? 

970  —  548 


5280 


970X4-77+548X8.45 


=5280 


422 

4626.9-1-4630.6 


=  240.7— feet  velocity. 


To    Compute    Advance     per     Revolution     of    Engine     in 
Feet    per    Minute. 


or  Current.    S+™      A.    5  280  +  240.7  X  8.45  =  73H 
R  970  970 

NOTE.  —  If  distance  is  given  in  knot  of  6080  feet,  add  15.  151  per  cent. 
To    Compute    Speed    of  Vessel    in    Feet    per    Minute. 


rT  +  R* 


=&    5*80 


548X8.454-970X4-77 


9257-5 


To   Compute  Number  of  Revolutions   to   Run  one   Mile 
in    Still    Water    and    the    Slip. 


_  N      548X8.45  +  970X4.77  =  9257.5  =  6  revolutio 

T+t  8.45  +  4.77  13-22 

700.37=U>li —700.26  =  58 .74  lost  in  slip  =  7. 7 5  per  cent. 


NOTE.— In  applying  these  formulae,  the  number  of  revolutions  in  the  run  should 
be  as  uniform  as  practicable. 

Between  runs,  a  variation  of  5  per  cent,  will  not  materially  affect  the  result. 
STEAM  SIPHON.     A.n    Independent    Lifting    P»ximp. 
Capacity  for  a  Discharge  Pipe  2  Ins.  in  Diameter,  per  Minute. 

Discharge. 

Gallons. 
119.68 


Water  raised. 

. 


Feet. 
14 
13 
13 


Ins. 
6 


ressure. 

Discharge. 

Water  raised. 

Pressure. 

Lbs. 
3° 
40 
50 

Gallons. 
63.54 
85-71 
100 

Feet. 
13 
13 
13 

Ins. 

2 
2 

2 

Lbs. 
60 
70 
80 

157.57 


.Friction.    Losses    and.    Distribution,    of*    3?ower    in 

Machinery. 
From    8    to    4OO    IP. 

Losses  Range  from  55  to  65  per  Cent. 


Per  Cent. 

Friction  of  Engine 10  to  1 1. 8 

"      of  Shafting 15  "17.7 

"      of  Belts  and  Gearing.  15  "17.7 


Per  Cent. 


Friction  of  Lathes  and  Ma- 
chinery   15  to  17.7 

Effective*  Operation 45  "35.1 


CAST   IKON,  DEW-POINT,   AND   COLUMNS.  IOI  I 


Strength    of  Cast    Iron. 

As  determined  by  Tests  on  a  Riehle  Instrument  at  Lexington,  Ky. 
Average  of  16  Tests. 


Tensile, 
per  Sq.  Inch. 

Elastic 
Limit. 

Modulus  of 
Elasticity. 

Transverse, 
per  Sq.  Inch. 

Elastic 
Limit. 

Modulus  of 
Elasticity. 

Lbs. 
24436 
Malleable. 
41582 

Lbs. 
21  469* 

31042 

Lbs. 
28240000 

13000000 

Lbs. 
Annealed. 
4425 
Refined. 
2435 

Lbs. 
2508 

(Jame, 

Lbs. 

21  OOOOOO 
19300000 

5  H.  Wtlls.) 

To   Ascertain    the    Degree   of  AJbsolnte    Dryness   in    the 
Air  and.  the  Dew-:Point. 

Mason"1  s  Hygrometer. 


ll 

11 

1  Excess  of 
|  Dryness. 

Absolute 
Dryness. 

Dryness 
Observed. 

If 

8  - 

Is 

11 

.2£> 
•<o 

1! 

Excess  of 
Dryness. 

Absolute 
Dryness. 

if 
11 

Excess  of 
Dryness. 

•2  « 

II 

0 

0 

0 

0 

o 

0 

0 

0 

0 

0 

0 

0 

-5 

.08 

1.17 

5-5 

.92 

12.83 

10.5 

i-75 

24-5 

15-5 

2.58 

36-17 

i 

•17 

2-33 

6 

I 

14 

II 

1.83 

25.67 

16 

2.67 

37-33 

i-5 

•25 

3-5 

6.5 

1.  08 

15-17 

"•5 

1.92 

26.83 

16.5 

2-75 

38.5 

2 

•33 

4-67 

7 

•'7 

16.33 

12 

2 

28 

17 

2.83 

39-67 

2-5 

,42 

5-83 

7-5 

•25 

I7/l 

12.5 

2.08 

29.17 

I7>'5 

2.92 

40.83 

3 

•5 

7 

8 

•33 

18.67 

13 

2.17 

30-33 

18 

3 

42 

3-5 

.58 

8.17 

8-5 

.42 

19.83 

'3>5 

2.25 

31-5 

18.5 

3-o8 

43-17 

4 

.67 

9-33 

9 

•5 

21 

14 

2.33 

32.67 

*9 

3-i7 

44-33 

45 

•75 

10.5 

9-5 

•58 

22.17 

i4-5 

2.42 

33-83 

i9-5 

3-25 

45-5 

5 

•83 

11.67 

10 

.67 

23-33 

15 

2-5 

35 

20 

3-33 

46.67 

To  Ascertain  the  Drynm.— OPERATION.— From  temperature  of  the  air  subtract 
that  of  the  wet  thermometer,  add  excess  of  dryness  from  the  table  for  the  differ- 
ence, multiply  sum  by  2,  and  the  result  will  give  absolute  dryness  in  degrees. 

ILLUSTRATION.— Temperature  of  air,  57;  wet  thermometer,  54.  Hence,  57—54 
=  3.  Add  .5,  from  table,  =  3. 5  which  X  2  =  7  degrees. 

To  Ascertain  the  Dew- Point.— From  temperature  of  the  air  subtract  Absolute 
Dryness  and  result  will  give  the  Dew-Point  in  degrees. 

ILLUSTRATION.— Temperature  of  air,  57 ;  Absolute  Dryness  =  7.  Hence,  57  —  7  = 
50°  Dew- Point. 

Safe  Gnashing  Strength  of  Columns  of  a  Height  not  ex- 
ceeding IS  times  their  Diameter. 
In  Pounds  per  Square  Inch  of  Transverse  Section. 


Material. 

Lbs. 

Material. 

Lbs. 

Material. 

Lbs. 

Basalt  

2  875 

Iron,  wrought  

14  400 

Mortar,  common  

36 

Brick  hard 

Limestone  hard 

7  20 

Oak  white  

4^2 

"      common  
Granite  hard 

58 

"          common 
Marble  hard 

432 

i  <ns 

u    common  

280 
1  295 

"       common 

enr 

"       common 

Spruce  red         .... 

54° 

Iron.  cast.  .  . 

287SO 

Mortar,  good  and  old 

58 

"       white  

240 

When  the  height  of  a  column  exceeds  12  times  its  least  diameter  in  feet,  or  area 
in  square  feet,  divide  the  tabular  weight  by  the  number  in  the  following  table 
corresponding  to  the  length. 


Height I    12 

Divisor .....     1.2 


18 


1.8 


27 
2.4 


3° 


0 
7.8 


60 


35        40 
3-9    I   4-8 

ILLUSTRATION. —Assume  height  of  a  column  of  white  oak  15  inches  in  diameter 
and  21  feet  in  length.     What  weight  will  it  support  ? 
432  -r- 1.8  =  240  Ibs. 

*  Tenaile  Refined  Ultimate,  33  695. 


IOI2 


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CEMENT,   HUMIDITY,  METRIC    MEASURES.         IOI3 

Tire  Cement. 

Mix  bisulphide  of  carbon,  160  parts;  gutta-percha,  29  parts;  caoutchouc,  40 
parts;  and  isinglass,  10  parts.  Pour  the  mass  into  the  crevices  of  a  rupture  after 
they  have  been  properly  cleaned.  If  the  rent  is  large,  apply  the  cement  in 
layers.  Bind  up  the  tire  lightly,  let  the  cement  dry  for  twenty-four  to  thirty-six 
hours,  remove  the  binding  and  the  protruding  cement  with  a  sharp  knife,  which 
must  previously  have  been  dipped  in  water. — German  practice. 

Relative    Humidity    and    Dew    !Point    of  the    Air. 
As  Determined  by  a  Dry  and  Wet  Thermometer. 

Difference  of  Temperature  between  the  Two  Thermometers  and  Degrees  of  Hu- 
midity. 

Saturation  being  100. 


*tf 

MB* 

sjj 

32° 

42s 

52° 

62° 

72° 

82° 

92C 

DiflF.  of 
Tempera- 
ture of  Air. 

4-° 

52° 

62° 

72° 

82° 

^ 

Diff.  of 
Tempera- 
ture of  Air. 

42C 

52° 

62° 

72C 

.2° 

92° 

0 

I 

2 

3 
4 

5 
6 

87 
75 

92 

§5 
78 
72 
66 
60 

93 

86 
So 

| 

04 

8 

82 
77 

% 

94 
89 
84 
79 
74 
69 

95 
90 
85 
80 
76 
72 

95 
90 

85 
81 

77 
73 

O 

9 
10 
ii 

12 

54 
49 
44 
40 

36 
33 

59 
54 

sj 

46 
*a 

39 

62 
58 
54 
5r-> 
47 
44 

65 
6  1 

57 

54 

8 

68 
64 
60 

57 

54 
51 

70 

^8 
62 

53 

0 
13 
H 
15 
*   16 

g 

3° 

27 

3* 
33 
30 
27 
25 

4i 

38 
35 
32 
30 
28 

45 
4* 

39 
36 

34 

32 

48 

45 
4-? 
4'-' 
3^ 
35 

50 
47 
45 
43 
4i 
33 

OPERATION.— If  temperature  of  air  is  72°  and  difference  of  temperature  between 
the  thermometers  is  7°     The  humidity  or  dew  point  is  65  degrees. 

(Greenwich  Observatory. ) 


Reduction,  of  Metric  IMeasures. 

As  enacted  by  the  Congress  of  the  United  States  and  in  United  States  Measures. 

In  addition  to  pp.  27-33,  &,  44,  47,  923,  934- 


Caloric  X  3- 968  =  B.T.U. 
(Centigrade  x  1.8)  +  32  —  degrees.  * 
Centimeters  X  .3937  =  inches. 
Centimeters  -r-  2. 54  =  inches. 
Cheval  vapeur  x  .9863  =  IP. 
Cube  Centimeters-r- 16. 383  —cube  inches. 
Cube  Centimeters -T-  3.69  =:  fl.  drachms. 
Cube  Centimeters  4-  29. 57  =  fluid  oz. 
Cube  Meters  X  35-315  —  cube  feet. 
Cube  Meters  X  1.308^1  cube  yards. 
Cube  Meters  X  264.2  =  gallons. 
Grams  X  15-432  —  grains. 
Grams  -=-  981.  =  dynes. 
Grams  (water)  -4-  29.57  =  fluid  ounces. 
Grams-:- 28.35  =  ounces  avoirdupois. 
Grams  per  cube  centimeter  -=-  27.7  =  Ibs. 

per  cube  inch. 
Gravity  Paris  =  980. 94  centimeters  per 

second. 

Hectare  x  2.471  =  acres. 
Hectoliters  x  3.531  =  cube  feet. 
Hectoliters  X  2.84  =  bushels. 
Hectoliters  x  -131  =cube  yards. 
Hectoliters  -f-  26. 42  =  gallons. 
Joule  X  .7373  =  f°ot  pounds. 
Kilo  per  Meter  x  .672  =  Ibs.  per  foot. 
Kilo  per  Cheval  x  2. 235  =  Ibs.  per  BP 
Kilo  per  Cu.  Meter  x  .026= Ibs.  per  cu  ft. 


Kilogram-meters  x  7.233  =  foot  Ibs. 
Kilogram  per  sq.  cent,  x  14-  223  —  Ibs.  per 

sq.  inch. 

Kilograms  X  2. 2046  =  pounds. 
Kilograms  x  35-  3  =  ounces  avoirdupois. 
Kilograms  -r- 1 102. 3  =  tons,  t 
Kilometers  x  .621  =  miles. 
Kilometers -r- 1.6093  =  miles. 
Kilometers  X  3280. 7  =  feet 
Kilo  Watts  X  i  34  =  IP. 
Liters  x  61.022  =  cube  inch. 
Liters  x  33-84  =  fluid  ounces. 
Liters  X  .2642  ==  gallons. 
Liters  -f-  3. 78 !  =  gallons. 
Liters  4-  28.316  =  cube  feet. 
Meters  x  39.37  =  inches. 
Meters  x  3-281  =  feet. 
Meters  X  1-094  =  yards. 
Millimeters  x  .03937  =  inches. 
Millimeters  -r-  25.4  =  inches. 
Square  Centimeters  x  .i55  =  sq.  inches. 
Square  Centimeters-:- 6. 451  =  sq.  inches. 
Square  Kilometers  x  247.1  =  acres. 
Square  Meters  X  10.764  =  sq.  feet. 
Square  Millimeters  x  -0155  =  sq.  inches. 
Square  Millimeters  -r-  645.  i  =  sq.  inches. 
Watts  -r-  746.  =  IP. 
Watts  -r-  .7373  =  foot  pounds  per  second. 


>  All  degrees  are  given  Fahrenheit. 


t  Tons  in  this  item  are  computed  at  2000  Ibs. 


IOI4          CHIMNEY    DRAUGHT,   STEAM    VESSELS,   ETC. 

To    Ascertain    the    Height    of*   a    Chimney    for    a    Re- 
quired.   Draught. 

Divide  7.6  by  the  absolute  temperature  of  the  externnl  air,  and  7.9  by  the  like 
temperature  of  the  gases  in  the  chimney  at  the  point  of  their  delivery  into  it;  sub- 
tract this  quotient  from  the  former,  divide  the  required  draught  by  the  difference, 
and  the  quotient  will  give  the  height  of  the  chimney  in  feet. 

Or,  -  =  h.    D  representing  the  draught  in  inches  of  water,  T  temperature 
7-6      7-9 
T         t 
of  air  -f-  460,  t  temperature  of  gases  -f-  460,  and  h  height  of  chimney  in  feet. 

ILLUSTRATION.  —  Assume  temperature  of  air  20°,  and  that  of  the  gases  600°,  and 
required  draught  .6  inch 

.6  .6  .  ,   . 

-  =  -  =  ?i.6feet. 
7.6  7.9  .00838 

20°  -f-  460°      600°  4-  460° 

To    Ascertain    the    Draught    of*  a    Chimney. 

In  Inches  of  Water. 

Proceed  as  above  to  determine  the  difference  of  temperature,  subtract  the  latter 
from  the  former,  and  multiply  the  remainder  by  the  height  of  the  chimney  in  feet. 
ILLUSTRATION.  —  Assume  like  temperature  and  height  of  chimney  as  abpve. 

7,.6  =  .6  i»c7, 


Resistance    of    Steam    Vessels.        :»  ' 

The  thrust  of  a  propeller  on  the  resisting  collars  of  a  propeller  shaft  is  the  meas- 
ure of  the  power  applied  to  the  propulsion  of  the  vessel. 

.66X33ooo_p       PXS      TO>       PX2232_TTr>       D  *  X  S*3    n1i<l  A  X  8,3  _ 

SXIIP  33<x*>~  33000    -"*•  C  ~^K~ 

IIP.  P  representing  the  thrust  of  the  propeller  in  Ibs.,  S  and  Sl  the  speed  of  the  ves- 
sel in  feet  per  minute  and  knots  per  hour,  D  displacement  in  tons  (2240),  A  area  of 
immersed  amidship  section  in  square  feet,  and  C  and  Ct  constants. 

Assume  the  following  elements  of  the  steamer  "El  Sol":  Length  between  per- 
pendiculars, 377.  2  feet  ;  amidship  section,  934  square  feet  ;  displacement,  6760  tons  ; 
S  and  Sj,  14.75  feet  and  14.5  knots  ;  C  and  C\,  310  and  813;  and  IIP  —  3500. 

.66  X  33  coo  =      6          p      5i  695  X  1475  =  61gp.     5»  *95  X  2232  = 
M75X3500                                     33000  33000 

6760*  X.4-  53  ^357-5  X3Q48  93^X^3 

310  310  813 

W  V  S3 

D.  K.  Clark  gives  —    —  =  EIP.     W  representing  wetted  surface  in  square  feet. 

Coefficients    of*   Radiation    of   Heat. 

For  a  Period  of  One  Hour  from  10.  76  Square  Feet  *  of  Surface. 


Silver,  polished 16 

Copper,  red 20 

Brass,  polished ->^-  32 

Sheet  iron,  polished. . .  56 


Sheet-iron,  leaded 81 

Sheet-iron,  black 345 

Glass,  polished 373 

Cast-iron,  rusted 419 


Paper 470 

Stone,  building 499 

Soot 500 

Water 662 


(^ Home  Study.") 

Heat    Radiated,    per    Sq.u.are    Foot    per    Hour. 

From  a  Temperature  of  180°  to  159°  in  Units. 

Tin-plate 1.37  |  Sheet-iron 2.24  |  Glass 2.18 

(Tredgeld.) 

*  One  square  Metre. 


FORCED   DRAUGHT,  GEOLOGICAL   STRATA,  ETC. 


Forced.    Draught    in    Marine    Boiler. 

Compressed  Air  Exhausting  Blast  in  the  S.  S.  "Resolute." 


Blowing 
.Engine. 

Engine. 

Coal 
per  hour. 

Coal* 
per  IIP 
per  hour. 

Water 
evapo- 
rated per 
Ib.  of  coal 

Blowing 
Engine. 

Engine. 

Coal 
per  hour. 

Coal 
perlH? 
per  hour. 

Water 

rated  per 
Ib.  of  coal 

Iff 

IIP 

Lbs. 

Lbs. 

Lbs. 

IH? 

IH? 

Lbs. 

Lbs. 

Lbs. 

Natural  ) 
draught.  j 

57-5 

213 

3-72 

10.77 

4.2 

118.8 

348 

2-93 

782 

.96 

88.8 

289 

3.26 

8.82 

5 

119.8 

374 

3-12 

7-53 

2 

100.5 

315 

3.12 

8 

6 

127.9  !    400 

3  .12 

7 

3 

106.1 

321 

3-04 

7.82 

7-4 

135-7  1    420 

3.10 

7-03 

(D.  K.  Clark.) 


Absorption  of  Q-eological   Strata. 
Per  Cent,  by  Volume. 


Formation. 

Location. 

Water  in 
loo  parts. 

Authority. 

Dolomite  

Joliet,  111  
Lemont  111      

1.  06 
1.  12 
4.76 
2-5 

5-55 
•       .29 
.42 
29 
4-4 
4.2 

2.1 
12.15 
.08 

{  £} 

•25 

i  '3  i5/ 

29 
23-95 
n.6 
4.81 
6.25 

12 

/33       } 

\4o      J 
Daniel  W. 

3S. 
Metal. 

G.  P.  Merrill. 

« 
M,  Delessee. 
D.  W.  Mead. 

E.  Wetherel. 
M  Delessee. 
G.  P.  Merrill. 

M.  Delessee= 

E.  Wetherel. 
G.  P.  Merrill 

D.  W.  Mead. 
R.  J.  Hinton. 

Mead,C.E.) 

ii 

Winona,  Minn  

ii 

Red  Wing  Minn 

ii 

Mantorville  "    

Gabbro 

Duluth           u 

Granite  Hornblende 

E  St  Cloud  " 

Freestone  Calcareous  

Grand  Beauchamp,  France 
Bedford  Ind 

Limestone 

"         Galena    

Rockfor'd  111 

"         Trenton  

u         Oolite 

Cheltenham  Eng 

"         Devonian  .... 

Boulogne  t  ranee 

Quincy  111 

u 

Big  Sturgeons  Bay,  Wis.  .  . 
Grand  Beauchamp,  France 

it                                   « 

Cheltenham,  Eng  

Gloucestershire,  Eng  , 
Fond  du  Lac  Wis 

Sandstone    

u         Quartzose  

"         Oolite 

"         Old  red  

tt 

Fort  Snelling,  Minn.  
Jordan,                " 
Berea  Ohio  

u 

11 

Clay  dry    

Sand  and  Gravel  

*H 
Cast-iron.    Abater    I?ip< 
To    Compxite    Thickness    of 

— (- .  25  =  T,  and  — \-  .25  =  T.     H  representing  head  of  pressure  of  water  in 

9600  4250 

feet,  d  internal  diameter  of  pipe,  and  T  thickness,  both  in  inches,  and  p  interior  press- 
ure in  Ibs.  per  sq.  inch. 

ILLUSTRATION. — Assume  head  of  water  200  feet,  diameter  of  pipe  8  ins.,  and  in- 
terior pressure  86.83  Ibs.  per  sq.  inch. 

200  X  8  ,  j  86.83  , 

— ^-.25  =.417  ins.,  and -\- .25  =.4134  ins. 

9600  4250 

For  faucet  ends,  the  equivalent  length  of  pipe,  equal  in  weight  to  that  of  the 
faucet,  7  -j-  d  -4- 15  =  ins. 
See  ante,  p.  147.  (D.  K.  Clark.) 

*  Anzin  briguettes.    The  fuel  consumed  and  the  power  were  doubled,  but  the  evaporative  efficiency 
was  reduced. 


IOl6   FRICTION  AND  FLOW  OF  WATER  IN  METAL  PIPES. 

Friction  of  Flow  of  Water  in  Smooth  Metal  Pipes, 

From  .5  to  3.5  Inches  in  Diameter. 
To    Compute    tlie    JLioss    of    Head. 

Per  ioo  Feet. 

.0126  +  —  5£i  ZJ  -  x     x  Ji__  __  H     d  internal  diameter,  H,  Zoss  o/  head  due 

V  o  a       2  p 

<o  friction  of  flow,  all  in  feet;  v,  velocity  of  flow  per  second,  and  I  representing 
length  of  pipe. 


ILLUSTRATION.  —  Assume  diameter  of  pipe  2.5  ins.,  length,  ioo  feet,  and  velocity 
f  flow  36  feet  per  secon 
flow,  influx,  and  efflux? 


.  .          .,  ,  , 

of  flow  36  feet  per  second,  what  will  be  the  loss  of  head  due  to  friction,  velocity  of 
ow,  influx, 


-   i   -°315  —  .06X2. 5-7- 12  ioo  1296 

•0126  -| — — X  — ^r —  X  7~~  —  -0126  -}-  -0032  =  .0158  X  480  X 

20.  i  =  152.44/66^  loss  due  to  friction. 

Loss  of  Head  Due  to  the  Influx  of  the   Water  into  the  Pipe. 

v2*       v2  1296      1206 

_+_x.505==_+_(_x.505=:30.34/^. 

Hence,  152.44  +  30.34  =  i82.68/ee«,  total  head. 

Friction   of  Flow   of  AVater   in    Cast-iron   Fipes, 

From  4  to  60  Inches  in  Diameter. 

.019892  -f-  • — -T —  X  3  X r  =  H-     Symbols  as  preceding. 

d          d      2  g 

ILLUSTRATION.— Assume  volume  of  water  required  20  ooo  ooo  gallons  per  24  hours, 
diameter  of  pipe  16  inches,  and  length  of  it  1000  feet.  What  will  be  the  loss  of 
head  due  to  friction  of  the  flow  and  what  the  loss  by  influx  into  the  pipe  in  feet? 

V      231 

—  X r-  720  =a  velocity  in  feet  per  minute.     V  representing  volume  of  discharge 

in  gallons,  t,  time  of  flow  in  minutes,  a,  area  of  section  of  pipe  in  sq.  inches,  and  720, 
lineal  inches  of  flow  per  minute. 

Gallon  =  231  and  area  of  pipe  =  201  cube  inches. 

-12X60  =13 889  X  1.14 

Or,  by  table,  p.  1016,  ^^ 
627 

t)a  22  it;2 

Hence,  22.15=  X  .2465,  from  table,  =  izo.g-^ftet  loss  of  head,  and  — ,  x  .505  =        3 

X  .  505  =  3. 86  feet  loss  by  influx  to  pipe. 

Loss  of  Head  Due  to  the  Influx  of  the  water  into  the  Pipe. 

V2  V2  22.  l62    .    22.  l62 

1 X-505  =  — |-  — x  .505  =  11. 5 feet.   Hence,  120. 94  -f-'n.  5  — 13 2. 44 

feet  total  head. 

20000000  gallons  per  24  hours  =13889  per  minute,  By  Coefficients  in  table,  p. 
1017,  for  a  pipe  of  16  ins.  13  889-:-  627  =  22.15  feet  velocity,  and  22.  is2  X  .2465  = 
120.94/66^  loss  of  head  due  to  friction. 

1?o    Compute  the    Flow   of   Water   from    a  Q-iven   Head. 

V  -i-  ^1  x  12X60  XT.     T  representing  time  of  flow  in  minutes. 

a 
ILLUSTRATION.  —Assume  the  elements  of  the  preceding  case. 

22. 15 1  -r-  —  X  720  X  60  x  24  =  19  987  430  gallons. 

2OI 


20  ooo  ooo      231       - 

•g        ,     X  --  '-  12X60  ==  13  889  x  1.  149  H-  720  =  22.  16  feet  velocity  per  second. 


—  representing  the  head  required  to  produce  the  velocity,  and  —  X  .505  the  loss  du«  to  the  en- 
trance of  the  water  into  the  pipe. 
t  By  Beardman,  p.  548,  v  would  =  23.6 /«««. 


FRICTION  AND  FLOW  OF  WATER  IN  METAL  PIPES.    IOI/ 

When    the    GUven    Length    is    Less    or    G-reater   than    the 
Length    of   1OOO    in    the    following    Table. 

The  ratio  of  the  given  length  to  the  length  in  the  table  is  ascertained  by  dividing 
the  length  in  the  table  (1000)  by  the  given  length,  and  the  inverse  ratio  to  the 
length  in  the  table  is  ascertained  by  dividing  the  given  length  by  1000. 

Assume,  as  in  the  second  of  the  preceding  cases  given,  the  length  to  be  1500  feet. 
—=1.5.     As  the  friction  head  (for  1000  feet)  of  120.94  corresponds  to  a  veloc- 
ity of  22. 16  feet  per  second,  120.94  x  1.5  =  181.41/6^,  the  frictional  head  in  a  pipe 
of  1500  feet  in  length. 

Application  of  the  Formulas  in  the  following  Table. 

Assume  a  lake  1500  feet  distant  to  discharge  water  through  a  cast-iron  pipe  10 
ins.  in  diameter  under  a  head  of  70  feet :  What  is  the  velocity  in  feet  per  second, 
the  loss  of  head  to  the  influx  into  and  flow  through  the  pipe,  and  the  discharge  in 
gallons  per  24  hours  ? 


.    .00167  v 2  g  h 

.01989-1 ^-=.02l89=:C.    • 

7     *   '     Tn-l-  TO  y  I 


V/.+.505+3XC  %  _____ 

v45°.   — =%a=ML4g/M«iMa.L  . 

Vi.  505 +  i8oo.7X. 02189        °-4 

C  =  352  512  X  10.48  =  3  694  326  gattons. 

Assume  a  discharge  of  water  of  2450  gallons  per  minute,  through  a  cast  iron  pipe 
jo  ins.  in  diameter  and  1500  feet  in  length:  what  will  be  the  loss  of  head  due  to 
friction,  and  what  the  discharge  in  24  hours? 

2450-7-245  (from  table)  =  io  feet  velocity  per  second,  io2  X  .4084  (from  table)  X 
—  =  61.26  feet  friction  head,  and  io  X  352  512  (from  table)  =  3  525 120  gallons. 


Hence,  i  +  .505  +  .02189  x  1800  X  -^—  =  40.807  X  1.708  =  69.698  feet  total 
head. 

Coefficients  for  Computations  of  Velocity  of  Flow,  Dis- 
charge, and  Loss  of  Head  due  to  Friction  of  Flow 
of  Water  in  Pipes,  1OOO  Feet  in  Length. 

Velocity  of  Flow. 

Discharge  -f-  Coefficient  =  mean  velocity  of  flow  in  feet  per  second,  and  velocity  x 
Coefficient  =  discharge  in  gallons  per  minute. 

Inches. 


Diameter..!    4    j    6    I    8    I    10    I    12    I    16   I    20    I    24   I   30   I    36 
Coefficient!    39   |    88    |  157  |  245  j  353  |  627  |  979  114101220313173 


48   I   60 
5640  |  8813 


Loss  of  Head  due  to  Friction  =  Coefficient  X  Square  of  Velocity. 

Inches. 


Diameter..)    4    j    6 
Coefficient  J  1. 161 1  .722 


. 5221] .4084! . 3352! .2465! .  1949 


24    I    3°    I    36    I    48    I    60 
.  i6u| .  12781 .  io6o|  .0789!  .0629 


Discharge  per  24  Hours  =  Coefficient  X  Velocity. 

Inches. 

Diameter I        4        I        6        I        8        I        io       I        12        I        16 

Coefficient. |     56402     |    126921    j    225608    |    352512    |    5076x7    |    902448 

Diameter. . , 
Coefficient. , 

Table,  and,  essentially,  the  computations  from  the  valuable  work  by  Edmund  B. 
Weston,  C.E.  (D.  Van  Nostrand  Co.,  1896). 


20        I        24  30        I        36  48        I        60 

1410048  |  2030490  I  3172600  I  4568568  I  8121859  I  12690400 


ioi8 


BELTS    AND    BELTING. 


To    Compnte    the    Actual    Discharge    of*  \Vater    through 
a    Conical    Xuloe    (Nozzle).     Coefficients  of  Velocity  and  of  Efflux. 

Angle.       Velocity.       Efflux.         Angle.       Velocity.       Efflux.         Angle.      Velocity.       Efflux. 


.829  .829  12°  4'          .955  .942         19°  28'         .97  .924 

5°  26  .919  .924  13°  24'        .963  .946         23°  .974  .914 

(Continued  from  page  443.)  (Home  study^ 

Pulleys  should  have  a  slight  convexity  of  surface.  Authorities  differ,  from  .  5  inch 
per  foot  of  breadth  to  .1  of  breadth.  Belts  run  at  a  high  speed  are  less  liable  to  blip 
than  at  low  speed. 

The  best  speeds  for  economy  are  from  1200  to  1500  feet  per  minute,  and  the  best 
for  result  not  to  exceed  1800. 

Belts.—  Leather,  hair-side i  I  Leather,  flesh-side. . .  .74  |  Rubber 51 

Gutta  percha 44  |  Canvas 35 

Coefficient  of  Friction  of  a  Belt  in  operation  is  assumed  to  be  from  .2  to  .4. 

Smooth-surface  belts  are  most  endurable  and  soft  most  adherent. 

Round  belts  .25  and  .5  inch  in  diameter  are  fully  equal  in  operation  to  flat  of  i 
and  3  ins.,  and  grooves  in  their  pulleys  should  be  angular  or  V  shaped. 

Long  belts  are  more  effective  than  short. 

The  neutral  point  of  a  rope  belt  is  at  .33  of  diameter  from  inside  surface. 

Friction  of  driving  and  pulley  bearings  is  about  .025. 

A  fan-blower  No.  6,*  driven  by  a  belt  3.875  ins.  in  width  and  .18  in  thickness,  at 
a  velocity  of  2820  revolutions  per  minute,  requires  power  of  9.7  horses. 

Area  of  belts  per  IP  varies  essentially,  ranging  from  25  to  100  square  feet;  the 
mean  is  75. 

The  average  "net  effective  stress"  of  a  belt  is  the  difference  of  tensional  stress 
between  its  driving  and  slack  surfaces  per  lineal  or  sq.  inch  of  section,  and  this 
stress  over  fast  and  loose  pulleys  was  but  .4  of  that  over  cones. 

"Idlers"  are  most  effective  on  the  slack  side  of  a  belt. 

Narrow  and  thick  belts  are  preferable  to  wide  and  thin.  The  joining  of  the  ends 
of  a  belt  should  be  by  splicing  and  cementing,  and  the  length  of  the  splice  the  same 
as  the  width  of  the  belt,  and  if  the  ends  are  cut  slightly  convex  and  so  connected 
the  effect  in  operation  will  be  that  of  equalizing  the  stress  on  the  centre  and  edges. 
The  final  stretching  of  leather  belts  is  6  per  cent. 

A  double  belt  with  an  arc  of  contact  of  180°  and  i  inch  in  width  will  sustain  a 
stress  of  35  Ibs.,  and  the  number  of  sq.  feet  of  a  double  belt  over  a  pulley  per  min- 
ute to  transmit  one  IP  is  80. 

The  transmitting  power  (resistance)  of  the  arc  of  contact  is  essentially  propor- 
tionate to  the  arc  of  180°. 

The  average  "working  load  "  on  fast  and  loose  pulleys  was  but  .4  that  of  on  cone 
pulleys,  and  the  "  net  working  load  "  is  the  difference  in  tension  between  the  driv- 
ing and  slack. 

The  diameter  of  a  pulley  should  be  increased  in  proportion  to  the  thickness  of, 
or  number  of  plus  of,  a  belt. 

A  band  wheel  at  the  Amoskeag  Mfg.  Co.,  N.  H.,  30  feet  in  diameter  and  no  ins. 
face,  drove  three  belts,  having  a  lineal  width  of  104  ins.,  at  a  speed  of  5750  feet  per 
minute.  Capacity  of  engine  1950  IP,  from  which  is  to  be  deducted  the  friction, 
which  is  assumed  largely  at  5  per  cent.,  leaving  1852  net  IP. 

Hence,  -^  =  17. 8  IP  per  inch  of  width  of  belt  and  5~j-=  323  feet  speed  of  belt 
per  IP  per  inch  of  width. 

If  a  belt  of  its  proper  length  slips,  the  under  surface  should  be  moistened  with 
boiled  linseed  oil. 

When  belts  have  become  dry  and  hard,  apply  neat's-foot  or  liver  oil,  mixed  with 
a  small  quantity  of  resin. 

Rubber  belts  are  improved  by  the  application  with  a  brush  of  a  composition  of 
litharge,  red  and  black  lead,  in  equal  parts,  mixed  with  boiled  linseed  oil,  and  var- 
nish sufficient  to  cause  it  to  dry  quickly.  They  are  less  liable  to  slip  than  leather, 
and  are  suited  for  service  when  exposed  to  moisture. 

Cement.  Gutta  percha  16  parts,  rubber  4,  pitch  2,  shellac  i,  and  linseed  oil  2; 
cut  in  small  parts,  melted,  and  well  mixed.  (Molesworth.) 

*For  a  table  «f  Belta  &r  Pan-blowers,  etc,,  see  J.  H.  Cooper,  in  "  Jour.  Franklin  Inst.,"  vol.  66,  p.  409- 


OIL  ENGINES.  —  BLAST  AND  EXHAUST  BLOWEKS. 


OIL    ENGINES. 

Oil  Engines  are  in  employment  as  Motors. 

In  the  Priestman,  mineral  oil  or  petroleum,  having  a  specific  gravity  of  .8  or  up- 
wards, with  a  flashing-point  from  75°  to  150°,  is  used. 

The  oil  is  mixed  with  air  under  a  pressure,  is  drawn  into  the  cylinder,  and  ig- 
nited by  an  electric  spark. 

The  consumption  of  oil  varies  from  1.25  pounds  per  brake  IP  per  hour  for  large 
engines  to  1.6  Ibs.  for  small. 

An  engine,  cylinder  8.5  ins.,  stroke  12  ins.,  and  180  revolutions  per  minute,  de- 
veloped 4.6  brake  HP,  with  a  consumption  of  1.2  Ibs.  of  oil  per  B?  per  hour. 

The  Hargreaves  motor  is  designed  for  the  use  of  coal-tar  or  creosote  as  fuel. 

It  consists  of  an  air-compressing  pump  and  motor  cylinder,  to  which  a  regenera- 
tor is  adapted,  which  absorbs  a  portion  of  the  heat  of  the  exhausted  gases,  and 
yields  it  to  the  incoming  charge. 

The  compressed  air  is  delivered  through  the  regenerator  into  the  motor  cylinder, 
where  it  is  exposed  to  a  jet  of  coal-tar  or  creosote,  and  being  heated  to  redness  ig- 
nites the  fuel. 

In  a  trial,  40  IB?  was  generated  by  the  consumption  of  .512  Ibs.  coal  tar  per  hour, 
and  32.4  per  cent,  of  heat  converted  into  work,  and  in  another  trial  with  a  smaller 
engine,  5.17  IB?  was  generated  by  the  consumption  of  1.2  Ibs.  of  creosote  per  hour, 
and  14.4  per  cent,  of  heat  converted  into  work.  (D.  K.  Clark.) 


Blast    and.    Exhaust    !Tan    Blowers, 

(In  addition  to  pp.  447-448  and  898. ) 

The  Blast  Area,  which  is  the  basis  of  all  computations,  is  the  diameter  of  the 
fan  (wheel)  multiplied  by  the  width  of  it  at  its  periphery. 

Exhaust  Fan. — The  area  of  its  discharge  should  be  about  equal  to  three  times 
the  blast  area,  or  equal  to  the  area  of  the  inlets,  and  the  width  of  the  fan  .25  its 
diameter  at  its  greatest  width. 

Volume  Blower.  —For  forced  draught  the  discharge  area  should  be  about  equal 
to  the  area  of  the  blast,  and  the  width  of  the  fan  .25  its  diameter  at  its  greatest 
width. 

Pressure  Blower.— As  for  a  Cupola,  the  discharge  area  should  be  .33  that  of  the 
blast,  and  one-half  the  area  of  the  inlet,  and  the  diameter  of  the  fan  should  be  pro- 
portionally great,  and  the  blades  of  the  fan  narrow  at  their  extremity. 

In  ordinary  practice  the  inlets  are  made  about  one-half  the  diameter  of  the  fan. 

(American  Blower  Co.) 

Dimensions    of*  Fan. 

PRESSURE. 
From  3  to  6  ounces  per  Sq.  Inch;  or  5.2  to  10.4  Inches  of  Water. 


Diameter  of 

Blades. 

Diameter  of 

Blades. 

Fan. 

Inlets. 

Width. 

Length. 

Fan. 

Inlets. 

Width. 

Length. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

3 

1.6 

•9 

•9 

4.6 

2-3 

I.I-5 

1.1.5 

f 

;•' 

.10.5 

I 

.10.5 

i 

6 

2.6 
3 

1:1 

1:1 

From  6  to  g  ounces  per  Sq.  Inch;  or  10.4  to  15.6  Inches  of  Water. 


Diameter  of 

Blades. 

Diameter  of 

Blades. 

Fan. 

Inlets. 

Width. 

Length. 

Fan. 

Inlets. 

Width.     |     Length. 

Ft.  Ins. 

Ft.  Ins 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

3 

i 

•7 

i 

4.6 

1.9 

.10.5 

1-4-5 

I6 

1:1 

•8.5 
-9-5 

1.1.5 

1-3-5 

I 

2 
2-4 

i 
1.2 

1.6 

I.  10 

(Mr.  Buckle's  Experiment*. ) 


IO2O          FLOOR   BEAMS,  GIRDERS,  COLUMNS,  ETC. 

To  Determine  the  Dimensions  of  Floor-Beams,  Q-ird- 
ers,  Colnmiis,  Foundations,  and  Filing  of  a  Build- 
ing to  Sustain.  Griven  Loads  on  tlie  Floors. 

Construction,  Dimensions,  and  Capacities  as  A  ssigned  by  the  Department  of 
Buildings,  City  of  New  York. 

Foundation. — Piles,  not  less  than  5  ins.  at  point,  spaced  not  to  exceed 
30  ins.  from  centres,  and  to  support  not  to  exceed  40000  Ibs.  —  5.714*  tons 
per  sq.  foot  with  two  lines  of  piles,  8.57  tons  with  three  lines,  and  11.43  tons 
with  four  lines,  etc.,  or  2.857  tons  P61*  additional  line. 

Walls  and  Piers. — Include  all  built  below  the  first  tier  of  beams,  at  or  be- 
low the  level  of  the  curb-stone.  Masonry,  of  stone  or  brick,  with  lime  mor- 
tar, not  to  be  subjected  to  a  stress  exceeding  16000  Ibs.,  with  lime  and 
cement  23000  Ibs.,  and  with  cement  30000  Ibs.  per  sq.  foot. 

Side  Walls. — Their  widths  as  determined  by  their  height  and  the  propor- 
tional area  of  flues  and  recesses  in  them.  If  of  stone,  at  least  8  ins.  wider 
than  the  wall  first  above  them,  to  a  depth  of  12  feet  below  the  curb,  and  for 
each  10  feet  or  part  thereof,  an  additional  4  ins.  If  of  brick,  for  8  ins.  put 
4 ;  other  requirements  same  as  for  stone.  In  buildings  where  the  beams 
are  25  feet  in  length  or  over,  an  addition  of  4  ins.  in  width  must  be  given  to 
the  side  walls  from  above  the  curb-stone. 

Front  and  Rear  Walls. — Except  where  supporting  a  girder,  for  half  the 
distance  between  it  and  the  column,  are  non-bearing,  and  may  be  4  ins.  less 
in  width. 

Footing  or  Base  Course  and  Piers. — Of  stone  or  concrete,  or  both,  and  at 
least  one  foot  wider  than  base  of  wall  or  pier.  Capacity  of  solid  primitive 
earth,  estimated  at  8  ooo  Ibs.  per  sq.  foot. 

When  the  instability  of  the  ground  is  such  as  to  render  additional  support 
to  piers  necessary,  they  are  to  be  connected  by  inverted  arches  of  the  full 
width  of  the  piers,  but  not  less  than  12  ins.  in  width. 


Width    of   \Valls    for    Q-iven    Heights. 

Height  Measured  from  Level  of  Curb-Stone. 


Height. 

Width. 

Feet. 
40  to    60 
60  to    75 
75  to    85 
85  to  ioo 

Ins.        Ft. 
1  6  to  4of 
20  to  25! 

24  tO  2O| 

28  to  zst 

Ins.        Ft. 

Then  12  to  top. 
'    16     " 
14    20  to  6of 
"     24  to  sof 

Ins.        Ft. 

Then  16  to  top. 

"     20  to  75t 

Ins.       Ft. 

Then  16  to  top. 

If  over  ioo  feet,  each  additional  25  feet,  or  part  thereof,  above  the  curb-stone  to 
be  increased  4  ins.,  and  if  there  is  a  clear  span  of  over  25  feet  between  the  walls,  4 
ins.  additional  width  for  every  12.5  feet,  or  fraction  thereof,  that  they  are  more 
than  25  feet  apart. 

NOTE.— For  other  and  fuller  details  of  dimensions,  see  Laws  relating  to  Construction  of  Buildings. 
Weight  of  Materials  Per  Cube  Foot. 


Materials. 

Lbs. 

Materials. 

Lbi. 

Materials. 

Lbs. 

Brick                       ) 

White  Marble  

1  60 

Spruce      

31 

"5 

Cast-iron 

Hemlock 

160 

Wrought  Iron  

480 

Georgia  or  Yellow) 

Granite    or    other) 

Rolled  Steel  

487 

Pine  / 

54 

stone... 

1  60 

White  Pine.... 

35 

White  Oak.  .  . 

54 

Roofs. — Weight  assigned  50  Ibs.  per  sq.  ft.  is  addition  to  weight  of  its  materials, 
assumed  in  the  following  computations  at  15  Ibs. 


*  2340  Ibs. 


t  Or  nearest  tier  of  beam*  to  that  height. 


FLOOR   BEAMS,  GIRDERS,  COLUMNS,  ETC. 


1021 


Header  and  Trimmer  Beams,  of  4  feet  or  less  in  length,  one  inch  deeper  than 
their  adjoining  floor  or  roof  beams;  when  over  4  feet  and  not  over  15  feet,  to  be 
proportionately  increased,  or  doubled  in  width;  and  when  over  15  feet  to  be  sup- 
plemented with  a  wrought  iron  Fitch  plate  of  suitable  thickness  securely  bolted  to 
beams. 

Crushing    and    Transverse     Strength    and.    Coefficients 
of    Safety. 


Crushing  per  Sq.  Inch. 

Transverse  One  Inch  Square 
and  Loaded  in  Centre 
between  Supports. 

Coefficients  or  Factors  of  Safety 
for  Crushing  and  Tensile. 

Lbs. 

Cast-iron  80000 

Lbs. 

Georgia   or  Yellow) 

Columns  and  Vertical  Sup- 

Rolled Iron                40000 

Pine               .   .   j  55° 

ports  of  Wrought  Iron 

Rolled  Steel        .  .     48  ooo 

White  Oak  550 

or  Rolled  Steel      Four 

White  Pine  3  500 

White  Pine  450 

Spruce                         3  500 

Spruce        450 

All  other  materials.  .Five. 

Georgia  or  Yellow  ) 

Hemlock  400 

For  Tie-rods  and  all  parts 

Ane  .....;!.}  Sooo 

subjected   to  a  Tensile 

White  Oak...          ..  6000 

If  uniformlv  loaded  doubled. 

stress...,               ...Six. 

ILLUSTRATION.— Assume  a  warehouse  of  stone  and  brick  masonry,  25  feet  in 
width,  4  stories  in  height,  with  a  cellar  and  sub-cellar,  with  one  line  of  girders  and 
columns  above  and  brick  piers  in  sub-cellar,  8  feet  apart  from  centres,  stairways  4 
feet  in  width  and  15  in  length;  ground,  wet  sand. 

Heights  between  Levels  of  Floors,  inclusive  of  Beams,  and  Required  Capacity  of 
Beams. 


Ft.    Lbs. 

Sub-cellar 8     — 

Cellar 10    300 


Ft. 
ist  Story 15 


300 


2d  Story 12.83  25° 


Ft.     Lbs. 

3d  Story 10.33  225 

4th  Story 9        200 


Height  of  Building  from  Curb  line,  as  determined  by  the  Heights  between  the  Floors. 

To  the  upper  side  or  level  of  the  4th  floor  39'  2",  and  to  the  under  side  of  the 
roof  48'  2",  hence  the  brick  walls  are  to  be  16"  in  width  from  ist  to  under  side  of 
3d  floor,  and  12"  above. 

Foundation  of  Side  Walls.— Sub-cellar,  i6"-f8"  for  stone  masonry  =  24"  for  a 
depth  of  12  feet  below  the  curb  line,  and  for  its  part  of  10  feet  below  this,  4"-}- 28" 
=  2  feet  4  ins. 

Cellar,  16"+  8"  for  stone  masonry  =  24",  or  2  feet. 

ist,  2d,  and  3d  floors  16"=  i'  4",  and  4th  floor  12"=  ifoot. 

Floor  beams,  half  lengths.     Cellar  12'  6" — 2'  4"=io.i6  feet,  2d,  3d,  and  4th 
stories  12'  6"—  i'  4"=  n.i6feet.     ist  story  12'  6"—  2'=  10.5  feet. 
NOTB. — The  sub-cellar  floor  is  of  masonry. 

All  Computations  made  for  a  Lineal  Section  of  8  feet  of  Length  of  Building. 
Operation. — Weight  to  be  supported  by  columns  under  roof.  Lbs.   Lbs. 

8  feet  apart  from  centres.    Area  of  section  n'  5"  x  8'=  92  sq.feet  x  65 
Ibs.  (stress  on  roof  and  weight  of  its  materials)  = 5  900 

Girder,  White  pine  s"x  5"  in  breadth,  depth  ==    /8  Xlte59°0  =  6. 48  (6.5)  ins.  63 

To  be  supported  by  column 5  963 

Column,  Yellow  pine,  9  feet,  less  oak  cap  and  girder  8".  5  =  Sfeet  3.5  ins.  85 

Diameter  by  approximation  from  table,  p.  769,  4.7  ins.,  area  =  17.4  sq. 


ins.,  and  by  formula,  p.  768, 


5000  X  17-34 


'—)    X.oo4 


6084 


Fourth  Story.—  Weight  required  to  be  supported  200  Ibs.  Area  of  sec- 
tion n'.i6  X  8'=  89. 33  sq.  feet  X  200  \bs.= 17866 

Floor  beams,  White  pine  2"x  io"and  19'. i"  apart  from  centres,  capac- 
ity not  less  than  200  Ibs.  =  257  Ibs.  per  sq.  foot * .  330 


*  Coefficient  for  White  pine  uniformlv  loaded. 


i  For  safety. 


IO22          FLOOR   BEAMS,  GIRDERS,  COLUMNS,  ETC. 

»  Lbs.       Lb«. 

Girder,  Yellow  pine,  7  ins.  breadth,  depth  =    /  X  ^8  ig6  =  8.7    (  8.5  ) 
ins.  and  cap  =  ....................................................  IQ4 

To  be  supported  by  column  .....................................  24438 

Column,  Yellow  pine,  9'  6"  less  girder  and  cap  —  8  feet  7  ins  ..........  24  438 

Diameter  by  approximation,  7.75  ins.,  area  =  47  sq.  in*.,  and  by 
formula,  p.  768,  -  5oooX47          +5=  ...........................  ^^ 

•+  (!£)'*•-, 

Weight  of  column  and  cap  ..........................................  I72 

Third  Story.  —  Weight  required  to  be  supported,  225  Ibs.    Area  of 
section,  89.33  X  225  Ibs.  =  ..........................................  eo  100 

Floor  beams,  White  pine,  same  as  preceding,—  257  Ibs.  per  sq.foot  ____  330 

Girder,  Yellow  pine,  8   ins.  breadth,  depth  =    /8-X-2°  43-  =  8.  5  ins. 

and  cap  =  ........................................................  220 

To  be  supported  by  column  .....................................  45260 

Column,  Cast-iron,  13',  less  girder,  cap,  and  sole  plate  i'=  12  feet  ......  _ 

Diameters  by  approximation,  as  preceding,  6  and  5.125  ins.  —  7.65 

sq.  ins.,  and  by  formula,  p.  768,  -  3       7'      —  =  .................  50579 

' 


Proceed  in  like  manner  for  remaining  columns  and  floors. 

Second  Story.  —  Weight  required  to  be  supported  ...................  22  333 

Floor  beams,  White  pine,  3  x  12  X  19'.  i  ins  ..........................  397 

Girder,  Yellow  pine,  8  ins.  in  breadth,  9  ins.  in  depth  ................  235 

To  be  supported  by  column  .....................................  68  225 

Column,  Cast-iron,  diameters  7X6  ins.,  length  15',  less  girder,  cap, 

and  sole  plate  =  14  feet  ...........................................  72  416 

First  Story.  —Weight  required  to  be  supported  .....................  26  800 

Floor  beams,  White  pine,  3  x  12  X  19.  i  ins  ..........................  397 

Girder,  Yellow  pine,  9  ins.  in  breadth,  9.5  ins.  in  depth  ...............  274 

To  be  supported  by  column  .............  .  ............  ...........  95696 

Column,  Cast-iron,  diameters  7X6  ins.,  length  9  feet,  less  girder,  cap, 

and  sole  plate  =  8  feet  ...........................................  112  070 

Cellar.  —Weight  required  to  be  supported  ..........................  25  200 

Floor  beams,  White  pine,  3X12X19.1  ins  ...........................  397 

To  be  supported  by  Pier  ........................................  121  293 

Sub-cellar,  Pier.  —Brick  masonry  in  lime  and  cement  mortar  30  ins. 
sq.  •=.  6.  25  sq.  feet,  which  at  23  ooo  Ibs.  per  sq.  foot  =  ................  143  750 

Weight.    6.25  X  6.33  feet  in  height  and  footing  stone,  8  ins.  in  depth  x 

42  ins.  square  =  ............................  .  ....................  5  800 

To  be  supported  in  addition  to  weight  of  piles  ....................  127093 

Requiring  4,  set  at  30  ins.  from  centres,  and  driven  at  least  to  a  refusal 

of  30000  Ibs.,  which,  when  they  are  in  a  quiescent  condition,  = 

40000  Ibs.  =  ...................  ................................  160000 

Side  Walls  and  Footings.—Weight  oi  each,  including  one  half  of 
weight  on  Pier  ...................................................  211  370 

Requiring  7  piles,  set  and  driven  in  like  manner  for  the  pier  ........  280000 

NOTK.  —  The  excess  of  bearing  capacity  of  the  piles  is  to  meet  any  extraordinary  loading  of  the  floors. 
and  the  possibility  of  adding  to  the  height  of  the  building. 

For  dimensions  of  Header  and  Trimmer  Beams,  see  pp.  834-841. 


*  Coefficient  for  Yellow  pine. 


NON-CONDUCTOKS    OF    HEAT. 


1023 


Loss    of    Pressure    of    ITlow    of  Air    for   "Varying    Diam- 
eters   of  IPipe    and.    Velocities. 

Loss  of  pressure  per  sq.  inch  for  varying  Diameter  of  pipe  and  Velocities  of  flow; 
computed  upon  the  basis  that  the  friction  is  directly  inverse  to  the  inner  surface 
of  the  circumference  of  the  pipe.  This  is  not  normally  correct,  inasmuch  as  whilst 
the  area  of  the  inner  surface  is  in  a  direct  ratio  with  the  diameter,  that  of  the 
transverse  area  of  the  pipe  is  as  the  square  of  its  diameter.  For  ordinary  reference 
and  for  pipes  of  approximate  diameters  it  is  suiflcieutly  correct. 

The  Table,  with  some  addition  to  the  velocity,  is  for  a  pipe  of  -2  ins.  in  diameter. 
For  other  Diameter  the  Loss  of  Pressure  is  directly  Inverse  to  the  Diameter. 


.0005 
.0008 


300 
400 
5oo 


.oo8 

.Oil 


750 
900 


.012 
.019 


III 


I2OO 
1500 
2000 


•05 

.078 

.104 


2400 
3000 
3600 


•312 

•45 


111 

i^ 


4800 
6000 


1.25 


(American  Blower  Co.) 


Relative    Efficiency    of  N"on-Cond.\actors    of    Heat. 

The  efficiency  of  substances  for  the  retention  of  heat  varies  generally  inversely 
to  their  power  of  conduction  of  it. 

By  experiment  it  was  ascertained  that  the  relative  condensation  of  steam  in  a 
metal  pipe  under  the  following  conditions  was  : 

Bare  ................  100  |  Cement  coated  ........  67  |  Hair-felt  covered  ......  27 

Relative    Cost    of    Steam    !Power    in    Engines    per    KP. 

Based  on  cost  of  one  of  1000  IP  1888. 


Plant,  Fuel,  and  Cost. 

Non 
Condensing. 

Condensing. 

(S3SSJ 

Engine  and  House  complete    

$  cts. 

$  cts. 

•J-3 

$  cts. 

Depreciation,  Repairs,  Interest,  Taxes,  and  In- 
surance 

Boilers,  House  and  Chimney  

24    8O 

18  36 

Depreciation,  Repairs,  Interest,  Taxes,  and   In- 
surance . 

o   08 

Total  yearly  cost  of  Coal  and  daily  attend- 
ance 308  days  

2C    CQC 

21   817 

2.50 

Total  yearly  cost 

•3-3    2/l8 

2  A     087 

Coal  per  i  IP  per  hour  in  Ibs.  .  . 

3 

2.  ^O 

Chas.  T.  Main,  A.  S.  M.  E. 

NOTE. — If  the  exhaust  steam  is  utilized,  the  cost  will  be  correspondingly  reduced. 

To  Grradiaate  the  Compass  of  a  Transit  Theod- 
olite to  Coincide  \vith  the  .Line  of  Sight 
of  the  Telescope. 

Bore  a  small  hole  in  centre  of  cover  of  object-glass  and  put  it  in  place ; 
depress  eye  end  of  telescope  to  the  graduated  circle  of  the  compass.  Place 
telescope  parallel  to  the  light,  as  that  from  a  window,  and  with  a  sheet  of 
white  glazed  paper,  or  like  surface  reflective  of  the  light,  a  distinct  view  of 
the  graduations  on  the  circle  will  be  observed  and  so  well  defined  that  the 
positions  of  180°  and  360°  with  regard  to  the  line  of  sight  can  be  obtained. 


IO24  APPENDIX. 

To  Determine  tlie  Diameter  of  Cylinder  of*  a  Non- 
Condensing  Steam-Engine,  and  the  Elements  of*  a 
Fire— Flue  JSt,eam  Boiler,  toy  Computation,  of  them 
from  the  required  Capacity  of*  Engine  and  of  its 
Assigned  Operations. 

HP  =  I50. 

Steam  pressure  by  gauge,  70  Ibs. ;  cut-off  at  one-third ;  stroke  of  piston, 
3.5  feet ;  revolutions,  50  per  minute ;  clearance,  or  volume  of  space  between 
mean  surface  of  piston  and  valve  seat  or  opening,  .1;  back  pressure  and 
friction  each  assumed  at  2  Ibs.  per  sq.  inch. 

Hence,  P  =  7o;  1  =  42-^-3-7-12  =  1.16  ;  Z'  =  i.i6-f  .1  =  1.26  ;  1^  =  3.5;  c  =  ,i; 
R=  3^^^=2.86;  6  =  2;  and/=2. 

Assume — Evaporation  of  water  =  10  Ibs.  per  Ib.  of  coal.  Combustion,  15  Ibs.  of 
coal  per  sq.  foot  of  grate  surface  per  hour.  Heating  to  grate  surface  as  35  to  i; 
and  transverse  area  of  tubes  to  grate  surface,  .  14. 

To    Compute    Mean.    Effective    Pressure    (p.   710-711). 
7o(i.26Xi+hyp.log.2.86-.x)      —^*73-9      4 

To  Compute  Diameter  of  Cylinder. 

150X33000  ___  ^  ^  which _._ 45  ^  lbs=30gs  in$_  ig  8s  (2o)  im  diameter. 
50X3-5X2 

Then,  diam.  20  ins.  =314.16  sq.  ins.  X  42  ins.  stroke  =  13 195,  steam  cut-off  at  33 
=  4354  X  50  X  2  =  strokes  X  60  minutes  =  26  124  ooo  cube  ins.  ;  which  -f- 1728  and 
again  by  5.05,  the  volume  of  one  Ib.  of  steam  at  the  absolute  pressure  z=  2994  Ibs. 

Assume  feedwater  at  50°.  Then  2994  Ibs.  steam  at  70  Ibs.  gauge  =  85°  absolute 
pressure,  by  Factor  of  Evaporation  (see  below),  =  1.201  (the  equivalent  value  of 
212°)  =  3596,  and  3596-7-34.5  (the  Ibs.  per  hour  evaporated  at  212°  at  70  Ibs.  gauge 
pressure*)  =  104  IIP. 

Hence,  2094  Ibs.  water  to  be  evaporated  per  hour ;  2994  -4- 10  Ibs.  water  evapo- 
rated per  Ib.  of  coal  =  299.4  Ibs.  coal  per  hour;  299.4-7- 15  Ibs.  of  coal  expended  per 
sq.  foot  of  grate  =  19.9  sq.  feet  of  grate  surface;  and  19.9  x  35  sq.  feet  of  heating 
surface  per  sq.  foot  of  grate  surface  —  696. 5  sq.  feet  of  heating  surface;  and  19.9  x 
.14,  proportionate  transverse  area  of  tubes  to  grate  surface,  =  2. 78  sq.  feet. 

If  engine  were  to  be  a  condensing  engine,  the  increase  in  the  mean  pressure  in 
the  cylinder  will  correspondingly  decrease  the  diameter  of  the  cylinder,  less  the 
increased  friction  of  the  operation  of  the  air-pump,  assumed  at  .7!  Ibs.  pressure 
per  sq.  inch. 

To    Compute    Factor    of   Evaporation. 

— ^—  x  F.    H  and  h  representing  total  heat  of  the  steam  at  given  pressure  and 
965.7 
temperature  of  feedwater,  in  degrees,  and  F  factor. 

ILLUSTRATION. — Assume  gauge  pressure  of  steam  70  Ibs.  per  sq.  inch,  and  tem- 
perature of  feedwater  no0. 

1209.8  —  no 

*-* — - —  =  x.  139. 

965-7 

Application.—  Assume  the  volume  of  water  evaporated  at  the  temperature  of  the 
feedwater  of  no0  to  be  40000  Ibs.  in  10  hours  ;  steam,  gauge  pressure  70  Ibs.  per 
sq.  inch ;  coal  consumed,  4000  Ibs.,  and  refuse  from  it  350  Ibs. 

Then,  40000-4-4000=10  Ibs.  water  evaporated  per  Ib.  of  coal  consumed,  and 
40000-1-4000  —  350=  10.96  Ibs.  water  evaporated  per  Ib.  of  combustible. 

For  this  pressure  of  steam  and  temperature  of  feedwater,  the  factor  of  evapora- 
tion t  =  1.139  '•>  which  X  40000  (volume  cf  water)  =  45  560  Ibs.  equivalent  evapora- 
tion at  212°  ;  and  45  560  -f-  4000  —  350  (Ibs.  of  combustible)  =  12.48  Ibs.  water  evapo- 
rated per  Ib.  of  combustible  from  and  at  212°. 

If  34.5  Ibs.  water  evaporated  from  and  at  212°  =  one  IIP,  a  boiler  or  boilers,  oper- 
ating with  the  given  elements,  will  have  developed  45 560-^-34.5  x  10=  132.1  H*. 

*  Am.  Soc.  M.  E  t  See  p.  478.  $  See  also  Am.  Soc.  M.  E.,  1884. 


APPENDIX. 


1025 


Heating    Surface. 
Of  a    Steam    Boiler,    etc. 

Heat  is  communicated  to  the  transmitting  surfaces  of  a  steam  boiler  in  the  fol- 
towing  order  of  eft'ect — viz.,  incandescence,  flame  and  gases  of  combustion ;  and  that 
cransmitted  by  radiation  of  it,  from  one  surface  to  another,  is  reduced,  in  the  ratio 
as  the  square  of  the  distance  between  the  surfaces,  and  it  is  also  reduced  by  a  de- 
pressed inclination  of  the  surface  upon  which  the  current  of  the  heat  impinges, 
and  contrariwise  increased  by  a  raised  inclination. 

Evaporative  Efficiency.— The  evaporative  efficiency  of  a  boiler,  or  of  an  assigned 
area  of  heating  surface,  as  one  sq.  foot,  depends  so  entirely  upon  the  thickness, 
position,  and  condition  of  it  that  it  is  wholly  impracticable  to  assign  a  determinate 
value  to  it.  It  is  also  measurably  affected  by  the  duration  of  the  time  of  the  trans- 
mission of  the  gases  of  combustion  over  it. 

Theoretical  and  Attainable  Evaporation. — If  all  the  heat  of  the  combustion  of 
coal  in  the  furnace  of  a  steam  boiler  was  utilized,  the  evaporation  from  one  pound 
of  best  anthracite  would  be  from  and  at  212°  about  15  Ibs.  of  water,  but  only  80  per 
cent,  of  that  has  been  attained. 

At  the  Centennial  Exhibition  in  Philadelphia  in  1876,  the  average  evaporation 
from  15  boilers  of  different  types,  with  grate  area  as  35  to  i,  was  10.27  Ibs.  of  water; 
and  the  averages  of  evaporation  per  sq.  foot  of  heating  surface  per  hour  was  2.99 
Ibs.,  varying  from  1.75,  to  9  ;  and  of  the  temperature  of  the  escaping  gases,  410°. 

Experiments  with  locomotive  boilers  by  D.  K.  Clark,  having  from  52  to  90  sq. 
feet  of  heating  surface  per  sq.  foot  of  grate,  gave  with  coke*  an  average  evaporation, 
at  the  ordinary  temperature  and  pressures.  9  Ibs.  of  water  per  Ib.  of  fuel. 

In  horizontal  tubular  boilers,  with  heating  to  grate  surface  as  25  to  i,  the  vol- 
ume of  water  evaporated  per  Ib.  of  fuel  decreased  as  the  fuel  consumed  per  sq.  foot 
of  grate  area  increased. 


Fuel 

Water  evaporated 
from  212° 

Tempera- 

Fuel 

Water  evaporated 
from  212° 

Tempera- 

per hour 
persq. 
foot 

persq. 
foot  of 
heating 

per  Ib. 
of  coal. 

persq. 
foot  of 
heating 

ture  of 

escaping 
gases. 

per  hour 
persq. 
foot 

persq. 
foot  of 
heating 

per  Ib. 
of  coal. 

persq. 
foot  of 
heating 

ture  of 
escaping 
gases. 

of  grate. 

surface. 

surface. 

of  grate. 

surface. 

surface. 

Lbs. 

Lbs. 

Lba. 

Lba. 

Deg. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Deg. 

6 

.24 

10.49 

2.52 

444 

16 

.64 

8.21 

5-25 

897 

8 

•32 

10.35 

3-31 

472 

18 

.72 

7-7 

5-54 

999 

IO 

•4 

10.05 

4.02 

532 

20 

.8 

7.32 

5-85 

1074 

12 

.48 

9-53 

4-57 

685 

22 

.88 

7.04 

6.19 

1130 

14 

.56 

8.87 

4.96 

766 

24 

.96 

6.82 

E»      7™!,., 

6-54 

"74 

r   c»    -\r 

Benj.  F.  Isherwood,  U. 

The  efficiency  is  also  dependent  upon  the  area  of  it  for  the  contact  of  furnace 
heat,  flame,  the  gases,  and  the  period  of  the  application  or  transmission  of  the 
heat  over  it ;  and  inasmuch  as  flame  imparts  more  heat  than  the  inflammable  gases, 
the  diameter  of  tubes  should  be,  so  far  as  practicable,  of  a  capacity  to  admit  of  the 
flow  of  it. 

Radiation  of  Heat  to  Surfaces.— If  the  thickness  and  surfaces  of  boiler  plates  and 
tubes  were  uniform,  or  progressively  reduced  in  thickness  and  resulting  capacity 
of  radiation,  their  progressive  effect  and  the  proper  temperature  of  the  gases  at  the 
point  of  delivery,  as  into  a  smoke-pipe  or  chimney-stack,  could  be  readily  obtained 
by  a  dividend,  of  the  difference  of  temperatures  between  that  of  the  entrance  of  the 
gases  of  combustion  into  the  flues  or  tubes,  and  that  of  212°,  the  divisor  being  that 
of  any  assumed  number. 

Thus,  assume  the  gases  at  the  bridge  wall  at  1700°,  which  is  the  temperature 
assigned  by  Chief  Engineer  Benj.  F.  Isherwood,  U.  S.  N.,  being  the  result  of  his 
observation  and  extended  experiments. 


Then,  at  six  locations  or  divisions  of  the  temperature, 


1700" 


-  =  248° 


and    1700°  — 2480=1452°   and 


1245 


,°  — 212° 


=  207°  ;    1452°  —  207°  =  1245°   and 

'3°  =  1072°  and  I0720~2I20  =  i44o;  I072o_I44o 


=  928°  and 


9280-2,20 
6 


o  _  I20o  =  8o8o  and 


8o8°~2I2° 


=  990. 


IO26  APPENDIX. 

Thus,  at  termination  of  6th  location,  99°  are  radiated  in  its  passage  from  the  sth, 
and  the  temperature  of  exits  808°  —  212°  =  596°. 

Professor  RanTdne  asserts  that  when  the  difference  of  temperature  between  the 
water  of  evaporation  and  the  gases  of  combustion  is  very  great,  that  the  rate  of 
conduction  increases  faster  than  the  ratio  of  the  difference,  and  is  nearly  propor- 
tional to  the  square  of  the  difference  of  temperature,  and  which  may  be  thus  ex- 
pressed. 

T  —  Z  -i-  C  =  R.  T  and  t  representing  the  temperatures  of  the  gases  and  the  water, 
C  a  constant  derived  from  experience,  160  to  200,  and  R  ratio  of  conduction  in  ther- 
mal units. 

Assume  temperature  of  gases  and  water  1500°  and  212°  and  C  =  180. 


1500 212  ,  _,,  , 

=0216  thermal  units. 

1 80 

Robert  Wilson,  London,  1896.  Gives  for  multitubular  and  other  boilers  with 
heating  surface  to  grate  area  from  30  to  40  to  i  :  9  sq.  feet  heating  surface  to  evap- 
orate i  cube  foot  fresh  water,  or  4.5  sq.  feet  of  total  heating  surface  per  H*.  In 
locomotive  and  like  boilers  with  a  blast  draugni,  with  heating  surface  to  grate  from 
60  to  80  to  i :  6  sq.  feet  heating  surface  to  evaporate  i  cube  foot  fresh  water,  or  3 
sq.  feet  of  total  heating  surface  per  IP  ;  and  the  highest  average  efficiency,  i  sq. 
foot  of  heating  surface  for  13.5  Ibs.  water,  or  4.66  sq.  feet  for  i  cube  foot  of  water. 
And  in  ordinary  externally  fired  boilers,  with  heating  surface  to  grate  10  to  16  to  i : 
18  sq.  feet  of  heating  surface  to  evaporate  i  cube  foot  watei,  or  9  feet  per  IP. 

Vertical  Boilers.—  Usually,  are  wasteful  of  fuel,  but  when  in  good  condition  and 
the  tubes  properly  spaced,  16  sq.  feet  of  heating  surface  have  evaporated  i  cube 
foot  of  water  or  8  sq.  feet  per  IP,  with  a  consumption  of  i  Ib.  coal  to  evaporate  8 
Ibs.  water  =  7.75  Ibs.  per  cube  foot.  Usually  10  to  12  sq.  feet  of  heating  surface  are 
required  per  IP. 

Tubes. — The  number  and  sectional  area  of  tubes  in  a  boiler  (horizontal  multi- 
tubular)  and  the  spaces  between  them  should  be  determined  by  the  transverse  area 
over  the  bridge  wall  or  the  grate  surface,  and  the  required  facility  of  the  ascending 
current  of  steam  from  over  furnace,  and  in  all  cases  should  be  set  in  direct  vertical 
lines,  and,  in  consideration  of  their  efficiency,  their  lower  or  inner  surface  kept  free 
from  accumulation  of  the  mechanical  deposit  of  ashes  and  soot. 

Inasmuch  as  the  surface  of  a  tube  increases  directly  with  its  diameter  and  its 
sectional  area,  or  capacity  as  the  square  of  it,  it  becomes  necessary  in  order  to  at- 
tain like  economy  of  evaporation  by  them,  when  of  different  diameters,  as  i  and  2 
ins.,  that  the  length  of  the  greater  diameter  must  be  twice  that  of  the  less. 

Thus,  the  2-inch  tube  having  four  times  the  sectional  area  or  capacity  for  the 
passage  of  flame  or  gases  than  the  less,  and  but  twice  its  heating  surface,  the  length 
of  the  greater  must  be  twice  that  of  the  less.* 

Length  of  Flues  and  Tubes. — The  greater  the  velocity  of  the  current  of  the  gases 
in  a  flue  or  tube,  the  greater  will  be  its  temperature  at  their  exit,  and  consequently 
the  greater  the  waste  of  it,  unless  the  length  of  them  is  proportional  to  the  velocity 
of  the  current. 

The  absorption  of  the  heat  of  the  gases  requires  time,  and  hence  the  longer  the 
course  of  them,  if  duly  proportioned,  the  greater  their  effect.! 

Peclet  assigns  the  proportion  of  radiating  heat  from  coal  in  its  condition  of  per- 
fect combustion  at  .5  of  its  latent  heat. 

Length  and  Area  of  Tubes.—  With  ordinary  smoke-pipe  or  chimney  draught,  the 
length  of  a  tube  should  not  exceed  40  times  its  diameter.  Experiment  with  a  tube 
2.5  ins.  in  diameter,  the  temperature  of  the  gases  at  their  delivery  being  500°,  the 
length  of  it  was  100  ins.,  or  40  times  its  diameter,  and  with  a  blast  draught,  as  in  a 
locomotive  boiler,  or  blower  draught ;  the  length  may  be  much  increased.  By 
late  experiments  on  a  railway  in  France  it  was  found  that  greater  economy  was 
attained  by  increasing  the  length  of  locomotive  tubes  above  12  feet.  In  conse- 
quence of  which  the  Baldwin  Locomotive  Works,  of  Philadelphia,  assign  a  length 
of  14  and  14.5  feet  for  a  2-inch  tube  =  84  and  87  times  the  diameter.  Prior  to  this 
2-inch  tubes  were  usually  but  10  and  12  feet  in  length.  In  all  cases,  however,  the 
length  should  be  in  direct  proportion  to  the  diameter. 

Vertical  Fire  Tubes.— Are  not  as  effective  as  horizontal  or  inclined,  as  the  gases 
*  Se$  P-  742-  t  For  the  evaporating  capacity  of  tubes  of  different  lengths,  see  also  p.  742. 


APPENDIX. 


IO27 


which  lose  some  of  their  temperature  by  contact  with  the  surface  of  the  tubes  are 
not  replaced  by  the  centre  current,  and  the  additional  temperature  of  it  imparted. 

Water  Tubes.— Whether  vertical  or  inclined,  enable  the  steam  which  is  gener- 
ated at  their  inner  surface  to  rise  as  fast  as  it  is  generated,  and,  as  a  consequence, 
the  velocity  of  the  current  of  the  water  is  increased. 

Water  or  Masonry  Bridge  Walls.— At  the  termination  of  the  grates,  by  diverting 
the  current  of  the  gases  of  combustion,  enable  them  to  be  better  commixed,  and 
effect  a  more  effective  combustion  and  consequent  economy. 

In  the  computation  of  the  area  of  heating  surface,  the  areas  of  the  furnace  above 
the  grates,  bridge  wall  if  water,  combustion  chamber,  flues,  tubes  (fire  or  water), 
and  connections  to  water-line  are  to  be  taken,  and  also  two-thirds  of  all  the  gas 
surfaces  above  it,  inasmuch  as  they,  as  in  the  case  of  a  steam  chimney,  are  sur- 
rounded by  steam  at  a  high  temperature. 

Steam    Heating. 

(In  addition  to  p.  527. ) 

To  Compute  Area  of  Plate  or  3?ipe  Surface  of  Cast- 
Iron,  to  Compensate  the  Reduction  of  Temperature 
in  an  Enclosed.  Space  by  the  Exposure  of  GJ-lass 
to  the  External  A-ir,  or  its  Equivalent  in  Exposed 
Walls. 

fp g 

—  =  A.    T  and  t  representing  degrees  of  required  temperature  of  space  and  of 

external  air,  H  temperature  of  heating  surface,  and  A  area  of  radiating  plate  or 
pipe  surface  in  sq.  feet. 

ILLUSTRATION.— Assume  temperature  of  external  air  70°,  of  heating  surface  160°, 
and  required  temperature  of  space  70°. 

7Oo 2Oo 

JL — —=,667  sq.feet  of  heating  surface  for  each  sq.foot  of  glass.     Hence,  if 

the  area  of  the  glass  in  windows  or  lights  of  a  room  is  80  sq.  feet,  then  .667  X  80  = 
53.36  sq.feet  additional  radiating  surface. 

Radiation. 

One  square  foot  of  Direct  Radiating  surface  will  heat 


Cube  Feet. 

In  an  ordinary  domestic  room 

with  glass  windows ,  35  to  45 

In  an  ordinary  public  room. ...  45  "55 

u  small  dormitories 50  "60 

"  large  dormitories 45  "55 

"  hallway  and  passages 60  "80 

"  school  and  low-ceiled  lecture- 
rooms  and  offices 60  "75 


Cube  Feet. 
In  churches  and  high -ceiled 

halls 65  to   95 

"  small  low -ceiled  factories 

and  workshops 50  "    60 

"  small  high  -ceiled  factories 

and  workshops 60  "    70 

"  large  high -ceiled  factories 
and  workshops 75  "  140 


Steam    Exhaust    Heating. 

Exhaust  steam,  according  to  the  condition  in  which  it  flows  from  an  engine, 
contains  water  and  oil,  varying  from  10  to  20  per  cent,  of  both  combined.  Conse- 
quently they  should  be  arrested  before  entering  a  heating  plant ;  and  as  any  restric- 
tion to  the  flow  of  the  steam  involves  a  resistance  to  it,  or  that  which  is  termed 
back  pressure,  the  receiving  and  distributing  pipes  should  have  the  greatest  prac- 
ticable capacity  and  least  restriction  to  the  flow  of  it  by  curves  and  angles. 

To  meet  an  emergent  requirement  for  heat,  a  direct  connection  to  the  boiler, 
through  an  automatic  reducing  pressure  valve,  must  be  furnished,  and,  contrari- 
wise, a  relief  valve  should  be  furnished,  in  order  that  when  a  reduced  temperature 
or  volume  of  steam  is  required  in  heating,  it  may  escape  into  the  air. 

To  Design  and  Proportion  an  Exhaust  Stenm  Plant. — It  is  necessary  first  to 
ascertain  the  area  of  radiating  surface  required  to  condense  the  exhaust  steam,  and 
to  obtain  this  the  volume  of  the  steam  which  the  engine  would  exhaust  must  be 
ascertained.  To  determine  which  (see  table,  p.  708),  give  weight  of  water  evaporated 
from  212°  per  sq.  foot  of  heating  surface. 


1028 


APPENDIX. 


ILLUSTRATION. — Assume  the  area  of  the  heating  surface  is  9000  sq.  feet,  of  grate 
30,  and  fuel  consumed  16  Ibs.  per  sq.  foot  of  grate  per  hour. 

The  water  evaporated  per  hour  per  sq.  foot  of  grate  =  8. 21  (see  table,  p.  1025)  X 
16  =  131.36  Ibs.  water,  which  x  26.36  (the  volume  of  i  Ib.  steam  at  212°)  =  3  462. 65 
cube  feet  of  steam. 

Water  evaporated  per  hour  per  IIP  in  a  non-condensing  engine  ranges  from  25 
to  40  Ibs.,  from  which,  for  condensation  and  leaks,  10  per  cent,  should  be  deducted 
to  obtain  the  volume  of  steam  available  for  heating. 

One  square  foot  of  Direct  Radiating  surface  will  heat 

Cube  Feet,  i  Cube  Feet. 

In  dwelling-houses 45  to  55    In  factories,  stores,  and  shops    90  to  100 

"  offices 65  "  75  I    "  churches,  auditoriums,  etc.  150  "  200 

For  Indirect  Radiation  deduct  20  per  cent.,  and  when  the  heat  is  transmitted  by 
blast  from  a  Blower  add  from  4  to  6  times,  in  accordance  with  its  volume  and  con- 
sequent velocity. 

Hot- Water    Heating. 

The  sectional  area  of  the  main  pipe  should,  in  all  cases,  exceed  that  of  its 
branches,  and  for  each  sq.  inch  of  its  section,  if  short,  indirect,  and  at  a  slight  in- 
clination, 50  sq.  feet,  and  if  long,  direct,  and  vertical,  100  sq.  feet. 

One  square  foot  of  Direct  Radiating  surface  will  heat,  the  average  temperature  of 
the  water  160°,  from  80  to  100  per  cent,  more  surface  than  by  Steam. 

Horse-Power    Required,    to    Drive    Machinery. 

In  addition  to  Frictional  Resistances,  etc.,  pp.  475-478.  See  a  very  full  table  of 
HP  required  in  American  Machinist,  April  12, 1894,  and  February  6,  1896. 

Referring  to  the  following  table,  it  will  be  noticed  that  the  loss  of  power  varies 
between  wide  limits,  but  in  all  cases  the  mechanical  loss  is  large,  averaging  over  41 
per  cent. 


Machinery. 

Work. 

Total 
H?. 

Horse 
Shafting. 

Power  to 
Machin- 
ery. 

Drive 

Shafting. 
per  cent. 

Union  Iron  Works  
Frontier  I.  &  B.  Works.  . 
Baldwin  Loc.  Works  
W  Sellers  &  Co  

Engines  and  Machinery. 
Marine  Engines,  etc  
Locomotives  
Heavy  Machinery  
Machine  Tools  
Cranes  and  Locks  
Presses  and  Dies  

406 
25 
2  5OO 
102.45 
1  80 
135-°5 

'       35 
150 

400 

9I 

2000 
40.89 

66.11 

II 

75 

100 

3°5 
i? 
500 
61.56 

'OS 
68.24 
24 
75 
qoo 

•23 

•32 
.80 
.40 
.41 
•49 
•3i 
•So 
.25 

Pond  Machine  Tool  Co.  .  . 
Yale  &  Towne  Co  

Ferracute  Machine  Co.  .  . 
Bridgeport  Forge  Co  
Hartford  Mch.  Screw  Co. 

Heavy  Forgings 

Machine  Screws.... 

(Prof.  J.  J.  Flather.) 
Refrigerating    Machinery. 

For  the  cooling  of  Brine  and  other  liquids  by  the  alternate  compression  and  ex- 
pansion of  air. 

,      T— t  P  T 


P  representing  power  required  in  foot-pounds,  T  absolute  maximum  temperature 
of  the  air  in  the  hot  or  compressive  end  of  the  refrigerator,  t  absolute  minimum  tem- 
perature of  the  air  in  the  cold  or  expansion  end,  and  C  cooling  work  in  thermal 
units.  (David  Thomson.) 

ILLUSTRATION.— Assume  T  =  80°,  t  =  30°,  and  C  =  80°  —  30°  =  50°. 

80 

772X5QX~8oJ 

=  31.25  X  1.6  =  50°. 

Hence,  the  most  economical  results,  as  regards  power  used,  are  obtained  when 
the  machine  is  operated  within  a  small  range  of  temperature,  as  in  a  brewery, 
where  the  temperature  of  the  water  is  frequently  reduced  to  but  10°. 

These  formula  are  applicable  to  all  refrigerating  machines,  whether  operated  by 


f  ?2. — 3°,=  3s  600  X  .625  =  24 125  —foot  -pounds,  and  ^-^  X- 


_  3O 


APPENDIX. 


IO29 


air,  ether,  ammonia,  or  any  other  liquid.  In  an  ammonia  machine,  or  any  other 
operated  on  the  same  principle,  in  which  mechanical  power  is  applied,  the  value 
of  P  is  the  heat  theoretically  required,  at  the  rate  of  i  heat-unit  for  772  foot-pounds 
or  power,  and  the  formula  i  becomes  (ammonia):  Heat  required  for  the  operation, 

r-'r     r 
_  c=c. 

The  ammonia  machine  is,  theoretically,  economically  superior,  as  heat  is  less  ex- 
pensive than  its  equivalent  in  mechanical  power. 
The  nature  of  the  vapor  operated  controls  the  capacity  of  the  machine. 

Relative  Capacities  of  Cylinder  Required. 


Ammonia i 

Carbonic  acid 16 

Methyl  Chloride 1.8 


Methyl  ether  ......................  1.8 

Sulphuric  acid  .....................  2.6 

Ether  .............................  15.  i 

(D.  K.  Clark.) 
Sewerage. 

In  order  that  an  estimate  of  the  volume  of  excessive  rainfalls  may  be  com- 
puted, the  following  data  are  derived  from  the  valuable  report  of  the  Sewerage  Com- 
mission of  Baltimore,  1897  : 

Philadelphia,  July  23,  1887  .....................  4.23  inches  in  13  minutes. 

Chestertown,  Md.,  Aug.  15,  1894  ...............  3.64      "      "30       *' 

Washington,  B.C.,  June  30,  1895  ...............  6.27       "      "  10       " 

The  average  of  26  falls  was  3  inches  in  10  minutes.        . 

In  the  city  of  New  York  a  fall  of  i  inch  in  10  minutes  has  frequently  occurred  — 
6  inches  per  hour. 

Flags. 

Safe  Transverse  Strength. 
Loaded  in  Middle.     Supported  at  Both  Ends. 

c. 

Freestone,  Little  Falls  ..............  121 

"         Belleville,  N.  J  ..........  101 

"         Connecticut...  ..........  65 

"         Dorchester,  Mass  ........  50 


c. 
Slate,  mean  of  242  and  537  ..........  390 

Glass  ........................  .  .....  210 

Bluestone  .........................  178 

Granite,  Quincy  ...................  131 

bd2 
To  Compute  Safe  Load,     -y-  X  C  =  Load  in  Ibs. 


C  representing  one-tenth  of 


breaking  weight. 

Assume  a  flag  or  block  of  Quincy  Granite,  6  feet  in  width,  6  ins.  in  depth,  and  3 
feet  in  length  between  its  supports. 

6  X  12  X  62 

X  131  =  72  X  131  =  9  432  Ibs. 


3X  12 

Average  weight  of  17  different  Sandstones,  a 


ascertained  by  Lieut.-Col.  Gilmore,  U.S.A.,  143  Ibs. 


CAST-STEEL    FLAT    ROPES. 

John  A.  Roebling's  Sons  Co.,  New  York. 


Dimensions. 

Weight 
per  Foot. 

Strength. 

DimeMion..       |    Jtftl 

Strength. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

•375  X  2 

1.19 

35700 

•5X3 

2.38 

71400 

•375  X  2.5 

1.86 

55^00 

•5X3-5 

2.97 

89000 

•375X3 

2 

60000 

•5X4 

3-3 

99000 

•375X3-5 

2-5 

75000 

•5X4-5 

4 

I2OOOO 

•375X4 

2.86 

85800 

•5X5 

4.27 

128000 

•375X4-5 

3.12 

93600 

•5X5-5 

4.82 

144600 

•375  X  5 

3-4 

100  000 

.5X6 

5-i 

153000 

•375X5-5 

3-9 

IIOOOO 

•5X7                   5-9 

177000 

Steel  Wire  Flat  Ropes  are  composed  of  a  number  of  strands,  alternately  twisted 
right  and  left,  laid  aside  of  each  other  and  sewed  together  with  soft  iron  wires. 
They  are  used  sometimes  in  place  of  round  ropes  in  shafts  of  mines:  wound  upon 
a  narrow  drum,  requiring  less  space  than  a  round  rope.  Soft  iron  sewing-wires 
wear  out  sooner  than  the  steel  strands,  and  then  it  is  necessary  to  replace  them 
with  new  iron  wires. 


1030 


APPENDIX. 


Illustrations    in.   Logarithms. 

To    Compute    the    Length    of   an    Arc    of   a    Circle    to 
Radius    1. 

RULE.— To  log.  of  degrees  in  the  arc,  add  2.241  877,  and  sum  is  log.  of  length. 

NOTE. — When  the  arc  is  in  minutes,  seconds,  etc.,  take  their  decimal  equivalents. 

ILLUSTRATION.— An  arc  of  a  circle  is  57°  17'  44"  48"  -\- ;  what  is  its  length? 

17°  44"  48"  =  .2957.  Log.  570-2957=_i-758l23 

Log.  3. 1416  -r-  180°  —  2.241877 

Log.  i  =   .  ooo  ooo 

To    Compute   the    Degrees    in.    an    Arc    of  a    Circle   -when 
the    Length    is    Griven. 

RULE.— To  log.  of  length  of  arc,  add  1.758 123,  and  from  the  sum  subtract  log.  of 
its  radius,  and  remainder  will  give  log.  of  degrees. 

ILLUSTRATION. — How  many  degrees  are  there  in  an  arc  when  the  length  is  2  and 
the  radius  i  ? 

Log.  2  =  .301 030 
Log.  180°  -r-  3. 1416  =  1.758123 

2.059153 
Log.  i  =  .000000 

"    2.059  J53  =  114- 59l66  =  "4°  35'  3°  '— • 


To    Compute    the    Angles    of  a    Triangle,   the    Length 
of  the    Sides    feeing    GHven. 

RULE  i. — To  the  logs,  of  the  differences  between  any  two  sides  and  half  the  sum 
of  the  sides,  add  the  arithmetical  complements  of  the  logs,  of  half  the  sum  of  the 
sides,  and  the  difference  between  it  and  the  remaining  side,  and  divide  the  sum  by  2, 
the  quotient  is  the  logarithmic  tangent  of  half  the  angle  opposite  to  the  latter  side. 

RULE  2. — To  the  logs,  of  half  the  sum  of  the  sides,  and  the  difference  between  it 
and  any  side,  add  the  arithmetical  complements  of  the  logs,  of  the  two  other  sides, 
divide  the  sum  by  2,  and  the  quotient  is  the  logarithmic  cosine  of  half  the  angle 
opposite  the  former  side. 

RULE  3.— To  the  logs,  of  the  differences  between  any  two  sides  and  half  the  sum 
of  the  sides,  add  the  arithmetical  complements  of  the  logs,  of  three  sides,  divide 
the  sum  by  2,  and  the  quotient  is  the  logarithmic  side  of  half  the  angle  opposite  to 
the  remaining  side. 

ILLUSTRATION.— The  sides  of  a  triangle  are  A  679,  B  537,  and  C  429.  What  are 
the  angles? 

—  =  822.5  and  822.5  —  679  =  143.5  =  difference  between  two  sides 

and  half  sum  of  sides.    822. 5  —  429  =  393. 5  and  822. 5  —  537  —  285. 5  =  differences  as 
above  obtained. 


Log.  143.5  —  2.156852 

"    393-5  =  2.594945 

Co.  Log.  822. 5  — 10  =  7.084  864 

'•'      "    285.5  —  10  =  7.544394 

2)19.381055 

9.690528 

Log.  143.5  =  2.156852 

"    285.5  =  2.455606 

Co.  Log.  679  —  10  =  7. 1 68  130 

"      "     537  —  10  =  7.270026 

2)19.050614 

9-525307 


2.  Log.  822. 5  =  2.915  136 

"    143-5  =  2.156852 

Co.  Log.  537  —  10  =  7.270026 

"      "    429  —  10  =  7.367543 

2)19.709557 

9-854779 

1.  9. 690  528  =  .  5  Log. e  Sin.  =  19°  35" 

2.  9. 854  779  =  .5      "     Cos.  =  44°  17' 30" 
3-  9-525 307  =  -5     "    Tan.  =  26°    7' 30 


90°    o'    o" 


APPENDIX. 


1031 


To    Compute    tlie    Volume    of  a    Pyramid    or    Cone. 

RULE.— To  logs,  of  area  of  base  and  height  add  7.522  879,  and  the  sum  is  the  log. 
of  the  volume. 

ILLUSTRATION.— The  largest  Pyramid  of  Egypt  has  a  base  of  700  feet  square,  a 
height  of  500  feet,  aud  assuming  its  faces  to  be  triangular,  and  to  be  constructed 
solid  of  granite  weighing  2654  oz.  per  cube  foot,  what  is  its  volume  and  weight? 

Log.  700  =  5.690196  Log.  volume  =  7.41 2 045 

"     500  =  2^698970  Log.  2054 •=  3. 423  901 

"  1-7-3  =  1.522879  Log.  2240  -f- 16  =  5. 445  632 

Log.  volume  7.912045  Log.  weight  =  6.781  578 

Log.  7.912  045  =  81  666  667  cube  feet 
"    6.781578=  6.047  528  tows. 


Centrifugal    Pumps. 

Morris  Machine  Worlcs,  Baldwinsville,  N.  T. 

Centrifugal  Pumps.— Are  simple  in  construction,  and  for  the  raising  of  great  vol- 
umes of  water,  to  or  from  a  low  elevation,  are  superior  to  a  piston  pump,  in  their 
dispensing  of  valves,  piston  and  its  packing,  etc. ; 
and  as  a  result  they  will  effectively  raise  gravel, 
sand,  paper  pulp,  sewage,  silt,  and  like  material : 
all  of  which  a  piston  pump  is  wholly  impractica- 
ble of  raising. 

The  efficiency  of  one  when  properly  propor- 
tioned and  constructed  is  fully  65  per  cent,  of 
the  power  expended  to  operate  it,  and  they  are 
also  applicable  to  furnish  surface-water  for  the 
condensers  of  marine  and  other  engines,  and 
brine  in  ice -making  machines,  in  dredging, 
wrecking,  etc.,  etc.,  at  a  less  cost  of  operation  than  any  other  pump  of  like 
capacity. 

Standard    Horizontal    Pump. 


No. 

Capacity  per  Minute. 

Minimum  Power  for 
each  Foot  of  Lift. 

Diameter  and 
Face  of  Pulley. 

Weight. 

Galls. 

IP. 

Ina. 

Lbs. 

J-5 

50  to        70 

.024 

6X    6 

168 

'•75 

75  to       loo 

•037 

7X    8 

232 

2 

no  to       150 

•054 

8X    8 

306 

2-5 

175  to      250 

.086 

8X    8 

348 

3 

2  5°   °      35° 

.124 

8X    8 

400 

4 

450  .0      600 

.223 

10  X  10 

545 

5 

750  to      900 

•372 

15  X  12 

826 

6 

i  xx)  to    T  400 

.496 

15  X  12 

965 

8 

I  700  tO      2  2OO 

.844 

20  X  12 

i  500 

10 

2  2OO  tO      3  OOO 

1.093 

24  X  12 

2170 

12 

3  ooo  to    4  ooo 

1.49 

30X14 

3050 

15 

4  800  to    6  ooo 

2.38 

40  X  15 

7  zoo 

*i5 

4  800  to    6  ooo 

2.38 

30X15 

3i5o 

18 

7  500  to  10  ooo 

3-73 

40  X  15 

9000 

*i8 

7  500  to  loooo 

3-73 

30  X  16 

350° 

22 

12000  tO  14000 

5-96 

48  X  20 

12000 

*  Refers  to  low-lift  pump.    The  number  of  pump  is  also  diameter  of  discharge  opening  in  inches. 
Where  more  than  25  feet  of  discharge  pipe  is  attached  to  pump,  one  or  two  sizes  larger  than  the  pump 
discharge  is  recommended. 

Railway. 

Speed.     Chicago,  Burlington  &  Quincy,  passenger  train,  Denver  to  Eckley,  14.8 
miles  in  9  minutes  =  98. 66  miles  per  hour. 


IO32  ASBESTOS    FELTINGS,   CEMENTS,   ETC. 

Asbestos    Fabrics,    Felts,     Cements,    Locomotive 
Lagging,   1C  to. 

H.  W.  Johns  ManvUle  Co..  New  York. 
Steam -Pipe    and.    Boiler    Coverings,    Paelzings,    Etc. 

In  the  protection  of  surfaces  from  loss  of  heat,  Hair-Felt  and  other  organic 
materials,  in  consequence  of  their  destructibility  at  high  temperatures,  have  been 
very  generally  superseded  by  Felts  made  from  Asbestos. 

Numerous  tests  of  the  relative  non-conductivity  of  materials  published  by  au- 
thorities have  given  an  impression  that  Asbestos  is  an  inferior  non  conductor  of 
heat.  This,  however,  is  an  error,  as  these  tests  are  made  with  the  dense  or  crude 
forms  of  Asbestos,  while  in  its  fibrous  state  it  contains  numerous  air  cells.  The 
best  insulator  known  is  air  confined  in  minute  cells,  so  that  heat  cannot  be  re- 
moved by  convection,  and  the  value  of  insulating  substances  depends  upon  the 
power  of  holding  minute  volumes  of  air  in  a  manner  that  precludes  circulation. 

Asbestos  Fibrous  Fabrics  are  claimed,  therefore,  to  be  the  very  best  and  most 
durable  non-conductors  of  heat. 

Asbestos    Fire-Felt 

Is  a  fabric  "felted"  from  Asbestos  fibres.  As  its  air-cells  are  innumerable  and 
microscopic  in  size,  Fire-Felt  is  a  successful  application  of  the  air-space  principle. 
In  addition  to  its  superior  insulating  properties,  it  is  fire-proof,  flexible,  light  in 
weight,  susceptible  of  any  desired  mechanical  arrangement,  and  indestructible.  It 
is  particularly  adapted  for  Marine,  Mine,  and  Railway  work,  as  moisture  and  vibra- 
tion will  not  disintegrate  it,  and  it  will  withstand  much  rough  usage.  It  is  sup- 
plied in  cylindrical  sections  for  pipes,  in  sheets  for  boilers,  drums,  flues,  etc.,  in 
rolls  for  grouped  pipes,  cylinders,  hot-air  pipes,  etc.,  and  in  blocks  for  Locomotive 
and  Boiler  Lagging,  etc. 

Asbestos     Fire-Felt,    Asbesto-Sponge    Felted,    and    As- 
besto- Sponge    Molded    Sectional     Pipe    Covering. 

These  are  formed  into  cylinders,  cut  lengthwise,  in  order  that  they  may  be  laid 
over  pipes,  and  are  furnished  with  a  canvas  jacket,  secured  by  metal  bands.  They 
are  suitable  for  both  high  and  low  steam  pressures.  The  Fire-Felt,  being  com- 
posed wholly  of  Asbestos,  is  especially  adapted  for  highest  pressures  and  super- 
heated steam. 

Champion,  Zero,  Brine,  and  Ammonia    Sectional    Pipe 
Covering. 

"Champion"  is  an  economical  covering  for  low-pressure  steam  and  hot- water 
pipes. 

"Zero"  effectually  prevents  water  and  gas  in  pipes  from  freezing. 

Brine  and  Ammonia  Pipe  Coverings  prevent  the  formation  of  ice  on  the  line  of 
pipe,  and  produce  important  economies  in  refrigerating  and  ice  plants. 

Asbestos    Cement    Felting. 

Composed  of  Asbestos  fibre,  infusorial  earth,  and  a  cementing  compound,  ap- 
plied to  pipes,  boilers,  etc.,  while  heated. 

Furnished  in  bags  or  barrels.  One  bag  contains  sufficient  material  to  cover 
about  40  square  feet  of  surface  i  in.  in  thickness,  and  weighs  about  120  Ibs.  net. 

A,sbestos    Lagging    for    Locomotives. 

Composed  wholly  of  pure  Asbestos,  suitable  for  all  styles  of  locomotives. 

In  slabs,  6  ins.  in  width  by  36  ins.  in  length,  from  .5  in.  to  2  ins.  in  thickness. 

Asbesto-Sponge    Hair-Felt 

Is  very  elastic,  and,  in  consequence  of  the  large  proportion  of  Asbestos  in  it,  it 
is  not  liable  to  injury  from  steam  heat. 
In  rolls  of  about  300  square  feet,  6  ft.  in  width,  and  .375  in.  in  thickness. 


ASBESTOS    FELTINGS,    CEMENTS,    ETC. 


1033 


Hair-Felt. 
Of  various  thicknesses. 
In  bales  of  300  square  feet,  72  ins.  in  width. 

Asbestos    Cloth. 

Pure  fibres  of  Asbestos  spun  into  threads  and  woven  into  cloths.    Produced  in 
various  weights  and  widths. 

Fine  cloth,  36  ins.  in  width,  weighs  3.33  oz.  per  square  foot. 
Medium  cloth,  36  ins  in  width,  weighs  4.66  oz.  per  square  foot. 
Heavy  cloth,  36  ins.  in  width,  weighs  6.25  oz.  per  square  foot. 


Asbestos    Packings    and    Asbesto-Metallio    Packings. 

These  are  especially  adapted  for  the  extreme  high-pressure  and  high-speed  en- 
gines of  modern  times.  They  are  supplied  in  flat,  round,  and  special  shapes  to 
meet  all  requirements. 

Asbestos    Mill-Board. 

Composed  of  pure  Asbestos  fibres.  Valuable  for  sheet  packing  and  general 
joint-work,  for  gas  fire-backs,  screens,  partitions,  and  general  fire-proofing  pur- 


In  sheets  40  by  40  ins.,  from  .03125  to  .5  in.  in  thickness. 


"Asbestos    Non-Burn    Paper    or    Building    Felt." 

Composed  of  pure  Asbestos  fibres.  Used  as  fire-proof  lining  between  floors,  side- 
walls,  etc.,  of  frame  and  other  structures  ;  also  for  railroad-car  partitions.  It  is 
vermin,  acid,  and  fire  proof,  and  is  also  made  damp-proof. 

Supplied  in  rolls  weighing  about  80  Ibs.,  36  ins.  in  width.  Three  weights— 
thin,  medium,  and  heavy. 

Nickel    Steel    and    Shafting. 

Nickel  Steel  is  well  adapted  for  shafting,  as  it  has  greater  elasticity  and  tensile 
strength  than  steel,  it  being  fully  30  per  cent,  greater,  the  latter  being  20  per 
cent. 

With  4.7  per  cent,  of  nickel  in  the  composition  of  steel,  the  elastic  strength  has 
been  increased  from  36000  to  41000  Ibs.  per  Q  inch,  and  the  transverse  from  67  ooo 
to  89  ooo  Ibs. 

Electrical    Expressions    and.    Equivalents. 


Rate  of  Operation. 


One    Watt. 

i          Ampere  per  sec. 

at  one  volt. 

•7373  foot-lbs.  per  sec. 

44.238         "         "  min. 

2654.28  "         *'  hour 

.50:*  mile-lbs.    "     • 

.00134  = -^ 'ff. 


One    tP. 

550    foot-lbs.   per   sec. 
33000         "          "    min. 
375  mile-lbs.     "    hoir 
746  Watts 

.746  Kilowatt 


One    Kilo-watt. 

737.3   foot-lbs.  per   sec. 

^yf-         "  "    min. 

502.7         "  "    hour 

«-34  H». 


Quantity  of  Operation. 
One    Watt-Hour. 

2654.28  foot-lbs. 
.503  mile-lbs. 
i        ampere  -  hour  X 
one  volt. 

—  IP-hour. 

746 


Quantity  of  Operation. 
One    IP-Hour. 

1980000.  foot  Ibs. 
375.  mile-lbs. 
746.  Watt-hour. 

.746  Kilowatt-hour 

Quantity  of  Current. 
'    One    Ampere- 
Hour. 

One  Ampere  flowing  for 
one  hour,  irrespective  ol 
the  voltage. 
Watt  hour-:- volts. 

Force  Moving  in  a  Circle. 
Torq.ue.     One  pounj 
at  a  radius  of  one  foot. 


1034 


CHAINS. 


Chains. 
ITor   Cables,    Cranes,    etc.,    of  AVrought   Iron   or    Steel. 

Cable  chains  are  designated  as  Open  or  Stud  link,  and  Crane,  Sus- 
pension, and  Hauling  as  Short  or  Open  link. 

Tensile  strength  of  fibrous  Wrought  Iron  and  of  soft  Steel  is  assumed  at  56  ooo 
Ibs.  per  square  inch. 

Short-link.*  The  average  ultimate  tensile  strength  of  the  link  of  a 
chain  is  ascertained  to  be  1.625  times  that  of  the  rod  or  bar  from  which  it  is 
forged,  and  to  avoid  injury  to  a  chain  in  testing  it  should  not  be  subjected  to 
a  stress  in  excess  of  one  half  of  its  tensile  strength  ;  nor,  in  consequence  of  the 
disastrous  results  of  the  rupture  of  a  cable  or  crane  chain,  should  it  be 
subjected  in  operation  to  a  stress  in  excess  of  one  half  of  its  testing  strength. 

The  average  tensile  strength  of  i  inch  round  and  chain-rolled  wrought  iron 
and  steel,  is  further  assumed  at  44  ooo  Ibs.,  and  a  link  of  such  chain  at  71  500 
Ibs.,  or  1.625  times  greater  than  that  of  a  rod  or  bar. 

Hence,  a  chain  of  i  inch  may  be  tested  to  35  750  Ibs.,  f  and  submitted  to 
a  working  stress  of  17  875  Ibs. 

The  Pencoyd  Iron  Works  gives  17920  Ibs.,  the  Pennsylvania  Railroad  15000,  and  Molesworth 
and  D.  K.  Clark,  both  of  England,  give  respectively  15  680  and  13  440  Ibs. 

When  the  lead  of  a  chain  is  inclined  to  the  stress,  as  when  it  encompasses  a 
weight  in  mass,  or  is  applied  to  cant-hooks,  the  greater  the  angle,  the  greater  the 
stress  with  a  given  load,  as  the  stress  on  each  chain  will  be  in  the  same  ratio  to 
half  the  load  that  the  length  of  one  half  or  side  of  the  chain  bears  to  the  vertical 
distance  in  a  line  between  the  point  of  suspension  of  the  chain  and  the  load. 

Thus,  Multiply  half  the  load  by  the  length  of  one  lead  of  the  chain,  divide 
the  product  by  the  vertical  distance  and  the  result  will  give  the  capacity  of  the 
chain. 

Or, X  sec.  .5  angle  of  spread  of  the  chain  =  Stress. 

ILLUSTRATION.— Assume  the  load  to  be  i  ooo  Ibs.,  the  length  of  one  side  of  the 
chain  to  be  5  ins.,  the  vertical  distance  4  ins.,  and  the  spread  of  the  chains  6  ins. 

Then  i  ooo  -4-2X5-^-4  =  625  Ibs.  on  each  chain. 

Stud-link.  Authorities  on  the  relative  strength  of  this  and  a  short  or  open 
link  are  very  materially  divided. 

D.  K.  Clark  gives  the  safe  working  load  of  the  two  links  approximately  as: 
Short-link  D*  -f- 10.7,  Stud-link  D«  -f-  7.07.  An  excess  of  strength  for  the  stud  link. 

D  representing  diameter  of  rod  in  eighths  of  an  inch. 

Again,  he  gives  the  ultimate  safe  working  stress  of  a  stud-link  chain  at  9  tons 
(20  160  Ibs.)  and  of  a  short-link  at  6  tons  (13440  Ibs.). 

This  is  wholly  at  variance  with  the  preceding  rule  for  the  determination  of  the 
stress  on  a  chain,  when  it  is  spread  or  diverging  outward  from  the  vertical;  for  as 
the  stress  of  the  load  increases  in  the  proportion  that  the  length  of  one  lead  of  a 
chain  is  to  the  vertical  distance,  as  here  illustrated,  the  length  of  the  stud  not 
only  spreads  the  span  of  the  link,  but  subjects  it  to  a  severe  combined  tensile  and 
transverse  stress  on  its  outer  surface,  in  the  direction  of  the  central  line  of  the 
stud,  as  the  tensile  and  transverse  strength  of  wrought  iron  or  steel  being  less  than 
that  of  their  crushing  strength,  the  neutral  axis  is  lowered  and  the  stress  on  the 
outer  surface  correspondingly  increased. 

Chaining   over    Inclined   Surface. 

1 1  cosin.  A  =  l.  L  representing  length  of  line  on  surface,  A  angle  of  inclination, 
and  I  length  of  line  reduced  to  the  horizontal. 


*  Crane  chains  are  usually  of  this  consiruction. 

fThe  table  on  p.  457  is  for  English  iron  and  is  for  31  360  Ibs. 


APPENDIX. 


1035 


Hydraulics   of*  a   Fire  -  Engine. 

With  a  ring  nozzle  the  Coefficient  of  discharge  is  about  .74. 

Loss  of  Pressure  by  Friction  in  Hose. — Loss  of  head  varied  as  the  square  of  the 
velocity  ot  the  flow  and  nearly  as  the  length  of  the  hose. 

The  effect  of  a  difference  in  diameter  of  hose,  even  of  .125  inch  for  2.5  ins.,  may 
cause  a  loss  of  25  per  cent. 

Wind.    Pressure. 

Normal  pressure  is  estimated  at  15  Ibs.  per  sq.  foot  and  maximum  at  30  Ibs. ;  but 
on  elevated  structures,  in  consequence  of  the  partial  vacuum  or  minus  pressure  in 
their  rear,  the  effect  of  the  wind  is  much  increased. 

At  the  summit  of  the  Eiffel  Tower,  1097  feet,  the  pressure  has  been  observed  to 
be  5  times  that  at  the  Central  Meteorological  Bureau,  at  its  height  of  70  feet  be- 
low.—R.  Kohfahl. 

From  observations  of  the  St.  Louis  tornado  in  1896,  the  pressure  was  computed 
to  vary  from  45  to  90  Ibs.  per  sq.  foot;  and  from  experiments  of  C.  F.  Martin  at  Mt. 
Washington,  U.  S.,  it  was  shown  that  rapid  and  intense  fluctuation  occurs ;  and  by 
Kernot,  that  a  marked  difference  results  from  the  presence  of  other  buildings.— 
T.  Bates. 

Effect  of   a   Low   Temperature    on    Iron    and    Steel. 
In  Tons  per  Sq.  Inch. 


Metal, 

Temperature. 

Elastic  Limit. 

Breaking  Stress. 

Teneile 
Strength. 

Ratio  of  Elastic 
and  Breaking. 

Wrought-   ( 
Iron  Bar.    ) 

-£ 

18.2 
18.5 

25.2 

26.5 

100 

.72 
•7 

( 

'9-3 

27.8 

107-5 

•  7 

Steel,       ( 

64° 

15.2 

25-7 

100 

•59 

Siemens'    ] 

-4° 

15.7 

27.7 

102.9 

Angle.      ( 

—112° 

18.9 

28.7 

123.8 

.66 

Malleable   ( 
Iron  Bar,    j 

-Jo 

19.6 
20 

25.5 
26.4 

100 

102.3 

$ 

( 

—  112° 

20.3 

27.4 

103.2 

•74 

The  conclusions  from  these  results  are: 

1.  Elastic  and  ultimate  limits  are  raised  by  low  temperatures. 

2.  The  variation  in  mechanical  properties  by  a  reduction  of  temperature  is  great- 
er in  steel  and  least  in  malleable  iron.     The  variations  between  the  extremes  at 
temperatures  given  are,  in  per  cent. : 


Metal, 

Elastic  Limit. 
Increase. 

Tensile  Strength. 
Increase. 

Elongation. 
Decrease. 

Siemens'  steel 

2^  8 

Wrought-iron  bar  
Malleable  iron  

5-4 
3-2 

7-5 
7-5 

14.1 
5-2 

3.  The  compression  by  impact  diminishes  with  a  reduction  of  temperature,  in 
like  manner  with  elongation  under  tension. 

4.  The  loss  of  malleability  is  8  per  cent,  at  4°,  and  23  at  112°.     The  change  being 
least  in  hammered  iron,  and  greatest  in  rivet  iron. 

5.  The  flexibility  of  iron  is  slightly  changed  in  soft  rivet  at  4°,  and  in  rolled  bar 
iron  at  112°;  but  all  other  qualities  were  more  or  less  injuriously  affected  at  the 
lowest  temperature. — M.  Rudeloff. 

Relative    Hardening   of*  Cement    and    Mortar    in    Fresh, 
and    Salt    Water. 

From  experiments  with  cement  with  varying  proportions  of  sand,  it  was  shown 
that,  when  it  was  mixed  with  and  submitted  to  fresh  water,  it  became  harder.— 
N.  M.  Koning  and  L.  Bienfort. 

Evaporative    I?o-wers    of*  Coke    and    Coal. 

From  experiments  at  Colmar.    The  calorific  values  were  i  and  .8933.—^,  Weber. 


1036  APPENDIX. 

The    Hiightest    Known.    Su.tostari.oe 

Is  the  pith  of  the  Sunflower;  its  specific  gravity  .028.  Elder  pith,  hitherto  held 
to  be  the  lightest,  is  .09  ;  Reindeer's  hair  .1,  and  Cork  .24.  Hence,  Reindeer's 
hair  has  a  buoyancy  of  i  to  10,  and  Sunflower  pith  i  to  ^.—Froitzheitn. 

Ratio   between    Surface   arid.    Mean  Velocity    of  the   Wet 
Section    of*  a  3Vtill-Raoe, 

As  determined  by  a  series  of  experiments,  is  .60  to  .65,  being  less  than  that  of 
.80,  usually  taken,  and  that  of  .71  to  .72  in  channels  with  earth  banks.—  R.  P.  T- 
Tutein  Northenius. 

Testing    of*  Stones. 

Granite,  Marble,  and  Sandstone  lose  strength  by  saturation  with  water,  and 
Sandstone  and  Granite  are  most  aflected  by  frosts  __  M.  Gary. 


Resistance   of  Wronght-Iron.  and  Steel  Rivets  in.  a  Lap 
of  not    I^ess    than.    Three. 

Per  Sq.  Inch. 


Elasticity 
per  sq.  inc^h. 

How 
Made 

Tempera- 
ture. 

Resistance  to 
Shearing. 

Elasticity 
per  sq.  inch. 

How 

Made. 

Tempera- 
ture. 

Resistance  to 
Shearing. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

IRON. 

STEEL.  . 

25536 
31360 

Hand.  ) 

Bright 
red  heat 

r  5800* 
i  6720 

31360 
32704 

Hand.) 

Bright 
red  heat. 

6384 
7168 

25  536  } 

Hydrau- 

White j 

7i68 

31  360 

Hydrau- 

White j 

8512 

31  360  J 

lic. 

heat,  j 

8288 

327°4 

lic. 

heat.    \ 

9408 

The  Iron  submitted  to  an  extension  of  12  per  cent,  before  fracture,  and  the  Steel 
18  per  cent.—Dupuy. 

IVEuzzle    Velocity-    of  the    G-errnan.    Infantry    Rifle. 

A  series  of  experiments  gave  the  following  results : 

Muzzle  velocity  2070  feet  per  second,  and  the  maximum  at  10  feet  from  the  muz- 
zle 2130  feet.—  InsVn.  C.  E. 

Effect  of  a   Diamond- Edged    Saw. 
Result  of  its  operation  at  the  Paris  Exposition. 

In  semi -hard  and  soft  stone,  n.8  ins.  per  minute.  Cost  a  cents  per  sq.  foot; 
cost  by  hand  sawing.  15  cents. — /.  Laftargue. 

Forced    Draught. 

For  non-caking  coal  it  is  necessary  to  reduce  the  width  of  the  air  space  between 
the  grates  of  a  furnace  to  .125  inch. 

The  greater  the  force  of  the  blast,  the  less  is  the  evaporative  effect  of  the  fuel, 
as  illustrated  in  a  steam-boiler  plant,  where  the  evaporation  with  natural  draught 
was  from  7  to  8  Ibs.  of  water  per  Ib.  of  coal;  it  fell  to  4  Ibs.  upon  the  introduction  of 
a  blast  draught,  and  although  there  was  a  length  of  flue  of  400  feet,  and  a  chimney 
130  feet  in  height,  flame  was  generated  at  the  top  of  the  chimney,  evidencing  that 
carbonic  oxide  left  the  furnace  unconsumed.  — D.  K.  C. 

Efficiency    of  liand    Uralses. 

From  a  series  of  experiments  on  the  tender  of  a  locomotive  on  the  Northern 
Railway  of  France,  it  was  deduced  that  the  frictional  resistance  absorbed  82.3 
per  cent  of  the  power  applied. 

In  general  it  is  assumed  that  the  efficacy  of  hand  brakes  does  not  exceed  20  pe* 
cent.— D.  K.  C. 

A.cetylene    Formula. 
C2  H2,  and  a  Specific  Gravity  .91. — M.  Hempel. 


APPENDIX. 

Effect   of  Kiln    IDrying    on.    Pine    and.    Hemlock. 

White  Pine.—  Weight  of  a  cube  foot,  36.4  Ibs.  ;  dried  at  212°,  22  Ibs.  RedPine.~ 
32.3  Ibs.  ;  at  212°,  31  Ibs.  Hemlock.—  53  Ibs.  ;  at  212°,  31.3  Ibs.—  Prof.  H.  T.  Bovey, 
LL.D. 

Test   of  an    Iron    \Vire    Rope    3.S    ins.  in    Diameter. 

Construction.—  Six  strands  on  a  core  of  hemp,  and  each  of  six  other  strands  on  a 
central  core  containing  108  wires,  .058  inch  in  diameter.  Tensile  strength  of  wire 
260  ooo  Ibs.  per  sq.  inch,  and  united  strength  of  all  740  ooo  Ibs.  Reduction  of  di- 
ameter with  a  stress  ot  150  tons  1.4  inch,  and  ultimate  strength  560000  Ibs.,  a  coeffi- 
cient of  75  percent.—  A.  Martens. 

Consolidation    of  Loose    or    Made    Q-ronnd 

May  be  successfully  attained  by  the  driving  of  piles  as  close  together  as  the  earth 
will  admit  withdrawing  of  them,  and  filling  their  holes  with  a  weak  concrete. 

In  an  instance  recited,  8  piles,  30  inches  apart,  driven  to  a  depth  of  23  feet  in 
made  ground,  supported  a  brick  chimney  213  feet  in  height  on  a  base  of  43  feet 
square.  —  Hoffman. 

For  the  foundation  of  the  buildings  of  the  Paris  Exposition,  the  ground  was 
rammed  by  conical  and  suitable  monkeys  from  a  pile-driver,  and  the  holes  filled 
with  hard  substances  rammed  down.  —  Dular. 

A.    New   <3-eneral    Formula   for   Train    Resistance. 

4-f-S  t.2-\  --  ~=J  =  R.  R  representing  resistance  in  Ibs.  per  ton  (2000),  S  ve- 
locity in  miles  per  hour,  and  T  weight  of  train  in  tons.—H.  L.  J. 

Resistance  of  Fast  Passenger  Trains  on  Straight  Road. 

Experiments  on  the  Northern  Railway  of  France  at  velocities  varying  15  to  35 
miles  per  hour.  i.  45  -f-  .  ooo8Va  =  R.  —  De  Laboriette. 

M!  agnail  vi  in. 

A  new  alloy  of  aluminum.  Spec,  gravity,  2-3;  Melting-point,  noo°  ;  Tensile 
strength  with  5  per  cent,  of  magnesium,  30000  Ibs.,  and,  with  an  addition  of  from 
5  to  20  per  cent,  of  it,  the  alloy  becomes  similar  to  brass  and  bronze,  and,  with  50 
per  cent,  it  loses  its  hardness  and  ceases  to  be  useful  for  mechanical  purposes; 
but  as  it  is  capable  of  receiving  a  very  high  polish,  it  is  eminently  suited  for  op- 
tical and  like  instruments.—  Mielke. 

IMaximite. 


its  susceptibil- 
of 


charge  of  projectiles,  viz.,  insensibility  to  heat  and  shock.  To  test  its  suscepti 
ity  to  chemical  change,  it  is  maintained  at  a  temperature  of  165°  for  a  period 
15  minutes. 

as 

ugh  a 
it  ex- 

, 00    ragmens,  an    a  r2-nc    se,  smary  exploded, 

burst  into  7000  fragments.    It  freezes  below  the  boiling-point  of  water,  and  possesses 
the  advantage  of  expanding  in  passing  from  a  fluid  to  a  solid.  —  A.  P.  H. 

Cylinder    Ratios    for    Compound    and   Triple    Expansion 
Steam  -  Engines. 

By  experiments  of  Mr.  Greacen.  of  Perth  Amboy,  N.  J.,  under  different  initial 
pressures  and  the  bushing  of  the  high-pressure  cylinder,  he  determined  the  ratios 
to  be:  With  60  Ibs.  pressure  per  sq.  inch,  i  to  4;  with  85  Ibs.,  i  to  5;  with  no  Ibs., 
i  to  6;  with  135  Ibs.,  i  to  7;  and  with  160  Ibs.,  i  to  8.—  B.  C.  Ball. 


1038 


APPENDIX. 


B  el  t  -  D  ri vi  ng. 


Belts  for  high  speed,  running  over  4000  feet  per  minute,  should  be  of  single, 
thin,  pliable,  and  tough  leather;  and  if  singly  compounded,  they  may  be  run  at 
9000  feet  per  minute,  with  less  loss  from  slipping. 

Narrow  pulleys  are  more  effective  than  wide;  thus,  two  belts  of  20  ins.  will 
transmit  more  power  than  one  of  40  ins.  over  a  wide  pulley.  Great  convexity  of 
pulley  increases  wear  of  belt,  and  induces  loss  of  power.  .0625  inch  in  convexity 
is  sufficient  for  a  pulley  of  6  ins.,  and  a  less  driven  pulley  may  be  flat  on  its  face. 
— /.  Tullis. 

Texxi.peratu.re    in    IVlines. 

From  observations  made  in  Australia,  the  mean  result  in  rock  was  an  increase 
of  i°  to  each  137  feet  »f  descent. — /.  Sterling. 

At  Lake  Superior,  U.  S.,  at  105  feet,  59°,  and  at  4580  feet,  79°,  a  difference  of  i° 
for  223.7  feet-  At  st-  Gothard  Tunnel  it  was  j°  for  60  feet.—  A.  Agassiz. 

Relative  Efficiency  of  a  Reciprocating  Piston   Pump,  a 
Rotary  ifump,  and.  a  Steam.  Sipnon. 

Water  raised  to  an  Elevation  of  17.66  feet,  and  Pounds  oj  Water  raised  per 
Pound  of  Steam. 

Reciprocating  Pump,  135.6  Ibs. ;  Rotary,  108.6  Ibs. ;  and  Steam  Siphon  (Giffard's), 
37.4  Ibs.—  B.  F.  Isherwood,  U.  S.  N. 

Tenacity   of  Nails   and   Drift    Bolts. 

Experiments  made  at  Sibley  College  furnish  the  following  results : 

Cut  Nails  are  superior  to  Wire  in  all  positions  ;  and,  as  the  pointing  of  a  nail  in- 
creases its  efficiency,  the  pointing  of  a  cut  nail  would  increase  its  tenacity  about 
30  per  cent.  Barbing  decreases  their  tenacity  about  32  per  cent. 

Wire  Nails.—  Their  tenacity  decreases  with  time  of  service.  Surface  of  a  nail 
should  be  slightly  rough.  Nails  should  be  wedge-shaped  in  both  directions,  where 
there  are  not  special  dangers  of  the  splitting  of  the  wood.  Nails  are  50  per  cent, 
more  effective  when  driven  perpendicular  to  the  grain  of  the  wood  than  with  it, 
and  most  effective  when  driven  perpendicular  to  the  surface,  and  when  submitted 
to  impact  they  hold  less  than  .083  the  stress  they  can  withstand  when  it  is  grad- 
ually applied. 

Drift  Bolts,  when  round,  are  superior  to  square,  and  the  holes  into  which  they 
are  to  be  driven  should  be  respectively,  .8125  and  .875  of  their  diameter. 

Relative  Tenacity  of  Woods. — White  pine,  i;  Basswood,  1.2;  Yellow  pine,  1.5; 
Chestnut,  1.6;  Elm  and  Sycamore,  2;  Beech,  3.2;  and  White  oak,  3.—  F.  W.  Clay. 

Lubrication    of  Metal    Bearings. 

From  results  of  extended  experiments  on  the  Paris- Lyons- Mediterranean  Railway, 
extending  from  1871  to  1890,  it  was  shown  that: 

Lubricating  Wicks  of  Wool  have  a  delivery  of  oil  over  that  of  cotton  of  from  50 
to  ioo  per  cent. ;  that 'their  renewals  were  but  as  68  to  100  of  the  cotton,  and  that 
they  were  less  liable  to  firing. 

Bearings.— The  wear  of  White  Metal  was  50  per  cent,  less  than  that  of  Bronze, 
and  bearings  of  it  diminished  the  resistance  of  trains  of  300  tons,  running  from  16 
to  26  miles  per  hour,  20  per  cent. ;  but  as  the  speed  was  increased,  this  gain  was 
diminished,  but  it  remained  always  at  5  per  cent.—  E.  Chabal. 

Execution    of    Masonry    or    Brickwork    during     Severe 
Frost. 

A  committee  of  the  Austrian  Union  of  Engineers  and  Architects,  after  extended 
experiments,  submitted  that  Portland  Cement  with  7  per  cent,  of  common  salt 
in  cold  water,  and  the  stone  or  brick  dry,  was  the  most  effective,  and  that  Lime 
mortar  was  useless.— Alfred  Greil 


APPENDIX. 


1039 


Talole  for  Reducing  Observed  Daily  Variation  of  Needle 
to    Mean   Variation   of  the   Day. 

U.  S.  Coast  and  Geodetic  Survey,  1878. 


SEASON. 

Nee 
A.M. 

iloEa 
netic 

A.M. 

at  of  Mean  2 
Meridian. 

A.M.   A.M. 

iag- 
A.M. 

A.M. 

Need 
NOON. 

eWei 
P.M. 

it  of  IV 
Meridi 

P.M. 

[ean& 
an. 

P.M. 

[agnet 
P.M. 

ic 
P.M. 

P  M, 

Spring     

6 

/ 

3 

4 

2 

I 

7 

4 
5 
3 

i 

8 
/ 

4 
5 
3 

2 

9 
t 

3 
4 

2 
•2 

IO 

f 

I 
I 

I 

A. 
II 
t 

I 
2 
2 

Noon. 

A. 

I 

h. 

2 

h. 
3 

A. 

4 

h. 

5 

A. 

6 

4 
4 
3 

2 

5 
6 
4 
3 

5 
5 
3 
5 

4 
4 

2 
2 

3 
3 

i 
i 

2 

2 

I 
I 

i 
i 

Summer  

Autumn 

Winter... 

Elevations,  at  Various  Locations,  of  Bench-Mar  Its 

.AJoove  Mean  Level  of  the  Ocean  at 

Sandy  Hook,  N.  «J. 

See  Treasury  Annual  Report  for  1899  of  Superintendent  U.  S.  Coast  and 
Geodetic  Survey. 


Location. 

Elevation  in 
Metres. 

Bench-Marks. 

Location. 

Elevation 
in  Metres. 

Bench- 
Marks. 

Albany,  N.  Y... 
Alex'dria  B.,N.Y. 
Altoona,  Pa  

5-013 

78.9705 

354-0357 
1.268 

6.7175 
20.0484 
162.0168 
96.8627 
187.595 
181.4580 
180.8124 
168.4273 
1.822.9081 

95-9903 
190.0727 
202.0164 
226.5461 
163.3446 
27.8699 
1.565.1693 
183.2189 
184.589 
191  .0642 
110.8262 
209.0303 
98.6150 
l36-399 
3.6951 
2.6313 
112.0816 
97.5692 
58.5092 
120.8052 
191.4484 
225.6832 
227  .  5684 
198.0135 

BM2 
PBMA 
PRRNo.i24 
a 

Ii 
A 
PBM  12 
PBM   i 
C 
BMi 
BMIX 

City  BM 
PBM  7 

PBM  182 
City  BM 
TBM  176 
No.  215 
N2 
PBM  275 
BMI.  (USE) 
PBM  8 
H 
685 
No.  V 
TBM  87 

T 

L 

PRR  No.  i 
BM  i 
No.  XXIII 
TBM  197 
TBM  484 
BM245 
PBM  194 

Lake  Michigan,  111.  . 
Leavenw'th,  Kan.  .  . 
Little  Rock,  Ark... 
Memphis,  Tenn  
Minneapolis,  Minn.  . 
Mobile,  Ala  

179.1269 
238.4976 

71-9737 
80.5465 
256.1462 
3.7448 
24.9946 
6.8144 
1.6034 

4.5927 
2.2991 
299.5705 
80.3769 
76.7016 
248.2064 
187.718 

18-5763 
196.0783 
11.7284 
7.7678 
10.8961 
.2659 
253.9426 

129.9745 
216.854 
7L5558 
1780.1063 
337.5317 
183-5342 
271.0406 
131.7546 
126.3358 
3L3525 
132.024 
2.936 
1013.8567 

PBM  100 
PBM  251 
BMi 
PBM 
TBM  13 
A 
BMPolka 
S 
H'way  H. 
E.42d  St. 
No.  5 
PBM   345 
BMA 
BMA 

8'o5A 

SGS 
PBM  231 
E 
City  BM 
No.  XIII 
TBM 
PBM  284 
PBM  14 
TBM   193 
BM  16 
Ov 
PBM  394 
Pt.  -Office 
M 

BM44 
No.  XL 
PBM  i 
A3 
BMCNo.4 
M2 

Annapolis,  Md... 
Biloxi,  Miss  
Brooklyn,  N.  Y.  .  . 
Burlington,  Iowa. 
Cairo,  111  
Cheyenne,  Wyo.  . 
Chicago,  111  

Cincinnati,  O  
Colorado  Sp's,  Col. 
Columbus,  Ky  
Cumberland,  Md. 
Dakota,  Minn  
Dayton,  O 

Natchez,  Miss  
Newport  News,  Va.. 
New  Orleans,  La.  .  . 
New  York,  N.Y.... 

Omaha  Neb 

Ontario,  Pt.  Dal,  Can. 
Oswego,  N.  Y  
Owego,  "  
Parkersburg,  W.Va. 
Perth  Amboy,  N.  J. 
Pr'rieduChien,Wis. 
Red  Bank,  N.  J  
Richmond,  Va  
Round  Brook,  N.  J. 
St.  Augustine,  Fla. 
St.  Joseph,  Mo  
St.  Louis,  Mo  
St.  Paul,  Minn  
Schenectady,  N.  Y.  . 
Sedalia  Col  

Decatur,  Ala  
Delta,  La  

Denver  (nr.),  Col.. 
Detroit,  Mich.... 
Dubuque,  Iowa... 
Duluth,  Minn  
Easton,  Penn  .... 
Erie,  Pa  ... 

Fort  Jefferson,  Ky 
Florence,  Ala  
Gov'nor's  I.,N.Y. 

Harrisburg,  Pa... 

Helena,  Ark.  .  .  ".  !  '. 
Jackson,  Tenn  
Jefferson  City,  Mo. 
Kansas  City,  Kan. 
"           Mo.. 
La  Crosse,  Wis..  . 

Sioux  Iowa  

Topeka  Kan  .... 

Utica  N  Y  

Van  Buren,  Ark.  ... 
Vicksburg,  Miss.  .  .  . 
Vincennes,  Ind.... 
Washington,  D.C... 
Winona,  Kan  

For  exact  location  and  description  of  Bench- Marks,  see  Report  as  above,  pp.  548-549. 


IO4O  APPENDIX. 

Insulation    of  Steam    Boilers    and    IMpes. 

From  the  experiments  of  several  parties  in  England,  St.  Petersburg,  and  Canada, 
the  following  results  were  obtained: 

With  steam  at  pressures  ranging  from  3  to  150  Ibs.  per  sq.  inch,  and  averaging 
75  Ibs.,  the  condensation  of  steam  in  pipes,  per  sq.  foot  per  hour,  was:  Uncovered, 
.60  Ibs. ;  with  mica  insulation,  with  steam  from  47  to  244  Ibs.,  averaging  150  Ibs., 
.143  Ibs. 

With  steam  at  150  Ibs.  permanent  in  the  pipes,  it  was  computed  that  each  sq. 
foot  of  uncovered  surface  involved  an  annual  loss  of  $2.11;  with  ordinary  and  good 
bagging,  55  cents;  and  with  mica  insulation,  28  cents. 

From  experiments  on  the  Canadian  Pacific  Railway,  the  rate  of  cooling  of  water- 
tanks  from  the  boiling  point  in  5  hours,  the  loss  of  temperature  varying  from  84° 
in  the  uncovered  to  20°  in  the  one  covered  with  mica;  and  from  other  experiments 
on  the  Grand  Junction  Railway,  with  5  locomotives,  with  steam  at  from  140  to  150 
Ibs.  pressure  per  sq.  inch,  the  observed  effects,  after  the  fires  were  drawn,  were: 
The  uncovered  boiler  lost  56  Ibs.  pressure  in  one  hour,  while  the  covered  lost,  re- 
spectively, 24,  20,  13,  and  6  Ibs.,  the  last  with  mica  covering. — Engineering,  1901, 
p.  234. 

Hdq.uid    Fuel. 

From  experiments  made  with  crude  Borneo  oil,  of  the  composition:  Carbon,  87.9 
per  cent;  hydrogen,  10.78;  oxygen,  1.24;  flash-point,  211°;  boiling-point,  395°; 
and  caloric  value,  18.831  B.  T.  U. 

The  constituents  of  fuel  oils  give  off  vapor  at  temperatures  from  100°  up  to  boil- 
ing-point of  the  oil,  near  which  point  a  residuum  of  dense  carbon  is  precipitated, 
tending  to  choke  pipes  and  to  accumulate  in  the  furnace. 

The  following  methods  of  using  it  are:  t.  Injecting  it  into  the  furnace  under 
pressure,  as  spray;  2.  Spraying  it  by  air  or  steam ;  3.  Vaporizing  it. 

The  evaporative  efficiency  under  the  first  was  12  Ibs.  water  from ;  and  at  212°  per 
Ib.  of  oil,  an  excess  of  air  and  a  large  furnace  being  required  for  combustion. 
Under  the  second,  13  to  14  Ibs.,  less  air  being  required.  Under  the  third,  15  to  16 
Ibs.,  a  minimum  of  air  being  required. 

The  conclusions  deduced  were:  ist.  A  reduction  in  consumption  of  fuel  with  the 
oil,  compared  with  coal,  of  about  40  per  cent.  2d.  A  reduction  in  bunker  space  of 
about  15  per  cent,  for  equal  weights  of  fuel.  3d.  A  reduction  in  furnace  labor  of 
at  least  50  per  cent. 

The  oil  should  be  filtered  before  being  used.—  E.  L.  Orde. 

Effects    of  Repeated  Stress   on   tlie   Strength  of  Wrought 

Iron. 

From  Tests  Made  on  a  Bridge  that  had  been  24  Years  in  Service. 
The  maximum  stress  being  6.64  tons  per  square  inch,  no  reduction  in  strength 
or  durability  from  its  service  was  observed.—  Zimmermann. 

Safe    Static    .Load    on    Ordinary    Foundations. 

In  Tons  per  Square  Foot. 


Alluvial  soil 5 

Clay  and  sand,  moist 1.33 

Clay,  hard  and  dry 1.5  to  3 

Earth,  firm x  to  1.5 


Sand,  sharp  and  clean i  to  1.5 

Gravel,  dry 2. 25 

Stones,  broken  and  concrete. .  3 

— Aide  Memoir e  and  Rankine 


APPENDIX.  IO4I 

Air-I?ixmps    of   Condensing    Steam-Eiigines. 

Are  more  effective  when  operated  independent  of  the  engine,  in  consequence  of 
possessing  the  advantage  of  their  operation  being  varied  to  meet  their  require- 
ments; and  vertical  single-acting,  at  a  velocity  of  piston  not  exceeding  400  feet 
per  minute,  are  more  effective  than  double-acting. 

The  required  dimensions  and  resulting  capacity  of  full  flowing  pumps  may  be 
computed  from  the  table  of  H.  R.  Worthington  on  p.  738. 

A  displacement  of  pump  cylinder  of  one-fifth  of  a  cube  foot  per  minute  is  held 
to  be  proper  for  one  IIP. 

Effect  of  the    TJse    of  Oil   or  Tallow  in   a   Steam-Boiler. 

Oil  or  grease  introduced  in  a  steam  boiler,  combining  with  alluvial  or  calcareous 
sediments  from  the  feed  water,  if  not  held  in  suspension  by  rapid  circulation  over 
the  heated  surfaces,  as  the  crowns  of  the  furnaces  and  tubes,  and  withdrawn  by 
pump  or  blown  out  as  it  subsides,  will  settle  upon  the  upper  surfaces  of  the  fur- 
nace and  tubes,  involving  the  burning  of  the  metal  and  their  consequent  disruption 
under  pressure. 

Stress    on    Trussed.    Beam    or    Rods. 
In  addition  to  pp.  621-623,  823. 

King   Truss.    To    Compute    Stress    on    Beam. 

.    *" "" "'*•: ~~,..->  _^  gec  t-_  _  cosec  t  — s.     W  repre- 

4 

i         j   I  \     senting  weight  uniformly  distributed,  I  length 

'     of  beam  between  supports,  d  depth  of  truss, 

both  in  feet,  and  i  angle. 

ILLUSTRATION. —Assume  W  =  2ooo  Ibs.,  Zc=20,  d  —  z.o^  feet.  t=  xi°  42',  and 
cotan.  1  =  5.1. 

2000X20  2000 

8x  =  2  495  Ibs.,  = X  5  i  =  2  550  &»• 

To    Compute    Stress    on    Rods. 


wi       .     w  w  v 

— - -  sec.  i  =  —  cosec.  t.  In  absence  of  angle  put . 

Bd  4  4          2d 

ILLUSTRATION. — Assume  as  preceding.     Sec.  t  =  1.02,  cosec.  i  =  5.1. 

—  X  1.02  =  2  445.4  Ibs.,  =  —  —  X  =  5- «  =  2  550 Ibs. 
o  x  2.03  4 

To    Compute    Stress   on    Centre. 
ILLUSTRATION. —  -  W=  -  x  2000=  i  250  Ibs. 

8  8 

Queen    Truss.      To    Compute    Stress    on    Beam. 
/         .„          -yy  j  ,, 

— —  sec.  t  =  |  W.  cotan.  t,  c  .-=  6.67  feet. 

o  d  8 


—  -^i  _  A_L^-^  r>~      ILLUSTRATION.  —  Assume     as     preceding. 

1     d=2  feet,  and  angle   i=i6°  42',  sec.  i  = 
1.044,  and  cotan.  i  =  3.42. 

I044=26loW,s.ja 

To    Compute     Stress    on    Rods. 


ILLUSTRATION.—  Assume  as  preceding,  cosec.  i  =  3.56. 

2000  X2°X  1.044^2610^.,  and  3 
8X2  _ 

3W 

In  absence  of  angle  put  ^- 


—  e 
3  a 

To    Compute    Stress    on    Centre. 


IO42  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Orthography  in  ordinary  use  of  following  words  and  terms  is  so  varied, 
that  they  are  here  given  for  the  purpose  of  aiding  in  the  establishment  oi 
d  uniformity  of  expression. 

Abut.    To  meet,  to  adjoin  to  at  the  end,  to  border  upon.    Abut  end  of  a  log,  etc., 
is  that  having  the  greatest  diameter  or  side. 
But  and  Butt  end,  when  applied  in  this  manner,  are  corruptions. 

Adit,    in  Mining,  the  opening  into  a  mine. 

Amidships.  The  middle  or  centre  of  a  vessel,  either  fore  and  aft  or  athwartships. 
The  amidship  frame  of  a  vessel  is  at  g&  and  is  termed  dead  flat. 

Arabesque.  Applied  to  painted  and  carved  or  sculptured  ornaments  of  imaginary 
foliage  and  animals,  in  which  there  are  no  perfect  figures  of  either.  Synonymous 
with  Moresque. 

Arbor.    The  principal  axis  or  spindle  of  a  machine  of  revolution 

Arris.  A  term  in  Mechanics,  the  line  in  which  the  two  straight  or  curved  sur- 
faces of  a  body,  forming  an  exterior  angle,  meet  each  other.  The  edges  of  a  body, 
as  a  brick,  are  arrises. 

Ashlar.    In  Masonry,  stones  roughly  squared,  or  when  faced. 

Athwart.  Across,  from  side  to  side,  transverse,  across  the  line  of  a  vessel's 
sourse. 

Athwartships,  reaching  across  a  vessel,  from  side  to  side. 

Bagasse.    Sugar-cane  in  its  crushed  state,  as  delivered  from  the  rollers  of  a  mill. 

Balk.    In  Carpentry,  a  piece  of  timber  from  4  to  10  ins.  square. 

Baluster.  A  small  column  or  pilaster ;  a  collection  of  them,  joined  by  a  rail,  forms 
a  balustrade. 

Banister  is  a  corruption  of  balustrade. 

Bark.    A  ship  without  a  mizzen-topsail,  and  formerly  a  small  ship. 
Bateau.    A  light  boat,  with  great  length  proportionate  to  its  beam,  and  wider  at 
its  centre  than  at  its  ends. 

Batten.  In  Carpentry,  a  piece  of  wood  from  i  to  2. 5  ins.  thick,  and  from  i  to  7 
ins.  in  breadth.  When  less  than  6  feet  in  length,  it  is  termed  a  deal-end. 

Berme.  In  Fortifications  and  Engineering,  a  space  of  ground  between  a  rampart 
and  a  moat  or  fosse,  to  arrest  the  ruins  of  a  rampart.  The  level  top  of  the  embank- 
ment of  a  canal,  opposite  to  and  alike  to  the  towpath. 

Bevel.    A  term  for  a  plane  having  any  other  angle  than  45°  or  90°. 

Binnacle.  The  case  in  which  the  compass,  or  compasses  (when  two  are  used),  ia 
set  on  board  of  a  vessel. 

Bit.  The  part  of  a  bridle  which  is  put  into  an  animal's  mouth.  In  Carpentry,  a 
boring  instrument. 

Bitter  End.    The  inboard  end  of  a  vessel's  cable  abaft  the  bitts. 

Bitts.  A  vertical  frame  upon  a  deck  of  a  vessel,  around  or  upon  which  is  secured 
cables,  hawsers,  sheets,  etc. 

Bogie.    Pivoted  truck,  to  ease  the  running  of  an  engine  or  car  around  a  curve. 

Boomkin.  A  short  spar  projecting  from  the  bow  or  quarter  of  a  vessel,  to  extend 
the  tack  of  a  sail  to  windward. 

Bowlder.  A  stone  rounded  by  natural  attrition ;  a  rounded  mass  of  rock  trans- 
ported from  its  original  bed. 

Buhr-stone.  A  stone  which  IB  nearly  pure  silex,  full  of  pores  and  cavities,  and 
used  for  Mill* 

Bunting.    Woolen  texture  of  which  colors  and  flags  are  made. 

Burden.    A  load.    The  quantity  that  a  ship  will  carry.     Hence  burdensome. 

Cog.    A  small  cask,  differing  from  a  barrel  only  in  size.    Commonly  written  Keg. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS.   1 043 

Caliber.  An  instrument  with  semi-circular  legs,  to  measure  diameters  of  spheres, 
or  exterior  and  interior  diameters  of  cylinders,  bores,  etc. 

A  pair  of  Calibtrs  is  superfluous  and  improper. 

Calk.  To  stop  seams  and  pay  them  with  pitch,  etc.  To  point  an  iron  shoe  so  as 
to  prevent  its  slipping. 

Cam.  An  irregular  curved  instrument,  having  its  axis  eccentric  to  the  shaft 
upon  which  it  is  fixed. 

Camber.  To  camber  is  to  cut  a  beam  or  mold  a  structure  archwise,  as  deck- 
beams  of  a  vessel. 

Camboose.  The  stove  or  range  in  which  the  cooking  in  a  vessel  is  effected.  The 
cooking- room  of  a  vessel;  this  term  is  usually  confined  to  merchant  vessels:  in 
vessels  of  war  it  is  termed  Galley. 

Camel.  In  Engineering,  a  decked  vessel,  having  great  stability,  designed  for  use 
in  the  lifting  of  sunken  vessels  or  structures.  Also  to  transport  loads  of  great 
weight  or  bulk. 

A  Scow  is  open  decked. 

Cantle.    A  fragment;  a  piece;  the  raised  portion  of  the  hind  part  of  a  saddle. 

Cantline.  The  space  between  the  sides  of  two  casks  stowed  aside  of  each  other. 
When  a  cask  is  laid  in  the  cantline  of  two  others,  it  is  said  to  be  stowed  bilge  and 
cantline. 

Capstan.    A  vertical  windlass. 

Caravel.  A  small  vessel  (of  25  or  30  tons'  burden)  used  upon  the  coast  of  France 
in  herring  fisheries. 

Carlings.  Pieces  of  timber  set  fore  and  aft  from  the  deck  beams  of  a  vessel,  to 
receive  the  ends  of  the  ledges  in  framing  a  deck. 

Carvel  built.— A  term  applied  to  the  manner  of  construction  of  small  boats,  to 
signify  that  the  edges  of  their  bottom  planks  are  laid  to  each  other  like  to  the  matt- 
ner  of  planking  vessels.  Opposed  to  the  term  Clincher. 

Caster.  A  small  phial  or  bottle  for  the  table.  Casters.  Small  wheels  placed 
upon  the  legs  of  tables,  etc.,  to  allow  them  to  be  moved  with  facility. 

Catamaran.  A  small  raft  of  logs,  usually  consisting  of  three,  the  centre  one  be- 
ing longer  and  wider  than  the  others,  and  designed  for  use  in  an  open  roadstead 
and  upon  a  sea-coast. 

Chamfer.    A  slope,  groove,  or  small  gutter  cut  in  wood,  metal,  or  stone. 

Chapetting.    Wearing  a  ship  around  without  bracing  her  fore  yards. 

Chimney.  The  flue  of  a  fireplace  or  furnace,  constructed  of  masonry  in  houses 
and  furnaces,  and  of  metal,  as  in  a  steam  boiler.  See  Pipe. 

Ckinse.     To  chinse  is  to  calk  slightly  with  a  knife  or  chisel. 

Chock.  In  Naval  Architecture,  small  pieces  of  wood  used  to  make  good  any  de- 
ficiency in  a  piece  of  timber,  frame,  etc.  See  Furrings. 

Choke.    To  stop,  to  obstruct,  to  block  up,  to  hinder,  etc. 

Cleats.  Pieces  of  wood  or  metal  of  various  shapes,  according  to  their  uses,  either 
to  belay  ropes  upon,  to  resist  or  support  weights  or  strains,  as  sheet,  shoar,  beam 
cleats,  etc. 

Clincher  built.  A  term  applied  to  the  construction  of  vessels'  bottoms,  when 
the  lower  edges  of  the  planks  overlay  the  next  under  them. 

Coak.  A  cylinder,  cube,  or  triangle  of  hard  wood  let  into  the  ends  or  faces  of  two 
pieces  of  timber  to  be  secured  together.  The  metallic  eyes  in  a  sheave  through 
which  the  pin  runs.  In  Naval  Architecture,  the  oblong  ridges  banded  on  the  mastfl 
of  ships. 

Coamingt.    Raised  borders  around  the  edges  of  hatches. 

Coble.     A  small  fishing  boat 

Cocoon.  The  case  which  certain  insects  make  for  a  covering  during  *he  period 
of  their  metamorphosis  to  the  pupa  state. 


IO44  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Cog.  In  Mechanics,  a  short  piece  of  wood  or  other  material  let  into  the  faces  of 
a  body  to  impart  motion  to  another.  A  term  applied  to  a  tooth  in  a  wheel  when  it 
is  made  of  a  different  material  than  that  of  the  wheel.  In  Mining,  an  intrusion  of 
matter  into  fissures  of  rocks,  as  when  a  mass  of  unstratitied  rocks  appears  to  be  in- 
jected into  a  rent  in  the  stratified  rocks. 

Cogging.  In  Carpentry,  the  cutting  of  a  piece  of  timber  so  as  to  leave  a  part 
alike  to  a  cog,  and  the  notching  of  the  upper  piece  so  as  to  conform  to  and  receive 
it.  Alike  to  indenting  or  tabling. 

Colter.    The  fore  iron  of  a  plough  that  cuts  earth  or  sod. 

Compass.    In  Geometry,  an  instrument  for  describing  circles,  measuring  figures,  etc. 

A  pair  of  Compasses  is  superfluous  and  improper. 

Connecting  Rod.  In  Mechanics,  the  connection  between  a  prime  and  secondary 
mover,  as  between  the  piston-rod  of  a  steam-engine  and  the  crank  of  a  water- wheel 
or  fly-wheel  shaft. 

The  term  Pitman  is  local,  and  altogether  inapplicable. 

Contrariwise.    Conversely,  opposite.     Orotsways  is  a  corruption. 

Corridor.  A  gallery  or  passage  in  or  around  a  building,  connected  with  various 
departments,  sometimes  running  within  a  quadrangle ;  it  may  be  opened  or  enclosed. 
In  Fortifications,  a  covert  way. 

Cyma.    A  molding  in  a  cornice. 
Damasquinerie.    Inlaying  in  metal. 
Davit.    A  short  boom  fitted  to  hoist  an  anchor  or  boat. 

Deals.    In  Carpentry,  the  pieces  of  timber  into  which  a  log  is  cut  or  sawed  up. 
Their  usual  thickness  is  3  by  9  ins.  and  exceeding  6  feet  in  length. 
Improperly  restricted  to  the  wood  of  fir-trees. 

Dike.  In  Engineering,  an  embankment  of  greater  length  than  breadth,  imper- 
vious to  water,  and  designed  as  a  wall  to  a  reservoir,  a  drain,  or  to  resist  the  influx 
of  a  river  or  sea. 

Dingey  (Nautical).    A  ship  or  vessel's  small  boat. 

Dock.  In  Marine  Architecture,  an  enclosure  in  a  harbor  or  shore  of  a  river,  for 
ihe  reception,  repair,  or  security  of  vessels  or  timber.  It  may  be  wholly  or  only 
partially  enclosed.  See  Pier. 

When  applied  to  a  single  pier  or  jetty,  it  is  a  misapplication. 

Dowel.  A  pin  of  wood  or  metal  inserted  in  the  edge  or  face  of  two  boards  or 
pieces,  so  as  to  secure  them  together. 

This  is  very  similar  to  coaking,  but  is  used  in  a  diminutive  sense.  An  illustration  of  it  is  had  in  the 
manner  a  cooper  secures  two  or  more  pieces  in  the  head  of  a  cask. 

Draught.  A  representation  by  delineation.  The  depth  which  a  vessel  or  any 
floating  body  sinks  into  water.  The  act  of  drawing.  A  detachment  of  men  from 
the  main  body,  etc. 

Ordinarily  written  draft. 

Dutchman.  In  Mechanics,  a  piece  of  like  material  with  the  structure,  let  into  a 
slack  place,  to  cover  slack  or  bad  work.  See  Shim. 

Edgewise.  An  edge  put  into  a  particular  direction.  Hence  endwise  and  sidewist 
have  similar  significations  with  reference  to  an  end  and  a  side. 

Edgtvays  is  a  corruption. 

Euphroe.    A  piece  of  wood  by  which  the  crowfoot  of  an  awning  is  extended. 

Fault.  In  Mining,  a  break  of  strata,  with  displacement,  which  interrupts  opera- 
tions. Also,  fissures  traversing  the  strata. 

Felloe,  Felloes.    The  pieces  of  wood  which  form  the  rim  of  a  wheel. 

Fetch.  Length  of  a  reservoir,  pond,  etc. ,  along  which  the  wind  may  blow  towards 
the  embankment  or  dam. 

Flange.  A  projection  from  an  end  or  from  the  body  of  an  instrument,  or  any 
part  composing  it,  for  the  purpose  of  receiving,  confining,  or  of  securing  it  to  a  suf> 
port  or  to  a  second  piece. 

Flier.    In  Carpentry,  a  straight  line  of  steps  in  a  stairway. 

Frap.    To  bind  together  with  a  rope,  as  tofrap  a  fall,  etc. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS.   1 045 

Frieze.  In  Architecture,  the  part  of  the  entablature  of  a  column  which  is  between 
the  architrave  and  the  cornice. 

Frustum.  The  part  of  a  solid  next  the  base,  left  by  the  removal  of  the  top  or 
segment. 

Frustrum,  although  used  by  some  lexicographers,  is  erroneous. 

Furrings.  Strips  of  timber  or  boards  fastened  to  frames,  joists,  etc.,  in  order  to 
bring  their  faces  to  the  required  shape  or  level. 

Galeting.    Putting  galets  into  pointing-mortar  or  cement. 

Galets.     Pieces  of  stone  chipped  off  by  the  stroke  of  a  chisel.     See  Spall. 

Galiot.  A  small  galley  built  for  speed,  having  one  mast,  and  from  16  to  20  thwarts 
for  rowers.  A  Dutch-constructed  brigantine. 

Gate.     In  Mechanics,  the  hole  through  which  molten  metal  is  poured  into  a  mold 

for  casting.      Geat  aud  Gett  are  corruptions. 

Gearing.  A  series  of  teeth  or  cogged  wheels  for  transmitting  motion.  To  gear  a 
machine  is  to  prepare  to  connect  its  parts  as  by  an  articulation. 

Gingle.    To  shake  so  as  to  produce  a  sharp,  clattering  noise,  commonly  Jingle. 

Girt.  The  circumference  of  a  tree  or  piece  of  timber.  Girth.  The  band  or  strap 
by  which  a  saddle  or  burden  is  secured  upon  the  back  of  an  animal,  by  passing 
around  his  belly.  In  Printing,  the  bands  of  a  press. 

Gnarled.     Knotty. 

Grave.    In  Nautical  language,  to  clean  a  vessel's  bottom  by  burning. 

Graving.  Burning  off  grass,  shells,  etc.,  from  a  ship's  bottom.  Synonymous 
with  Breaming. 

Grommet.    A  wreath  or  ring  of  rope. 

Gymbal  Ring.  A  circular  rynd  for  the  connection  of  the  upper  mill-stone  to  the 
spindle  by  which  the  stone  is  suspended,  so  that  it  may  vibrate  upon  all  sides. 

Harpings.  The  fore  part  of  *,he  wales  of  a  vessel  which  encompass  her  bows, 
and  are  fastened  to  the  stem.  Cat  harpings,  ropes  which  brace  in  the  shrouds  of 
the  lower  masts  of  a  vessel. 

Hogging.  A  term  applied  to  the  hull  of  a  vessel  when  her  ends  drop  below  her 
centre.  See  Sagging. 

Horsing.     In  Naval  Architecture,  calking  with  a  large  maul  or  beetle. 

Jam.     To  press,  to  crowd,  to  wedge  in.     In  Nautical  language,  to  squeeze  tight. 

Jamb.    A  pier;  the  sides  of  an  opening  in  a  wall. 

Jib.  The  projecting  beam  of  a  crane  from  which  the  pulleys  and  weight  are  sus- 
pended. A  sail  in  a  vessel. 

Jibe.  To  shift  a  boom-sail  from  one  tack  to  another;  hence  Jibing,  the  shifting 
of  a  boom. 

Jigging.     Washing  minerals  in  a  sieve. 

Keelson.  The  timber  within  a  vessel  laid  upon  the  middle  of  the  floor  timbers, 
and  exactly  over  the  keel.  When  located  on  the  floors  or  at  the  sides,  it  is  termed  a 
sisters  or  a  side  keelson. 

Kerf.    Slit  made  by  cut  of  a  saw. 

Kevel.     Large  wooden  cleats  to  belay  hawsers  and  ropes  to,  commonly  Cavil. 

Lacquer.     A  spirituous  solution  of  lac.     To  varnish  with  lacquer. 

Lagan.    Articles  sunk  in  the  water  with  a  buoy  attached. 

Laitance.  A  pulpy,  gelatinous  fluid  washed  from  the  cement  of  concrete  depos- 
ited in  water. 

Lap-sided.  A  term  expressive  of  the  condition  of  a  vessel  or  any  body  when  it 
will  not  float  or  sit  upright. 

Lay-to.  To  arrest  headway  of  a  vessel,  without  anchoring  or  securing  her  to  a 
buoy,  etc.,  as  by  counterbracing  her  yards,  or  stopping  her  engine. 

Leaf.    A  trench  to  conduct  water  to  or  from  a  mill-wheel. 

Leech.  In  Nautical  language,  the  perpendicular  or  slanting  edge  of  a  sail  wher 
not  secured  to  a  spar  or  stay. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Luf.    The  fullest  part  of  the  bow  of  a  vessel. 

Mad.    A  large  double-headed  wooden  hammer. 

Mantle.  To  expand,  to  spread.  Mantelpiece.  The  shelf  over  a  fireplace  in  front 
of  a  chimney. 

Marquetry.    Checkered  or  inlaid  work  in  wood. 

Matrass.    A  chemical  vessel  with  a  body  alike  to  an  egg,  and  a  tapering  neck. 

Mattress.    A  quilted  bed;  a  bed  stuffed  with  hair,  moss,  etc.,  and  quilted. 

Mitred.  In  Mechanics,  cut  to  an  angle  of  45°,  or  two  pieces  joined  so  as  to  make 
a  right  angle. 

Mizzen-mast.    The  aftermost  mast  in  a  three-masted  vessel. 

Mold.  In  Mechanics,  a  matrix  in  which  a  casting  is  formed.  A  number  of  pieces 
of  vellum  or  like  substance,  between  which  gold  and  silver  are  laid  for  the  purpose 
of  being  beaten.  Thin  pieces  of  materials  cut  to  curves  or  any  required  figure.  In 
Naval  Architecture,  pieces  of  thin  board  cut  to  the  lines  of  a  vessel's  timbers,  etc. 

Fine  earth,  such  as  constitutes  soil.  A  substance  which  forms  upon  bodies  in 
warm  and  confined  damp  air. 

This  orthography  is  by  analogy,  as  gold,  told,  old,  bold,  cold,  fold,  etc. 

Molding.    In  Architecture,  a  projection  beyond  a  wall,  from  a  column,  wainscot,  etc, 

Moresque.    See  Arabesque. 

Mortise.    A  hole  cut  in  any  material  to  receive  the  end  or  tenon  of  another  piece, 

Muck.    A  mass  of  dung  in  a  moist  state,  or  of  dung  and  putrefied  vegetable  matter. 

Mul  I  ion.  A  vertical  bar  dividing  the  lights  in  a  window ;  the  horizontal  are 
termed  transoms. 

Net.    Clear  of  deductions,  as  net  weight 

Newel.    An  upright  post,  around  which  winding  stairs  turn. 

Nigged.    Stone  hewed  with  a  pick  or  pointed  hammer  instead  of  a  chisel. 

Ogee.  A  molding  with  a  concave  and  convex  outline,  like  to  an  S.  See  Cymo 
and  Talon. 

Paillasse.    Masonry  raised  upon  a  floor.    A  bed. 

Pargeting.    In  Architecture,  rough  plastering,  alike  to  that  upon  chimney* 

Parquetry.    Inlaying  of  wood  in  figures.    See  Marquetry. 

Parral.    The  rope  by  which  a  yard  is  secured  to  a  mast  at  its  centre. 

Pawl.    The  catch  which  stops,  or  holds,  or  falls  on  to  a  ratchet  wheel. 

Peek.  The  upper  or  pointed  corner  of  a  sail  extended  by  a  gaff,  or  a  yard  set  ob- 
liquely to  a  mast.  To  peek  a  yard  is  to  point  it  perpendicularly  to  a  mast. 

Pendant.  A  short  rope  over  the  head  of  a  mast  for  the  attachment  of  tackles 
thereto;  a  tackle,  etc. 

Pennant.    A  small  pointed  flag. 

Pier.  In  Marine  Architecture,  a  mole  or  jetty,  projecting  into  a  river  or  sea,  to 
protect  vessels  from  the  sea,  or  for  convenience  of  their  lading.  See  Dock. 

Erroneously  termed  a  Dock. 

Pile.    In  Engineering,  spars  pointed  at  one  end  and  driven  into  soil  to  support  a 

Superstructure  or  holdfast.      Spile  is  a  corruption. 

Pipe.  In  Mechanics,  a  metallic  tube.  The  flue  of  a  fireplace  or  furnace  when 
constructed  of  metal ;  usually  of  a  cylindrical  form. 

The  term  or  application  of  Stack  (which  refers  solely  to  masonry)  to  »  metallic  pipe  is  a  misappli- 
cation. 

Piragua.    A  small  vessel  with  two  masts  and  two  boom-sails. 

Commonly  termed  Perry-augur. 

Pirogue.  A  canoe  formed  from  a  single  log,  propelled  by  paddles  or  by  a  sail, 
with  the  aid  of  an  outrigger. 

Plastering.  In  Architecture,  covering  with  plaster  cement  or  mortar  upon  walla 
or  laths.  In  England,  termed  laying,  if  in  one  or  two  coat  work;  and  pricking  up, 
If  in  three-coat  work.  ,  . 

Plumber  block.    A  bearing  to  receive  and  support  the  journal  of  a  shaft* 

JWocre.   Maata  of  one  piece,  without  tope. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Poppets.  In  Naval  Architecture,  pieces  of  timber  set  perpendicular  to  a  vessel's 
bilge  ways,  and  extending  to  her  bottom,  to  support  her  in  launching. 

Porch.  An  arched  vestibule  at  the  entrance  of  a  building.  A  vestibule  supported 
by  columns.  A  portico. 

Portico.  A  gallery  near  to  the  ground,  the  sides  being  open.  A  piazza  encom- 
passed with  arches  supported  by  columns,  where  persons  may  walk;  the  roof  may 
be  flat  or  vaulted. 

Pozzuolana.  A  loose,  porous,  volcanic  substance,  composed  of  silicious,  argilla- 
ceous, and  calcareous  earths  and  iron. 

Prize.      In  Mechanics,  to  raise  With  a  lever.      To  pry  and  a  pry  are  corruption* 

Proa,  Flying.  A  narrow  canoe,  the  outer  or  lee  side  being  nearly  flat.  A  frame- 
work, projecting  several  feet  to  the  windward  side,  supports  a  solid  bearing,  in  the 
form  of  a  canoe.  Used  in  the  Ladrone  Islands. 

Purlin.  In  Carpentry,  a  piece  of  timber  laid  horizontal  upon  the  rafters  of  a 
roof,  to  support  the  covering. 

Bump.  In  Architecture,  a  flight  of  steps  on  a  line  tangential  to  the  steps.  A 
concave  sweep  connecting  a  higher  and  lower  portion  of  a  railing,  wall,  etc.  A 
•loping  line  of  a  surface,  as  an  inclined  platform. 

Rarefaction.  The  act  or  process  of  distending  bodies,  by  separating  their  parts 
and  rendering  them  more  rare  or  porous.  It  is  opposed  to  Condensation. 

Rebate.  In  Mechanics,  to  pare  down  an  edge  of  a  board  or  a  plate  for  the  purpose 
of  receiving  another  board  or  plate  by  lapping.  To  lap  and  unite  edges  of  boards 
and  plates.  In  Xaval  Architecture,  the  grooves  in  the  side  of  the  keel  for  receiving 
the  garboard  strake  of  plank. 

Commonly  written  Rabbet. 

Remou.  Eddy  water  without  progressive  action,  in  bed  of  a  river;  a  return  of 
water  against  direction  of  flow  of  a  river. 

Rendering.  In  Architecture,  laying  plaster  or  mortar  upon  mortar  or  walls. 
Rendered  and  Set  refers  to  two  coats  or  layers,  and  Rendered,  floated,  and  Set,  to 
three  coats  or  layers. 

Reniform.     Kidney-shaped. 

Resin.    The  residuum  of  the  distillation  of  turpentine,     itorin  u  a  corruption. 

Riband.     In  Xaval  Architecture,  a  long,  narrow,  flexible  piece  of  timber. 

Rimer.  A  bit  or  boring  tool  for  making  a  tapering  hole.  In  Mechanics,  to  Rime 
in  to  bevel  out  a  hole.  Riming.  The  opening  of  the  seams  between  the  planks  of  a 
vessel  for  the  purpose  of  calking  them. 

Rotary.    Turning  upon  an  axis,  as  a  wheel. 

Rynd.  The  metallic  collar  in  the  upper  mill-stone  by  which  it  is  connected  to 
the  spindle. 

Sagging.  A  term  applied  to  the  hull  of  a  vessel  when  her  centre  drops  below  her 
ends.  The  converse  of  Hogging. 

Scallop.    To  mark  or  cut  an  edge  into  segments  of  circles. 

Scarcement.    A  set  back  in  the  face  of  a  wall  or  in  a  bank  of  earth.    A  footing. 

Scarf.  To  join;  to  piece;  to  unite  two  pieces  of  timber  at  their  ends  by  running 
the  end  of  one  over  and  upon  the  other,  and  bolting  or  securing  them  together. 

Scend.    The  settling  of  a  vessel  below  the  level  of  her  keel 

Selvagee.  A  strap  made  of  rope-yarns,  without  being  twisted  or  laid  up,  and  re- 
tained in  form  by  knotting  it  at  intervals. 

Sennit.    Braided  cordage. 

Sewage.    The  matter  borne  off  by  a  sewer. 

Sewed.  In  nautical  language,  the  condition  of  a  vessel  aground ;  she  is  said  to  be 
tewed  by  as  much  as  the  difference  in  depth  of  water  around  her  and  her  floating 
depth. 

Sewerage.    The  system  of  sewers. 

Shaky.    Cracked  or  split,  or  as  timber  loosely  put  together. 

Shammy.    Leather  prepared  from  the  skin  of  a  chamois  goat 


IO48  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Sheer.  In  Naval  Architecture,  the  curve  or  bend  of  a  ship's  deck  or  sides.  To 
she.!r,  to  slip  or  move  aside. 

Sheers.  Elevated  spars  connected  at  the  upper  ends,  and  used  to  elevate  heavy 
bodies,  as  masts,  etc. 

Shim.  In  Naval  Architecture,  a  piece  of  wood  or  iron  let  into  a  slack  place  in  s 
frame,  plank,  or  plate  to  fill  out  to  a  fair  surface  or  line. 

Shoal.    A  great  multitude;  a  crowd;  a  multitude  of  fish. 

School  is  a  corruption. 

Shoar.  An  oblique  brace,  the  upper  end  resting  against  the  substance  to  be  sup- 
ported. 

Sholes.    Pieces  of  plank  under  the  heels  of  shears,  etc. 

Shoot.  A  passage  way  on  the  side  of  a  steep  hill,  down  which  wood,  coal,  etc.,  are 
thrown  or  slid.  The  artificial  or  natural  contraction  of  a  river.  A  young  pig. 

Sidewise.    See  Edgewise. 

Signalled.    Communicated  by  signals. 

Signalized,  when  applied  to  signals,  is  a  misapplication  of  words. 

Sill.  A  piece  of  timber  upon  which  a  building  rests ;  the  horizontal  piece  of  tim- 
ber or  stone  at  the  bottom  of  a  framed  case. 

Siphon.    A  curved  tube  or  pipe  designed  to  draw  fluids  out  of  vessels. 

Skeg.  The  extreme  after-part  of  the  keel  of  a  vessel;  the  portion  that  supports 
the  rudder-post. 

Slantwise.    Oblique;  not  perpendicular. 

Sleek.    To  make  smooth.    Refuse ;  small  coal. 

Sleeker.  A  spherical-shaped,  curved,  or  plane-surfaced  instrument  with  which  to 
smooth  surfaces. 

Slue.    The  turning  of  a  substance  upon  an  axis  within  its  figure. 

Snying.  A  term  applied  to  planks  when  their  edges  at  their  ends  are  curved  or 
rounded  upward,  as  a  strake  at  the  ends  of  a  full-modelled  vessel. 

Spall.  A  piece  of  stone,  etc.,  chipped  off  by  the  stroke  of  a  hammer  or  the  force 
of  a  blow.  Spoiling,  breaking  up  of  ore  into  small  pieces. 

Spandrel  In  Architecture,  the  irregular  triangular  space  between  the  outer  lines' 
or  extrados  of  an  arch,  a  horizontal  line  drawn  from  its  apex,  and  a  vertical  line 
from  its  springing. 

Sponson.  An  addition  to  the  outer  side  of  the  hull  of  a  steam  vessel,  commencing 
near  the  light  water-line  and  running  up  to  the  wheel  guards;  applied  for  the  pur- 
pose of  shielding  the  deck-beams  from  the  shock  of  a  sea. 

Spnnson- sided.  The  hull  of  a  vessel  is  so  termed  when  her  frames  have  the  out- 
line of  a  sponson,  and  the  spaee  afforded  by  the  curvature  is  included  in  the  hold. 

Spending,  Sponsing,  etc.,  are  corruptions. 

Squilgee.  A  wooden  instrument,  alike  to  a  hoe,  its  edge  faced  with  leather  or 
vulcanized  rubber,  used  to  facilitate  the  drying  of  wet  floors,  or  decks  of  a  vessel. 

Stack.  In  Masonry,  a  number  of  chimneys  or  pipes  standing  together.  The 
chimney  of  a  blast  furnace. 

The  application  of  this  word  to  the  smoke-pipe  of  a  steam-boiler  is  wholly  erroneous. 

Stage.  In  Engineering,  the  interval  or  distance  between  two  elevations,  in  shovel- 
ling, throwing,  or  lifting. 

Steeving.    The  elevation  of  a  vessel's  bowsprit}  cathead,  etc. 

Strake.     A  breadth  of  plank. 

Strut.    An  oblique  brace  to  support  a  rafter. 

Style.    The  gnomon  of  a  sun-dial. 

Sump.    In  Mining,  a  pit  or  well  into  which  water  may  be  led  from  a  mine  or  work. 

Surcingle.  A  belt,  band,  or  girth,  which  passes  over  a  saddle  or  blanket  upon  a 
horse's  back. 

Stooge.  To  bear  or  force  down.  An  instrument  having  a  groove  on  its  under 
•ide  for  the  purpose  of  giving  shape  to  any  piece  mbjected  to  it  when  receiving  a 
blow  from  a  hammer. 


ORTHOGRAPHY  OF  TECHNICAL  \VORDS  AND  TERMS.  IO49 

i\ 

Sypkered.  Overlapping  the  chamfered  edge  of  one  plank  upon  the  chamfered 
edge  of  another  in  such  a  manner  that  the  joint  shall  be  a  plane  surface. 

Talus.  In  Architecture,  the  slope  or  batter  of  a  wall,  parapet,  etc.  In  Geology, 
a  sloping  heap  of  rubble  at  foot  of  a  cliff. 

Template.  In  Architecture,  a  wooden  bearing  to  receive  the  end  of  a  girder  to 
distribute  its  weight. 

Templet.    A  mold  cut  to  an  exact  section  of  any  piece  or  structure. 

Tenon.  The  end  of  a  piece  of  wood,  cut  into  the  form  of  a  rectangular  prism,  de- 
signed to  be  set  into  a  cavity  of  a  like  form  in  another  piece,  which  is  termed  the 
mortise. 

Terring.    The  earth  overlying  a  quarry. 

Tester.    The  top  covering  of  a  bedstead. 

Tholes.    The  pins  in  the  gunwale  of  a  boat  which  are  used  as  rowlocka 

Thwarts.    The  athwartship  seats  in  a  boat. 

Tide-rode.    The  situation  of  a  vessel  at  anchor,  when  she  rides  in  direction  of  tht 
current  instead  of  the  wind. 
Tire.    The  metal  hoop  that  binds  the  felloes  of  a  wheel 

Tompion.  The  stopper  of  a  piece  of  ordnance.  The  iron  bottom  to  which  grape- 
shot  are  secured. 

Treenails.  Wooden  pins  employed  to  secure  the  planking  of  a  vessel  to  the 
frames. 

Trepan.  In  Mining,  the  instrument  used  in  the  comminution  of  rock  in  earth- 
boring  at  great  depths. 

Trestle.  The  frame  of  a  table  ;  a  movable  form  of  support.  In  Mast-making,  two 
pieces  of  timber  set  horizontally  upon  opposite  sides  of  a  mast-head. 

Trice.    In  Seamanship,  to  haul  or  tie  up  by  means  of  a  rope  or  tricing-line. 

Tue-iron  or  Tuyere.  The  nozzle  of  a  bellows  or  blast-pipe  in  a  forge  or  smelting. 
furnaca 

Vice.    In  Mechanics,  a  press  to  hold  fast  anything  to  be  worked  upon. 

Voyal.  In  Seamanship,  a  purchase  applied  to  the  weighing  of  an  anchor,  leading 
to  a  capstan. 

Wagon.  An  open  or  partially  enclosed  four-wheeled  vehicle,  adapted  for  the 
transportation  of  persons,  goods,  etc. 

Wear.  I*  nautical  language,  to  put  a  vessel  upon  a  contrary  tack  by  turning 
her  around  stern  to  the  wind. 

Weir.  A  dam  across  a  river  or  stream  to  arrest  the  water;  a  fence  of  twigs  or 
stakes  in  a  stream  to  divert  the  run  of  fish. 

Whipple-tree.    The  bar  to  which  the  traces  of  harness  are  fastened. 

Wind-rode.  The  situation  of  a  vessel  at  anchor,  when  she  rides  in  direction  ol 
the  wind  instead  of  the  current. 

Windrow.    A  row  or  line  of  hay,  etc.,  raked  together. 

Withe.  An  instrument  fitted  to  the  end  of  a  boom  or  mast,  with  a  ring,  through 
which  a  boom  is  rigged  out  or  mast  set  up. 

Woold.    To  wind;  particularly  to  bind  a  rope  around  a  spar,  etc. 


Astragal*  In  Architecture,  a  round  molding,  surrounding  the  head  or  base  of  a 
colunm.  In  Gunnery,  a  like  molding  on  cannon  near  the  mouth. 

Creusote.    An  oily  colorless  liquid,  procured  from  coal-tar. 

Flume,  a  channel  for  conducting  water,  as  that  by  which  the  surplus  water  of  a 
canal  is  led  to  a  lower  level. 

Forebay.  The  part  of  a  Mill-race  or  Penstock,  from  which  water  flows  upon  a 
water-wheel. 

Grillage.  A  frame,  constructed  of  beams  laid  in  parallel  rows,  and  crossed  at 
right  angles,  with  others  notched  over  them. 

Designed  to  uniformly  distribute  or  extend  the  area  of  a  foundation. 


IO5O  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

Hypotenuse.    Commonly,  but  incorrectly,  hypothenuse. 

ArcMtectUre>  a  pier  that  juts  out  or  P"***"  into  a  river  or  a 


Kibble.    In  Mining,  a  metallic  bucket  in  which  ore  is  drawn  up  to  the  surface 
Lewis     One  or  two  frustums  of  a  right-angled  metallic  wedge  set  inverted  or 

«W*»^  «*  MraSU; 

orTfaduct.  ^  En9ineerin^  a  cylindrjcal  pillar  terminating  a  wing  wall  of  a  bridge 


Nautical    Wra?Ped  with  canvas  or  tarred  rope,  to  resist  wear  from 

Payed.    Nautical.    Painted,  tarred,  or  greased,  to  resist  moisture  and  wear. 
floodgate*'    An  artificial  conduit  for  water  to  a  water-wheel,  and  furnished  with  a 

Ravel.    To  disentangle,  untwist,  or  unweave.   The  usual  prefix  Un  i.  wholly  superfluous. 

Roil.    To  render  turbid,  to  stir  or  mix. 

Scabble.    The  dressing  of  the  faces  of  rough  stones,  as  with  a  broad  chisel. 

Served,  Service.    Nautical.    The  layer  of  wrapping,  as  spun  yarn,  lines,  etc. 
around  a  stay  or  rope,  to  resist  friction  and  wear. 

Shackle,  or  Clevis.    An  open  link  set  in  a  chain,  secured  by  a  pin  running  through 
eyes  m  its  ends,  which,  when  withdrawn,  admits  chain  to  be  parted  at  that  point. 

Soffit.     In  Architecture,  the  under  side  of  an  opening  ;  the  lower  surface  of  a 
vault  or  arch;  also  the  under  surface  of  an  arch  between  columns 

*le  with 


Strike,  in  Geology,  is  the  compass  direction  of  the  intersection  of  the  plane  of 
stratified  rock  with  the  plane  of  the  horizon. 

Altars.    In  Naval  Architecture,  the  steps  on  the  sides  and  end  of  a  marine  dock. 

Gin.  An  instrument  operated  by  men  or  animals  for  the  raising  or  drawing  of 
heavy  bodies;  usually  a  vertical  revolving  windlass  and  lever. 

Sump.  In  Salt-works,  a  pond  in  which  the  sea  or  saline  water  is  retained  for  use 
in  the  future. 

Skeet.  Nautical.  A  scoop  with  a  long  handle,  for  use  in  wetting  the  sails  or  the 
sides  of  a  vessel. 

Wyes.  The  vertical  standards  on  which  the  telescope  of  a  Theodolite  or  Level  is 
supported,  and  which  admits  of  their  being  reversed  by  a  reversal  of  its  ends 
When  the  telescope  is  reversed  by  rotation  on  its  trunnions,  the  instrument  is 
termed  a  Transit. 

Cantalcver.  An  angular  lever,  as  a  projecting  bracket  under  a  balcony,  the  eaves 
of  a  building  or  the  span  of  a  bridge,  when  the  intrados  is  defined  by  lines  from  the 
abutments,  at  an  angle  elevating  from  the  horizon. 

Camel.  In  Naval  Architecture.  A  decked  and  flat-bottomed  vessel,  alike  to  a 
scow;  adapted  for  transportation  of  heavy  material,  in  the  raising  of  sunken  ves- 
vels,  etc.,  and  for  the  transportation  of  heavy  materials,  as  armaments  from  vessels. 
to  a  shore,  etc.,  commonly,  but  erroneously,  a  scow. 

Scow.    An  open  and  flat-bottomed  vessel,  adapted  for  operation  in  shallow  water. 

Sprocket.  A  radial  projection  on  the  circumference  of  a  wheel;  to  engage  the 
links  of  a  chain,  as  those  on  the  wheel  base  of  a  capstan. 

Spud.  In  Mechanics.  A  spade-like  instrument  for  recovering  a  tool  in  a  tuba 
well.  In  Surveying.  A  nail  driven  in  a  monument  or  stake,  to  designate  a  line  or 
point. 

Beam.  In  Mechanics.  When  vibrating,  as  in  a  Vertical  or  Side-lever  Steam  or 
other  Engine,  it  is  simply  a  Beam,  as  Main,  Overhead,  Side-lever,  Air-pump,  etc. 

Working  is  superfluous,  and  Walking  is  a  local  vulgarism. 

Size.  In  Mining.  A  separation  of  coarse  and  fine  grains  or  parts.  In  Mechanics 
or  Arts.  A  weak,  viscous,  and  glutinous  substance  or  varnish.  In  Geometry  or 
Volume,  the  application  of  it  to  areas,  structures,  etc.  ,  is  objectionable. 

Corruption  of  Assize,  a  Statute  of  measure  and  price, 


ORTHOGRAPHY  OP  TECHNICAL  WORDS  AND  TERMS.    10$  I 

Adjutage.    An  opening  in  a  vessel  for  tne  efflux  of  a  fluid. 

Archean.  Oldest  period  of  geological  time.  A  term  given  to  crystalline  schists 
and  massive  rocks  underneath  the  oldest  fossiliferous  stratified  rocks. 

Bascule  Bridge.  A  bridge  structure  for  the  passage  of  vessels  in  a  river;  by 
which  a  single  or  divided  floor  is  counterpoised  by  the  weight  of  the  inner  end  or 
ends.  The  whole  of  the  movable  floor  or  floors  resting  on  a  transverse  shaft. 

Beton.  Artificial  stone  made  by  the  admixture  of  broken  stone,  shingle,  gravel, 
etc.,  with  hydraulic  cement  and  water.  When  mixed  with  lime  it  is  termed  Con- 
crete. 

Breast- summer.  A  beam  of  metal,  stone,  or  wood,  designed  to  sustain  a  wall  over 
a  doorway  or  like  opening  or  floor;  a  lintel. 

Chaplet.  A  metallic  support  or  Stud,  set  in  a  mold  to  sustain  a  core  against  the 
pressure  of  the  metal  when  fluid. 

Crab.  A  shaft,  vertical  or  horizontal,  constructed  as  a  rope  drum;  for  the  draw- 
ing or  raising  of  heavy  bodies,  and  operated  by  a  winch  or  handspikes  in  the 
manner  of  a  windlass  or  capstan. 

Dolmen  or  Tolmen.  (Celtic.)  A  Druidical  monument  consisting  of  a  large  stone 
set  horizontal  on  two  vertical  stones  at  a  short  distance  apart,  and  a  few  feet  in 
height.  (Breton.)  An  excavated  stone  containing  human  remains.  Also  Cromleeh. 
A  large  flat  and  crooked  stone,  set  horizontal  upon  four  others  set  vertical,  alike  to 
a  table. 

Firmer  Tools.  Short  chisels  and  gouges,  as  distinguished  from  ordinary  long- 
bladed,  and  usually  operated  by  hand.  The  gouge  is  ground  upon  its  outer  side; 
the  ordinary  upon  its  inner. 

Flitch.  In  Construction.— The  combination  of  wood  with  iron,  either  in  plates 
or  a  flanged  beam. 

Gantry.  A  frame  of  posts  and  header,  to  support  a  travelling  winch,  wherewith 
heavy  weights,  as  stone  for  foundation  walls,  or  machines,  may  be  raised  and  trans 
ported. 

Jag.    A  rough  point  or  barb  on  the  projecting  surfaces  of  metal;  produced  by 
nicking  it,  as  with  a  chisel,  or  by  casting.      Jaggers.     The  rough  projections. 
Jagging.    The  insertion  of  a  jagged  or  serrated  bar,  shaft,  or  eye-bolt  in  a  casting, 
.  to  prevent  its  being  easily  withdrawn. 

Key.  In  Mechanics,  a  tapered  wedge  used  in  connection  with  a  gib  and  strap, 
and  also  with  brasses;  to  adjust  the  length  of  the  rod  to  which  they  are  attached. 
A  Cottar. 

Lacustrine.  Pertaining  to  a  lake,  and  applied  to  deposits  which  are  present  in 
lake  basins. 

Lewis.  A  combination  of  one  or  two  dovetailed  iron  pieces,  with  a  shackle  eye 
and  bolt;  set  into  a  dovetailed  under  cut  in  a  body  of  stone,  marble,  or  cement 
block,  and  set  out  and  secured  by  the  insertion  of  a  wedge.  Lewis  Bolt.  A  bolt 
with  a  jagged  end,  for  insertion  in  a  block  of  stone,  etc.,  and  leaded  in. 

Luting.  The  laying  or  insertion  of  a  paste,  cement,  or  adhesive  material  of  plas- 
tic consistency,  in  or  over  a  crevice  or  between  joints  of  a  pipe,  etc. 

Mandrel.  A  metal  spindle  for  chucking  lathe  work.  A  tapered  metal  rod  or 
former  on  which  nuts,  etc.,  etc.,  are  dressed  to  shape. 

Moraine.    Material  as  rocks,  earth,  etc.,  pushed  or  deposited  by  glaciers. 

Pein.  The  lesser  head  of  a  hammer,  and  is  termed  Ball  when  it  is  spherical; 
Cross  when  in  the  form  of  a  round-edged  ridge,  at  right  angles  to  the  axis  of  the 
handle;  and  Straight,  when  a  like  ridge  is  in  the  plane  of  the  handle. 

Pierre  Perdue.  Lost  or  random  stone  projected  in  water,  usually  for  a  base  to  a 
superstructure,  or  to  construct  a  Breakwater.  Riprap. 

Scrim.  A  screen  or  shade;  a  thin,  coarse  cloth,  used  for  temporary  windows  or 
doors  in  a  building  in  progress  of  completion. 

Seepage.  Percolation ;  oozing  fluid  or  moisture;  also,  the  volume  of  a  fluid  that 
percolates. 

Slope.  A  step,  an  excavation  in  a  mine  to  enable  ore  to  be  rendered  accessible 
by  a  shaft  or  drift.  To  remove  the  contents  of  a  vein. 

Sullage.    Scoriae,  cinder,  scurf,  etc.,  which  floats  on  the  surface  of  molten  metal. 

Tackle.    The  connection  of  two  or  more  blocks  and  a  rope. 

THE    END. 


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